src/HOL/Complex.thy
 changeset 44724 0b900a9d8023 parent 44715 1a17d8913976 child 44748 7f6838b3474a
```     1.1 --- a/src/HOL/Complex.thy	Mon Sep 05 14:42:31 2011 +0200
1.2 +++ b/src/HOL/Complex.thy	Mon Sep 05 08:38:50 2011 -0700
1.3 @@ -12,15 +12,11 @@
1.4
1.5  datatype complex = Complex real real
1.6
1.7 -primrec
1.8 -  Re :: "complex \<Rightarrow> real"
1.9 -where
1.10 -  Re: "Re (Complex x y) = x"
1.11 +primrec Re :: "complex \<Rightarrow> real"
1.12 +  where Re: "Re (Complex x y) = x"
1.13
1.14 -primrec
1.15 -  Im :: "complex \<Rightarrow> real"
1.16 -where
1.17 -  Im: "Im (Complex x y) = y"
1.18 +primrec Im :: "complex \<Rightarrow> real"
1.19 +  where Im: "Im (Complex x y) = y"
1.20
1.21  lemma complex_surj [simp]: "Complex (Re z) (Im z) = z"
1.22    by (induct z) simp
1.23 @@ -37,17 +33,17 @@
1.25  begin
1.26
1.27 -definition
1.28 -  complex_zero_def: "0 = Complex 0 0"
1.29 +definition complex_zero_def:
1.30 +  "0 = Complex 0 0"
1.31
1.32 -definition
1.33 -  complex_add_def: "x + y = Complex (Re x + Re y) (Im x + Im y)"
1.35 +  "x + y = Complex (Re x + Re y) (Im x + Im y)"
1.36
1.37 -definition
1.38 -  complex_minus_def: "- x = Complex (- Re x) (- Im x)"
1.39 +definition complex_minus_def:
1.40 +  "- x = Complex (- Re x) (- Im x)"
1.41
1.42 -definition
1.43 -  complex_diff_def: "x - (y\<Colon>complex) = x + - y"
1.44 +definition complex_diff_def:
1.45 +  "x - (y\<Colon>complex) = x + - y"
1.46
1.47  lemma Complex_eq_0 [simp]: "Complex a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
1.49 @@ -94,25 +90,23 @@
1.50  end
1.51
1.52
1.53 -
1.54  subsection {* Multiplication and Division *}
1.55
1.56  instantiation complex :: field_inverse_zero
1.57  begin
1.58
1.59 -definition
1.60 -  complex_one_def: "1 = Complex 1 0"
1.61 +definition complex_one_def:
1.62 +  "1 = Complex 1 0"
1.63
1.64 -definition
1.65 -  complex_mult_def: "x * y =
1.66 -    Complex (Re x * Re y - Im x * Im y) (Re x * Im y + Im x * Re y)"
1.67 +definition complex_mult_def:
1.68 +  "x * y = Complex (Re x * Re y - Im x * Im y) (Re x * Im y + Im x * Re y)"
1.69
1.70 -definition
1.71 -  complex_inverse_def: "inverse x =
1.72 +definition complex_inverse_def:
1.73 +  "inverse x =
1.74      Complex (Re x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)) (- Im x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>))"
1.75
1.76 -definition
1.77 -  complex_divide_def: "x / (y\<Colon>complex) = x * inverse y"
1.78 +definition complex_divide_def:
1.79 +  "x / (y\<Colon>complex) = x * inverse y"
1.80
1.81  lemma Complex_eq_1 [simp]: "(Complex a b = 1) = (a = 1 \<and> b = 0)"
1.83 @@ -147,10 +141,10 @@
1.84
1.85  instance
1.86    by intro_classes (simp_all add: complex_mult_def
1.87 -  right_distrib left_distrib right_diff_distrib left_diff_distrib
1.88 -  complex_inverse_def complex_divide_def
1.90 -  complex_eq_iff)
1.91 +    right_distrib left_distrib right_diff_distrib left_diff_distrib
1.92 +    complex_inverse_def complex_divide_def
1.94 +    complex_eq_iff)
1.95
1.96  end
1.97
1.98 @@ -160,8 +154,8 @@
