src/HOL/Complex.thy
author huffman
Fri Mar 30 12:32:35 2012 +0200 (2012-03-30)
changeset 47220 52426c62b5d0
parent 47108 2a1953f0d20d
child 49962 a8cc904a6820
permissions -rw-r--r--
replace lemmas eval_nat_numeral with a simpler reformulation
     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)\<twosuperior> + (Im x)\<twosuperior>)) (- Im x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>))"
   107 
   108 definition complex_divide_def:
   109   "x / (y\<Colon>complex) = x * inverse y"
   110 
   111 lemma Complex_eq_1 [simp]: "(Complex a b = 1) = (a = 1 \<and> b = 0)"
   112   by (simp add: complex_one_def)
   113 
   114 lemma complex_Re_one [simp]: "Re 1 = 1"
   115   by (simp add: complex_one_def)
   116 
   117 lemma complex_Im_one [simp]: "Im 1 = 0"
   118   by (simp add: complex_one_def)
   119 
   120 lemma complex_mult [simp]:
   121   "Complex a b * Complex c d = Complex (a * c - b * d) (a * d + b * c)"
   122   by (simp add: complex_mult_def)
   123 
   124 lemma complex_Re_mult [simp]: "Re (x * y) = Re x * Re y - Im x * Im y"
   125   by (simp add: complex_mult_def)
   126 
   127 lemma complex_Im_mult [simp]: "Im (x * y) = Re x * Im y + Im x * Re y"
   128   by (simp add: complex_mult_def)
   129 
   130 lemma complex_inverse [simp]:
   131   "inverse (Complex a b) = Complex (a / (a\<twosuperior> + b\<twosuperior>)) (- b / (a\<twosuperior> + b\<twosuperior>))"
   132   by (simp add: complex_inverse_def)
   133 
   134 lemma complex_Re_inverse:
   135   "Re (inverse x) = Re x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)"
   136   by (simp add: complex_inverse_def)
   137 
   138 lemma complex_Im_inverse:
   139   "Im (inverse x) = - Im x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)"
   140   by (simp add: complex_inverse_def)
   141 
   142 instance
   143   by intro_classes (simp_all add: complex_mult_def
   144     right_distrib left_distrib right_diff_distrib left_diff_distrib
   145     complex_inverse_def complex_divide_def
   146     power2_eq_square add_divide_distrib [symmetric]
   147     complex_eq_iff)
   148 
   149 end
   150 
   151 
   152 subsection {* Numerals and Arithmetic *}
   153 
   154 lemma complex_Re_of_nat [simp]: "Re (of_nat n) = of_nat n"
   155   by (induct n) simp_all
   156 
   157 lemma complex_Im_of_nat [simp]: "Im (of_nat n) = 0"
   158   by (induct n) simp_all
   159 
   160 lemma complex_Re_of_int [simp]: "Re (of_int z) = of_int z"
   161   by (cases z rule: int_diff_cases) simp
   162 
   163 lemma complex_Im_of_int [simp]: "Im (of_int z) = 0"
   164   by (cases z rule: int_diff_cases) simp
   165 
   166 lemma complex_Re_numeral [simp]: "Re (numeral v) = numeral v"
   167   using complex_Re_of_int [of "numeral v"] by simp
   168 
   169 lemma complex_Re_neg_numeral [simp]: "Re (neg_numeral v) = neg_numeral v"
   170   using complex_Re_of_int [of "neg_numeral v"] by simp
   171 
   172 lemma complex_Im_numeral [simp]: "Im (numeral v) = 0"
   173   using complex_Im_of_int [of "numeral v"] by simp
   174 
   175 lemma complex_Im_neg_numeral [simp]: "Im (neg_numeral v) = 0"
   176   using complex_Im_of_int [of "neg_numeral v"] by simp
   177 
   178 lemma Complex_eq_numeral [simp]:
   179   "(Complex a b = numeral w) = (a = numeral w \<and> b = 0)"
   180   by (simp add: complex_eq_iff)
