src/HOL/Complex.thy
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tuned proofs;
     1 (*  Title:       HOL/Complex.thy

     2     Author:      Jacques D. Fleuriot, 2001 University of Edinburgh

     3     Author:      Lawrence C Paulson, 2003/4

     4 *)

     5

     6 section \<open>Complex Numbers: Rectangular and Polar Representations\<close>

     7

     8 theory Complex

     9 imports Transcendental

    10 begin

    11

    12 text \<open>

    13   We use the \<^theory_text>\<open>codatatype\<close> command to define the type of complex numbers. This

    14   allows us to use \<^theory_text>\<open>primcorec\<close> to define complex functions by defining their

    15   real and imaginary result separately.

    16 \<close>

    17

    18 codatatype complex = Complex (Re: real) (Im: real)

    19

    20 lemma complex_surj: "Complex (Re z) (Im z) = z"

    21   by (rule complex.collapse)

    22

    23 lemma complex_eqI [intro?]: "Re x = Re y \<Longrightarrow> Im x = Im y \<Longrightarrow> x = y"

    24   by (rule complex.expand) simp

    25

    26 lemma complex_eq_iff: "x = y \<longleftrightarrow> Re x = Re y \<and> Im x = Im y"

    27   by (auto intro: complex.expand)

    28

    29

    30 subsection \<open>Addition and Subtraction\<close>

    31

    32 instantiation complex :: ab_group_add

    33 begin

    34

    35 primcorec zero_complex

    36   where

    37     "Re 0 = 0"

    38   | "Im 0 = 0"

    39

    40 primcorec plus_complex

    41   where

    42     "Re (x + y) = Re x + Re y"

    43   | "Im (x + y) = Im x + Im y"

    44

    45 primcorec uminus_complex

    46   where

    47     "Re (- x) = - Re x"

    48   | "Im (- x) = - Im x"

    49

    50 primcorec minus_complex

    51   where

    52     "Re (x - y) = Re x - Re y"

    53   | "Im (x - y) = Im x - Im y"

    54

    55 instance

    56   by standard (simp_all add: complex_eq_iff)

    57

    58 end

    59

    60

    61 subsection \<open>Multiplication and Division\<close>

    62

    63 instantiation complex :: field

    64 begin

    65

    66 primcorec one_complex

    67   where

    68     "Re 1 = 1"

    69   | "Im 1 = 0"

    70

    71 primcorec times_complex

    72   where

    73     "Re (x * y) = Re x * Re y - Im x * Im y"

    74   | "Im (x * y) = Re x * Im y + Im x * Re y"

    75

    76 primcorec inverse_complex

    77   where

    78     "Re (inverse x) = Re x / ((Re x)\<^sup>2 + (Im x)\<^sup>2)"

    79   | "Im (inverse x) = - Im x / ((Re x)\<^sup>2 + (Im x)\<^sup>2)"

    80

    81 definition "x div y = x * inverse y" for x y :: complex

    82

    83 instance

    84   by standard

    85      (simp_all add: complex_eq_iff divide_complex_def

    86       distrib_left distrib_right right_diff_distrib left_diff_distrib

    87       power2_eq_square add_divide_distrib [symmetric])

    88

    89 end

    90

    91 lemma Re_divide: "Re (x / y) = (Re x * Re y + Im x * Im y) / ((Re y)\<^sup>2 + (Im y)\<^sup>2)"

    92   by (simp add: divide_complex_def add_divide_distrib)

    93

    94 lemma Im_divide: "Im (x / y) = (Im x * Re y - Re x * Im y) / ((Re y)\<^sup>2 + (Im y)\<^sup>2)"

    95   unfolding divide_complex_def times_complex.sel inverse_complex.sel

    96   by (simp add: divide_simps)

    97

    98 lemma Re_power2: "Re (x ^ 2) = (Re x)^2 - (Im x)^2"

    99   by (simp add: power2_eq_square)

   100

   101 lemma Im_power2: "Im (x ^ 2) = 2 * Re x * Im x"

   102   by (simp add: power2_eq_square)

   103

   104 lemma Re_power_real [simp]: "Im x = 0 \<Longrightarrow> Re (x ^ n) = Re x ^ n "

   105   by (induct n) simp_all

   106

   107 lemma Im_power_real [simp]: "Im x = 0 \<Longrightarrow> Im (x ^ n) = 0"

   108   by (induct n) simp_all

   109

   110

   111 subsection \<open>Scalar Multiplication\<close>

   112

   113 instantiation complex :: real_field

   114 begin

   115

   116 primcorec scaleR_complex

   117   where

   118     "Re (scaleR r x) = r * Re x"

   119   | "Im (scaleR r x) = r * Im x"

