src/HOL/Complex.thy
 author eberlm Mon Nov 02 11:56:28 2015 +0100 (2015-11-02) changeset 61531 ab2e862263e7 parent 61104 3c2d4636cebc child 61552 980dd46a03fb permissions -rw-r--r--
Rounding function, uniform limits, cotangent, binomial identities
     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::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 lemma complex_Re_fact [simp]: "Re (fact n) = fact n"

   175 proof -

   176   have "(fact n :: complex) = of_real (fact n)" by simp

   177   also have "Re \<dots> = fact n" by (subst Re_complex_of_real) simp_all

   178   finally show ?thesis .

   179 qed

   180

   181 lemma complex_Im_fact [simp]: "Im (fact n) = 0"

   182   by (subst of_nat_fact [symmetric]) (simp only: complex_Im_of_nat)

   183

   184

   185 subsection \<open>The Complex Number $i$\<close>

   186

   187 primcorec "ii" :: complex  ("\<i>") where

   188   "Re ii = 0"

   189 | "Im ii = 1"

   190

   191 lemma Complex_eq[simp]: "Complex a b = a + \<i> * b"

   192   by (simp add: complex_eq_iff)

   193

   194 lemma complex_eq: "a = Re a + \<i> * Im a"

   195   by (simp add: complex_eq_iff)

   196

   197 lemma fun_complex_eq: "f = (\<lambda>x. Re (f x) + \<i> * Im (f x))"

   198   by (simp add: fun_eq_iff complex_eq)

   199

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

   201   by (simp add: complex_eq_iff)

   202

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

   204   by (simp add: power2_eq_square)

   205

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

   207   by (rule inverse_unique) simp

   208

   209 lemma divide_i [simp]: "x / ii = - ii * x"

   210   by (simp add: divide_complex_def)

   211

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

   213   by (simp add: mult.assoc [symmetric])

   214

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

   216   by (simp add: complex_eq_iff)

   217

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

   219   by (simp add: complex_eq_iff)

   220

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

   222   by (simp add: complex_eq_iff)

   223

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

   225   by (simp add: complex_eq_iff)

   226

   227 lemma complex_split_polar: "\<exists>r a. z = complex_of_real r * (cos a + \<i> * sin a)"

   228   by (simp add: complex_eq_iff polar_Ex)

   229

   230 lemma i_even_power [simp]: "\<i> ^ (n * 2) = (-1) ^ n"

   231   by (metis mult.commute power2_i power_mult)

   232

   233 lemma Re_ii_times [simp]: "Re (ii*z) = - Im z"

   234   by simp

   235

   236 lemma Im_ii_times [simp]: "Im (ii*z) = Re z"

   237   by simp

   238

   239 lemma ii_times_eq_iff: "ii*w = z \<longleftrightarrow> w = -(ii*z)"

   240   by auto

   241

   242 lemma divide_numeral_i [simp]: "z / (numeral n * ii) = -(ii*z) / numeral n"

   243   by (metis divide_divide_eq_left divide_i mult.commute mult_minus_right)

   244

   245 subsection \<open>Vector Norm\<close>

   246

   247 instantiation complex :: real_normed_field

   248 begin

   249

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

   251

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

   253   where "cmod \<equiv> norm"

   254

   255 definition complex_sgn_def:

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

   257

   258 definition dist_complex_def:

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

   260

   261 definition open_complex_def:

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

   263

   264 instance proof

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

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

   267     by (simp add: norm_complex_def complex_eq_iff)

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

   269     by (simp add: norm_complex_def complex_eq_iff real_sqrt_sum_squares_triangle_ineq)

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

   271     by (simp add: norm_complex_def complex_eq_iff power_mult_distrib distrib_left [symmetric] real_sqrt_mult)

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

   273     by (simp add: norm_complex_def complex_eq_iff real_sqrt_mult [symmetric] power2_eq_square algebra_simps)

   274 qed (rule complex_sgn_def dist_complex_def open_complex_def)+

   275

   276 end

   277

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

   279   by (simp add: norm_complex_def)

   280

   281 lemma cmod_unit_one: "cmod (cos a + \<i> * sin a) = 1"

   282   by (simp add: norm_complex_def)

   283

   284 lemma cmod_complex_polar: "cmod (r * (cos a + \<i> * sin a)) = \<bar>r\<bar>"

   285   by (simp add: norm_mult cmod_unit_one)

   286

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

   288   unfolding norm_complex_def

   289   by (rule real_sqrt_sum_squares_ge1)

