src/HOL/Complex.thy
author wenzelm
Mon Aug 31 21:28:08 2015 +0200 (2015-08-31)
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     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 \<open>Complex Numbers: Rectangular and Polar Representations\<close>
     8 
     9 theory Complex
    10 imports Transcendental
    11 begin
    12 
    13 text \<open>
    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 \<close>
    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 \<open>Addition and Subtraction\<close>
    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 \<open>Multiplication and Division\<close>
    57 
    58 instantiation complex :: field
    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 div (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 [simp]: "Im x = 0 \<Longrightarrow> Re (x ^ n) = Re x ^ n "
    97   by (induct n) simp_all
    98 
    99 lemma Im_power_real [simp]: "Im x = 0 \<Longrightarrow> Im (x ^ n) = 0"
   100   by (induct n) simp_all
   101 
   102 subsection \<open>Scalar Multiplication\<close>
   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 \<open>Numerals, Arithmetic, and Embedding from Reals\<close>
   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 lemma of_real_Re [simp]:
   171     "z \<in> \<real> \<Longrightarrow> of_real (Re z) = z"
   172   by (auto simp: Reals_def)
   173 
   174 subsection \<open>The Complex Number $i$\<close>
   175 
   176 primcorec "ii" :: complex  ("\<i>") where
   177   "Re ii = 0"
   178 | "Im ii = 1"
   179 
   180 lemma Complex_eq[simp]: "Complex a b = a + \<i> * b"
   181   by (simp add: complex_eq_iff)
   182 
   183 lemma complex_eq: "a = Re a + \<i> * Im a"
   184   by (simp add: complex_eq_iff)
   185 
   186 lemma fun_complex_eq: "f = (\<lambda>x. Re (f x) + \<i> * Im (f x))"
   187   by (simp add: fun_eq_iff complex_eq)
   188 
   189 lemma i_squared [simp]: "ii * ii = -1"
   190   by (simp add: complex_eq_iff)
   191 
   192 lemma power2_i [simp]: "ii\<^sup>2 = -1"
   193   by (simp add: power2_eq_square)
   194 
   195 lemma inverse_i [simp]: "inverse ii = - ii"
   196   by (rule inverse_unique) simp
   197 
   198 lemma divide_i [simp]: "x / ii = - ii * x"
   199   by (simp add: divide_complex_def)
   200 
   201 lemma complex_i_mult_minus [simp]: "ii * (ii * x) = - x"
   202   by (simp add: mult.assoc [symmetric])
   203 
   204 lemma complex_i_not_zero [simp]: "ii \<noteq> 0"
   205   by (simp add: complex_eq_iff)
   206 
   207 lemma complex_i_not_one [simp]: "ii \<noteq> 1"
   208   by (simp add: complex_eq_iff)
   209 
   210 lemma complex_i_not_numeral [simp]: "ii \<noteq> numeral w"
   211   by (simp add: complex_eq_iff)
   212 
   213 lemma complex_i_not_neg_numeral [simp]: "ii \<noteq> - numeral w"
   214   by (simp add: complex_eq_iff)
   215 
   216 lemma complex_split_polar: "\<exists>r a. z = complex_of_real r * (cos a + \<i> * sin a)"
   217   by (simp add: complex_eq_iff polar_Ex)
   218 
   219 lemma i_even_power [simp]: "\<i> ^ (n * 2) = (-1) ^ n"
   220   by (metis mult.commute power2_i power_mult)
   221 
   222 lemma Re_ii_times [simp]: "Re (ii*z) = - Im z"
   223   by simp
   224 
   225 lemma Im_ii_times [simp]: "Im (ii*z) = Re z"
   226   by simp
   227 
   228 lemma ii_times_eq_iff: "ii*w = z \<longleftrightarrow> w = -(ii*z)"
   229   by auto
   230 
   231 lemma divide_numeral_i [simp]: "z / (numeral n * ii) = -(ii*z) / numeral n"
   232   by (metis divide_divide_eq_left divide_i mult.commute mult_minus_right)
   233 
   234 subsection \<open>Vector Norm\<close>
   235 
   236 instantiation complex :: real_normed_field
   237 begin
   238 
   239 definition "norm z = sqrt ((Re z)\<^sup>2 + (Im z)\<^sup>2)"
   240 
   241 abbreviation cmod :: "complex \<Rightarrow> real"
   242   where "cmod \<equiv> norm"
   243 
   244 definition complex_sgn_def:
   245   "sgn x = x /\<^sub>R cmod x"
   246 
   247 definition dist_complex_def:
   248   "dist x y = cmod (x - y)"
   249 
   250 definition open_complex_def:
   251   "open (S :: complex set) \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
   252 
   253 instance proof
   254   fix r :: real and x y :: complex and S :: "complex set"
   255   show "(norm x = 0) = (x = 0)"
   256     by (simp add: norm_complex_def complex_eq_iff)
   257   show "norm (x + y) \<le> norm x + norm y"
   258     by (simp add: norm_complex_def complex_eq_iff real_sqrt_sum_squares_triangle_ineq)
   259   show "norm (scaleR r x) = \<bar>r\<bar> * norm x"
   260     by (simp add: norm_complex_def complex_eq_iff power_mult_distrib distrib_left [symmetric] real_sqrt_mult)
   261   show "norm (x * y) = norm x * norm y"
   262     by (simp add: norm_complex_def complex_eq_iff real_sqrt_mult [symmetric] power2_eq_square algebra_simps)
   263 qed (rule complex_sgn_def dist_complex_def open_complex_def)+
   264 
   265 end
   266 
   267 lemma norm_ii [simp]: "norm ii = 1"
   268   by (simp add: norm_complex_def)
   269 
   270 lemma cmod_unit_one: "cmod (cos a + \<i> * sin a) = 1"
   271   by (simp add: norm_complex_def)
   272 
   273 lemma cmod_complex_polar: "cmod (r * (cos a + \<i> * sin a)) = \<bar>r\<bar>"