1.99  instantiation complex :: number_ring
1.100  begin
1.101
1.102 -definition number_of_complex where
1.103 -  complex_number_of_def: "number_of w = (of_int w \<Colon> complex)"
1.104 +definition complex_number_of_def:
1.105 +  "number_of w = (of_int w \<Colon> complex)"
1.106
1.107  instance
1.108    by intro_classes (simp only: complex_number_of_def)
1.109 @@ -169,26 +163,26 @@
1.110  end
1.111
1.112  lemma complex_Re_of_nat [simp]: "Re (of_nat n) = of_nat n"
1.113 -by (induct n) simp_all
1.114 +  by (induct n) simp_all
1.115
1.116  lemma complex_Im_of_nat [simp]: "Im (of_nat n) = 0"
1.117 -by (induct n) simp_all
1.118 +  by (induct n) simp_all
1.119
1.120  lemma complex_Re_of_int [simp]: "Re (of_int z) = of_int z"
1.121 -by (cases z rule: int_diff_cases) simp
1.122 +  by (cases z rule: int_diff_cases) simp
1.123
1.124  lemma complex_Im_of_int [simp]: "Im (of_int z) = 0"
1.125 -by (cases z rule: int_diff_cases) simp
1.126 +  by (cases z rule: int_diff_cases) simp
1.127
1.128  lemma complex_Re_number_of [simp]: "Re (number_of v) = number_of v"
1.129 -unfolding number_of_eq by (rule complex_Re_of_int)
1.130 +  unfolding number_of_eq by (rule complex_Re_of_int)
1.131
1.132  lemma complex_Im_number_of [simp]: "Im (number_of v) = 0"
1.133 -unfolding number_of_eq by (rule complex_Im_of_int)
1.134 +  unfolding number_of_eq by (rule complex_Im_of_int)
1.135
1.136  lemma Complex_eq_number_of [simp]:
1.137    "(Complex a b = number_of w) = (a = number_of w \<and> b = 0)"
1.139 +  by (simp add: complex_eq_iff)
1.140
1.141
1.142  subsection {* Scalar Multiplication *}
1.143 @@ -196,8 +190,8 @@
1.144  instantiation complex :: real_field
1.145  begin
1.146
1.147 -definition
1.148 -  complex_scaleR_def: "scaleR r x = Complex (r * Re x) (r * Im x)"
1.149 +definition complex_scaleR_def:
1.150 +  "scaleR r x = Complex (r * Re x) (r * Im x)"
1.151
1.152  lemma complex_scaleR [simp]:
1.153    "scaleR r (Complex a b) = Complex (r * a) (r * b)"
1.154 @@ -231,34 +225,33 @@
1.155
1.156  subsection{* Properties of Embedding from Reals *}
1.157
1.158 -abbreviation
1.159 -  complex_of_real :: "real \<Rightarrow> complex" where
1.160 -    "complex_of_real \<equiv> of_real"
1.161 +abbreviation complex_of_real :: "real \<Rightarrow> complex"
1.162 +  where "complex_of_real \<equiv> of_real"
1.163
1.164  lemma complex_of_real_def: "complex_of_real r = Complex r 0"
1.165 -by (simp add: of_real_def complex_scaleR_def)
1.166 +  by (simp add: of_real_def complex_scaleR_def)
1.167
1.168  lemma Re_complex_of_real [simp]: "Re (complex_of_real z) = z"
1.170 +  by (simp add: complex_of_real_def)
1.171
1.172  lemma Im_complex_of_real [simp]: "Im (complex_of_real z) = 0"
1.174 +  by (simp add: complex_of_real_def)
1.175
1.177 -     "Complex x y + complex_of_real r = Complex (x+r) y"
1.179 +  shows "Complex x y + complex_of_real r = Complex (x+r) y"
1.180 +  by (simp add: complex_of_real_def)
1.181
1.183 -     "complex_of_real r + Complex x y = Complex (r+x) y"
1.185 +  shows "complex_of_real r + Complex x y = Complex (r+x) y"