   290

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

   292   by (rule order_trans [OF _ norm_ge_zero]) simp

   293

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

   295   by (rule ord_le_eq_trans [OF norm_triangle_ineq2]) simp

   296

   297 lemma abs_Re_le_cmod: "\<bar>Re x\<bar> \<le> cmod x"

   298   by (simp add: norm_complex_def)

   299

   300 lemma abs_Im_le_cmod: "\<bar>Im x\<bar> \<le> cmod x"

   301   by (simp add: norm_complex_def)

   302

   303 lemma cmod_le: "cmod z \<le> \<bar>Re z\<bar> + \<bar>Im z\<bar>"

   304   apply (subst complex_eq)

   305   apply (rule order_trans)

   306   apply (rule norm_triangle_ineq)

   307   apply (simp add: norm_mult)

   308   done

   309

   310 lemma cmod_eq_Re: "Im z = 0 \<Longrightarrow> cmod z = \<bar>Re z\<bar>"

   311   by (simp add: norm_complex_def)

   312

   313 lemma cmod_eq_Im: "Re z = 0 \<Longrightarrow> cmod z = \<bar>Im z\<bar>"

   314   by (simp add: norm_complex_def)

   315

   316 lemma cmod_power2: "cmod z ^ 2 = (Re z)^2 + (Im z)^2"

   317   by (simp add: norm_complex_def)

   318

   319 lemma cmod_plus_Re_le_0_iff: "cmod z + Re z \<le> 0 \<longleftrightarrow> Re z = - cmod z"

   320   using abs_Re_le_cmod[of z] by auto

   321

   322 lemma Im_eq_0: "\<bar>Re z\<bar> = cmod z \<Longrightarrow> Im z = 0"

   323   by (subst (asm) power_eq_iff_eq_base[symmetric, where n=2])

   324      (auto simp add: norm_complex_def)

   325

   326 lemma abs_sqrt_wlog:

   327   fixes x::"'a::linordered_idom"

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

   329 by (metis abs_ge_zero assms power2_abs)

   330

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

   332   unfolding norm_complex_def

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

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

   335   apply (rule power2_le_imp_le)

   336   apply (simp_all add: power2_sum add.commute sum_squares_bound real_sqrt_mult [symmetric])

   337   done

   338

   339 lemma complex_unit_circle: "z \<noteq> 0 \<Longrightarrow> (Re z / cmod z)\<^sup>2 + (Im z / cmod z)\<^sup>2 = 1"

   340   by (simp add: norm_complex_def divide_simps complex_eq_iff)

   341

   342

   343 text \<open>Properties of complex signum.\<close>

   344

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

   346   by (simp add: sgn_div_norm divide_inverse scaleR_conv_of_real mult.commute)

   347

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

   349   by (simp add: complex_sgn_def divide_inverse)

   350

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

   352   by (simp add: complex_sgn_def divide_inverse)

   353

   354

   355 subsection \<open>Completeness of the Complexes\<close>

   356

   357 lemma bounded_linear_Re: "bounded_linear Re"

   358   by (rule bounded_linear_intro [where K=1], simp_all add: norm_complex_def)

   359

   360 lemma bounded_linear_Im: "bounded_linear Im"

   361   by (rule bounded_linear_intro [where K=1], simp_all add: norm_complex_def)

   362

   363 lemmas Cauchy_Re = bounded_linear.Cauchy [OF bounded_linear_Re]

   364 lemmas Cauchy_Im = bounded_linear.Cauchy [OF bounded_linear_Im]

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

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

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

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

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

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

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

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

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

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

   375 lemmas sums_Re = bounded_linear.sums [OF bounded_linear_Re]

   376 lemmas sums_Im = bounded_linear.sums [OF bounded_linear_Im]

   377

   378 lemma tendsto_Complex [tendsto_intros]:

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

   380   by (auto intro!: tendsto_intros)

   381

   382 lemma tendsto_complex_iff:

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

   384 proof safe

   385   assume "((\<lambda>x. Re (f x)) ---> Re x) F" "((\<lambda>x. Im (f x)) ---> Im x) F"

   386   from tendsto_Complex[OF this] show "(f ---> x) F"

   387     unfolding complex.collapse .

   388 qed (auto intro: tendsto_intros)

   389

   390 lemma continuous_complex_iff: "continuous F f \<longleftrightarrow>

   391     continuous F (\<lambda>x. Re (f x)) \<and> continuous F (\<lambda>x. Im (f x))"

   392   unfolding continuous_def tendsto_complex_iff ..