   274   by (simp add: norm_mult cmod_unit_one)
   275 
   276 lemma complex_Re_le_cmod: "Re x \<le> cmod x"
   277   unfolding norm_complex_def
   278   by (rule real_sqrt_sum_squares_ge1)
   279 
   280 lemma complex_mod_minus_le_complex_mod: "- cmod x \<le> cmod x"
   281   by (rule order_trans [OF _ norm_ge_zero]) simp
   282 
   283 lemma complex_mod_triangle_ineq2: "cmod (b + a) - cmod b \<le> cmod a"
   284   by (rule ord_le_eq_trans [OF norm_triangle_ineq2]) simp
   285 
   286 lemma abs_Re_le_cmod: "\<bar>Re x\<bar> \<le> cmod x"
   287   by (simp add: norm_complex_def)
   288 
   289 lemma abs_Im_le_cmod: "\<bar>Im x\<bar> \<le> cmod x"
   290   by (simp add: norm_complex_def)
   291 
   292 lemma cmod_le: "cmod z \<le> \<bar>Re z\<bar> + \<bar>Im z\<bar>"
   293   apply (subst complex_eq)
   294   apply (rule order_trans)
   295   apply (rule norm_triangle_ineq)
   296   apply (simp add: norm_mult)
   297   done
   298 
   299 lemma cmod_eq_Re: "Im z = 0 \<Longrightarrow> cmod z = \<bar>Re z\<bar>"
   300   by (simp add: norm_complex_def)
   301 
   302 lemma cmod_eq_Im: "Re z = 0 \<Longrightarrow> cmod z = \<bar>Im z\<bar>"
   303   by (simp add: norm_complex_def)
   304 
   305 lemma cmod_power2: "cmod z ^ 2 = (Re z)^2 + (Im z)^2"
   306   by (simp add: norm_complex_def)
   307 
   308 lemma cmod_plus_Re_le_0_iff: "cmod z + Re z \<le> 0 \<longleftrightarrow> Re z = - cmod z"
   309   using abs_Re_le_cmod[of z] by auto
   310 
   311 lemma Im_eq_0: "\<bar>Re z\<bar> = cmod z \<Longrightarrow> Im z = 0"
   312   by (subst (asm) power_eq_iff_eq_base[symmetric, where n=2])
   313      (auto simp add: norm_complex_def)
   314 
   315 lemma abs_sqrt_wlog:
   316   fixes x::"'a::linordered_idom"
   317   assumes "\<And>x::'a. x \<ge> 0 \<Longrightarrow> P x (x\<^sup>2)" shows "P \<bar>x\<bar> (x\<^sup>2)"
   318 by (metis abs_ge_zero assms power2_abs)
   319 
   320 lemma complex_abs_le_norm: "\<bar>Re z\<bar> + \<bar>Im z\<bar> \<le> sqrt 2 * norm z"
   321   unfolding norm_complex_def
   322   apply (rule abs_sqrt_wlog [where x="Re z"])
   323   apply (rule abs_sqrt_wlog [where x="Im z"])
   324   apply (rule power2_le_imp_le)
   325   apply (simp_all add: power2_sum add.commute sum_squares_bound real_sqrt_mult [symmetric])
   326   done
   327 
   328 lemma complex_unit_circle: "z \<noteq> 0 \<Longrightarrow> (Re z / cmod z)\<^sup>2 + (Im z / cmod z)\<^sup>2 = 1"
   329   by (simp add: norm_complex_def divide_simps complex_eq_iff)
   330 
   331 
   332 text \<open>Properties of complex signum.\<close>
   333 
   334 lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)"
   335   by (simp add: sgn_div_norm divide_inverse scaleR_conv_of_real mult.commute)
   336 
   337 lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z"
   338   by (simp add: complex_sgn_def divide_inverse)
   339 
   340 lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z"
   341   by (simp add: complex_sgn_def divide_inverse)
   342 
   343 
   344 subsection \<open>Completeness of the Complexes\<close>
   345 
   346 lemma bounded_linear_Re: "bounded_linear Re"
   347   by (rule bounded_linear_intro [where K=1], simp_all add: norm_complex_def)
   348 
   349 lemma bounded_linear_Im: "bounded_linear Im"
   350   by (rule bounded_linear_intro [where K=1], simp_all add: norm_complex_def)
   351 
   352 lemmas Cauchy_Re = bounded_linear.Cauchy [OF bounded_linear_Re]
   353 lemmas Cauchy_Im = bounded_linear.Cauchy [OF bounded_linear_Im]
   354 lemmas tendsto_Re [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_Re]
   355 lemmas tendsto_Im [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_Im]
   356 lemmas isCont_Re [simp] = bounded_linear.isCont [OF bounded_linear_Re]
   357 lemmas isCont_Im [simp] = bounded_linear.isCont [OF bounded_linear_Im]
   358 lemmas continuous_Re [simp] = bounded_linear.continuous [OF bounded_linear_Re]
   359 lemmas continuous_Im [simp] = bounded_linear.continuous [OF bounded_linear_Im]
   360 lemmas continuous_on_Re [continuous_intros] = bounded_linear.continuous_on[OF bounded_linear_Re]
   361 lemmas continuous_on_Im [continuous_intros] = bounded_linear.continuous_on[OF bounded_linear_Im]
   362 lemmas has_derivative_Re [derivative_intros] = bounded_linear.has_derivative[OF bounded_linear_Re]
   363 lemmas has_derivative_Im [derivative_intros] = bounded_linear.has_derivative[OF bounded_linear_Im]
   364 lemmas sums_Re = bounded_linear.sums [OF bounded_linear_Re]
   365 lemmas sums_Im = bounded_linear.sums [OF bounded_linear_Im]
   366 
   367 lemma tendsto_Complex [tendsto_intros]:
   368   "(f ---> a) F \<Longrightarrow> (g ---> b) F \<Longrightarrow> ((\<lambda>x. Complex (f x) (g x)) ---> Complex a b) F"
   369   by (auto intro!: tendsto_intros)
   370 
   371 lemma tendsto_complex_iff:
   372   "(f ---> x) F \<longleftrightarrow> (((\<lambda>x. Re (f x)) ---> Re x) F \<and> ((\<lambda>x. Im (f x)) ---> Im x) F)"
   373 proof safe
   374   assume "((\<lambda>x. Re (f x)) ---> Re x) F" "((\<lambda>x. Im (f x)) ---> Im x) F"
   375   from tendsto_Complex[OF this] show "(f ---> x) F"
   376     unfolding complex.collapse .