   120

   121 instance

   122 proof

   123   fix a b :: real and x y :: complex

   124   show "scaleR a (x + y) = scaleR a x + scaleR a y"

   125     by (simp add: complex_eq_iff distrib_left)

   126   show "scaleR (a + b) x = scaleR a x + scaleR b x"

   127     by (simp add: complex_eq_iff distrib_right)

   128   show "scaleR a (scaleR b x) = scaleR (a * b) x"

   129     by (simp add: complex_eq_iff mult.assoc)

   130   show "scaleR 1 x = x"

   131     by (simp add: complex_eq_iff)

   132   show "scaleR a x * y = scaleR a (x * y)"

   133     by (simp add: complex_eq_iff algebra_simps)

   134   show "x * scaleR a y = scaleR a (x * y)"

   135     by (simp add: complex_eq_iff algebra_simps)

   136 qed

   137

   138 end

   139

   140

   141 subsection \<open>Numerals, Arithmetic, and Embedding from Reals\<close>

   142

   143 abbreviation complex_of_real :: "real \<Rightarrow> complex"

   144   where "complex_of_real \<equiv> of_real"

   145

   146 declare [[coercion "of_real :: real \<Rightarrow> complex"]]

   147 declare [[coercion "of_rat :: rat \<Rightarrow> complex"]]

   148 declare [[coercion "of_int :: int \<Rightarrow> complex"]]

   149 declare [[coercion "of_nat :: nat \<Rightarrow> complex"]]

   150

   151 lemma complex_Re_of_nat [simp]: "Re (of_nat n) = of_nat n"

   152   by (induct n) simp_all

   153

   154 lemma complex_Im_of_nat [simp]: "Im (of_nat n) = 0"

   155   by (induct n) simp_all

   156

   157 lemma complex_Re_of_int [simp]: "Re (of_int z) = of_int z"

   158   by (cases z rule: int_diff_cases) simp

   159

   160 lemma complex_Im_of_int [simp]: "Im (of_int z) = 0"

   161   by (cases z rule: int_diff_cases) simp

   162

   163 lemma complex_Re_numeral [simp]: "Re (numeral v) = numeral v"

   164   using complex_Re_of_int [of "numeral v"] by simp

   165

   166 lemma complex_Im_numeral [simp]: "Im (numeral v) = 0"

   167   using complex_Im_of_int [of "numeral v"] by simp

   168

   169 lemma Re_complex_of_real [simp]: "Re (complex_of_real z) = z"

   170   by (simp add: of_real_def)

   171

   172 lemma Im_complex_of_real [simp]: "Im (complex_of_real z) = 0"

   173   by (simp add: of_real_def)

   174

   175 lemma Re_divide_numeral [simp]: "Re (z / numeral w) = Re z / numeral w"

   176   by (simp add: Re_divide sqr_conv_mult)

   177

   178 lemma Im_divide_numeral [simp]: "Im (z / numeral w) = Im z / numeral w"

   179   by (simp add: Im_divide sqr_conv_mult)

   180

   181 lemma Re_divide_of_nat [simp]: "Re (z / of_nat n) = Re z / of_nat n"

   182   by (cases n) (simp_all add: Re_divide divide_simps power2_eq_square del: of_nat_Suc)

   183

   184 lemma Im_divide_of_nat [simp]: "Im (z / of_nat n) = Im z / of_nat n"

   185   by (cases n) (simp_all add: Im_divide divide_simps power2_eq_square del: of_nat_Suc)

   186

   187 lemma of_real_Re [simp]: "z \<in> \<real> \<Longrightarrow> of_real (Re z) = z"

   188   by (auto simp: Reals_def)

   189

   190 lemma complex_Re_fact [simp]: "Re (fact n) = fact n"

   191 proof -

   192   have "(fact n :: complex) = of_real (fact n)"

   193     by simp

   194   also have "Re \<dots> = fact n"

   195     by (subst Re_complex_of_real) simp_all

   196   finally show ?thesis .

   197 qed

   198

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

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

   201

   202

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

   204

   205 primcorec "ii" :: complex  ("\<i>")

   206   where

   207     "Re \<i> = 0"

   208   | "Im \<i> = 1"

   209

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

   211   by (simp add: complex_eq_iff)

   212

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

   214   by (simp add: complex_eq_iff)

   215

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

   217   by (simp add: fun_eq_iff complex_eq)

   218

   219 lemma i_squared [simp]: "\<i> * \<i> = -1"

   220   by (simp add: complex_eq_iff)

   221

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

   223   by (simp add: power2_eq_square)

   224

   225 lemma inverse_i [simp]: "inverse \<i> = - \<i>"

   226   by (rule inverse_unique) simp

   227

   228 lemma divide_i [simp]: "x / \<i> = - \<i> * x"