   393

   394 lemma has_vector_derivative_complex_iff: "(f has_vector_derivative x) F \<longleftrightarrow>

   395     ((\<lambda>x. Re (f x)) has_field_derivative (Re x)) F \<and>

   396     ((\<lambda>x. Im (f x)) has_field_derivative (Im x)) F"

   397   unfolding has_vector_derivative_def has_field_derivative_def has_derivative_def tendsto_complex_iff

   398   by (simp add: field_simps bounded_linear_scaleR_left bounded_linear_mult_right)

   399

   400 lemma has_field_derivative_Re[derivative_intros]:

   401   "(f has_vector_derivative D) F \<Longrightarrow> ((\<lambda>x. Re (f x)) has_field_derivative (Re D)) F"

   402   unfolding has_vector_derivative_complex_iff by safe

   403

   404 lemma has_field_derivative_Im[derivative_intros]:

   405   "(f has_vector_derivative D) F \<Longrightarrow> ((\<lambda>x. Im (f x)) has_field_derivative (Im D)) F"

   406   unfolding has_vector_derivative_complex_iff by safe

   407

   408 instance complex :: banach

   409 proof

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

   411   assume X: "Cauchy X"

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

   413     by (intro tendsto_Complex convergent_LIMSEQ_iff[THEN iffD1] Cauchy_convergent_iff[THEN iffD1] Cauchy_Re Cauchy_Im)

   414   then show "convergent X"

   415     unfolding complex.collapse by (rule convergentI)

   416 qed

   417

   418 declare

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

   420

   421 subsection \<open>Complex Conjugation\<close>

   422

   423 primcorec cnj :: "complex \<Rightarrow> complex" where

   424   "Re (cnj z) = Re z"

   425 | "Im (cnj z) = - Im z"

   426

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

   428   by (simp add: complex_eq_iff)

   429

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

   431   by (simp add: complex_eq_iff)

   432

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

   434   by (simp add: complex_eq_iff)

   435

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

   437   by (simp add: complex_eq_iff)

   438

   439 lemma complex_cnj_add [simp]: "cnj (x + y) = cnj x + cnj y"

   440   by (simp add: complex_eq_iff)

   441

   442 lemma cnj_setsum [simp]: "cnj (setsum f s) = (\<Sum>x\<in>s. cnj (f x))"

   443   by (induct s rule: infinite_finite_induct) auto

   444

   445 lemma complex_cnj_diff [simp]: "cnj (x - y) = cnj x - cnj y"

   446   by (simp add: complex_eq_iff)

   447

   448 lemma complex_cnj_minus [simp]: "cnj (- x) = - cnj x"

   449   by (simp add: complex_eq_iff)

   450

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

   452   by (simp add: complex_eq_iff)

   453

   454 lemma complex_cnj_mult [simp]: "cnj (x * y) = cnj x * cnj y"

   455   by (simp add: complex_eq_iff)

   456

   457 lemma cnj_setprod [simp]: "cnj (setprod f s) = (\<Prod>x\<in>s. cnj (f x))"

   458   by (induct s rule: infinite_finite_induct) auto

   459

   460 lemma complex_cnj_inverse [simp]: "cnj (inverse x) = inverse (cnj x)"

   461   by (simp add: complex_eq_iff)

   462

   463 lemma complex_cnj_divide [simp]: "cnj (x / y) = cnj x / cnj y"

   464   by (simp add: divide_complex_def)

   465

   466 lemma complex_cnj_power [simp]: "cnj (x ^ n) = cnj x ^ n"

   467   by (induct n) simp_all

   468

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

   470   by (simp add: complex_eq_iff)

   471

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

   473   by (simp add: complex_eq_iff)

   474

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

   476   by (simp add: complex_eq_iff)

   477

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

   479   by (simp add: complex_eq_iff)

   480

   481 lemma complex_cnj_scaleR [simp]: "cnj (scaleR r x) = scaleR r (cnj x)"

   482   by (simp add: complex_eq_iff)

   483

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

   485   by (simp add: norm_complex_def)

   486

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

   488   by (simp add: complex_eq_iff)

   489

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

   491   by (simp add: complex_eq_iff)

   492

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

   494   by (simp add: complex_eq_iff)

   495

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

   497   by (simp add: complex_eq_iff)

   498

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

   500   by (simp add: complex_eq_iff power2_eq_square)

   501

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

   503   by (simp add: norm_mult power2_eq_square)

   504

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

   506   by (simp add: norm_complex_def power2_eq_square)

