src/HOL/Complex.thy
 author paulson Fri Mar 21 15:36:00 2014 +0000 (2014-03-21) changeset 56238 5d147e1e18d1 parent 56217 dc429a5b13c4 child 56331 bea2196627cb permissions -rw-r--r--
a few new lemmas and generalisations of old ones
     1 (*  Title:       HOL/Complex.thy

     2     Author:      Jacques D. Fleuriot

     3     Copyright:   2001 University of Edinburgh

     4     Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4

     5 *)

     6

     7 header {* Complex Numbers: Rectangular and Polar Representations *}

     8

     9 theory Complex

    10 imports Transcendental

    11 begin

    12

    13 datatype complex = Complex real real

    14

    15 primrec Re :: "complex \<Rightarrow> real"

    16   where Re: "Re (Complex x y) = x"

    17

    18 primrec Im :: "complex \<Rightarrow> real"

    19   where Im: "Im (Complex x y) = y"

    20

    21 lemma complex_surj [simp]: "Complex (Re z) (Im z) = z"

    22   by (induct z) simp

    23

    24 lemma complex_eqI [intro?]: "\<lbrakk>Re x = Re y; Im x = Im y\<rbrakk> \<Longrightarrow> x = y"

    25   by (induct x, induct y) simp

    26

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

    28   by (induct x, induct y) simp

    29

    30

    31 subsection {* Addition and Subtraction *}

    32

    33 instantiation complex :: ab_group_add

    34 begin

    35

    36 definition complex_zero_def:

    37   "0 = Complex 0 0"

    38

    39 definition complex_add_def:

    40   "x + y = Complex (Re x + Re y) (Im x + Im y)"

    41

    42 definition complex_minus_def:

    43   "- x = Complex (- Re x) (- Im x)"

    44

    45 definition complex_diff_def:

    46   "x - (y\<Colon>complex) = x + - y"

    47

    48 lemma Complex_eq_0 [simp]: "Complex a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"

    49   by (simp add: complex_zero_def)

    50

    51 lemma complex_Re_zero [simp]: "Re 0 = 0"

    52   by (simp add: complex_zero_def)

    53

    54 lemma complex_Im_zero [simp]: "Im 0 = 0"

    55   by (simp add: complex_zero_def)

    56

    57 lemma complex_add [simp]:

    58   "Complex a b + Complex c d = Complex (a + c) (b + d)"

    59   by (simp add: complex_add_def)

    60

    61 lemma complex_Re_add [simp]: "Re (x + y) = Re x + Re y"

    62   by (simp add: complex_add_def)

    63

    64 lemma complex_Im_add [simp]: "Im (x + y) = Im x + Im y"

    65   by (simp add: complex_add_def)

    66

    67 lemma complex_minus [simp]:

    68   "- (Complex a b) = Complex (- a) (- b)"

    69   by (simp add: complex_minus_def)

    70

    71 lemma complex_Re_minus [simp]: "Re (- x) = - Re x"

    72   by (simp add: complex_minus_def)

    73

    74 lemma complex_Im_minus [simp]: "Im (- x) = - Im x"

    75   by (simp add: complex_minus_def)

    76

    77 lemma complex_diff [simp]:

    78   "Complex a b - Complex c d = Complex (a - c) (b - d)"

    79   by (simp add: complex_diff_def)

    80

    81 lemma complex_Re_diff [simp]: "Re (x - y) = Re x - Re y"

    82   by (simp add: complex_diff_def)

    83

    84 lemma complex_Im_diff [simp]: "Im (x - y) = Im x - Im y"

    85   by (simp add: complex_diff_def)

    86

    87 instance

    88   by intro_classes (simp_all add: complex_add_def complex_diff_def)

    89

    90 end

    91

    92

    93 subsection {* Multiplication and Division *}

    94

    95 instantiation complex :: field_inverse_zero

    96 begin

    97

    98 definition complex_one_def:

    99   "1 = Complex 1 0"

   100

   101 definition complex_mult_def:

   102   "x * y = Complex (Re x * Re y - Im x * Im y) (Re x * Im y + Im x * Re y)"

   103

   104 definition complex_inverse_def:

   105   "inverse x =

   106     Complex (Re x / ((Re x)\<^sup>2 + (Im x)\<^sup>2)) (- Im x / ((Re x)\<^sup>2 + (Im x)\<^sup>2))"

   107

   108 definition complex_divide_def:

   109   "x / (y\<Colon>complex) = x * inverse y"

   110

   111 lemma Complex_eq_1 [simp]:

   112   "Complex a b = 1 \<longleftrightarrow> a = 1 \<and> b = 0"

   113   by (simp add: complex_one_def)

   114

   115 lemma Complex_eq_neg_1 [simp]:

   116   "Complex a b = - 1 \<longleftrightarrow> a = - 1 \<and> b = 0"

   117   by (simp add: complex_one_def)

   118

   119 lemma complex_Re_one [simp]: "Re 1 = 1"

   120   by (simp add: complex_one_def)

   121

   122 lemma complex_Im_one [simp]: "Im 1 = 0"

   123   by (simp add: complex_one_def)

   124

   125 lemma complex_mult [simp]:

   126   "Complex a b * Complex c d = Complex (a * c - b * d) (a * d + b * c)"

   127   by (simp add: complex_mult_def)

   128

   129 lemma complex_Re_mult [simp]: "Re (x * y) = Re x * Re y - Im x * Im y"

   130   by (simp add: complex_mult_def)

   131

   132 lemma complex_Im_mult [simp]: "Im (x * y) = Re x * Im y + Im x * Re y"

   133   by (simp add: complex_mult_def)

   134

   135 lemma complex_inverse [simp]:

   136   "inverse (Complex a b) = Complex (a / (a\<^sup>2 + b\<^sup>2)) (- b / (a\<^sup>2 + b\<^sup>2))"

   137   by (simp add: complex_inverse_def)

   138

   139 lemma complex_Re_inverse:

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

   141   by (simp add: complex_inverse_def)

   142

   143 lemma complex_Im_inverse:

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

   145   by (simp add: complex_inverse_def)

   146

   147 instance

   148   by intro_classes (simp_all add: complex_mult_def

   149     distrib_left distrib_right right_diff_distrib left_diff_distrib

   150     complex_inverse_def complex_divide_def

   151     power2_eq_square add_divide_distrib [symmetric]

   152     complex_eq_iff)

   153

   154 end

   155

   156

   157 subsection {* Numerals and Arithmetic *}

   158

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

   160   by (induct n) simp_all

   161

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

   163   by (induct n) simp_all

   164

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

   166   by (cases z rule: int_diff_cases) simp

   167

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

   169   by (cases z rule: int_diff_cases) simp

   170

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

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

   173

   174 lemma complex_Re_neg_numeral [simp]: "Re (- numeral v) = - numeral v"

   175   using complex_Re_of_int [of "- numeral v"] by simp

   176

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

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

   179

   180 lemma complex_Im_neg_numeral [simp]: "Im (- numeral v) = 0"

   181   using complex_Im_of_int [of "- numeral v"] by simp

   182

   183 lemma Complex_eq_numeral [simp]:

   184   "Complex a b = numeral w \<longleftrightarrow> a = numeral w \<and> b = 0"

   185   by (simp add: complex_eq_iff)

   186

   187 lemma Complex_eq_neg_numeral [simp]:

   188   "Complex a b = - numeral w \<longleftrightarrow> a = - numeral w \<and> b = 0"

   189   by (simp add: complex_eq_iff)

   190

   191

   192 subsection {* Scalar Multiplication *}

   193

   194 instantiation complex :: real_field

   195 begin

   196

   197 definition complex_scaleR_def:

   198   "scaleR r x = Complex (r * Re x) (r * Im x)"

   199

   200 lemma complex_scaleR [simp]:

   201   "scaleR r (Complex a b) = Complex (r * a) (r * b)"

   202   unfolding complex_scaleR_def by simp

   203

   204 lemma complex_Re_scaleR [simp]: "Re (scaleR r x) = r * Re x"

   205   unfolding complex_scaleR_def by simp

   206

   207 lemma complex_Im_scaleR [simp]: "Im (scaleR r x) = r * Im x"

   208   unfolding complex_scaleR_def by simp

   209

   210 instance

   211 proof

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

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

   214     by (simp add: complex_eq_iff distrib_left)