   229   by (simp add: divide_complex_def)

   230

   231 lemma complex_i_mult_minus [simp]: "\<i> * (\<i> * x) = - x"

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

   233

   234 lemma complex_i_not_zero [simp]: "\<i> \<noteq> 0"

   235   by (simp add: complex_eq_iff)

   236

   237 lemma complex_i_not_one [simp]: "\<i> \<noteq> 1"

   238   by (simp add: complex_eq_iff)

   239

   240 lemma complex_i_not_numeral [simp]: "\<i> \<noteq> numeral w"

   241   by (simp add: complex_eq_iff)

   242

   243 lemma complex_i_not_neg_numeral [simp]: "\<i> \<noteq> - numeral w"

   244   by (simp add: complex_eq_iff)

   245

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

   247   by (simp add: complex_eq_iff polar_Ex)

   248

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

   250   by (metis mult.commute power2_i power_mult)

   251

   252 lemma Re_ii_times [simp]: "Re (\<i> * z) = - Im z"

   253   by simp

   254

   255 lemma Im_ii_times [simp]: "Im (\<i> * z) = Re z"

   256   by simp

   257

   258 lemma ii_times_eq_iff: "\<i> * w = z \<longleftrightarrow> w = - (\<i> * z)"

   259   by auto

   260

   261 lemma divide_numeral_i [simp]: "z / (numeral n * \<i>) = - (\<i> * z) / numeral n"

   262   by (metis divide_divide_eq_left divide_i mult.commute mult_minus_right)

   263

   264

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

   266

   267 instantiation complex :: real_normed_field

   268 begin

   269

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

   271

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

   273   where "cmod \<equiv> norm"

   274

   275 definition complex_sgn_def: "sgn x = x /\<^sub>R cmod x"

   276

   277 definition dist_complex_def: "dist x y = cmod (x - y)"

   278

   279 definition uniformity_complex_def [code del]:

   280   "(uniformity :: (complex \<times> complex) filter) = (INF e:{0 <..}. principal {(x, y). dist x y < e})"

   281

   282 definition open_complex_def [code del]:

   283   "open (U :: complex set) \<longleftrightarrow> (\<forall>x\<in>U. eventually (\<lambda>(x', y). x' = x \<longrightarrow> y \<in> U) uniformity)"

   284

   285 instance

   286 proof

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

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

   289     by (simp add: norm_complex_def complex_eq_iff)

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

   291     by (simp add: norm_complex_def complex_eq_iff real_sqrt_sum_squares_triangle_ineq)

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

   293     by (simp add: norm_complex_def complex_eq_iff power_mult_distrib distrib_left [symmetric]

   294         real_sqrt_mult)

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

   296     by (simp add: norm_complex_def complex_eq_iff real_sqrt_mult [symmetric]

   297         power2_eq_square algebra_simps)

   298 qed (rule complex_sgn_def dist_complex_def open_complex_def uniformity_complex_def)+

   299

   300 end

   301

   302 declare uniformity_Abort[where 'a = complex, code]

   303

   304 lemma norm_ii [simp]: "norm \<i> = 1"

   305   by (simp add: norm_complex_def)

   306

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

   308   by (simp add: norm_complex_def)

   309

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

   311   by (simp add: norm_mult cmod_unit_one)

   312

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

   314   unfolding norm_complex_def by (rule real_sqrt_sum_squares_ge1)

   315

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

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

   318

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

   320   by (rule ord_le_eq_trans [OF norm_triangle_ineq2]) simp

   321

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

   323   by (simp add: norm_complex_def)

   324

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

   326   by (simp add: norm_complex_def)

   327

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

   329   apply (subst complex_eq)

   330   apply (rule order_trans)

   331    apply (rule norm_triangle_ineq)

   332   apply (simp add: norm_mult)

   333   done

   334

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

   336   by (simp add: norm_complex_def)

   337

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

   339   by (simp add: norm_complex_def)

   340

   341 lemma cmod_power2: "(cmod z)\<^sup>2 = (Re z)\<^sup>2 + (Im z)\<^sup>2"

   342   by (simp add: norm_complex_def)

   343

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

   345   using abs_Re_le_cmod[of z] by auto

   346

   347 lemma cmod_Re_le_iff: "Im x = Im y \<Longrightarrow> cmod x \<le> cmod y \<longleftrightarrow> \<bar>Re x\<bar> \<le> \<bar>Re y\<bar>"

   348   by (metis add.commute add_le_cancel_left norm_complex_def real_sqrt_abs real_sqrt_le_iff)