   377 qed (auto intro: tendsto_intros)
   378 
   379 lemma continuous_complex_iff: "continuous F f \<longleftrightarrow>
   380     continuous F (\<lambda>x. Re (f x)) \<and> continuous F (\<lambda>x. Im (f x))"
   381   unfolding continuous_def tendsto_complex_iff ..
   382 
   383 lemma has_vector_derivative_complex_iff: "(f has_vector_derivative x) F \<longleftrightarrow>
   384     ((\<lambda>x. Re (f x)) has_field_derivative (Re x)) F \<and>
   385     ((\<lambda>x. Im (f x)) has_field_derivative (Im x)) F"
   386   unfolding has_vector_derivative_def has_field_derivative_def has_derivative_def tendsto_complex_iff
   387   by (simp add: field_simps bounded_linear_scaleR_left bounded_linear_mult_right)
   388 
   389 lemma has_field_derivative_Re[derivative_intros]:
   390   "(f has_vector_derivative D) F \<Longrightarrow> ((\<lambda>x. Re (f x)) has_field_derivative (Re D)) F"
   391   unfolding has_vector_derivative_complex_iff by safe
   392 
   393 lemma has_field_derivative_Im[derivative_intros]:
   394   "(f has_vector_derivative D) F \<Longrightarrow> ((\<lambda>x. Im (f x)) has_field_derivative (Im D)) F"
   395   unfolding has_vector_derivative_complex_iff by safe
   396 
   397 instance complex :: banach
   398 proof
   399   fix X :: "nat \<Rightarrow> complex"
   400   assume X: "Cauchy X"
   401   then have "(\<lambda>n. Complex (Re (X n)) (Im (X n))) ----> Complex (lim (\<lambda>n. Re (X n))) (lim (\<lambda>n. Im (X n)))"
   402     by (intro tendsto_Complex convergent_LIMSEQ_iff[THEN iffD1] Cauchy_convergent_iff[THEN iffD1] Cauchy_Re Cauchy_Im)
   403   then show "convergent X"
   404     unfolding complex.collapse by (rule convergentI)
   405 qed
   406 
   407 declare
   408   DERIV_power[where 'a=complex, unfolded of_nat_def[symmetric], derivative_intros]
   409 
   410 subsection \<open>Complex Conjugation\<close>
   411 
   412 primcorec cnj :: "complex \<Rightarrow> complex" where
   413   "Re (cnj z) = Re z"
   414 | "Im (cnj z) = - Im z"
   415 
   416 lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)"
   417   by (simp add: complex_eq_iff)
   418 
   419 lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z"
   420   by (simp add: complex_eq_iff)
   421 
   422 lemma complex_cnj_zero [simp]: "cnj 0 = 0"
   423   by (simp add: complex_eq_iff)
   424 
   425 lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)"
   426   by (simp add: complex_eq_iff)
   427 
   428 lemma complex_cnj_add [simp]: "cnj (x + y) = cnj x + cnj y"
   429   by (simp add: complex_eq_iff)
   430 
   431 lemma cnj_setsum [simp]: "cnj (setsum f s) = (\<Sum>x\<in>s. cnj (f x))"
   432   by (induct s rule: infinite_finite_induct) auto
   433 
   434 lemma complex_cnj_diff [simp]: "cnj (x - y) = cnj x - cnj y"
   435   by (simp add: complex_eq_iff)
   436 
   437 lemma complex_cnj_minus [simp]: "cnj (- x) = - cnj x"
   438   by (simp add: complex_eq_iff)
   439 
   440 lemma complex_cnj_one [simp]: "cnj 1 = 1"
   441   by (simp add: complex_eq_iff)
   442 
   443 lemma complex_cnj_mult [simp]: "cnj (x * y) = cnj x * cnj y"
   444   by (simp add: complex_eq_iff)
   445 
   446 lemma cnj_setprod [simp]: "cnj (setprod f s) = (\<Prod>x\<in>s. cnj (f x))"
   447   by (induct s rule: infinite_finite_induct) auto
   448 
   449 lemma complex_cnj_inverse [simp]: "cnj (inverse x) = inverse (cnj x)"
   450   by (simp add: complex_eq_iff)
   451 
   452 lemma complex_cnj_divide [simp]: "cnj (x / y) = cnj x / cnj y"
   453   by (simp add: divide_complex_def)
   454 
   455 lemma complex_cnj_power [simp]: "cnj (x ^ n) = cnj x ^ n"
   456   by (induct n) simp_all
   457 
   458 lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n"
   459   by (simp add: complex_eq_iff)
   460 
   461 lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z"
   462   by (simp add: complex_eq_iff)
   463 
   464 lemma complex_cnj_numeral [simp]: "cnj (numeral w) = numeral w"
   465   by (simp add: complex_eq_iff)
   466 
   467 lemma complex_cnj_neg_numeral [simp]: "cnj (- numeral w) = - numeral w"
   468   by (simp add: complex_eq_iff)
   469 
   470 lemma complex_cnj_scaleR [simp]: "cnj (scaleR r x) = scaleR r (cnj x)"
   471   by (simp add: complex_eq_iff)
   472 
   473 lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z"
   474   by (simp add: norm_complex_def)
   475 
   476 lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x"
   477   by (simp add: complex_eq_iff)
   478 
   479 lemma complex_cnj_i [simp]: "cnj ii = - ii"
   480   by (simp add: complex_eq_iff)
   481 
   482 lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)"
   483   by (simp add: complex_eq_iff)
   484 
   485 lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im z) * ii"