1.186 +  by (simp add: complex_of_real_def)
1.187
1.188  lemma Complex_mult_complex_of_real:
1.189 -     "Complex x y * complex_of_real r = Complex (x*r) (y*r)"
1.191 +  shows "Complex x y * complex_of_real r = Complex (x*r) (y*r)"
1.192 +  by (simp add: complex_of_real_def)
1.193
1.194  lemma complex_of_real_mult_Complex:
1.195 -     "complex_of_real r * Complex x y = Complex (r*x) (r*y)"
1.197 +  shows "complex_of_real r * Complex x y = Complex (r*x) (r*y)"
1.198 +  by (simp add: complex_of_real_def)
1.199
1.200
1.201  subsection {* Vector Norm *}
1.202 @@ -269,9 +262,8 @@
1.203  definition complex_norm_def:
1.204    "norm z = sqrt ((Re z)\<twosuperior> + (Im z)\<twosuperior>)"
1.205
1.206 -abbreviation
1.207 -  cmod :: "complex \<Rightarrow> real" where
1.208 -  "cmod \<equiv> norm"
1.209 +abbreviation cmod :: "complex \<Rightarrow> real"
1.210 +  where "cmod \<equiv> norm"
1.211
1.212  definition complex_sgn_def:
1.213    "sgn x = x /\<^sub>R cmod x"
1.214 @@ -313,29 +305,30 @@
1.215  end
1.216
1.217  lemma cmod_unit_one [simp]: "cmod (Complex (cos a) (sin a)) = 1"
1.218 -by simp
1.219 +  by simp
1.220
1.221  lemma cmod_complex_polar [simp]:
1.222 -     "cmod (complex_of_real r * Complex (cos a) (sin a)) = abs r"
1.224 +  "cmod (complex_of_real r * Complex (cos a) (sin a)) = abs r"
1.225 +  by (simp add: norm_mult)
1.226
1.227  lemma complex_Re_le_cmod: "Re x \<le> cmod x"
1.228 -unfolding complex_norm_def
1.229 -by (rule real_sqrt_sum_squares_ge1)
1.230 +  unfolding complex_norm_def
1.231 +  by (rule real_sqrt_sum_squares_ge1)
1.232
1.233  lemma complex_mod_minus_le_complex_mod [simp]: "- cmod x \<le> cmod x"
1.234 -by (rule order_trans [OF _ norm_ge_zero], simp)
1.235 +  by (rule order_trans [OF _ norm_ge_zero], simp)
1.236
1.237  lemma complex_mod_triangle_ineq2 [simp]: "cmod(b + a) - cmod b \<le> cmod a"
1.238 -by (rule ord_le_eq_trans [OF norm_triangle_ineq2], simp)
1.239 +  by (rule ord_le_eq_trans [OF norm_triangle_ineq2], simp)
1.240
1.241  lemmas real_sum_squared_expand = power2_sum [where 'a=real]
1.242
1.243  lemma abs_Re_le_cmod: "\<bar>Re x\<bar> \<le> cmod x"
1.244 -by (cases x) simp
1.245 +  by (cases x) simp
1.246
1.247  lemma abs_Im_le_cmod: "\<bar>Im x\<bar> \<le> cmod x"
1.248 -by (cases x) simp
1.249 +  by (cases x) simp
1.250 +
1.251
1.252  subsection {* Completeness of the Complexes *}
1.253
1.254 @@ -357,25 +350,25 @@
1.255  lemmas Cauchy_Im = bounded_linear.Cauchy [OF bounded_linear_Im]
1.256
1.257  lemma tendsto_Complex [tendsto_intros]:
1.258 -  assumes "(f ---> a) net" and "(g ---> b) net"
1.259 -  shows "((\<lambda>x. Complex (f x) (g x)) ---> Complex a b) net"
1.260 +  assumes "(f ---> a) F" and "(g ---> b) F"
1.261 +  shows "((\<lambda>x. Complex (f x) (g x)) ---> Complex a b) F"
1.262  proof (rule tendstoI)
1.263    fix r :: real assume "0 < r"
1.264    hence "0 < r / sqrt 2" by (simp add: divide_pos_pos)
1.265 -  have "eventually (\<lambda>x. dist (f x) a < r / sqrt 2) net"
1.266 -    using `(f ---> a) net` and `0 < r / sqrt 2` by (rule tendstoD)