   349

   350 lemma cmod_Im_le_iff: "Re x = Re y \<Longrightarrow> cmod x \<le> cmod y \<longleftrightarrow> \<bar>Im x\<bar> \<le> \<bar>Im y\<bar>"

   351   by (metis add_le_cancel_left norm_complex_def real_sqrt_abs real_sqrt_le_iff)

   352

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

   354   by (subst (asm) power_eq_iff_eq_base[symmetric, where n=2]) (auto simp add: norm_complex_def)

   355

   356 lemma abs_sqrt_wlog: "(\<And>x. x \<ge> 0 \<Longrightarrow> P x (x\<^sup>2)) \<Longrightarrow> P \<bar>x\<bar> (x\<^sup>2)"

   357   for x::"'a::linordered_idom"

   358   by (metis abs_ge_zero power2_abs)

   359

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

   361   unfolding norm_complex_def

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

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

   364   apply (rule power2_le_imp_le)

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

   366   done

   367

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

   369   by (simp add: norm_complex_def divide_simps complex_eq_iff)

   370

   371

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

   373

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

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

   376

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

   378   by (simp add: complex_sgn_def divide_inverse)

   379

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

   381   by (simp add: complex_sgn_def divide_inverse)

   382

   383

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

   385

   386 lemma bounded_linear_Re: "bounded_linear Re"

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

   388

   389 lemma bounded_linear_Im: "bounded_linear Im"

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

   391

   392 lemmas Cauchy_Re = bounded_linear.Cauchy [OF bounded_linear_Re]

   393 lemmas Cauchy_Im = bounded_linear.Cauchy [OF bounded_linear_Im]

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

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

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

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

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

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

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

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

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

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

   404 lemmas sums_Re = bounded_linear.sums [OF bounded_linear_Re]

   405 lemmas sums_Im = bounded_linear.sums [OF bounded_linear_Im]

   406

   407 lemma tendsto_Complex [tendsto_intros]:

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

   409   by (auto intro!: tendsto_intros)

   410

   411 lemma tendsto_complex_iff:

   412   "(f \<longlongrightarrow> x) F \<longleftrightarrow> (((\<lambda>x. Re (f x)) \<longlongrightarrow> Re x) F \<and> ((\<lambda>x. Im (f x)) \<longlongrightarrow> Im x) F)"

   413 proof safe

   414   assume "((\<lambda>x. Re (f x)) \<longlongrightarrow> Re x) F" "((\<lambda>x. Im (f x)) \<longlongrightarrow> Im x) F"

   415   from tendsto_Complex[OF this] show "(f \<longlongrightarrow> x) F"

   416     unfolding complex.collapse .

   417 qed (auto intro: tendsto_intros)

   418

   419 lemma continuous_complex_iff:

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

   421   by (simp only: continuous_def tendsto_complex_iff)

   422

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

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

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

   426   by (simp add: has_vector_derivative_def has_field_derivative_def has_derivative_def

   427       tendsto_complex_iff field_simps bounded_linear_scaleR_left bounded_linear_mult_right)

   428

   429 lemma has_field_derivative_Re[derivative_intros]:

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

   431   unfolding has_vector_derivative_complex_iff by safe

   432

   433 lemma has_field_derivative_Im[derivative_intros]:

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

   435   unfolding has_vector_derivative_complex_iff by safe

   436

   437 instance complex :: banach

   438 proof

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

   440   assume X: "Cauchy X"

   441   then have "(\<lambda>n. Complex (Re (X n)) (Im (X n))) \<longlonglongrightarrow>

   442     Complex (lim (\<lambda>n. Re (X n))) (lim (\<lambda>n. Im (X n)))"

   443     by (intro tendsto_Complex convergent_LIMSEQ_iff[THEN iffD1]

   444         Cauchy_convergent_iff[THEN iffD1] Cauchy_Re Cauchy_Im)

   445   then show "convergent X"

   446     unfolding complex.collapse by (rule convergentI)

   447 qed

   448

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

   450

   451

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

   453

   454 primcorec cnj :: "complex \<Rightarrow> complex"

   455   where

   456     "Re (cnj z) = Re z"

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

   458

   459 lemma complex_cnj_cancel_iff [simp]: "cnj x = cnj y \<longleftrightarrow> x = y"

   460   by (simp add: complex_eq_iff)

   461

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

   463   by (simp add: complex_eq_iff)

   464

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

   466   by (simp add: complex_eq_iff)

   467

   468 lemma complex_cnj_zero_iff [iff]: "cnj z = 0 \<longleftrightarrow> z = 0"

   469   by (simp add: complex_eq_iff)

   470

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

   472   by (simp add: complex_eq_iff)

   473

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

   475   by (induct s rule: infinite_finite_induct) auto

   476

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

   478   by (simp add: complex_eq_iff)