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

   216     by (simp add: complex_eq_iff distrib_right)

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

   218     by (simp add: complex_eq_iff mult_assoc)

   219   show "scaleR 1 x = x"

   220     by (simp add: complex_eq_iff)

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

   222     by (simp add: complex_eq_iff algebra_simps)

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

   224     by (simp add: complex_eq_iff algebra_simps)

   225 qed

   226

   227 end

   228

   229

   230 subsection{* Properties of Embedding from Reals *}

   231

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

   233   where "complex_of_real \<equiv> of_real"

   234

   235 lemma complex_of_real_def: "complex_of_real r = Complex r 0"

   236   by (simp add: of_real_def complex_scaleR_def)

   237

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

   239   by (simp add: complex_of_real_def)

   240

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

   242   by (simp add: complex_of_real_def)

   243

   244 lemma Complex_add_complex_of_real [simp]:

   245   shows "Complex x y + complex_of_real r = Complex (x+r) y"

   246   by (simp add: complex_of_real_def)

   247

   248 lemma complex_of_real_add_Complex [simp]:

   249   shows "complex_of_real r + Complex x y = Complex (r+x) y"

   250   by (simp add: complex_of_real_def)

   251

   252 lemma Complex_mult_complex_of_real:

   253   shows "Complex x y * complex_of_real r = Complex (x*r) (y*r)"

   254   by (simp add: complex_of_real_def)

   255

   256 lemma complex_of_real_mult_Complex:

   257   shows "complex_of_real r * Complex x y = Complex (r*x) (r*y)"

   258   by (simp add: complex_of_real_def)

   259

   260 lemma complex_eq_cancel_iff2 [simp]:

   261   shows "(Complex x y = complex_of_real xa) = (x = xa & y = 0)"

   262   by (simp add: complex_of_real_def)

   263

   264 lemma complex_split_polar:

   265      "\<exists>r a. z = complex_of_real r * (Complex (cos a) (sin a))"

   266   by (simp add: complex_eq_iff polar_Ex)

   267

   268

   269 subsection {* Vector Norm *}

   270

   271 instantiation complex :: real_normed_field

   272 begin

   273

   274 definition complex_norm_def:

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

   276

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

   278   where "cmod \<equiv> norm"

   279

   280 definition complex_sgn_def:

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

   282

   283 definition dist_complex_def:

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

   285

   286 definition open_complex_def:

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

   288

   289 lemmas cmod_def = complex_norm_def

   290

   291 lemma complex_norm [simp]: "cmod (Complex x y) = sqrt (x\<^sup>2 + y\<^sup>2)"

   292   by (simp add: complex_norm_def)

   293

   294 instance proof

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

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

   297     by (induct x) simp

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

   299     by (induct x, induct y)

   300        (simp add: real_sqrt_sum_squares_triangle_ineq)

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

   302     by (induct x)

   303        (simp add: power_mult_distrib distrib_left [symmetric] real_sqrt_mult)

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

   305     by (induct x, induct y)

   306        (simp add: real_sqrt_mult [symmetric] power2_eq_square algebra_simps)

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

   308     by (rule complex_sgn_def)

   309   show "dist x y = cmod (x - y)"

   310     by (rule dist_complex_def)

   311   show "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"

   312     by (rule open_complex_def)

   313 qed

   314

   315 end

   316

   317 lemma cmod_unit_one: "cmod (Complex (cos a) (sin a)) = 1"

   318   by simp

   319

   320 lemma cmod_complex_polar:

   321   "cmod (complex_of_real r * Complex (cos a) (sin a)) = abs r"

   322   by (simp add: norm_mult)

   323

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

   325   unfolding complex_norm_def

   326   by (rule real_sqrt_sum_squares_ge1)

   327

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

   329   by (rule order_trans [OF _ norm_ge_zero], simp)

   330

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

   332   by (rule ord_le_eq_trans [OF norm_triangle_ineq2], simp)

   333

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

   335   by (cases x) simp

   336

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

   338   by (cases x) simp

   339

   340 text {* Properties of complex signum. *}

   341

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

   343   by (simp add: sgn_div_norm divide_inverse scaleR_conv_of_real mult_commute)