   181 
   182 lemma Complex_eq_neg_numeral [simp]:
   183   "(Complex a b = neg_numeral w) = (a = neg_numeral w \<and> b = 0)"
   184   by (simp add: complex_eq_iff)
   185 
   186 
   187 subsection {* Scalar Multiplication *}
   188 
   189 instantiation complex :: real_field
   190 begin
   191 
   192 definition complex_scaleR_def:
   193   "scaleR r x = Complex (r * Re x) (r * Im x)"
   194 
   195 lemma complex_scaleR [simp]:
   196   "scaleR r (Complex a b) = Complex (r * a) (r * b)"
   197   unfolding complex_scaleR_def by simp
   198 
   199 lemma complex_Re_scaleR [simp]: "Re (scaleR r x) = r * Re x"
   200   unfolding complex_scaleR_def by simp
   201 
   202 lemma complex_Im_scaleR [simp]: "Im (scaleR r x) = r * Im x"
   203   unfolding complex_scaleR_def by simp
   204 
   205 instance
   206 proof
   207   fix a b :: real and x y :: complex
   208   show "scaleR a (x + y) = scaleR a x + scaleR a y"
   209     by (simp add: complex_eq_iff right_distrib)
   210   show "scaleR (a + b) x = scaleR a x + scaleR b x"
   211     by (simp add: complex_eq_iff left_distrib)
   212   show "scaleR a (scaleR b x) = scaleR (a * b) x"
   213     by (simp add: complex_eq_iff mult_assoc)
   214   show "scaleR 1 x = x"
   215     by (simp add: complex_eq_iff)
   216   show "scaleR a x * y = scaleR a (x * y)"
   217     by (simp add: complex_eq_iff algebra_simps)
   218   show "x * scaleR a y = scaleR a (x * y)"
   219     by (simp add: complex_eq_iff algebra_simps)
   220 qed
   221 
   222 end
   223 
   224 
   225 subsection{* Properties of Embedding from Reals *}
   226 
   227 abbreviation complex_of_real :: "real \<Rightarrow> complex"
   228   where "complex_of_real \<equiv> of_real"
   229 
   230 lemma complex_of_real_def: "complex_of_real r = Complex r 0"
   231   by (simp add: of_real_def complex_scaleR_def)
   232 
   233 lemma Re_complex_of_real [simp]: "Re (complex_of_real z) = z"
   234   by (simp add: complex_of_real_def)
   235 
   236 lemma Im_complex_of_real [simp]: "Im (complex_of_real z) = 0"
   237   by (simp add: complex_of_real_def)
   238 
   239 lemma Complex_add_complex_of_real [simp]:
   240   shows "Complex x y + complex_of_real r = Complex (x+r) y"
   241   by (simp add: complex_of_real_def)
   242 
   243 lemma complex_of_real_add_Complex [simp]:
   244   shows "complex_of_real r + Complex x y = Complex (r+x) y"
   245   by (simp add: complex_of_real_def)
   246 
   247 lemma Complex_mult_complex_of_real:
   248   shows "Complex x y * complex_of_real r = Complex (x*r) (y*r)"
   249   by (simp add: complex_of_real_def)
   250 
   251 lemma complex_of_real_mult_Complex:
   252   shows "complex_of_real r * Complex x y = Complex (r*x) (r*y)"
   253   by (simp add: complex_of_real_def)
   254 
   255 lemma complex_eq_cancel_iff2 [simp]:
   256   shows "(Complex x y = complex_of_real xa) = (x = xa & y = 0)"
   257   by (simp add: complex_of_real_def)
   258 
   259 lemma complex_split_polar:
   260      "\<exists>r a. z = complex_of_real r * (Complex (cos a) (sin a))"
   261   by (simp add: complex_eq_iff polar_Ex)
   262 
   263 
   264 subsection {* Vector Norm *}
   265 
   266 instantiation complex :: real_normed_field
   267 begin
   268 
   269 definition complex_norm_def:
   270   "norm z = sqrt ((Re z)\<twosuperior> + (Im z)\<twosuperior>)"