1.267 +  have "eventually (\<lambda>x. dist (f x) a < r / sqrt 2) F"
1.268 +    using `(f ---> a) F` and `0 < r / sqrt 2` by (rule tendstoD)
1.269    moreover
1.270 -  have "eventually (\<lambda>x. dist (g x) b < r / sqrt 2) net"
1.271 -    using `(g ---> b) net` and `0 < r / sqrt 2` by (rule tendstoD)
1.272 +  have "eventually (\<lambda>x. dist (g x) b < r / sqrt 2) F"
1.273 +    using `(g ---> b) F` and `0 < r / sqrt 2` by (rule tendstoD)
1.274    ultimately
1.275 -  show "eventually (\<lambda>x. dist (Complex (f x) (g x)) (Complex a b) < r) net"
1.276 +  show "eventually (\<lambda>x. dist (Complex (f x) (g x)) (Complex a b) < r) F"
1.277      by (rule eventually_elim2)
1.279  qed
1.280
1.281  lemma LIMSEQ_Complex:
1.282    "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. Complex (X n) (Y n)) ----> Complex a b"
1.283 -by (rule tendsto_Complex)
1.284 +  by (rule tendsto_Complex)
1.285
1.286  instance complex :: banach
1.287  proof
1.288 @@ -394,133 +387,131 @@
1.289
1.290  subsection {* The Complex Number @{term "\<i>"} *}
1.291
1.292 -definition
1.293 -  "ii" :: complex  ("\<i>") where
1.294 -  i_def: "ii \<equiv> Complex 0 1"
1.295 +definition "ii" :: complex  ("\<i>")
1.296 +  where i_def: "ii \<equiv> Complex 0 1"
1.297
1.298  lemma complex_Re_i [simp]: "Re ii = 0"
1.300 +  by (simp add: i_def)
1.301
1.302  lemma complex_Im_i [simp]: "Im ii = 1"
1.304 +  by (simp add: i_def)
1.305
1.306  lemma Complex_eq_i [simp]: "(Complex x y = ii) = (x = 0 \<and> y = 1)"
1.308 +  by (simp add: i_def)
1.309
1.310  lemma complex_i_not_zero [simp]: "ii \<noteq> 0"
1.312 +  by (simp add: complex_eq_iff)
1.313
1.314  lemma complex_i_not_one [simp]: "ii \<noteq> 1"
1.316 +  by (simp add: complex_eq_iff)
1.317
1.318  lemma complex_i_not_number_of [simp]: "ii \<noteq> number_of w"
1.320 +  by (simp add: complex_eq_iff)
1.321
1.322  lemma i_mult_Complex [simp]: "ii * Complex a b = Complex (- b) a"
1.324 +  by (simp add: complex_eq_iff)
1.325
1.326  lemma Complex_mult_i [simp]: "Complex a b * ii = Complex (- b) a"
1.328 +  by (simp add: complex_eq_iff)
1.329
1.330  lemma i_complex_of_real [simp]: "ii * complex_of_real r = Complex 0 r"
1.331 -by (simp add: i_def complex_of_real_def)
1.332 +  by (simp add: i_def complex_of_real_def)
1.333
1.334  lemma complex_of_real_i [simp]: "complex_of_real r * ii = Complex 0 r"
1.335 -by (simp add: i_def complex_of_real_def)
1.336 +  by (simp add: i_def complex_of_real_def)
1.337
1.338  lemma i_squared [simp]: "ii * ii = -1"
1.340 +  by (simp add: i_def)
1.341
1.342  lemma power2_i [simp]: "ii\<twosuperior> = -1"
1.344 +  by (simp add: power2_eq_square)
1.345
1.346  lemma inverse_i [simp]: "inverse ii = - ii"
1.347 -by (rule inverse_unique, simp)
1.348 +  by (rule inverse_unique, simp)
1.349
1.350
1.351  subsection {* Complex Conjugation *}
1.352
1.353 -definition
1.354 -  cnj :: "complex \<Rightarrow> complex" where
1.355 +definition cnj :: "complex \<Rightarrow> complex" where
1.356    "cnj z = Complex (Re z) (- Im z)"
1.357