   344

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

   346   by (simp add: complex_sgn_def divide_inverse)

   347

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

   349   by (simp add: complex_sgn_def divide_inverse)

   350

   351

   352 subsection {* Completeness of the Complexes *}

   353

   354 lemma bounded_linear_Re: "bounded_linear Re"

   355   by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def)

   356

   357 lemma bounded_linear_Im: "bounded_linear Im"

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

   359

   360 lemmas tendsto_Re [tendsto_intros] =

   361   bounded_linear.tendsto [OF bounded_linear_Re]

   362

   363 lemmas tendsto_Im [tendsto_intros] =

   364   bounded_linear.tendsto [OF bounded_linear_Im]

   365

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

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

   368 lemmas Cauchy_Re = bounded_linear.Cauchy [OF bounded_linear_Re]

   369 lemmas Cauchy_Im = bounded_linear.Cauchy [OF bounded_linear_Im]

   370

   371 lemma tendsto_Complex [tendsto_intros]:

   372   assumes "(f ---> a) F" and "(g ---> b) F"

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

   374 proof (rule tendstoI)

   375   fix r :: real assume "0 < r"

   376   hence "0 < r / sqrt 2" by (simp add: divide_pos_pos)

   377   have "eventually (\<lambda>x. dist (f x) a < r / sqrt 2) F"

   378     using (f ---> a) F and 0 < r / sqrt 2 by (rule tendstoD)

   379   moreover

   380   have "eventually (\<lambda>x. dist (g x) b < r / sqrt 2) F"

   381     using (g ---> b) F and 0 < r / sqrt 2 by (rule tendstoD)

   382   ultimately

   383   show "eventually (\<lambda>x. dist (Complex (f x) (g x)) (Complex a b) < r) F"

   384     by (rule eventually_elim2)

   385        (simp add: dist_norm real_sqrt_sum_squares_less)

   386 qed

   387

   388 instance complex :: banach

   389 proof

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

   391   assume X: "Cauchy X"

   392   from Cauchy_Re [OF X] have 1: "(\<lambda>n. Re (X n)) ----> lim (\<lambda>n. Re (X n))"

   393     by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)

   394   from Cauchy_Im [OF X] have 2: "(\<lambda>n. Im (X n)) ----> lim (\<lambda>n. Im (X n))"

   395     by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)

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

   397     using tendsto_Complex [OF 1 2] by simp

   398   thus "convergent X"

   399     by (rule convergentI)

   400 qed

   401

   402 declare

   403   DERIV_power[where 'a=complex, THEN DERIV_cong,

   404               unfolded of_nat_def[symmetric], DERIV_intros]

   405

   406

   407 subsection {* The Complex Number $i$ *}

   408

   409 definition "ii" :: complex  ("\<i>")

   410   where i_def: "ii \<equiv> Complex 0 1"

   411

   412 lemma complex_Re_i [simp]: "Re ii = 0"

   413   by (simp add: i_def)

   414

   415 lemma complex_Im_i [simp]: "Im ii = 1"

   416   by (simp add: i_def)

   417

   418 lemma Complex_eq_i [simp]: "(Complex x y = ii) = (x = 0 \<and> y = 1)"

   419   by (simp add: i_def)

   420

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

   422   by (simp add: i_def)

   423

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

   425   by (simp add: complex_eq_iff)

   426

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

   428   by (simp add: complex_eq_iff)

   429

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

   431   by (simp add: complex_eq_iff)

   432

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

   434   by (simp add: complex_eq_iff)

   435

   436 lemma i_mult_Complex [simp]: "ii * Complex a b = Complex (- b) a"

   437   by (simp add: complex_eq_iff)

   438

   439 lemma Complex_mult_i [simp]: "Complex a b * ii = Complex (- b) a"

   440   by (simp add: complex_eq_iff)

   441

   442 lemma i_complex_of_real [simp]: "ii * complex_of_real r = Complex 0 r"

   443   by (simp add: i_def complex_of_real_def)

   444

   445 lemma complex_of_real_i [simp]: "complex_of_real r * ii = Complex 0 r"

   446   by (simp add: i_def complex_of_real_def)

   447

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

   449   by (simp add: i_def)