   507

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

   509   by simp

   510

   511 lemma complex_cnj_fact [simp]: "cnj (fact n) = fact n"

   512   by (subst of_nat_fact [symmetric], subst complex_cnj_of_nat) simp

   513

   514 lemma complex_cnj_pochhammer [simp]: "cnj (pochhammer z n) = pochhammer (cnj z) n"

   515   by (induction n arbitrary: z) (simp_all add: pochhammer_rec)

   516

   517 lemma bounded_linear_cnj: "bounded_linear cnj"

   518   using complex_cnj_add complex_cnj_scaleR

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

   520

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

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

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

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

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

   526

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

   528   by (simp add: tendsto_iff dist_complex_def complex_cnj_diff [symmetric] del: complex_cnj_diff)

   529

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

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

   532

   533

   534 subsection\<open>Basic Lemmas\<close>

   535

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

   537   by (metis zero_complex.sel complex_eqI sum_power2_eq_zero_iff)

   538

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

   540   by (metis complex_eq_0 less_numeral_extra(3) sum_power2_gt_zero_iff)

   541

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

   543 by (cases z)

   544    (auto simp: complex_eq_iff norm_complex_def power2_eq_square[symmetric] of_real_power[symmetric]

   545          simp del: of_real_power)

   546

   547 lemma complex_div_cnj: "a / b = (a * cnj b) / (norm b)^2"

   548   using complex_norm_square by auto

   549

   550 lemma Re_complex_div_eq_0: "Re (a / b) = 0 \<longleftrightarrow> Re (a * cnj b) = 0"

   551   by (auto simp add: Re_divide)

   552

   553 lemma Im_complex_div_eq_0: "Im (a / b) = 0 \<longleftrightarrow> Im (a * cnj b) = 0"

   554   by (auto simp add: Im_divide)

   555

   556 lemma complex_div_gt_0:

   557   "(Re (a / b) > 0 \<longleftrightarrow> Re (a * cnj b) > 0) \<and> (Im (a / b) > 0 \<longleftrightarrow> Im (a * cnj b) > 0)"

   558 proof cases

   559   assume "b = 0" then show ?thesis by auto

   560 next

   561   assume "b \<noteq> 0"

   562   then have "0 < (Re b)\<^sup>2 + (Im b)\<^sup>2"

   563     by (simp add: complex_eq_iff sum_power2_gt_zero_iff)

   564   then show ?thesis

   565     by (simp add: Re_divide Im_divide zero_less_divide_iff)

   566 qed

   567

   568 lemma Re_complex_div_gt_0: "Re (a / b) > 0 \<longleftrightarrow> Re (a * cnj b) > 0"

   569   and Im_complex_div_gt_0: "Im (a / b) > 0 \<longleftrightarrow> Im (a * cnj b) > 0"

   570   using complex_div_gt_0 by auto

   571

   572 lemma Re_complex_div_ge_0: "Re(a / b) \<ge> 0 \<longleftrightarrow> Re(a * cnj b) \<ge> 0"

   573   by (metis le_less Re_complex_div_eq_0 Re_complex_div_gt_0)

   574

   575 lemma Im_complex_div_ge_0: "Im(a / b) \<ge> 0 \<longleftrightarrow> Im(a * cnj b) \<ge> 0"

   576   by (metis Im_complex_div_eq_0 Im_complex_div_gt_0 le_less)

   577

   578 lemma Re_complex_div_lt_0: "Re(a / b) < 0 \<longleftrightarrow> Re(a * cnj b) < 0"

   579   by (metis less_asym neq_iff Re_complex_div_eq_0 Re_complex_div_gt_0)

   580

   581 lemma Im_complex_div_lt_0: "Im(a / b) < 0 \<longleftrightarrow> Im(a * cnj b) < 0"

   582   by (metis Im_complex_div_eq_0 Im_complex_div_gt_0 less_asym neq_iff)

   583

   584 lemma Re_complex_div_le_0: "Re(a / b) \<le> 0 \<longleftrightarrow> Re(a * cnj b) \<le> 0"

   585   by (metis not_le Re_complex_div_gt_0)

   586

   587 lemma Im_complex_div_le_0: "Im(a / b) \<le> 0 \<longleftrightarrow> Im(a * cnj b) \<le> 0"

   588   by (metis Im_complex_div_gt_0 not_le)

   589

   590 lemma Re_divide_of_real [simp]: "Re (z / of_real r) = Re z / r"

   591   by (simp add: Re_divide power2_eq_square)