1.358  lemma complex_cnj [simp]: "cnj (Complex a b) = Complex a (- b)"
1.360 +  by (simp add: cnj_def)
1.361
1.362  lemma complex_Re_cnj [simp]: "Re (cnj x) = Re x"
1.364 +  by (simp add: cnj_def)
1.365
1.366  lemma complex_Im_cnj [simp]: "Im (cnj x) = - Im x"
1.368 +  by (simp add: cnj_def)
1.369
1.370  lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)"
1.372 +  by (simp add: complex_eq_iff)
1.373
1.374  lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z"
1.376 +  by (simp add: cnj_def)
1.377
1.378  lemma complex_cnj_zero [simp]: "cnj 0 = 0"
1.380 +  by (simp add: complex_eq_iff)
1.381
1.382  lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)"
1.384 +  by (simp add: complex_eq_iff)
1.385
1.386  lemma complex_cnj_add: "cnj (x + y) = cnj x + cnj y"
1.388 +  by (simp add: complex_eq_iff)
1.389
1.390  lemma complex_cnj_diff: "cnj (x - y) = cnj x - cnj y"
1.392 +  by (simp add: complex_eq_iff)
1.393
1.394  lemma complex_cnj_minus: "cnj (- x) = - cnj x"
1.396 +  by (simp add: complex_eq_iff)
1.397
1.398  lemma complex_cnj_one [simp]: "cnj 1 = 1"
1.400 +  by (simp add: complex_eq_iff)
1.401
1.402  lemma complex_cnj_mult: "cnj (x * y) = cnj x * cnj y"
1.404 +  by (simp add: complex_eq_iff)
1.405
1.406  lemma complex_cnj_inverse: "cnj (inverse x) = inverse (cnj x)"
1.408 +  by (simp add: complex_inverse_def)
1.409
1.410  lemma complex_cnj_divide: "cnj (x / y) = cnj x / cnj y"
1.411 -by (simp add: complex_divide_def complex_cnj_mult complex_cnj_inverse)
1.412 +  by (simp add: complex_divide_def complex_cnj_mult complex_cnj_inverse)
1.413
1.414  lemma complex_cnj_power: "cnj (x ^ n) = cnj x ^ n"
1.415 -by (induct n, simp_all add: complex_cnj_mult)
1.416 +  by (induct n, simp_all add: complex_cnj_mult)
1.417
1.418  lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n"
1.420 +  by (simp add: complex_eq_iff)
1.421
1.422  lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z"
1.424 +  by (simp add: complex_eq_iff)
1.425
1.426  lemma complex_cnj_number_of [simp]: "cnj (number_of w) = number_of w"
1.428 +  by (simp add: complex_eq_iff)
1.429
1.430  lemma complex_cnj_scaleR: "cnj (scaleR r x) = scaleR r (cnj x)"
1.432 +  by (simp add: complex_eq_iff)
1.433
1.434  lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z"
1.436 +  by (simp add: complex_norm_def)
1.437
1.438  lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x"
1.440 +  by (simp add: complex_eq_iff)
1.441
1.442  lemma complex_cnj_i [simp]: "cnj ii = - ii"
1.444 +  by (simp add: complex_eq_iff)
1.445
1.446  lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)"
1.448 +  by (simp add: complex_eq_iff)
1.449
1.450  lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im z) * ii"
1.452 +  by (simp add: complex_eq_iff)
1.453
1.454  lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\<twosuperior> + (Im z)\<twosuperior>)"
1.455 -by (simp add: complex_eq_iff power2_eq_square)
1.456 +  by (simp add: complex_eq_iff power2_eq_square)
1.457
1.458  lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<twosuperior>"