   450

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

   452   by (simp add: power2_eq_square)

   453

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

   455   by (rule inverse_unique, simp)

   456

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

   458   by (simp add: mult_assoc [symmetric])

   459

   460

   461 subsection {* Complex Conjugation *}

   462

   463 definition cnj :: "complex \<Rightarrow> complex" where

   464   "cnj z = Complex (Re z) (- Im z)"

   465

   466 lemma complex_cnj [simp]: "cnj (Complex a b) = Complex a (- b)"

   467   by (simp add: cnj_def)

   468

   469 lemma complex_Re_cnj [simp]: "Re (cnj x) = Re x"

   470   by (simp add: cnj_def)

   471

   472 lemma complex_Im_cnj [simp]: "Im (cnj x) = - Im x"

   473   by (simp add: cnj_def)

   474

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

   476   by (simp add: complex_eq_iff)

   477

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

   479   by (simp add: cnj_def)

   480

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

   482   by (simp add: complex_eq_iff)

   483

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

   485   by (simp add: complex_eq_iff)

   486

   487 lemma complex_cnj_add: "cnj (x + y) = cnj x + cnj y"

   488   by (simp add: complex_eq_iff)

   489

   490 lemma complex_cnj_diff: "cnj (x - y) = cnj x - cnj y"

   491   by (simp add: complex_eq_iff)

   492

   493 lemma complex_cnj_minus: "cnj (- x) = - cnj x"

   494   by (simp add: complex_eq_iff)

   495

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

   497   by (simp add: complex_eq_iff)

   498

   499 lemma complex_cnj_mult: "cnj (x * y) = cnj x * cnj y"

   500   by (simp add: complex_eq_iff)

   501

   502 lemma complex_cnj_inverse: "cnj (inverse x) = inverse (cnj x)"

   503   by (simp add: complex_inverse_def)

   504

   505 lemma complex_cnj_divide: "cnj (x / y) = cnj x / cnj y"

   506   by (simp add: complex_divide_def complex_cnj_mult complex_cnj_inverse)

   507

   508 lemma complex_cnj_power: "cnj (x ^ n) = cnj x ^ n"

   509   by (induct n, simp_all add: complex_cnj_mult)

   510

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

   512   by (simp add: complex_eq_iff)

   513

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

   515   by (simp add: complex_eq_iff)

   516

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

   518   by (simp add: complex_eq_iff)

   519

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

   521   by (simp add: complex_eq_iff)

   522

   523 lemma complex_cnj_scaleR: "cnj (scaleR r x) = scaleR r (cnj x)"

   524   by (simp add: complex_eq_iff)

   525

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

   527   by (simp add: complex_norm_def)

   528

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

   530   by (simp add: complex_eq_iff)

   531

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

   533   by (simp add: complex_eq_iff)

   534

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

   536   by (simp add: complex_eq_iff)

   537

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

   539   by (simp add: complex_eq_iff)

   540

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

   542   by (simp add: complex_eq_iff power2_eq_square)

   543

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

   545   by (simp add: norm_mult power2_eq_square)

   546

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

   548   by (simp add: cmod_def power2_eq_square)

   549

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

   551   by simp

   552

   553 lemma bounded_linear_cnj: "bounded_linear cnj"

   554   using complex_cnj_add complex_cnj_scaleR

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

   556

   557 lemmas tendsto_cnj [tendsto_intros] =

   558   bounded_linear.tendsto [OF bounded_linear_cnj]

   559

   560 lemmas isCont_cnj [simp] =

   561   bounded_linear.isCont [OF bounded_linear_cnj]

   562

   563

   564 subsection{*Basic Lemmas*}

   565

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

   567   by (metis complex_Im_zero complex_Re_zero complex_eqI sum_power2_eq_zero_iff)

   568

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

   570 by (metis complex_eq_0 less_numeral_extra(3) sum_power2_gt_zero_iff)

   571

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

   573 apply (cases z, auto)

   574 by (metis complex_of_real_def of_real_add of_real_power power2_eq_square)

   575

   576 lemma complex_div_eq_0:

   577     "(Re(a / b) = 0 \<longleftrightarrow> Re(a * cnj b) = 0) & (Im(a / b) = 0 \<longleftrightarrow> Im(a * cnj b) = 0)"