   271 
   272 abbreviation cmod :: "complex \<Rightarrow> real"
   273   where "cmod \<equiv> norm"
   274 
   275 definition complex_sgn_def:
   276   "sgn x = x /\<^sub>R cmod x"
   277 
   278 definition dist_complex_def:
   279   "dist x y = cmod (x - y)"
   280 
   281 definition open_complex_def:
   282   "open (S :: complex set) \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
   283 
   284 lemmas cmod_def = complex_norm_def
   285 
   286 lemma complex_norm [simp]: "cmod (Complex x y) = sqrt (x\<twosuperior> + y\<twosuperior>)"
   287   by (simp add: complex_norm_def)
   288 
   289 instance proof
   290   fix r :: real and x y :: complex and S :: "complex set"
   291   show "0 \<le> norm x"
   292     by (induct x) simp
   293   show "(norm x = 0) = (x = 0)"
   294     by (induct x) simp
   295   show "norm (x + y) \<le> norm x + norm y"
   296     by (induct x, induct y)
   297        (simp add: real_sqrt_sum_squares_triangle_ineq)
   298   show "norm (scaleR r x) = \<bar>r\<bar> * norm x"
   299     by (induct x)
   300        (simp add: power_mult_distrib right_distrib [symmetric] real_sqrt_mult)
   301   show "norm (x * y) = norm x * norm y"
   302     by (induct x, induct y)
   303        (simp add: real_sqrt_mult [symmetric] power2_eq_square algebra_simps)
   304   show "sgn x = x /\<^sub>R cmod x"
   305     by (rule complex_sgn_def)
   306   show "dist x y = cmod (x - y)"
   307     by (rule dist_complex_def)
   308   show "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
   309     by (rule open_complex_def)
   310 qed
   311 
   312 end
   313 
   314 lemma cmod_unit_one: "cmod (Complex (cos a) (sin a)) = 1"
   315   by simp
   316 
   317 lemma cmod_complex_polar:
   318   "cmod (complex_of_real r * Complex (cos a) (sin a)) = abs r"
   319   by (simp add: norm_mult)
   320 
   321 lemma complex_Re_le_cmod: "Re x \<le> cmod x"
   322   unfolding complex_norm_def
   323   by (rule real_sqrt_sum_squares_ge1)
   324 
   325 lemma complex_mod_minus_le_complex_mod: "- cmod x \<le> cmod x"
   326   by (rule order_trans [OF _ norm_ge_zero], simp)
   327 
   328 lemma complex_mod_triangle_ineq2: "cmod(b + a) - cmod b \<le> cmod a"
   329   by (rule ord_le_eq_trans [OF norm_triangle_ineq2], simp)
   330 
   331 lemma abs_Re_le_cmod: "\<bar>Re x\<bar> \<le> cmod x"
   332   by (cases x) simp
   333 
   334 lemma abs_Im_le_cmod: "\<bar>Im x\<bar> \<le> cmod x"
   335   by (cases x) simp
   336 
   337 text {* Properties of complex signum. *}
   338 
   339 lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)"
   340   by (simp add: sgn_div_norm divide_inverse scaleR_conv_of_real mult_commute)
   341 
   342 lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z"
   343   by (simp add: complex_sgn_def divide_inverse)
   344 
   345 lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z"
   346   by (simp add: complex_sgn_def divide_inverse)
   347 
   348 
   349 subsection {* Completeness of the Complexes *}
   350 
   351 lemma bounded_linear_Re: "bounded_linear Re"
   352   by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def)
   353 
   354 lemma bounded_linear_Im: "bounded_linear Im"
   355   by (rule bounded_linear_intro [where K=1], simp_all add: complex_norm_def)
   356 
   357 lemmas tendsto_Re [tendsto_intros] =