1.459 -by (simp add: norm_mult power2_eq_square)
1.460 +  by (simp add: norm_mult power2_eq_square)
1.461
1.462  lemma bounded_linear_cnj: "bounded_linear cnj"
1.464 @@ -537,34 +528,33 @@
1.465
1.466  text {*------------ Argand -------------*}
1.467
1.468 -definition
1.469 -  arg :: "complex => real" where
1.470 +definition arg :: "complex => real" where
1.471    "arg z = (SOME a. Re(sgn z) = cos a & Im(sgn z) = sin a & -pi < a & a \<le> pi)"
1.472
1.473  lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)"
1.474 -by (simp add: complex_sgn_def divide_inverse scaleR_conv_of_real mult_commute)
1.475 +  by (simp add: sgn_div_norm divide_inverse scaleR_conv_of_real mult_commute)
1.476
1.477  lemma i_mult_eq: "ii * ii = complex_of_real (-1)"
1.478 -by (simp add: i_def complex_of_real_def)
1.479 +  by (simp add: i_def complex_of_real_def)
1.480
1.481  lemma i_mult_eq2 [simp]: "ii * ii = -(1::complex)"
1.482 -by (simp add: i_def complex_one_def)
1.483 +  by (simp add: i_def complex_one_def)
1.484
1.485  lemma complex_eq_cancel_iff2 [simp]:
1.486 -     "(Complex x y = complex_of_real xa) = (x = xa & y = 0)"
1.488 +  shows "(Complex x y = complex_of_real xa) = (x = xa & y = 0)"
1.489 +  by (simp add: complex_of_real_def)
1.490
1.491  lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z"
1.492 -by (simp add: complex_sgn_def divide_inverse)
1.493 +  by (simp add: complex_sgn_def divide_inverse)
1.494
1.495  lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z"
1.496 -by (simp add: complex_sgn_def divide_inverse)
1.497 +  by (simp add: complex_sgn_def divide_inverse)
1.498
1.499  lemma complex_inverse_complex_split:
1.500       "inverse(complex_of_real x + ii * complex_of_real y) =
1.501        complex_of_real(x/(x ^ 2 + y ^ 2)) -
1.502        ii * complex_of_real(y/(x ^ 2 + y ^ 2))"
1.503 -by (simp add: complex_of_real_def i_def diff_minus divide_inverse)
1.504 +  by (simp add: complex_of_real_def i_def diff_minus divide_inverse)
1.505
1.506  (*----------------------------------------------------------------------------*)
1.507  (* Many of the theorems below need to be moved elsewhere e.g. Transc. Also *)
1.508 @@ -638,10 +628,10 @@
1.509  done
1.510
1.511  lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a"
1.512 -by (simp add: rcis_def cis_def)
1.513 +  by (simp add: rcis_def cis_def)
1.514
1.515  lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a"
1.516 -by (simp add: rcis_def cis_def)
1.517 +  by (simp add: rcis_def cis_def)
1.518
1.519  lemma sin_cos_squared_add2_mult: "(r * cos a)\<twosuperior> + (r * sin a)\<twosuperior> = r\<twosuperior>"
1.520  proof -
1.521 @@ -651,44 +641,44 @@
1.522  qed
1.523
1.524  lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r"
1.527
1.528  lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))"
1.529 -by (simp add: cmod_def power2_eq_square)
1.530 +  by (simp add: cmod_def power2_eq_square)
1.531
1.532  lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0"
1.533 -by simp
1.534 +  by simp
1.535
1.536  lemma cis_rcis_eq: "cis a = rcis 1 a"