   486   by (simp add: complex_eq_iff)
   487 
   488 lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\<^sup>2 + (Im z)\<^sup>2)"
   489   by (simp add: complex_eq_iff power2_eq_square)
   490 
   491 lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<^sup>2"
   492   by (simp add: norm_mult power2_eq_square)
   493 
   494 lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))"
   495   by (simp add: norm_complex_def power2_eq_square)
   496 
   497 lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0"
   498   by simp
   499 
   500 lemma bounded_linear_cnj: "bounded_linear cnj"
   501   using complex_cnj_add complex_cnj_scaleR
   502   by (rule bounded_linear_intro [where K=1], simp)
   503 
   504 lemmas tendsto_cnj [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_cnj]
   505 lemmas isCont_cnj [simp] = bounded_linear.isCont [OF bounded_linear_cnj]
   506 lemmas continuous_cnj [simp, continuous_intros] = bounded_linear.continuous [OF bounded_linear_cnj]
   507 lemmas continuous_on_cnj [simp, continuous_intros] = bounded_linear.continuous_on [OF bounded_linear_cnj]
   508 lemmas has_derivative_cnj [simp, derivative_intros] = bounded_linear.has_derivative [OF bounded_linear_cnj]
   509 
   510 lemma lim_cnj: "((\<lambda>x. cnj(f x)) ---> cnj l) F \<longleftrightarrow> (f ---> l) F"
   511   by (simp add: tendsto_iff dist_complex_def complex_cnj_diff [symmetric] del: complex_cnj_diff)
   512 
   513 lemma sums_cnj: "((\<lambda>x. cnj(f x)) sums cnj l) \<longleftrightarrow> (f sums l)"
   514   by (simp add: sums_def lim_cnj cnj_setsum [symmetric] del: cnj_setsum)
   515 
   516 
   517 subsection\<open>Basic Lemmas\<close>
   518 
   519 lemma complex_eq_0: "z=0 \<longleftrightarrow> (Re z)\<^sup>2 + (Im z)\<^sup>2 = 0"
   520   by (metis zero_complex.sel complex_eqI sum_power2_eq_zero_iff)
   521 
   522 lemma complex_neq_0: "z\<noteq>0 \<longleftrightarrow> (Re z)\<^sup>2 + (Im z)\<^sup>2 > 0"
   523   by (metis complex_eq_0 less_numeral_extra(3) sum_power2_gt_zero_iff)
   524 
   525 lemma complex_norm_square: "of_real ((norm z)\<^sup>2) = z * cnj z"
   526 by (cases z)
   527    (auto simp: complex_eq_iff norm_complex_def power2_eq_square[symmetric] of_real_power[symmetric]
   528          simp del: of_real_power)
   529 
   530 lemma Re_complex_div_eq_0: "Re (a / b) = 0 \<longleftrightarrow> Re (a * cnj b) = 0"
   531   by (auto simp add: Re_divide)
   532 
   533 lemma Im_complex_div_eq_0: "Im (a / b) = 0 \<longleftrightarrow> Im (a * cnj b) = 0"
   534   by (auto simp add: Im_divide)
   535 
   536 lemma complex_div_gt_0:
   537   "(Re (a / b) > 0 \<longleftrightarrow> Re (a * cnj b) > 0) \<and> (Im (a / b) > 0 \<longleftrightarrow> Im (a * cnj b) > 0)"
   538 proof cases
   539   assume "b = 0" then show ?thesis by auto
   540 next
   541   assume "b \<noteq> 0"
   542   then have "0 < (Re b)\<^sup>2 + (Im b)\<^sup>2"
   543     by (simp add: complex_eq_iff sum_power2_gt_zero_iff)
   544   then show ?thesis
   545     by (simp add: Re_divide Im_divide zero_less_divide_iff)
   546 qed
   547 
   548 lemma Re_complex_div_gt_0: "Re (a / b) > 0 \<longleftrightarrow> Re (a * cnj b) > 0"
   549   and Im_complex_div_gt_0: "Im (a / b) > 0 \<longleftrightarrow> Im (a * cnj b) > 0"
   550   using complex_div_gt_0 by auto
   551 
   552 lemma Re_complex_div_ge_0: "Re(a / b) \<ge> 0 \<longleftrightarrow> Re(a * cnj b) \<ge> 0"
   553   by (metis le_less Re_complex_div_eq_0 Re_complex_div_gt_0)
   554 
   555 lemma Im_complex_div_ge_0: "Im(a / b) \<ge> 0 \<longleftrightarrow> Im(a * cnj b) \<ge> 0"
   556   by (metis Im_complex_div_eq_0 Im_complex_div_gt_0 le_less)
   557 
   558 lemma Re_complex_div_lt_0: "Re(a / b) < 0 \<longleftrightarrow> Re(a * cnj b) < 0"
   559   by (metis less_asym neq_iff Re_complex_div_eq_0 Re_complex_div_gt_0)
   560 
   561 lemma Im_complex_div_lt_0: "Im(a / b) < 0 \<longleftrightarrow> Im(a * cnj b) < 0"
   562   by (metis Im_complex_div_eq_0 Im_complex_div_gt_0 less_asym neq_iff)
   563 
   564 lemma Re_complex_div_le_0: "Re(a / b) \<le> 0 \<longleftrightarrow> Re(a * cnj b) \<le> 0"
   565   by (metis not_le Re_complex_div_gt_0)
   566 
   567 lemma Im_complex_div_le_0: "Im(a / b) \<le> 0 \<longleftrightarrow> Im(a * cnj b) \<le> 0"
   568   by (metis Im_complex_div_gt_0 not_le)
   569 
   570 lemma Re_setsum[simp]: "Re (setsum f s) = (\<Sum>x\<in>s. Re (f x))"
   571   by (induct s rule: infinite_finite_induct) auto
   572 
   573 lemma Im_setsum[simp]: "Im (setsum f s) = (\<Sum>x\<in>s. Im(f x))"
   574   by (induct s rule: infinite_finite_induct) auto
   575 
   576 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)"
   577   unfolding sums_def tendsto_complex_iff Im_setsum Re_setsum ..