   358   bounded_linear.tendsto [OF bounded_linear_Re]
   359 
   360 lemmas tendsto_Im [tendsto_intros] =
   361   bounded_linear.tendsto [OF bounded_linear_Im]
   362 
   363 lemmas isCont_Re [simp] = bounded_linear.isCont [OF bounded_linear_Re]
   364 lemmas isCont_Im [simp] = bounded_linear.isCont [OF bounded_linear_Im]
   365 lemmas Cauchy_Re = bounded_linear.Cauchy [OF bounded_linear_Re]
   366 lemmas Cauchy_Im = bounded_linear.Cauchy [OF bounded_linear_Im]
   367 
   368 lemma tendsto_Complex [tendsto_intros]:
   369   assumes "(f ---> a) F" and "(g ---> b) F"
   370   shows "((\<lambda>x. Complex (f x) (g x)) ---> Complex a b) F"
   371 proof (rule tendstoI)
   372   fix r :: real assume "0 < r"
   373   hence "0 < r / sqrt 2" by (simp add: divide_pos_pos)
   374   have "eventually (\<lambda>x. dist (f x) a < r / sqrt 2) F"
   375     using `(f ---> a) F` and `0 < r / sqrt 2` by (rule tendstoD)
   376   moreover
   377   have "eventually (\<lambda>x. dist (g x) b < r / sqrt 2) F"
   378     using `(g ---> b) F` and `0 < r / sqrt 2` by (rule tendstoD)
   379   ultimately
   380   show "eventually (\<lambda>x. dist (Complex (f x) (g x)) (Complex a b) < r) F"
   381     by (rule eventually_elim2)
   382        (simp add: dist_norm real_sqrt_sum_squares_less)
   383 qed
   384 
   385 instance complex :: banach
   386 proof
   387   fix X :: "nat \<Rightarrow> complex"
   388   assume X: "Cauchy X"
   389   from Cauchy_Re [OF X] have 1: "(\<lambda>n. Re (X n)) ----> lim (\<lambda>n. Re (X n))"
   390     by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
   391   from Cauchy_Im [OF X] have 2: "(\<lambda>n. Im (X n)) ----> lim (\<lambda>n. Im (X n))"
   392     by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)
   393   have "X ----> Complex (lim (\<lambda>n. Re (X n))) (lim (\<lambda>n. Im (X n)))"
   394     using tendsto_Complex [OF 1 2] by simp
   395   thus "convergent X"
   396     by (rule convergentI)
   397 qed
   398 
   399 
   400 subsection {* The Complex Number $i$ *}
   401 
   402 definition "ii" :: complex  ("\<i>")
   403   where i_def: "ii \<equiv> Complex 0 1"
   404 
   405 lemma complex_Re_i [simp]: "Re ii = 0"
   406   by (simp add: i_def)
   407 
   408 lemma complex_Im_i [simp]: "Im ii = 1"
   409   by (simp add: i_def)
   410 
   411 lemma Complex_eq_i [simp]: "(Complex x y = ii) = (x = 0 \<and> y = 1)"
   412   by (simp add: i_def)
   413 
   414 lemma norm_ii [simp]: "norm ii = 1"
   415   by (simp add: i_def)
   416 
   417 lemma complex_i_not_zero [simp]: "ii \<noteq> 0"
   418   by (simp add: complex_eq_iff)
   419 
   420 lemma complex_i_not_one [simp]: "ii \<noteq> 1"
   421   by (simp add: complex_eq_iff)
   422 
   423 lemma complex_i_not_numeral [simp]: "ii \<noteq> numeral w"
   424   by (simp add: complex_eq_iff)
   425 
   426 lemma complex_i_not_neg_numeral [simp]: "ii \<noteq> neg_numeral w"
   427   by (simp add: complex_eq_iff)
   428 
   429 lemma i_mult_Complex [simp]: "ii * Complex a b = Complex (- b) a"
   430   by (simp add: complex_eq_iff)
   431 
   432 lemma Complex_mult_i [simp]: "Complex a b * ii = Complex (- b) a"
   433   by (simp add: complex_eq_iff)
   434 
   435 lemma i_complex_of_real [simp]: "ii * complex_of_real r = Complex 0 r"
   436   by (simp add: i_def complex_of_real_def)
   437 