   479

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

   481   by (simp add: complex_eq_iff)

   482

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

   484   by (simp add: complex_eq_iff)

   485

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

   487   by (simp add: complex_eq_iff)

   488

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

   490   by (induct s rule: infinite_finite_induct) auto

   491

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

   493   by (simp add: complex_eq_iff)

   494

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

   496   by (simp add: divide_complex_def)

   497

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

   499   by (induct n) simp_all

   500

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

   502   by (simp add: complex_eq_iff)

   503

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

   505   by (simp add: complex_eq_iff)

   506

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

   508   by (simp add: complex_eq_iff)

   509

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

   511   by (simp add: complex_eq_iff)

   512

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

   514   by (simp add: complex_eq_iff)

   515

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

   517   by (simp add: norm_complex_def)

   518

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

   520   by (simp add: complex_eq_iff)

   521

   522 lemma complex_cnj_i [simp]: "cnj \<i> = - \<i>"

   523   by (simp add: complex_eq_iff)

   524

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

   526   by (simp add: complex_eq_iff)

   527

   528 lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im z) * \<i>"

   529   by (simp add: complex_eq_iff)

   530

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

   532   by (simp add: complex_eq_iff power2_eq_square)

   533

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

   535   by (simp add: norm_mult power2_eq_square)

   536

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

   538   by (simp add: norm_complex_def power2_eq_square)

   539

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

   541   by simp

   542

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

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

   545

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

   547   by (induct n arbitrary: z) (simp_all add: pochhammer_rec)

   548

   549 lemma bounded_linear_cnj: "bounded_linear cnj"

   550   using complex_cnj_add complex_cnj_scaleR by (rule bounded_linear_intro [where K=1]) simp

   551

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

   553   and isCont_cnj [simp] = bounded_linear.isCont [OF bounded_linear_cnj]

   554   and continuous_cnj [simp, continuous_intros] = bounded_linear.continuous [OF bounded_linear_cnj]

   555   and continuous_on_cnj [simp, continuous_intros] = bounded_linear.continuous_on [OF bounded_linear_cnj]

   556   and has_derivative_cnj [simp, derivative_intros] = bounded_linear.has_derivative [OF bounded_linear_cnj]

   557

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

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

   560

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

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

   563

   564

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

   566

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

   568   by (metis zero_complex.sel complex_eqI sum_power2_eq_zero_iff)

   569

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

   571   by (metis complex_eq_0 less_numeral_extra(3) sum_power2_gt_zero_iff)

   572

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

   574   by (cases z)

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

   576       simp del: of_real_power)

   577

   578 lemma complex_div_cnj: "a / b = (a * cnj b) / (norm b)\<^sup>2"

   579   using complex_norm_square by auto

   580

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

   582   by (auto simp add: Re_divide)

   583

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

   585   by (auto simp add: Im_divide)

   586

   587 lemma complex_div_gt_0: "(Re (a / b) > 0 \<longleftrightarrow> Re (a * cnj b) > 0) \<and> (Im (a / b) > 0 \<longleftrightarrow> Im (a * cnj b) > 0)"

   588 proof (cases "b = 0")

   589   case True

   590   then show ?thesis by auto

   591 next

   592   case False

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

   594     by (simp add: complex_eq_iff sum_power2_gt_zero_iff)

   595   then show ?thesis

   596     by (simp add: Re_divide Im_divide zero_less_divide_iff)

   597 qed

   598

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

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

   601   using complex_div_gt_0 by auto

   602

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

   604   by (metis le_less Re_complex_div_eq_0 Re_complex_div_gt_0)

   605

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

   607   by (metis Im_complex_div_eq_0 Im_complex_div_gt_0 le_less)

   608

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

   610   by (metis less_asym neq_iff Re_complex_div_eq_0 Re_complex_div_gt_0)

   611

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

   613   by (metis Im_complex_div_eq_0 Im_complex_div_gt_0 less_asym neq_iff)

   614

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

   616   by (metis not_le Re_complex_div_gt_0)

   617

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

   619   by (metis Im_complex_div_gt_0 not_le)

   620

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

   622   by (simp add: Re_divide power2_eq_square)

   623

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

   625   by (simp add: Im_divide power2_eq_square)

   626

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

   628   by (metis Re_divide_of_real of_real_Re)

   629

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

   631   by (metis Im_divide_of_real of_real_Re)

   632

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

   634   by (induct s rule: infinite_finite_induct) auto

   635

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

   637   by (induct s rule: infinite_finite_induct) auto

   638

   639 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)"

   640   unfolding sums_def tendsto_complex_iff Im_setsum Re_setsum ..