   578 
   579 lemma summable_complex_iff: "summable f \<longleftrightarrow> summable (\<lambda>x. Re (f x)) \<and>  summable (\<lambda>x. Im (f x))"
   580   unfolding summable_def sums_complex_iff[abs_def] by (metis complex.sel)
   581 
   582 lemma summable_complex_of_real [simp]: "summable (\<lambda>n. complex_of_real (f n)) \<longleftrightarrow> summable f"
   583   unfolding summable_complex_iff by simp
   584 
   585 lemma summable_Re: "summable f \<Longrightarrow> summable (\<lambda>x. Re (f x))"
   586   unfolding summable_complex_iff by blast
   587 
   588 lemma summable_Im: "summable f \<Longrightarrow> summable (\<lambda>x. Im (f x))"
   589   unfolding summable_complex_iff by blast
   590 
   591 lemma complex_is_Real_iff: "z \<in> \<real> \<longleftrightarrow> Im z = 0"
   592   by (auto simp: Reals_def complex_eq_iff)
   593 
   594 lemma Reals_cnj_iff: "z \<in> \<real> \<longleftrightarrow> cnj z = z"
   595   by (auto simp: complex_is_Real_iff complex_eq_iff)
   596 
   597 lemma in_Reals_norm: "z \<in> \<real> \<Longrightarrow> norm(z) = abs(Re z)"
   598   by (simp add: complex_is_Real_iff norm_complex_def)
   599 
   600 lemma series_comparison_complex:
   601   fixes f:: "nat \<Rightarrow> 'a::banach"
   602   assumes sg: "summable g"
   603      and "\<And>n. g n \<in> \<real>" "\<And>n. Re (g n) \<ge> 0"
   604      and fg: "\<And>n. n \<ge> N \<Longrightarrow> norm(f n) \<le> norm(g n)"
   605   shows "summable f"
   606 proof -
   607   have g: "\<And>n. cmod (g n) = Re (g n)" using assms
   608     by (metis abs_of_nonneg in_Reals_norm)
   609   show ?thesis
   610     apply (rule summable_comparison_test' [where g = "\<lambda>n. norm (g n)" and N=N])
   611     using sg
   612     apply (auto simp: summable_def)
   613     apply (rule_tac x="Re s" in exI)
   614     apply (auto simp: g sums_Re)
   615     apply (metis fg g)
   616     done
   617 qed
   618 
   619 subsection\<open>Polar Form for Complex Numbers\<close>
   620 
   621 lemma complex_unimodular_polar: "(norm z = 1) \<Longrightarrow> \<exists>x. z = Complex (cos x) (sin x)"
   622   using sincos_total_2pi [of "Re z" "Im z"]
   623   by auto (metis cmod_power2 complex_eq power_one)
   624 
   625 subsubsection \<open>$\cos \theta + i \sin \theta$\<close>
   626 
   627 primcorec cis :: "real \<Rightarrow> complex" where
   628   "Re (cis a) = cos a"
   629 | "Im (cis a) = sin a"
   630 
   631 lemma cis_zero [simp]: "cis 0 = 1"
   632   by (simp add: complex_eq_iff)
   633 
   634 lemma norm_cis [simp]: "norm (cis a) = 1"
   635   by (simp add: norm_complex_def)
   636 
   637 lemma sgn_cis [simp]: "sgn (cis a) = cis a"
   638   by (simp add: sgn_div_norm)
   639 
   640 lemma cis_neq_zero [simp]: "cis a \<noteq> 0"
   641   by (metis norm_cis norm_zero zero_neq_one)
   642 
   643 lemma cis_mult: "cis a * cis b = cis (a + b)"
   644   by (simp add: complex_eq_iff cos_add sin_add)
   645 
   646 lemma DeMoivre: "(cis a) ^ n = cis (real n * a)"
   647   by (induct n, simp_all add: real_of_nat_Suc algebra_simps cis_mult)
   648 
   649 lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)"
   650   by (simp add: complex_eq_iff)
   651 
   652 lemma cis_divide: "cis a / cis b = cis (a - b)"
   653   by (simp add: divide_complex_def cis_mult)
   654 
   655 lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)"
   656   by (auto simp add: DeMoivre)
   657 
   658 lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)"
   659   by (auto simp add: DeMoivre)
   660 
   661 lemma cis_pi: "cis pi = -1"
   662   by (simp add: complex_eq_iff)
   663 
   664 subsubsection \<open>$r(\cos \theta + i \sin \theta)$\<close>
   665 
   666 definition rcis :: "real \<Rightarrow> real \<Rightarrow> complex" where
   667   "rcis r a = complex_of_real r * cis a"
   668 
   669 lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a"
   670   by (simp add: rcis_def)
   671 
   672 lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a"
   673   by (simp add: rcis_def)
   674 
   675 lemma rcis_Ex: "\<exists>r a. z = rcis r a"
   676   by (simp add: complex_eq_iff polar_Ex)
   677 
   678 lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r"
   679   by (simp add: rcis_def norm_mult)
   680 
   681 lemma cis_rcis_eq: "cis a = rcis 1 a"
   682   by (simp add: rcis_def)
   683 
   684 lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)"