   438 lemma complex_of_real_i [simp]: "complex_of_real r * ii = Complex 0 r"
   439   by (simp add: i_def complex_of_real_def)
   440 
   441 lemma i_squared [simp]: "ii * ii = -1"
   442   by (simp add: i_def)
   443 
   444 lemma power2_i [simp]: "ii\<twosuperior> = -1"
   445   by (simp add: power2_eq_square)
   446 
   447 lemma inverse_i [simp]: "inverse ii = - ii"
   448   by (rule inverse_unique, simp)
   449 
   450 lemma complex_i_mult_minus [simp]: "ii * (ii * x) = - x"
   451   by (simp add: mult_assoc [symmetric])
   452 
   453 
   454 subsection {* Complex Conjugation *}
   455 
   456 definition cnj :: "complex \<Rightarrow> complex" where
   457   "cnj z = Complex (Re z) (- Im z)"
   458 
   459 lemma complex_cnj [simp]: "cnj (Complex a b) = Complex a (- b)"
   460   by (simp add: cnj_def)
   461 
   462 lemma complex_Re_cnj [simp]: "Re (cnj x) = Re x"
   463   by (simp add: cnj_def)
   464 
   465 lemma complex_Im_cnj [simp]: "Im (cnj x) = - Im x"
   466   by (simp add: cnj_def)
   467 
   468 lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)"
   469   by (simp add: complex_eq_iff)
   470 
   471 lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z"
   472   by (simp add: cnj_def)
   473 
   474 lemma complex_cnj_zero [simp]: "cnj 0 = 0"
   475   by (simp add: complex_eq_iff)
   476 
   477 lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)"
   478   by (simp add: complex_eq_iff)
   479 
   480 lemma complex_cnj_add: "cnj (x + y) = cnj x + cnj y"
   481   by (simp add: complex_eq_iff)
   482 
   483 lemma complex_cnj_diff: "cnj (x - y) = cnj x - cnj y"
   484   by (simp add: complex_eq_iff)
   485 
   486 lemma complex_cnj_minus: "cnj (- x) = - cnj x"
   487   by (simp add: complex_eq_iff)
   488 
   489 lemma complex_cnj_one [simp]: "cnj 1 = 1"
   490   by (simp add: complex_eq_iff)
   491 
   492 lemma complex_cnj_mult: "cnj (x * y) = cnj x * cnj y"
   493   by (simp add: complex_eq_iff)
   494 
   495 lemma complex_cnj_inverse: "cnj (inverse x) = inverse (cnj x)"
   496   by (simp add: complex_inverse_def)
   497 
   498 lemma complex_cnj_divide: "cnj (x / y) = cnj x / cnj y"
   499   by (simp add: complex_divide_def complex_cnj_mult complex_cnj_inverse)
   500 
   501 lemma complex_cnj_power: "cnj (x ^ n) = cnj x ^ n"
   502   by (induct n, simp_all add: complex_cnj_mult)
   503 
   504 lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n"
   505   by (simp add: complex_eq_iff)
   506 
   507 lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z"
   508   by (simp add: complex_eq_iff)
   509 
   510 lemma complex_cnj_numeral [simp]: "cnj (numeral w) = numeral w"
   511   by (simp add: complex_eq_iff)
   512 
   513 lemma complex_cnj_neg_numeral [simp]: "cnj (neg_numeral w) = neg_numeral w"
   514   by (simp add: complex_eq_iff)
   515 
   516 lemma complex_cnj_scaleR: "cnj (scaleR r x) = scaleR r (cnj x)"
   517   by (simp add: complex_eq_iff)
   518 
   519 lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z"
   520   by (simp add: complex_norm_def)
   521 
   522 lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x"
   523   by (simp add: complex_eq_iff)
   524 
   525 lemma complex_cnj_i [simp]: "cnj ii = - ii"
   526   by (simp add: complex_eq_iff)
   527 
   528 lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)"