   592

   593 lemma Im_divide_of_real [simp]: "Im (z / of_real r) = Im z / r"

   594   by (simp add: Im_divide power2_eq_square)

   595

   596 lemma Re_divide_Reals: "r \<in> Reals \<Longrightarrow> Re (z / r) = Re z / Re r"

   597   by (metis Re_divide_of_real of_real_Re)

   598

   599 lemma Im_divide_Reals: "r \<in> Reals \<Longrightarrow> Im (z / r) = Im z / Re r"

   600   by (metis Im_divide_of_real of_real_Re)

   601

   602 lemma Re_setsum[simp]: "Re (setsum f s) = (\<Sum>x\<in>s. Re (f x))"

   603   by (induct s rule: infinite_finite_induct) auto

   604

   605 lemma Im_setsum[simp]: "Im (setsum f s) = (\<Sum>x\<in>s. Im(f x))"

   606   by (induct s rule: infinite_finite_induct) auto

   607

   608 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)"

   609   unfolding sums_def tendsto_complex_iff Im_setsum Re_setsum ..

   610

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

   612   unfolding summable_def sums_complex_iff[abs_def] by (metis complex.sel)

   613

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

   615   unfolding summable_complex_iff by simp

   616

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

   618   unfolding summable_complex_iff by blast

   619

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

   621   unfolding summable_complex_iff by blast

   622

   623 lemma complex_is_Nat_iff: "z \<in> \<nat> \<longleftrightarrow> Im z = 0 \<and> (\<exists>i. Re z = of_nat i)"

   624   by (auto simp: Nats_def complex_eq_iff)

   625

   626 lemma complex_is_Int_iff: "z \<in> \<int> \<longleftrightarrow> Im z = 0 \<and> (\<exists>i. Re z = of_int i)"

   627   by (auto simp: Ints_def complex_eq_iff)

   628

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

   630   by (auto simp: Reals_def complex_eq_iff)

   631

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

   633   by (auto simp: complex_is_Real_iff complex_eq_iff)

   634

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

   636   by (simp add: complex_is_Real_iff norm_complex_def)

   637

   638 lemma series_comparison_complex:

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

   640   assumes sg: "summable g"

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

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

   643   shows "summable f"

   644 proof -

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

   646     by (metis abs_of_nonneg in_Reals_norm)

   647   show ?thesis

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

   649     using sg

   650     apply (auto simp: summable_def)

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

   652     apply (auto simp: g sums_Re)

   653     apply (metis fg g)

   654     done

   655 qed

   656

   657 subsection\<open>Polar Form for Complex Numbers\<close>

   658

   659 lemma complex_unimodular_polar: "(norm z = 1) \<Longrightarrow> \<exists>x. z = Complex (cos x) (sin x)"

   660   using sincos_total_2pi [of "Re z" "Im z"]

   661   by auto (metis cmod_power2 complex_eq power_one)

   662

   663 subsubsection \<open>$\cos \theta + i \sin \theta$\<close>

   664

   665 primcorec cis :: "real \<Rightarrow> complex" where

   666   "Re (cis a) = cos a"

   667 | "Im (cis a) = sin a"

   668

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

   670   by (simp add: complex_eq_iff)

   671

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

   673   by (simp add: norm_complex_def)

   674

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

   676   by (simp add: sgn_div_norm)

   677

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

   679   by (metis norm_cis norm_zero zero_neq_one)

   680

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

   682   by (simp add: complex_eq_iff cos_add sin_add)

   683

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

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

   686

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

   688   by (simp add: complex_eq_iff)

   689

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

   691   by (simp add: divide_complex_def cis_mult)

   692

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

   694   by (auto simp add: DeMoivre)

   695

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

   697   by (auto simp add: DeMoivre)

   698

   699 lemma cis_pi: "cis pi = -1"

   700   by (simp add: complex_eq_iff)

   701

   702 subsubsection \<open>$r(\cos \theta + i \sin \theta)$\<close>

   703

   704 definition rcis :: "real \<Rightarrow> real \<Rightarrow> complex" where

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

   706

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

   708   by (simp add: rcis_def)

   709

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

   711   by (simp add: rcis_def)

   712

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

   714   by (simp add: complex_eq_iff polar_Ex)

   715

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

   717   by (simp add: rcis_def norm_mult)

   718

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

   720   by (simp add: rcis_def)

   721

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

   723   by (simp add: rcis_def cis_mult)

   724

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

   726   by (simp add: rcis_def)