   685   by (simp add: rcis_def cis_mult)
   686 
   687 lemma rcis_zero_mod [simp]: "rcis 0 a = 0"
   688   by (simp add: rcis_def)
   689 
   690 lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r"
   691   by (simp add: rcis_def)
   692 
   693 lemma rcis_eq_zero_iff [simp]: "rcis r a = 0 \<longleftrightarrow> r = 0"
   694   by (simp add: rcis_def)
   695 
   696 lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)"
   697   by (simp add: rcis_def power_mult_distrib DeMoivre)
   698 
   699 lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)"
   700   by (simp add: divide_inverse rcis_def)
   701 
   702 lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)"
   703   by (simp add: rcis_def cis_divide [symmetric])
   704 
   705 subsubsection \<open>Complex exponential\<close>
   706 
   707 abbreviation Exp :: "complex \<Rightarrow> complex"
   708   where "Exp \<equiv> exp"
   709 
   710 lemma cis_conv_exp: "cis b = exp (\<i> * b)"
   711 proof -
   712   { fix n :: nat
   713     have "\<i> ^ n = fact n *\<^sub>R (cos_coeff n + \<i> * sin_coeff n)"
   714       by (induct n)
   715          (simp_all add: sin_coeff_Suc cos_coeff_Suc complex_eq_iff Re_divide Im_divide field_simps
   716                         power2_eq_square real_of_nat_Suc add_nonneg_eq_0_iff
   717                         real_of_nat_def[symmetric])
   718     then have "(\<i> * complex_of_real b) ^ n /\<^sub>R fact n =
   719         of_real (cos_coeff n * b^n) + \<i> * of_real (sin_coeff n * b^n)"
   720       by (simp add: field_simps) }
   721   then show ?thesis using sin_converges [of b] cos_converges [of b]
   722     by (auto simp add: cis.ctr exp_def simp del: of_real_mult
   723              intro!: sums_unique sums_add sums_mult sums_of_real)
   724 qed
   725 
   726 lemma Exp_eq_polar: "Exp z = exp (Re z) * cis (Im z)"
   727   unfolding cis_conv_exp exp_of_real [symmetric] mult_exp_exp by (cases z) simp
   728 
   729 lemma Re_exp: "Re (exp z) = exp (Re z) * cos (Im z)"
   730   unfolding Exp_eq_polar by simp
   731 
   732 lemma Im_exp: "Im (exp z) = exp (Re z) * sin (Im z)"
   733   unfolding Exp_eq_polar by simp
   734 
   735 lemma norm_cos_sin [simp]: "norm (Complex (cos t) (sin t)) = 1"
   736   by (simp add: norm_complex_def)
   737 
   738 lemma norm_exp_eq_Re [simp]: "norm (exp z) = exp (Re z)"
   739   by (simp add: cis.code cmod_complex_polar Exp_eq_polar)
   740 
   741 lemma complex_Exp_Ex: "\<exists>a r. z = complex_of_real r * Exp a"
   742   apply (insert rcis_Ex [of z])
   743   apply (auto simp add: Exp_eq_polar rcis_def mult.assoc [symmetric])
   744   apply (rule_tac x = "ii * complex_of_real a" in exI, auto)
   745   done
   746 
   747 lemma Exp_two_pi_i [simp]: "Exp((2::complex) * complex_of_real pi * ii) = 1"
   748   by (simp add: Exp_eq_polar complex_eq_iff)
   749 
   750 subsubsection \<open>Complex argument\<close>
   751 
   752 definition arg :: "complex \<Rightarrow> real" where
   753   "arg z = (if z = 0 then 0 else (SOME a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi))"
   754 
   755 lemma arg_zero: "arg 0 = 0"
   756   by (simp add: arg_def)
   757 
   758 lemma arg_unique:
   759   assumes "sgn z = cis x" and "-pi < x" and "x \<le> pi"
   760   shows "arg z = x"
   761 proof -
   762   from assms have "z \<noteq> 0" by auto
   763   have "(SOME a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi) = x"
   764   proof
   765     fix a def d \<equiv> "a - x"
   766     assume a: "sgn z = cis a \<and> - pi < a \<and> a \<le> pi"
   767     from a assms have "- (2*pi) < d \<and> d < 2*pi"
   768       unfolding d_def by simp
   769     moreover from a assms have "cos a = cos x" and "sin a = sin x"
   770       by (simp_all add: complex_eq_iff)
   771     hence cos: "cos d = 1" unfolding d_def cos_diff by simp
   772     moreover from cos have "sin d = 0" by (rule cos_one_sin_zero)
   773     ultimately have "d = 0"
   774       unfolding sin_zero_iff
   775       by (auto elim!: evenE dest!: less_2_cases)
   776     thus "a = x" unfolding d_def by simp
   777   qed (simp add: assms del: Re_sgn Im_sgn)
   778   with \<open>z \<noteq> 0\<close> show "arg z = x"
   779     unfolding arg_def by simp
   780 qed
   781 
   782 lemma arg_correct:
   783   assumes "z \<noteq> 0" shows "sgn z = cis (arg z) \<and> -pi < arg z \<and> arg z \<le> pi"