   529   by (simp add: complex_eq_iff)
   530 
   531 lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im z) * ii"
   532   by (simp add: complex_eq_iff)
   533 
   534 lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\<twosuperior> + (Im z)\<twosuperior>)"
   535   by (simp add: complex_eq_iff power2_eq_square)
   536 
   537 lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<twosuperior>"
   538   by (simp add: norm_mult power2_eq_square)
   539 
   540 lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))"
   541   by (simp add: cmod_def power2_eq_square)
   542 
   543 lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0"
   544   by simp
   545 
   546 lemma bounded_linear_cnj: "bounded_linear cnj"
   547   using complex_cnj_add complex_cnj_scaleR
   548   by (rule bounded_linear_intro [where K=1], simp)
   549 
   550 lemmas tendsto_cnj [tendsto_intros] =
   551   bounded_linear.tendsto [OF bounded_linear_cnj]
   552 
   553 lemmas isCont_cnj [simp] =
   554   bounded_linear.isCont [OF bounded_linear_cnj]
   555 
   556 
   557 subsection{*Finally! Polar Form for Complex Numbers*}
   558 
   559 subsubsection {* $\cos \theta + i \sin \theta$ *}
   560 
   561 definition cis :: "real \<Rightarrow> complex" where
   562   "cis a = Complex (cos a) (sin a)"
   563 
   564 lemma Re_cis [simp]: "Re (cis a) = cos a"
   565   by (simp add: cis_def)
   566 
   567 lemma Im_cis [simp]: "Im (cis a) = sin a"
   568   by (simp add: cis_def)
   569 
   570 lemma cis_zero [simp]: "cis 0 = 1"
   571   by (simp add: cis_def)
   572 
   573 lemma norm_cis [simp]: "norm (cis a) = 1"
   574   by (simp add: cis_def)
   575 
   576 lemma sgn_cis [simp]: "sgn (cis a) = cis a"
   577   by (simp add: sgn_div_norm)
   578 
   579 lemma cis_neq_zero [simp]: "cis a \<noteq> 0"
   580   by (metis norm_cis norm_zero zero_neq_one)
   581 
   582 lemma cis_mult: "cis a * cis b = cis (a + b)"
   583   by (simp add: cis_def cos_add sin_add)
   584 
   585 lemma DeMoivre: "(cis a) ^ n = cis (real n * a)"
   586   by (induct n, simp_all add: real_of_nat_Suc algebra_simps cis_mult)
   587 
   588 lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)"
   589   by (simp add: cis_def)
   590 
   591 lemma cis_divide: "cis a / cis b = cis (a - b)"
   592   by (simp add: complex_divide_def cis_mult diff_minus)
   593 
   594 lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)"
   595   by (auto simp add: DeMoivre)
   596 
   597 lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)"
   598   by (auto simp add: DeMoivre)
   599 
   600 subsubsection {* $r(\cos \theta + i \sin \theta)$ *}
   601 
   602 definition rcis :: "[real, real] \<Rightarrow> complex" where
   603   "rcis r a = complex_of_real r * cis a"
   604 
   605 lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a"
   606   by (simp add: rcis_def)
   607 
   608 lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a"
   609   by (simp add: rcis_def)
   610 
   611 lemma rcis_Ex: "\<exists>r a. z = rcis r a"
   612   by (simp add: complex_eq_iff polar_Ex)
   613 
   614 lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r"
   615   by (simp add: rcis_def norm_mult)
   616 
   617 lemma cis_rcis_eq: "cis a = rcis 1 a"
   618   by (simp add: rcis_def)
   619 
   620 lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)"