   641

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

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

   644

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

   646   unfolding summable_complex_iff by simp

   647

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

   649   unfolding summable_complex_iff by blast

   650

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

   652   unfolding summable_complex_iff by blast

   653

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

   655   by (auto simp: Nats_def complex_eq_iff)

   656

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

   658   by (auto simp: Ints_def complex_eq_iff)

   659

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

   661   by (auto simp: Reals_def complex_eq_iff)

   662

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

   664   by (auto simp: complex_is_Real_iff complex_eq_iff)

   665

   666 lemma in_Reals_norm: "z \<in> \<real> \<Longrightarrow> norm z = \<bar>Re z\<bar>"

   667   by (simp add: complex_is_Real_iff norm_complex_def)

   668

   669 lemma series_comparison_complex:

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

   671   assumes sg: "summable g"

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

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

   674   shows "summable f"

   675 proof -

   676   have g: "\<And>n. cmod (g n) = Re (g n)"

   677     using assms by (metis abs_of_nonneg in_Reals_norm)

   678   show ?thesis

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

   680     using sg

   681      apply (auto simp: summable_def)

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

   683      apply (auto simp: g sums_Re)

   684     apply (metis fg g)

   685     done

   686 qed

   687

   688

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

   690

   691 lemma complex_unimodular_polar:

   692   assumes "norm z = 1"

   693   obtains t where "0 \<le> t" "t < 2 * pi" "z = Complex (cos t) (sin t)"

   694   by (metis cmod_power2 one_power2 complex_surj sincos_total_2pi [of "Re z" "Im z"] assms)

   695

   696

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

   698

   699 primcorec cis :: "real \<Rightarrow> complex"

   700   where

   701     "Re (cis a) = cos a"

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

   703

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

   705   by (simp add: complex_eq_iff)

   706

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

   708   by (simp add: norm_complex_def)

   709

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

   711   by (simp add: sgn_div_norm)

   712

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

   714   by (metis norm_cis norm_zero zero_neq_one)

   715

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

   717   by (simp add: complex_eq_iff cos_add sin_add)

   718

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

   720   by (induct n) (simp_all add: algebra_simps cis_mult)

   721

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

   723   by (simp add: complex_eq_iff)

   724

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

   726   by (simp add: divide_complex_def cis_mult)

   727

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

   729   by (auto simp add: DeMoivre)

   730

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

   732   by (auto simp add: DeMoivre)

   733

   734 lemma cis_pi: "cis pi = -1"

   735   by (simp add: complex_eq_iff)

   736

   737

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

   739

   740 definition rcis :: "real \<Rightarrow> real \<Rightarrow> complex"

   741   where "rcis r a = complex_of_real r * cis a"

   742

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

   744   by (simp add: rcis_def)

   745

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

   747   by (simp add: rcis_def)

   748

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

   750   by (simp add: complex_eq_iff polar_Ex)

   751

   752 lemma complex_mod_rcis [simp]: "cmod (rcis r a) = \<bar>r\<bar>"

   753   by (simp add: rcis_def norm_mult)

   754

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

   756   by (simp add: rcis_def)

   757

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

   759   by (simp add: rcis_def cis_mult)

   760

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

   762   by (simp add: rcis_def)

   763

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

   765   by (simp add: rcis_def)

   766

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

   768   by (simp add: rcis_def)

   769

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

   771   by (simp add: rcis_def power_mult_distrib DeMoivre)

   772

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

   774   by (simp add: divide_inverse rcis_def)

   775

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

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

   778

   779

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

   781

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

   783 proof -

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

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

   786     for n :: nat

   787   proof -

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

   789       by (induct n)

   790         (simp_all add: sin_coeff_Suc cos_coeff_Suc complex_eq_iff Re_divide Im_divide field_simps

   791           power2_eq_square add_nonneg_eq_0_iff)

   792     then show ?thesis

   793       by (simp add: field_simps)

   794   qed

   795   then show ?thesis

   796     using sin_converges [of b] cos_converges [of b]

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

   798         intro!: sums_unique sums_add sums_mult sums_of_real)

   799 qed

   800

   801 lemma exp_eq_polar: "exp z = exp (Re z) * cis (Im z)"

   802   unfolding cis_conv_exp exp_of_real [symmetric] mult_exp_exp

   803   by (cases z) simp

   804

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

   806   unfolding exp_eq_polar by simp

   807

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

   809   unfolding exp_eq_polar by simp

   810

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

   812   by (simp add: norm_complex_def)

   813

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

   815   by (simp add: cis.code cmod_complex_polar exp_eq_polar)

   816

   817 lemma complex_exp_exists: "\<exists>a r. z = complex_of_real r * exp a"

   818   apply (insert rcis_Ex [of z])

   819   apply (auto simp add: exp_eq_polar rcis_def mult.assoc [symmetric])