   784 proof (simp add: arg_def assms, rule someI_ex)
   785   obtain r a where z: "z = rcis r a" using rcis_Ex by fast
   786   with assms have "r \<noteq> 0" by auto
   787   def b \<equiv> "if 0 < r then a else a + pi"
   788   have b: "sgn z = cis b"
   789     unfolding z b_def rcis_def using \<open>r \<noteq> 0\<close>
   790     by (simp add: of_real_def sgn_scaleR sgn_if complex_eq_iff)
   791   have cis_2pi_nat: "\<And>n. cis (2 * pi * real_of_nat n) = 1"
   792     by (induct_tac n) (simp_all add: distrib_left cis_mult [symmetric] complex_eq_iff)
   793   have cis_2pi_int: "\<And>x. cis (2 * pi * real_of_int x) = 1"
   794     by (case_tac x rule: int_diff_cases)
   795        (simp add: right_diff_distrib cis_divide [symmetric] cis_2pi_nat)
   796   def c \<equiv> "b - 2*pi * of_int \<lceil>(b - pi) / (2*pi)\<rceil>"
   797   have "sgn z = cis c"
   798     unfolding b c_def
   799     by (simp add: cis_divide [symmetric] cis_2pi_int)
   800   moreover have "- pi < c \<and> c \<le> pi"
   801     using ceiling_correct [of "(b - pi) / (2*pi)"]
   802     by (simp add: c_def less_divide_eq divide_le_eq algebra_simps)
   803   ultimately show "\<exists>a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi" by fast
   804 qed
   805 
   806 lemma arg_bounded: "- pi < arg z \<and> arg z \<le> pi"
   807   by (cases "z = 0") (simp_all add: arg_zero arg_correct)
   808 
   809 lemma cis_arg: "z \<noteq> 0 \<Longrightarrow> cis (arg z) = sgn z"
   810   by (simp add: arg_correct)
   811 
   812 lemma rcis_cmod_arg: "rcis (cmod z) (arg z) = z"
   813   by (cases "z = 0") (simp_all add: rcis_def cis_arg sgn_div_norm of_real_def)
   814 
   815 lemma cos_arg_i_mult_zero [simp]: "y \<noteq> 0 \<Longrightarrow> Re y = 0 \<Longrightarrow> cos (arg y) = 0"
   816   using cis_arg [of y] by (simp add: complex_eq_iff)
   817 
   818 subsection \<open>Square root of complex numbers\<close>
   819 
   820 primcorec csqrt :: "complex \<Rightarrow> complex" where
   821   "Re (csqrt z) = sqrt ((cmod z + Re z) / 2)"
   822 | "Im (csqrt z) = (if Im z = 0 then 1 else sgn (Im z)) * sqrt ((cmod z - Re z) / 2)"
   823 
   824 lemma csqrt_of_real_nonneg [simp]: "Im x = 0 \<Longrightarrow> Re x \<ge> 0 \<Longrightarrow> csqrt x = sqrt (Re x)"
   825   by (simp add: complex_eq_iff norm_complex_def)
   826 
   827 lemma csqrt_of_real_nonpos [simp]: "Im x = 0 \<Longrightarrow> Re x \<le> 0 \<Longrightarrow> csqrt x = \<i> * sqrt \<bar>Re x\<bar>"
   828   by (simp add: complex_eq_iff norm_complex_def)
   829 
   830 lemma of_real_sqrt: "x \<ge> 0 \<Longrightarrow> of_real (sqrt x) = csqrt (of_real x)"
   831   by (simp add: complex_eq_iff norm_complex_def)
   832 
   833 lemma csqrt_0 [simp]: "csqrt 0 = 0"
   834   by simp
   835 
   836 lemma csqrt_1 [simp]: "csqrt 1 = 1"
   837   by simp
   838 
   839 lemma csqrt_ii [simp]: "csqrt \<i> = (1 + \<i>) / sqrt 2"
   840   by (simp add: complex_eq_iff Re_divide Im_divide real_sqrt_divide real_div_sqrt)
   841 
   842 lemma power2_csqrt[simp,algebra]: "(csqrt z)\<^sup>2 = z"
   843 proof cases
   844   assume "Im z = 0" then show ?thesis
   845     using real_sqrt_pow2[of "Re z"] real_sqrt_pow2[of "- Re z"]
   846     by (cases "0::real" "Re z" rule: linorder_cases)
   847        (simp_all add: complex_eq_iff Re_power2 Im_power2 power2_eq_square cmod_eq_Re)
   848 next
   849   assume "Im z \<noteq> 0"
   850   moreover
   851   have "cmod z * cmod z - Re z * Re z = Im z * Im z"
   852     by (simp add: norm_complex_def power2_eq_square)
   853   moreover
   854   have "\<bar>Re z\<bar> \<le> cmod z"
   855     by (simp add: norm_complex_def)
   856   ultimately show ?thesis
   857     by (simp add: Re_power2 Im_power2 complex_eq_iff real_sgn_eq
   858                   field_simps real_sqrt_mult[symmetric] real_sqrt_divide)
   859 qed
   860 
   861 lemma csqrt_eq_0 [simp]: "csqrt z = 0 \<longleftrightarrow> z = 0"
   862   by auto (metis power2_csqrt power_eq_0_iff)
   863 
   864 lemma csqrt_eq_1 [simp]: "csqrt z = 1 \<longleftrightarrow> z = 1"
   865   by auto (metis power2_csqrt power2_eq_1_iff)
   866 
   867 lemma csqrt_principal: "0 < Re (csqrt z) \<or> Re (csqrt z) = 0 \<and> 0 \<le> Im (csqrt z)"
   868   by (auto simp add: not_less cmod_plus_Re_le_0_iff Im_eq_0)
   869 
   870 lemma Re_csqrt: "0 \<le> Re (csqrt z)"
   871   by (metis csqrt_principal le_less)