   621   by (simp add: rcis_def cis_mult)
   622 
   623 lemma rcis_zero_mod [simp]: "rcis 0 a = 0"
   624   by (simp add: rcis_def)
   625 
   626 lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r"
   627   by (simp add: rcis_def)
   628 
   629 lemma rcis_eq_zero_iff [simp]: "rcis r a = 0 \<longleftrightarrow> r = 0"
   630   by (simp add: rcis_def)
   631 
   632 lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)"
   633   by (simp add: rcis_def power_mult_distrib DeMoivre)
   634 
   635 lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)"
   636   by (simp add: divide_inverse rcis_def)
   637 
   638 lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)"
   639   by (simp add: rcis_def cis_divide [symmetric])
   640 
   641 subsubsection {* Complex exponential *}
   642 
   643 abbreviation expi :: "complex \<Rightarrow> complex"
   644   where "expi \<equiv> exp"
   645 
   646 lemma cis_conv_exp: "cis b = exp (Complex 0 b)"
   647 proof (rule complex_eqI)
   648   { fix n have "Complex 0 b ^ n =
   649     real (fact n) *\<^sub>R Complex (cos_coeff n * b ^ n) (sin_coeff n * b ^ n)"
   650       apply (induct n)
   651       apply (simp add: cos_coeff_def sin_coeff_def)
   652       apply (simp add: sin_coeff_Suc cos_coeff_Suc del: mult_Suc)
   653       done } note * = this
   654   show "Re (cis b) = Re (exp (Complex 0 b))"
   655     unfolding exp_def cis_def cos_def
   656     by (subst bounded_linear.suminf[OF bounded_linear_Re summable_exp_generic],
   657       simp add: * mult_assoc [symmetric])
   658   show "Im (cis b) = Im (exp (Complex 0 b))"
   659     unfolding exp_def cis_def sin_def
   660     by (subst bounded_linear.suminf[OF bounded_linear_Im summable_exp_generic],
   661       simp add: * mult_assoc [symmetric])
   662 qed
   663 
   664 lemma expi_def: "expi z = complex_of_real (exp (Re z)) * cis (Im z)"
   665   unfolding cis_conv_exp exp_of_real [symmetric] mult_exp_exp by simp
   666 
   667 lemma Re_exp: "Re (exp z) = exp (Re z) * cos (Im z)"
   668   unfolding expi_def by simp
   669 
   670 lemma Im_exp: "Im (exp z) = exp (Re z) * sin (Im z)"
   671   unfolding expi_def by simp
   672 
   673 lemma complex_expi_Ex: "\<exists>a r. z = complex_of_real r * expi a"
   674 apply (insert rcis_Ex [of z])
   675 apply (auto simp add: expi_def rcis_def mult_assoc [symmetric])
   676 apply (rule_tac x = "ii * complex_of_real a" in exI, auto)
   677 done
   678 
   679 lemma expi_two_pi_i [simp]: "expi((2::complex) * complex_of_real pi * ii) = 1"
   680   by (simp add: expi_def cis_def)
   681 
   682 subsubsection {* Complex argument *}
   683 
   684 definition arg :: "complex \<Rightarrow> real" where
   685   "arg z = (if z = 0 then 0 else (SOME a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi))"
   686 
   687 lemma arg_zero: "arg 0 = 0"
   688   by (simp add: arg_def)
   689 
   690 lemma of_nat_less_of_int_iff: (* TODO: move *)
   691   "(of_nat n :: 'a::linordered_idom) < of_int x \<longleftrightarrow> int n < x"
   692   by (metis of_int_of_nat_eq of_int_less_iff)
   693 
   694 lemma real_of_nat_less_numeral_iff [simp]: (* TODO: move *)
   695   "real (n::nat) < numeral w \<longleftrightarrow> n < numeral w"
   696   using of_nat_less_of_int_iff [of n "numeral w", where 'a=real]