   820   apply (rule_tac x = "\<i> * complex_of_real a" in exI)

   821   apply auto

   822   done

   823

   824 lemma exp_pi_i [simp]: "exp (of_real pi * \<i>) = -1"

   825   by (metis cis_conv_exp cis_pi mult.commute)

   826

   827 lemma exp_pi_i' [simp]: "exp (\<i> * of_real pi) = -1"

   828   using cis_conv_exp cis_pi by auto

   829

   830 lemma exp_two_pi_i [simp]: "exp (2 * of_real pi * \<i>) = 1"

   831   by (simp add: exp_eq_polar complex_eq_iff)

   832

   833 lemma exp_two_pi_i' [simp]: "exp (\<i> * (of_real pi * 2)) = 1"

   834   by (metis exp_two_pi_i mult.commute)

   835

   836

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

   838

   839 definition arg :: "complex \<Rightarrow> real"

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

   841

   842 lemma arg_zero: "arg 0 = 0"

   843   by (simp add: arg_def)

   844

   845 lemma arg_unique:

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

   847   shows "arg z = x"

   848 proof -

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

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

   851   proof

   852     fix a

   853     define d where "d = a - x"

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

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

   856       unfolding d_def by simp

   857     moreover

   858     from a assms have "cos a = cos x" and "sin a = sin x"

   859       by (simp_all add: complex_eq_iff)

   860     then have cos: "cos d = 1"

   861       by (simp add: d_def cos_diff)

   862     moreover from cos have "sin d = 0"

   863       by (rule cos_one_sin_zero)

   864     ultimately have "d = 0"

   865       by (auto simp: sin_zero_iff elim!: evenE dest!: less_2_cases)

   866     then show "a = x"

   867       by (simp add: d_def)

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

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

   870     by (simp add: arg_def)

   871 qed

   872

   873 lemma arg_correct:

   874   assumes "z \<noteq> 0"

   875   shows "sgn z = cis (arg z) \<and> -pi < arg z \<and> arg z \<le> pi"

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

   877   obtain r a where z: "z = rcis r a"

   878     using rcis_Ex by fast

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

   880   define b where "b = (if 0 < r then a else a + pi)"

   881   have b: "sgn z = cis b"

   882     using \<open>r \<noteq> 0\<close> by (simp add: z b_def rcis_def of_real_def sgn_scaleR sgn_if complex_eq_iff)

   883   have cis_2pi_nat: "cis (2 * pi * real_of_nat n) = 1" for n

   884     by (induct n) (simp_all add: distrib_left cis_mult [symmetric] complex_eq_iff)

   885   have cis_2pi_int: "cis (2 * pi * real_of_int x) = 1" for x

   886     by (cases x rule: int_diff_cases)

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

   888   define c where "c = b - 2 * pi * of_int \<lceil>(b - pi) / (2 * pi)\<rceil>"

   889   have "sgn z = cis c"

   890     by (simp add: b c_def cis_divide [symmetric] cis_2pi_int)

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

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

   893     by (simp add: c_def less_divide_eq divide_le_eq algebra_simps del: le_of_int_ceiling)

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

   895     by fast

   896 qed

   897

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

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

   900

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

   902   by (simp add: arg_correct)

   903

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

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

   906

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

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

   909

   910

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

   912

   913 primcorec csqrt :: "complex \<Rightarrow> complex"

   914   where

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

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

   917

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

   919   by (simp add: complex_eq_iff norm_complex_def)

   920

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

   922   by (simp add: complex_eq_iff norm_complex_def)

   923

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

   925   by (simp add: complex_eq_iff norm_complex_def)

   926

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

   928   by simp

   929

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

   931   by simp

   932

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

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

   935

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

   937 proof (cases "Im z = 0")

   938   case True

   939   then show ?thesis

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

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

   942       (simp_all add: complex_eq_iff Re_power2 Im_power2 power2_eq_square cmod_eq_Re)

   943 next

   944   case False

   945   moreover have "cmod z * cmod z - Re z * Re z = Im z * Im z"

   946     by (simp add: norm_complex_def power2_eq_square)

   947   moreover have "\<bar>Re z\<bar> \<le> cmod z"

   948     by (simp add: norm_complex_def)

   949   ultimately show ?thesis

   950     by (simp add: Re_power2 Im_power2 complex_eq_iff real_sgn_eq

   951         field_simps real_sqrt_mult[symmetric] real_sqrt_divide)

   952 qed

   953

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

   955   by auto (metis power2_csqrt power_eq_0_iff)

   956

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

   958   by auto (metis power2_csqrt power2_eq_1_iff)

   959

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

   961   by (auto simp add: not_less cmod_plus_Re_le_0_iff Im_eq_0)