   727

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

   729   by (simp add: rcis_def)

   730

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

   732   by (simp add: rcis_def)

   733

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

   735   by (simp add: rcis_def power_mult_distrib DeMoivre)

   736

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

   738   by (simp add: divide_inverse rcis_def)

   739

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

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

   742

   743 subsubsection \<open>Complex exponential\<close>

   744

   745 abbreviation Exp :: "complex \<Rightarrow> complex"

   746   where "Exp \<equiv> exp"

   747

   748 lemma cis_conv_exp: "cis b = exp (\<i> * b)"

   749 proof -

   750   { fix n :: nat

   751     have "\<i> ^ n = fact n *\<^sub>R (cos_coeff n + \<i> * sin_coeff n)"

   752       by (induct n)

   753          (simp_all add: sin_coeff_Suc cos_coeff_Suc complex_eq_iff Re_divide Im_divide field_simps

   754                         power2_eq_square real_of_nat_Suc add_nonneg_eq_0_iff

   755                         real_of_nat_def[symmetric])

   756     then have "(\<i> * complex_of_real b) ^ n /\<^sub>R fact n =

   757         of_real (cos_coeff n * b^n) + \<i> * of_real (sin_coeff n * b^n)"

   758       by (simp add: field_simps) }

   759   then show ?thesis using sin_converges [of b] cos_converges [of b]

   760     by (auto simp add: cis.ctr exp_def simp del: of_real_mult

   761              intro!: sums_unique sums_add sums_mult sums_of_real)

   762 qed

   763

   764 lemma Exp_eq_polar: "Exp z = exp (Re z) * cis (Im z)"

   765   unfolding cis_conv_exp exp_of_real [symmetric] mult_exp_exp by (cases z) simp

   766

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

   768   unfolding Exp_eq_polar by simp

   769

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

   771   unfolding Exp_eq_polar by simp

   772

   773 lemma norm_cos_sin [simp]: "norm (Complex (cos t) (sin t)) = 1"

   774   by (simp add: norm_complex_def)

   775

   776 lemma norm_exp_eq_Re [simp]: "norm (exp z) = exp (Re z)"

   777   by (simp add: cis.code cmod_complex_polar Exp_eq_polar)

   778

   779 lemma complex_Exp_Ex: "\<exists>a r. z = complex_of_real r * Exp a"

   780   apply (insert rcis_Ex [of z])

   781   apply (auto simp add: Exp_eq_polar rcis_def mult.assoc [symmetric])

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

   783   done

   784

   785 lemma Exp_two_pi_i [simp]: "Exp((2::complex) * complex_of_real pi * ii) = 1"

   786   by (simp add: Exp_eq_polar complex_eq_iff)

   787

   788 subsubsection \<open>Complex argument\<close>

   789

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

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

   792

   793 lemma arg_zero: "arg 0 = 0"

   794   by (simp add: arg_def)

   795

   796 lemma arg_unique:

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

   798   shows "arg z = x"

   799 proof -

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

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

   802   proof

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

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

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

   806       unfolding d_def by simp

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

   808       by (simp_all add: complex_eq_iff)

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

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

   811     ultimately have "d = 0"

   812       unfolding sin_zero_iff

   813       by (auto elim!: evenE dest!: less_2_cases)

   814     thus "a = x" unfolding d_def by simp

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

   816   with \<open>z \<noteq> 0\<close> show "arg z = x"

   817     unfolding arg_def by simp

   818 qed

   819

   820 lemma arg_correct:

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

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

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

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

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

   826   have b: "sgn z = cis b"

   827     unfolding z b_def rcis_def using \<open>r \<noteq> 0\<close>

   828     by (simp add: of_real_def sgn_scaleR sgn_if complex_eq_iff)

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

   830     by (induct_tac n) (simp_all add: distrib_left cis_mult [symmetric] complex_eq_iff)

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

   832     by (case_tac x rule: int_diff_cases)

   833        (simp add: right_diff_distrib cis_divide [symmetric] cis_2pi_nat)

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

   835   have "sgn z = cis c"

   836     unfolding b c_def

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

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

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

   840     by (simp add: c_def less_divide_eq divide_le_eq algebra_simps)

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

   842 qed

   843

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

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

   846

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

   848   by (simp add: arg_correct)

   849

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

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

   852

   853 lemma cos_arg_i_mult_zero [simp]: "y \<noteq> 0 \<Longrightarrow> Re y = 0 \<Longrightarrow> cos (arg y) = 0"

   854   using cis_arg [of y] by (simp add: complex_eq_iff)