   578 proof (cases "b=0")

   579   case True then show ?thesis by auto

   580 next

   581   case False

   582   show ?thesis

   583   proof (cases b)

   584     case (Complex x y)

   585     then have "x\<^sup>2 + y\<^sup>2 > 0"

   586       by (metis Complex_eq_0 False sum_power2_gt_zero_iff)

   587     then have "!!u v. u / (x\<^sup>2 + y\<^sup>2) + v / (x\<^sup>2 + y\<^sup>2) = (u + v) / (x\<^sup>2 + y\<^sup>2)"

   588       by (metis add_divide_distrib)

   589     with Complex False show ?thesis

   590       by (auto simp: complex_divide_def)

   591   qed

   592 qed

   593

   594 lemma re_complex_div_eq_0: "Re(a / b) = 0 \<longleftrightarrow> Re(a * cnj b) = 0"

   595   and im_complex_div_eq_0: "Im(a / b) = 0 \<longleftrightarrow> Im(a * cnj b) = 0"

   596 using complex_div_eq_0 by auto

   597

   598

   599 lemma complex_div_gt_0:

   600     "(Re(a / b) > 0 \<longleftrightarrow> Re(a * cnj b) > 0) & (Im(a / b) > 0 \<longleftrightarrow> Im(a * cnj b) > 0)"

   601 proof (cases "b=0")

   602   case True then show ?thesis by auto

   603 next

   604   case False

   605   show ?thesis

   606   proof (cases b)

   607     case (Complex x y)

   608     then have "x\<^sup>2 + y\<^sup>2 > 0"

   609       by (metis Complex_eq_0 False sum_power2_gt_zero_iff)

   610     moreover have "!!u v. u / (x\<^sup>2 + y\<^sup>2) + v / (x\<^sup>2 + y\<^sup>2) = (u + v) / (x\<^sup>2 + y\<^sup>2)"

   611       by (metis add_divide_distrib)

   612     ultimately show ?thesis using Complex False 0 < x\<^sup>2 + y\<^sup>2

   613       apply (simp add: complex_divide_def  zero_less_divide_iff less_divide_eq)

   614       apply (metis less_divide_eq mult_zero_left diff_conv_add_uminus minus_divide_left)

   615       done

   616   qed

   617 qed

   618

   619 lemma re_complex_div_gt_0: "Re(a / b) > 0 \<longleftrightarrow> Re(a * cnj b) > 0"

   620   and im_complex_div_gt_0: "Im(a / b) > 0 \<longleftrightarrow> Im(a * cnj b) > 0"

   621 using complex_div_gt_0 by auto

   622

   623 lemma re_complex_div_ge_0: "Re(a / b) \<ge> 0 \<longleftrightarrow> Re(a * cnj b) \<ge> 0"

   624   by (metis le_less re_complex_div_eq_0 re_complex_div_gt_0)

   625

   626 lemma im_complex_div_ge_0: "Im(a / b) \<ge> 0 \<longleftrightarrow> Im(a * cnj b) \<ge> 0"

   627   by (metis im_complex_div_eq_0 im_complex_div_gt_0 le_less)

   628

   629 lemma re_complex_div_lt_0: "Re(a / b) < 0 \<longleftrightarrow> Re(a * cnj b) < 0"

   630   by (metis less_asym neq_iff re_complex_div_eq_0 re_complex_div_gt_0)

   631

   632 lemma im_complex_div_lt_0: "Im(a / b) < 0 \<longleftrightarrow> Im(a * cnj b) < 0"

   633   by (metis im_complex_div_eq_0 im_complex_div_gt_0 less_asym neq_iff)

   634

   635 lemma re_complex_div_le_0: "Re(a / b) \<le> 0 \<longleftrightarrow> Re(a * cnj b) \<le> 0"

   636   by (metis not_le re_complex_div_gt_0)

   637

   638 lemma im_complex_div_le_0: "Im(a / b) \<le> 0 \<longleftrightarrow> Im(a * cnj b) \<le> 0"

   639   by (metis im_complex_div_gt_0 not_le)

   640

   641 lemma Re_setsum: "Re(setsum f s) = setsum (%x. Re(f x)) s"