   962

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

   964   by (metis csqrt_principal le_less)

   965

   966 lemma csqrt_square:

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

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

   969 proof -

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

   971     by (simp add: power2_eq_iff[symmetric])

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

   973     using csqrt_principal[of "b ^ 2"] assms

   974     by (intro disjCI notI) (auto simp: complex_eq_iff)

   975   ultimately show ?thesis

   976     by auto

   977 qed

   978

   979 lemma csqrt_unique: "w\<^sup>2 = z \<Longrightarrow> 0 < Re w \<or> Re w = 0 \<and> 0 \<le> Im w \<Longrightarrow> csqrt z = w"

   980   by (auto simp: csqrt_square)

   981

   982 lemma csqrt_minus [simp]:

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

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

   985 proof -

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

   987   proof (rule csqrt_square)

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

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

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

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

   992   qed

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

   994     by (simp add: power_mult_distrib)

   995   finally show ?thesis .

   996 qed

   997

   998

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

  1000

  1001 lemmas expand_complex_eq = complex_eq_iff

  1002 lemmas complex_Re_Im_cancel_iff = complex_eq_iff

  1003 lemmas complex_equality = complex_eqI

  1004 lemmas cmod_def = norm_complex_def

  1005 lemmas complex_norm_def = norm_complex_def

  1006 lemmas complex_divide_def = divide_complex_def

  1007

  1008 lemma legacy_Complex_simps:

  1009   shows Complex_eq_0: "Complex a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"

  1010     and complex_add: "Complex a b + Complex c d = Complex (a + c) (b + d)"

  1011     and complex_minus: "- (Complex a b) = Complex (- a) (- b)"

  1012     and complex_diff: "Complex a b - Complex c d = Complex (a - c) (b - d)"

  1013     and Complex_eq_1: "Complex a b = 1 \<longleftrightarrow> a = 1 \<and> b = 0"

  1014     and Complex_eq_neg_1: "Complex a b = - 1 \<longleftrightarrow> a = - 1 \<and> b = 0"

  1015     and complex_mult: "Complex a b * Complex c d = Complex (a * c - b * d) (a * d + b * c)"

  1016     and complex_inverse: "inverse (Complex a b) = Complex (a / (a\<^sup>2 + b\<^sup>2)) (- b / (a\<^sup>2 + b\<^sup>2))"

  1017     and Complex_eq_numeral: "Complex a b = numeral w \<longleftrightarrow> a = numeral w \<and> b = 0"

  1018     and Complex_eq_neg_numeral: "Complex a b = - numeral w \<longleftrightarrow> a = - numeral w \<and> b = 0"

  1019     and complex_scaleR: "scaleR r (Complex a b) = Complex (r * a) (r * b)"

  1020     and Complex_eq_i: "Complex x y = \<i> \<longleftrightarrow> x = 0 \<and> y = 1"

  1021     and i_mult_Complex: "\<i> * Complex a b = Complex (- b) a"

  1022     and Complex_mult_i: "Complex a b * \<i> = Complex (- b) a"

  1023     and i_complex_of_real: "\<i> * complex_of_real r = Complex 0 r"

  1024     and complex_of_real_i: "complex_of_real r * \<i> = Complex 0 r"

  1025     and Complex_add_complex_of_real: "Complex x y + complex_of_real r = Complex (x+r) y"

  1026     and complex_of_real_add_Complex: "complex_of_real r + Complex x y = Complex (r+x) y"

  1027     and Complex_mult_complex_of_real: "Complex x y * complex_of_real r = Complex (x*r) (y*r)"

  1028     and complex_of_real_mult_Complex: "complex_of_real r * Complex x y = Complex (r*x) (r*y)"

  1029     and complex_eq_cancel_iff2: "(Complex x y = complex_of_real xa) = (x = xa \<and> y = 0)"

  1030     and complex_cn: "cnj (Complex a b) = Complex a (- b)"

  1031     and Complex_setsum': "setsum (\<lambda>x. Complex (f x) 0) s = Complex (setsum f s) 0"

  1032     and Complex_setsum: "Complex (setsum f s) 0 = setsum (\<lambda>x. Complex (f x) 0) s"

  1033     and complex_of_real_def: "complex_of_real r = Complex r 0"

  1034     and complex_norm: "cmod (Complex x y) = sqrt (x\<^sup>2 + y\<^sup>2)"

  1035   by (simp_all add: norm_complex_def field_simps complex_eq_iff Re_divide Im_divide del: Complex_eq)

  1036

  1037 lemma Complex_in_Reals: "Complex x 0 \<in> \<real>"

  1038   by (metis Reals_of_real complex_of_real_def)

  1039

  1040 end