   855

   856 subsection \<open>Square root of complex numbers\<close>

   857

   858 primcorec csqrt :: "complex \<Rightarrow> complex" where

   859   "Re (csqrt z) = sqrt ((cmod z + Re z) / 2)"

   860 | "Im (csqrt z) = (if Im z = 0 then 1 else sgn (Im z)) * sqrt ((cmod z - Re z) / 2)"

   861

   862 lemma csqrt_of_real_nonneg [simp]: "Im x = 0 \<Longrightarrow> Re x \<ge> 0 \<Longrightarrow> csqrt x = sqrt (Re x)"

   863   by (simp add: complex_eq_iff norm_complex_def)

   864

   865 lemma csqrt_of_real_nonpos [simp]: "Im x = 0 \<Longrightarrow> Re x \<le> 0 \<Longrightarrow> csqrt x = \<i> * sqrt \<bar>Re x\<bar>"

   866   by (simp add: complex_eq_iff norm_complex_def)

   867

   868 lemma of_real_sqrt: "x \<ge> 0 \<Longrightarrow> of_real (sqrt x) = csqrt (of_real x)"

   869   by (simp add: complex_eq_iff norm_complex_def)

   870

   871 lemma csqrt_0 [simp]: "csqrt 0 = 0"

   872   by simp

   873

   874 lemma csqrt_1 [simp]: "csqrt 1 = 1"

   875   by simp

   876

   877 lemma csqrt_ii [simp]: "csqrt \<i> = (1 + \<i>) / sqrt 2"

   878   by (simp add: complex_eq_iff Re_divide Im_divide real_sqrt_divide real_div_sqrt)

   879

   880 lemma power2_csqrt[simp,algebra]: "(csqrt z)\<^sup>2 = z"

   881 proof cases

   882   assume "Im z = 0" then show ?thesis

   883     using real_sqrt_pow2[of "Re z"] real_sqrt_pow2[of "- Re z"]

   884     by (cases "0::real" "Re z" rule: linorder_cases)

   885        (simp_all add: complex_eq_iff Re_power2 Im_power2 power2_eq_square cmod_eq_Re)

   886 next

   887   assume "Im z \<noteq> 0"

   888   moreover

   889   have "cmod z * cmod z - Re z * Re z = Im z * Im z"

   890     by (simp add: norm_complex_def power2_eq_square)

   891   moreover

   892   have "\<bar>Re z\<bar> \<le> cmod z"

   893     by (simp add: norm_complex_def)

   894   ultimately show ?thesis

   895     by (simp add: Re_power2 Im_power2 complex_eq_iff real_sgn_eq

   896                   field_simps real_sqrt_mult[symmetric] real_sqrt_divide)

   897 qed

   898

   899 lemma csqrt_eq_0 [simp]: "csqrt z = 0 \<longleftrightarrow> z = 0"

   900   by auto (metis power2_csqrt power_eq_0_iff)

   901

   902 lemma csqrt_eq_1 [simp]: "csqrt z = 1 \<longleftrightarrow> z = 1"

   903   by auto (metis power2_csqrt power2_eq_1_iff)

   904

   905 lemma csqrt_principal: "0 < Re (csqrt z) \<or> Re (csqrt z) = 0 \<and> 0 \<le> Im (csqrt z)"

   906   by (auto simp add: not_less cmod_plus_Re_le_0_iff Im_eq_0)

   907

   908 lemma Re_csqrt: "0 \<le> Re (csqrt z)"

   909   by (metis csqrt_principal le_less)

   910

   911 lemma csqrt_square:

   912   assumes "0 < Re b \<or> (Re b = 0 \<and> 0 \<le> Im b)"

   913   shows "csqrt (b^2) = b"

   914 proof -

   915   have "csqrt (b^2) = b \<or> csqrt (b^2) = - b"

   916     unfolding power2_eq_iff[symmetric] by (simp add: power2_csqrt)

   917   moreover have "csqrt (b^2) \<noteq> -b \<or> b = 0"

   918     using csqrt_principal[of "b ^ 2"] assms by (intro disjCI notI) (auto simp: complex_eq_iff)

   919   ultimately show ?thesis

   920     by auto

   921 qed

   922

   923 lemma csqrt_unique:

   924     "w^2 = z \<Longrightarrow> (0 < Re w \<or> Re w = 0 \<and> 0 \<le> Im w) \<Longrightarrow> csqrt z = w"

   925   by (auto simp: csqrt_square)