   697   by (simp add: real_of_nat_def zless_nat_eq_int_zless [symmetric])
   698 
   699 lemma arg_unique:
   700   assumes "sgn z = cis x" and "-pi < x" and "x \<le> pi"
   701   shows "arg z = x"
   702 proof -
   703   from assms have "z \<noteq> 0" by auto
   704   have "(SOME a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi) = x"
   705   proof
   706     fix a def d \<equiv> "a - x"
   707     assume a: "sgn z = cis a \<and> - pi < a \<and> a \<le> pi"
   708     from a assms have "- (2*pi) < d \<and> d < 2*pi"
   709       unfolding d_def by simp
   710     moreover from a assms have "cos a = cos x" and "sin a = sin x"
   711       by (simp_all add: complex_eq_iff)
   712     hence "cos d = 1" unfolding d_def cos_diff by simp
   713     moreover hence "sin d = 0" by (rule cos_one_sin_zero)
   714     ultimately have "d = 0"
   715       unfolding sin_zero_iff even_mult_two_ex
   716       by (safe, auto simp add: numeral_2_eq_2 less_Suc_eq)
   717     thus "a = x" unfolding d_def by simp
   718   qed (simp add: assms del: Re_sgn Im_sgn)
   719   with `z \<noteq> 0` show "arg z = x"
   720     unfolding arg_def by simp
   721 qed
   722 
   723 lemma arg_correct:
   724   assumes "z \<noteq> 0" shows "sgn z = cis (arg z) \<and> -pi < arg z \<and> arg z \<le> pi"
   725 proof (simp add: arg_def assms, rule someI_ex)
   726   obtain r a where z: "z = rcis r a" using rcis_Ex by fast
   727   with assms have "r \<noteq> 0" by auto
   728   def b \<equiv> "if 0 < r then a else a + pi"
   729   have b: "sgn z = cis b"
   730     unfolding z b_def rcis_def using `r \<noteq> 0`
   731     by (simp add: of_real_def sgn_scaleR sgn_if, simp add: cis_def)
   732   have cis_2pi_nat: "\<And>n. cis (2 * pi * real_of_nat n) = 1"
   733     by (induct_tac n, simp_all add: right_distrib cis_mult [symmetric],
   734       simp add: cis_def)
   735   have cis_2pi_int: "\<And>x. cis (2 * pi * real_of_int x) = 1"
   736     by (case_tac x rule: int_diff_cases,
   737       simp add: right_diff_distrib cis_divide [symmetric] cis_2pi_nat)
   738   def c \<equiv> "b - 2*pi * of_int \<lceil>(b - pi) / (2*pi)\<rceil>"
   739   have "sgn z = cis c"
   740     unfolding b c_def
   741     by (simp add: cis_divide [symmetric] cis_2pi_int)
   742   moreover have "- pi < c \<and> c \<le> pi"
   743     using ceiling_correct [of "(b - pi) / (2*pi)"]
   744     by (simp add: c_def less_divide_eq divide_le_eq algebra_simps)
   745   ultimately show "\<exists>a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi" by fast
   746 qed
   747 
   748 lemma arg_bounded: "- pi < arg z \<and> arg z \<le> pi"
   749   by (cases "z = 0", simp_all add: arg_zero arg_correct)
   750 
   751 lemma cis_arg: "z \<noteq> 0 \<Longrightarrow> cis (arg z) = sgn z"
   752   by (simp add: arg_correct)
   753 
   754 lemma rcis_cmod_arg: "rcis (cmod z) (arg z) = z"
   755   by (cases "z = 0", simp_all add: rcis_def cis_arg sgn_div_norm of_real_def)
   756 
   757 lemma cos_arg_i_mult_zero [simp]:
   758      "y \<noteq> 0 ==> cos (arg(Complex 0 y)) = 0"
   759   using cis_arg [of "Complex 0 y"] by (simp add: complex_eq_iff)
   760 
   761 text {* Legacy theorem names *}
   762 
   763 lemmas expand_complex_eq = complex_eq_iff
   764 lemmas complex_Re_Im_cancel_iff = complex_eq_iff
   765 lemmas complex_equality = complex_eqI
   766 
   767 end