   872 
   873 lemma csqrt_square:
   874   assumes "0 < Re b \<or> (Re b = 0 \<and> 0 \<le> Im b)"
   875   shows "csqrt (b^2) = b"
   876 proof -
   877   have "csqrt (b^2) = b \<or> csqrt (b^2) = - b"
   878     unfolding power2_eq_iff[symmetric] by (simp add: power2_csqrt)
   879   moreover have "csqrt (b^2) \<noteq> -b \<or> b = 0"
   880     using csqrt_principal[of "b ^ 2"] assms by (intro disjCI notI) (auto simp: complex_eq_iff)
   881   ultimately show ?thesis
   882     by auto
   883 qed
   884 
   885 lemma csqrt_unique:
   886     "w^2 = z \<Longrightarrow> (0 < Re w \<or> Re w = 0 \<and> 0 \<le> Im w) \<Longrightarrow> csqrt z = w"
   887   by (auto simp: csqrt_square)
   888 
   889 lemma csqrt_minus [simp]:
   890   assumes "Im x < 0 \<or> (Im x = 0 \<and> 0 \<le> Re x)"
   891   shows "csqrt (- x) = \<i> * csqrt x"
   892 proof -
   893   have "csqrt ((\<i> * csqrt x)^2) = \<i> * csqrt x"
   894   proof (rule csqrt_square)
   895     have "Im (csqrt x) \<le> 0"
   896       using assms by (auto simp add: cmod_eq_Re mult_le_0_iff field_simps complex_Re_le_cmod)
   897     then show "0 < Re (\<i> * csqrt x) \<or> Re (\<i> * csqrt x) = 0 \<and> 0 \<le> Im (\<i> * csqrt x)"
   898       by (auto simp add: Re_csqrt simp del: csqrt.simps)
   899   qed
   900   also have "(\<i> * csqrt x)^2 = - x"
   901     by (simp add: power_mult_distrib)
   902   finally show ?thesis .
   903 qed
   904 
   905 text \<open>Legacy theorem names\<close>
   906 
   907 lemmas expand_complex_eq = complex_eq_iff
   908 lemmas complex_Re_Im_cancel_iff = complex_eq_iff
   909 lemmas complex_equality = complex_eqI
   910 lemmas cmod_def = norm_complex_def
   911 lemmas complex_norm_def = norm_complex_def
   912 lemmas complex_divide_def = divide_complex_def
   913 
   914 lemma legacy_Complex_simps:
   915   shows Complex_eq_0: "Complex a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
   916     and complex_add: "Complex a b + Complex c d = Complex (a + c) (b + d)"
   917     and complex_minus: "- (Complex a b) = Complex (- a) (- b)"
   918     and complex_diff: "Complex a b - Complex c d = Complex (a - c) (b - d)"
   919     and Complex_eq_1: "Complex a b = 1 \<longleftrightarrow> a = 1 \<and> b = 0"
   920     and Complex_eq_neg_1: "Complex a b = - 1 \<longleftrightarrow> a = - 1 \<and> b = 0"
   921     and complex_mult: "Complex a b * Complex c d = Complex (a * c - b * d) (a * d + b * c)"
   922     and complex_inverse: "inverse (Complex a b) = Complex (a / (a\<^sup>2 + b\<^sup>2)) (- b / (a\<^sup>2 + b\<^sup>2))"
   923     and Complex_eq_numeral: "Complex a b = numeral w \<longleftrightarrow> a = numeral w \<and> b = 0"
   924     and Complex_eq_neg_numeral: "Complex a b = - numeral w \<longleftrightarrow> a = - numeral w \<and> b = 0"
   925     and complex_scaleR: "scaleR r (Complex a b) = Complex (r * a) (r * b)"
   926     and Complex_eq_i: "(Complex x y = ii) = (x = 0 \<and> y = 1)"
   927     and i_mult_Complex: "ii * Complex a b = Complex (- b) a"
   928     and Complex_mult_i: "Complex a b * ii = Complex (- b) a"
   929     and i_complex_of_real: "ii * complex_of_real r = Complex 0 r"
   930     and complex_of_real_i: "complex_of_real r * ii = Complex 0 r"
   931     and Complex_add_complex_of_real: "Complex x y + complex_of_real r = Complex (x+r) y"
   932     and complex_of_real_add_Complex: "complex_of_real r + Complex x y = Complex (r+x) y"
   933     and Complex_mult_complex_of_real: "Complex x y * complex_of_real r = Complex (x*r) (y*r)"
   934     and complex_of_real_mult_Complex: "complex_of_real r * Complex x y = Complex (r*x) (r*y)"
   935     and complex_eq_cancel_iff2: "(Complex x y = complex_of_real xa) = (x = xa & y = 0)"
   936     and complex_cn: "cnj (Complex a b) = Complex a (- b)"
   937     and Complex_setsum': "setsum (%x. Complex (f x) 0) s = Complex (setsum f s) 0"
   938     and Complex_setsum: "Complex (setsum f s) 0 = setsum (%x. Complex (f x) 0) s"
   939     and complex_of_real_def: "complex_of_real r = Complex r 0"
   940     and complex_norm: "cmod (Complex x y) = sqrt (x\<^sup>2 + y\<^sup>2)"
   941   by (simp_all add: norm_complex_def field_simps complex_eq_iff Re_divide Im_divide del: Complex_eq)
   942 
   943 lemma Complex_in_Reals: "Complex x 0 \<in> \<real>"
   944   by (metis Reals_of_real complex_of_real_def)
   945 
   946 end