   926

   927 lemma csqrt_minus [simp]:

   928   assumes "Im x < 0 \<or> (Im x = 0 \<and> 0 \<le> Re x)"

   929   shows "csqrt (- x) = \<i> * csqrt x"

   930 proof -

   931   have "csqrt ((\<i> * csqrt x)^2) = \<i> * csqrt x"

   932   proof (rule csqrt_square)

   933     have "Im (csqrt x) \<le> 0"

   934       using assms by (auto simp add: cmod_eq_Re mult_le_0_iff field_simps complex_Re_le_cmod)

   935     then show "0 < Re (\<i> * csqrt x) \<or> Re (\<i> * csqrt x) = 0 \<and> 0 \<le> Im (\<i> * csqrt x)"

   936       by (auto simp add: Re_csqrt simp del: csqrt.simps)

   937   qed

   938   also have "(\<i> * csqrt x)^2 = - x"

   939     by (simp add: power_mult_distrib)

   940   finally show ?thesis .

   941 qed

   942

   943 text \<open>Legacy theorem names\<close>

   944

   945 lemmas expand_complex_eq = complex_eq_iff

   946 lemmas complex_Re_Im_cancel_iff = complex_eq_iff

   947 lemmas complex_equality = complex_eqI

   948 lemmas cmod_def = norm_complex_def

   949 lemmas complex_norm_def = norm_complex_def

   950 lemmas complex_divide_def = divide_complex_def

   951

   952 lemma legacy_Complex_simps:

   953   shows Complex_eq_0: "Complex a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"

   954     and complex_add: "Complex a b + Complex c d = Complex (a + c) (b + d)"

   955     and complex_minus: "- (Complex a b) = Complex (- a) (- b)"

   956     and complex_diff: "Complex a b - Complex c d = Complex (a - c) (b - d)"

   957     and Complex_eq_1: "Complex a b = 1 \<longleftrightarrow> a = 1 \<and> b = 0"

   958     and Complex_eq_neg_1: "Complex a b = - 1 \<longleftrightarrow> a = - 1 \<and> b = 0"

   959     and complex_mult: "Complex a b * Complex c d = Complex (a * c - b * d) (a * d + b * c)"

   960     and complex_inverse: "inverse (Complex a b) = Complex (a / (a\<^sup>2 + b\<^sup>2)) (- b / (a\<^sup>2 + b\<^sup>2))"

   961     and Complex_eq_numeral: "Complex a b = numeral w \<longleftrightarrow> a = numeral w \<and> b = 0"

   962     and Complex_eq_neg_numeral: "Complex a b = - numeral w \<longleftrightarrow> a = - numeral w \<and> b = 0"

   963     and complex_scaleR: "scaleR r (Complex a b) = Complex (r * a) (r * b)"

   964     and Complex_eq_i: "(Complex x y = ii) = (x = 0 \<and> y = 1)"

   965     and i_mult_Complex: "ii * Complex a b = Complex (- b) a"

   966     and Complex_mult_i: "Complex a b * ii = Complex (- b) a"

   967     and i_complex_of_real: "ii * complex_of_real r = Complex 0 r"

   968     and complex_of_real_i: "complex_of_real r * ii = Complex 0 r"

   969     and Complex_add_complex_of_real: "Complex x y + complex_of_real r = Complex (x+r) y"

   970     and complex_of_real_add_Complex: "complex_of_real r + Complex x y = Complex (r+x) y"

   971     and Complex_mult_complex_of_real: "Complex x y * complex_of_real r = Complex (x*r) (y*r)"

   972     and complex_of_real_mult_Complex: "complex_of_real r * Complex x y = Complex (r*x) (r*y)"

   973     and complex_eq_cancel_iff2: "(Complex x y = complex_of_real xa) = (x = xa & y = 0)"

   974     and complex_cn: "cnj (Complex a b) = Complex a (- b)"

   975     and Complex_setsum': "setsum (%x. Complex (f x) 0) s = Complex (setsum f s) 0"

   976     and Complex_setsum: "Complex (setsum f s) 0 = setsum (%x. Complex (f x) 0) s"

   977     and complex_of_real_def: "complex_of_real r = Complex r 0"

   978     and complex_norm: "cmod (Complex x y) = sqrt (x\<^sup>2 + y\<^sup>2)"

   979   by (simp_all add: norm_complex_def field_simps complex_eq_iff Re_divide Im_divide del: Complex_eq)

   980

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

   982   by (metis Reals_of_real complex_of_real_def)

   983

   984 end