   642 apply (cases "finite s")

   643   by (induct s rule: finite_induct) auto

   644

   645 lemma Im_setsum: "Im(setsum f s) = setsum (%x. Im(f x)) s"

   646 apply (cases "finite s")

   647   by (induct s rule: finite_induct) auto

   648

   649 lemma Complex_setsum': "setsum (%x. Complex (f x) 0) s = Complex (setsum f s) 0"

   650 apply (cases "finite s")

   651   by (induct s rule: finite_induct) auto

   652

   653 lemma Complex_setsum: "Complex (setsum f s) 0 = setsum (%x. Complex (f x) 0) s"

   654   by (metis Complex_setsum')

   655

   656 lemma cnj_setsum: "cnj (setsum f s) = setsum (%x. cnj (f x)) s"

   657 apply (cases "finite s")

   658   by (induct s rule: finite_induct) (auto simp: complex_cnj_add)

   659

   660 lemma of_real_setsum: "of_real (setsum f s) = setsum (%x. of_real(f x)) s"

   661 apply (cases "finite s")

   662   by (induct s rule: finite_induct) auto

   663

   664 lemma of_real_setprod: "of_real (setprod f s) = setprod (%x. of_real(f x)) s"

   665 apply (cases "finite s")

   666   by (induct s rule: finite_induct) auto

   667

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

   669 by (metis Reals_cases Reals_of_real complex.exhaust complex.inject complex_cnj

   670           complex_of_real_def equal_neg_zero)

   671

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

   673   by (metis Reals_of_real complex_of_real_def)

   674

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

   676   by (metis Re_complex_of_real Reals_cases norm_of_real)

   677

   678

   679 subsection{*Finally! Polar Form for Complex Numbers*}

   680

   681 subsubsection {* $\cos \theta + i \sin \theta$ *}

   682

   683 definition cis :: "real \<Rightarrow> complex" where

   684   "cis a = Complex (cos a) (sin a)"

   685

   686 lemma Re_cis [simp]: "Re (cis a) = cos a"

   687   by (simp add: cis_def)

   688

   689 lemma Im_cis [simp]: "Im (cis a) = sin a"

   690   by (simp add: cis_def)

   691

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

   693   by (simp add: cis_def)

   694

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

   696   by (simp add: cis_def)

   697

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

   699   by (simp add: sgn_div_norm)

   700

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

   702   by (metis norm_cis norm_zero zero_neq_one)

   703

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

   705   by (simp add: cis_def cos_add sin_add)

   706

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

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

   709

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

   711   by (simp add: cis_def)

   712

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

   714   by (simp add: complex_divide_def cis_mult)

   715

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

   717   by (auto simp add: DeMoivre)

   718

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

   720   by (auto simp add: DeMoivre)

   721

   722 subsubsection {* $r(\cos \theta + i \sin \theta)$ *}

   723

   724 definition rcis :: "[real, real] \<Rightarrow> complex" where

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

   726

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

   728   by (simp add: rcis_def)

   729

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

   731   by (simp add: rcis_def)

   732

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

   734   by (simp add: complex_eq_iff polar_Ex)

   735

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

   737   by (simp add: rcis_def norm_mult)

   738

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

   740   by (simp add: rcis_def)

   741

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

   743   by (simp add: rcis_def cis_mult)

   744

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

   746   by (simp add: rcis_def)

   747

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

   749   by (simp add: rcis_def)

   750

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

   752   by (simp add: rcis_def)

   753

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

   755   by (simp add: rcis_def power_mult_distrib DeMoivre)

   756

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

   758   by (simp add: divide_inverse rcis_def)

   759

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

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

   762

   763 subsubsection {* Complex exponential *}

   764

   765 abbreviation expi :: "complex \<Rightarrow> complex"

   766   where "expi \<equiv> exp"

   767

   768 lemma cis_conv_exp: "cis b = exp (Complex 0 b)"

   769 proof (rule complex_eqI)

   770   { fix n have "Complex 0 b ^ n =

   771     real (fact n) *\<^sub>R Complex (cos_coeff n * b ^ n) (sin_coeff n * b ^ n)"

   772       apply (induct n)

   773       apply (simp add: cos_coeff_def sin_coeff_def)