1.538 +  by (simp add: rcis_def)
1.539
1.540  lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)"
1.542 -              complex_of_real_def)
1.544 +    right_diff_distrib complex_of_real_def)
1.545
1.546  lemma cis_mult: "cis a * cis b = cis (a + b)"
1.547 -by (simp add: cis_rcis_eq rcis_mult)
1.548 +  by (simp add: cis_rcis_eq rcis_mult)
1.549
1.550  lemma cis_zero [simp]: "cis 0 = 1"
1.551 -by (simp add: cis_def complex_one_def)
1.552 +  by (simp add: cis_def complex_one_def)
1.553
1.554  lemma rcis_zero_mod [simp]: "rcis 0 a = 0"
1.556 +  by (simp add: rcis_def)
1.557
1.558  lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r"
1.560 +  by (simp add: rcis_def)
1.561
1.562  lemma complex_of_real_minus_one:
1.563     "complex_of_real (-(1::real)) = -(1::complex)"
1.564 -by (simp add: complex_of_real_def complex_one_def)
1.565 +  by (simp add: complex_of_real_def complex_one_def)
1.566
1.567  lemma complex_i_mult_minus [simp]: "ii * (ii * x) = - x"
1.568 -by (simp add: mult_assoc [symmetric])
1.569 +  by (simp add: mult_assoc [symmetric])
1.570
1.571
1.572  lemma cis_real_of_nat_Suc_mult:
1.573     "cis (real (Suc n) * a) = cis a * cis (real n * a)"
1.576
1.577  lemma DeMoivre: "(cis a) ^ n = cis (real n * a)"
1.578  apply (induct_tac "n")
1.579 @@ -696,16 +686,16 @@
1.580  done
1.581
1.582  lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)"
1.583 -by (simp add: rcis_def power_mult_distrib DeMoivre)
1.584 +  by (simp add: rcis_def power_mult_distrib DeMoivre)
1.585
1.586  lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)"
1.587 -by (simp add: cis_def complex_inverse_complex_split diff_minus)
1.588 +  by (simp add: cis_def complex_inverse_complex_split diff_minus)
1.589
1.590  lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)"
1.591 -by (simp add: divide_inverse rcis_def)
1.592 +  by (simp add: divide_inverse rcis_def)
1.593
1.594  lemma cis_divide: "cis a / cis b = cis (a - b)"
1.595 -by (simp add: complex_divide_def cis_mult diff_minus)
1.596 +  by (simp add: complex_divide_def cis_mult diff_minus)
1.597
1.598  lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)"
1.600 @@ -714,16 +704,16 @@
1.601  done
1.602
1.603  lemma Re_cis [simp]: "Re(cis a) = cos a"
1.605 +  by (simp add: cis_def)
1.606
1.607  lemma Im_cis [simp]: "Im(cis a) = sin a"
1.609 +  by (simp add: cis_def)
1.610
1.611  lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)"
1.612 -by (auto simp add: DeMoivre)
1.613 +  by (auto simp add: DeMoivre)
1.614
1.615  lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)"
1.616 -by (auto simp add: DeMoivre)
1.617 +  by (auto simp add: DeMoivre)
1.618
1.619  lemma complex_expi_Ex: "\<exists>a r. z = complex_of_real r * expi a"
1.620  apply (insert rcis_Ex [of z])
1.621 @@ -732,7 +722,7 @@
1.622  done
1.623
1.624  lemma expi_two_pi_i [simp]: "expi((2::complex) * complex_of_real pi * ii) = 1"
1.625 -by (simp add: expi_def cis_def)
1.626 +  by (simp add: expi_def cis_def)
1.627
1.628  text {* Legacy theorem names *}
1.629
```