   774       apply (simp add: sin_coeff_Suc cos_coeff_Suc del: mult_Suc)

   775       done } note * = this

   776   show "Re (cis b) = Re (exp (Complex 0 b))"

   777     unfolding exp_def cis_def cos_def

   778     by (subst bounded_linear.suminf[OF bounded_linear_Re summable_exp_generic],

   779       simp add: * mult_assoc [symmetric])

   780   show "Im (cis b) = Im (exp (Complex 0 b))"

   781     unfolding exp_def cis_def sin_def

   782     by (subst bounded_linear.suminf[OF bounded_linear_Im summable_exp_generic],

   783       simp add: * mult_assoc [symmetric])

   784 qed

   785

   786 lemma expi_def: "expi z = complex_of_real (exp (Re z)) * cis (Im z)"

   787   unfolding cis_conv_exp exp_of_real [symmetric] mult_exp_exp by simp

   788

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

   790   unfolding expi_def by simp

   791

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

   793   unfolding expi_def by simp

   794

   795 lemma complex_expi_Ex: "\<exists>a r. z = complex_of_real r * expi a"

   796 apply (insert rcis_Ex [of z])

   797 apply (auto simp add: expi_def rcis_def mult_assoc [symmetric])

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

   799 done

   800

   801 lemma expi_two_pi_i [simp]: "expi((2::complex) * complex_of_real pi * ii) = 1"

   802   by (simp add: expi_def cis_def)

   803

   804 subsubsection {* Complex argument *}

   805

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

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

   808

   809 lemma arg_zero: "arg 0 = 0"

   810   by (simp add: arg_def)

   811

   812 lemma of_nat_less_of_int_iff: (* TODO: move *)

   813   "(of_nat n :: 'a::linordered_idom) < of_int x \<longleftrightarrow> int n < x"

   814   by (metis of_int_of_nat_eq of_int_less_iff)

   815

   816 lemma real_of_nat_less_numeral_iff [simp]: (* TODO: move *)

   817   "real (n::nat) < numeral w \<longleftrightarrow> n < numeral w"

   818   using of_nat_less_of_int_iff [of n "numeral w", where 'a=real]

   819   by (simp add: real_of_nat_def zless_nat_eq_int_zless [symmetric])

   820

   821 lemma arg_unique:

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

   823   shows "arg z = x"

   824 proof -

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

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

   827   proof

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

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

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

   831       unfolding d_def by simp

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

   833       by (simp_all add: complex_eq_iff)

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

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

   836     ultimately have "d = 0"

   837       unfolding sin_zero_iff even_mult_two_ex

   838       by (auto simp add: numeral_2_eq_2 less_Suc_eq)

   839     thus "a = x" unfolding d_def by simp

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

   841   with z \<noteq> 0 show "arg z = x"

   842     unfolding arg_def by simp

   843 qed

   844

   845 lemma arg_correct:

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

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

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

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

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

   851   have b: "sgn z = cis b"

   852     unfolding z b_def rcis_def using r \<noteq> 0

   853     by (simp add: of_real_def sgn_scaleR sgn_if, simp add: cis_def)

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

   855     by (induct_tac n, simp_all add: distrib_left cis_mult [symmetric],

   856       simp add: cis_def)

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

   858     by (case_tac x rule: int_diff_cases,

   859       simp add: right_diff_distrib cis_divide [symmetric] cis_2pi_nat)

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

   861   have "sgn z = cis c"

   862     unfolding b c_def

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

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

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

   866     by (simp add: c_def less_divide_eq divide_le_eq algebra_simps)

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

   868 qed

   869

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

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

   872

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

   874   by (simp add: arg_correct)

   875

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

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

   878

   879 lemma cos_arg_i_mult_zero [simp]:

   880      "y \<noteq> 0 ==> cos (arg(Complex 0 y)) = 0"

   881   using cis_arg [of "Complex 0 y"] by (simp add: complex_eq_iff)

   882

   883 text {* Legacy theorem names *}

   884

   885 lemmas expand_complex_eq = complex_eq_iff

   886 lemmas complex_Re_Im_cancel_iff = complex_eq_iff

   887 lemmas complex_equality = complex_eqI

   888

   889 end