src/HOL/Complex.thy
 author paulson Mon Feb 22 14:37:56 2016 +0000 (2016-02-22) changeset 62379 340738057c8c parent 62102 877463945ce9 child 62620 d21dab28b3f9 permissions -rw-r--r--
An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
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 *)
7 section \<open>Complex Numbers: Rectangular and Polar Representations\<close>
9 theory Complex
10 imports Transcendental
11 begin
13 text \<open>
14 We use the \<open>codatatype\<close> command to define the type of complex numbers. This allows us to use
15 \<open>primcorec\<close> to define complex functions by defining their real and imaginary result
16 separately.
17 \<close>
19 codatatype complex = Complex (Re: real) (Im: real)
21 lemma complex_surj: "Complex (Re z) (Im z) = z"
22   by (rule complex.collapse)
24 lemma complex_eqI [intro?]: "\<lbrakk>Re x = Re y; Im x = Im y\<rbrakk> \<Longrightarrow> x = y"
25   by (rule complex.expand) simp
27 lemma complex_eq_iff: "x = y \<longleftrightarrow> Re x = Re y \<and> Im x = Im y"
28   by (auto intro: complex.expand)
30 subsection \<open>Addition and Subtraction\<close>
32 instantiation complex :: ab_group_add
33 begin
35 primcorec zero_complex where
36   "Re 0 = 0"
37 | "Im 0 = 0"
39 primcorec plus_complex where
40   "Re (x + y) = Re x + Re y"
41 | "Im (x + y) = Im x + Im y"
43 primcorec uminus_complex where
44   "Re (- x) = - Re x"
45 | "Im (- x) = - Im x"
47 primcorec minus_complex where
48   "Re (x - y) = Re x - Re y"
49 | "Im (x - y) = Im x - Im y"
51 instance
52   by intro_classes (simp_all add: complex_eq_iff)
54 end
56 subsection \<open>Multiplication and Division\<close>
58 instantiation complex :: field
59 begin
61 primcorec one_complex where
62   "Re 1 = 1"
63 | "Im 1 = 0"
65 primcorec times_complex where
66   "Re (x * y) = Re x * Re y - Im x * Im y"
67 | "Im (x * y) = Re x * Im y + Im x * Re y"
69 primcorec inverse_complex where
70   "Re (inverse x) = Re x / ((Re x)\<^sup>2 + (Im x)\<^sup>2)"
71 | "Im (inverse x) = - Im x / ((Re x)\<^sup>2 + (Im x)\<^sup>2)"
73 definition "x div (y::complex) = x * inverse y"
75 instance
76   by intro_classes
77      (simp_all add: complex_eq_iff divide_complex_def
78       distrib_left distrib_right right_diff_distrib left_diff_distrib
79       power2_eq_square add_divide_distrib [symmetric])
81 end
83 lemma Re_divide: "Re (x / y) = (Re x * Re y + Im x * Im y) / ((Re y)\<^sup>2 + (Im y)\<^sup>2)"
84   unfolding divide_complex_def by (simp add: add_divide_distrib)
86 lemma Im_divide: "Im (x / y) = (Im x * Re y - Re x * Im y) / ((Re y)\<^sup>2 + (Im y)\<^sup>2)"
87   unfolding divide_complex_def times_complex.sel inverse_complex.sel
88   by (simp_all add: divide_simps)
90 lemma Re_power2: "Re (x ^ 2) = (Re x)^2 - (Im x)^2"
91   by (simp add: power2_eq_square)
93 lemma Im_power2: "Im (x ^ 2) = 2 * Re x * Im x"
94   by (simp add: power2_eq_square)
96 lemma Re_power_real [simp]: "Im x = 0 \<Longrightarrow> Re (x ^ n) = Re x ^ n "
97   by (induct n) simp_all
99 lemma Im_power_real [simp]: "Im x = 0 \<Longrightarrow> Im (x ^ n) = 0"
100   by (induct n) simp_all
102 subsection \<open>Scalar Multiplication\<close>
104 instantiation complex :: real_field
105 begin
107 primcorec scaleR_complex where
108   "Re (scaleR r x) = r * Re x"
109 | "Im (scaleR r x) = r * Im x"
111 instance
112 proof
113   fix a b :: real and x y :: complex
114   show "scaleR a (x + y) = scaleR a x + scaleR a y"
115     by (simp add: complex_eq_iff distrib_left)
116   show "scaleR (a + b) x = scaleR a x + scaleR b x"
117     by (simp add: complex_eq_iff distrib_right)
118   show "scaleR a (scaleR b x) = scaleR (a * b) x"
119     by (simp add: complex_eq_iff mult.assoc)
120   show "scaleR 1 x = x"
121     by (simp add: complex_eq_iff)
122   show "scaleR a x * y = scaleR a (x * y)"
123     by (simp add: complex_eq_iff algebra_simps)
124   show "x * scaleR a y = scaleR a (x * y)"
125     by (simp add: complex_eq_iff algebra_simps)
126 qed
128 end
130 subsection \<open>Numerals, Arithmetic, and Embedding from Reals\<close>
132 abbreviation complex_of_real :: "real \<Rightarrow> complex"
133   where "complex_of_real \<equiv> of_real"
135 declare [[coercion "of_real :: real \<Rightarrow> complex"]]
136 declare [[coercion "of_rat :: rat \<Rightarrow> complex"]]
137 declare [[coercion "of_int :: int \<Rightarrow> complex"]]
138 declare [[coercion "of_nat :: nat \<Rightarrow> complex"]]
140 lemma complex_Re_of_nat [simp]: "Re (of_nat n) = of_nat n"
141   by (induct n) simp_all
143 lemma complex_Im_of_nat [simp]: "Im (of_nat n) = 0"
144   by (induct n) simp_all
146 lemma complex_Re_of_int [simp]: "Re (of_int z) = of_int z"
147   by (cases z rule: int_diff_cases) simp
149 lemma complex_Im_of_int [simp]: "Im (of_int z) = 0"
150   by (cases z rule: int_diff_cases) simp
152 lemma complex_Re_numeral [simp]: "Re (numeral v) = numeral v"
153   using complex_Re_of_int [of "numeral v"] by simp
155 lemma complex_Im_numeral [simp]: "Im (numeral v) = 0"
156   using complex_Im_of_int [of "numeral v"] by simp
158 lemma Re_complex_of_real [simp]: "Re (complex_of_real z) = z"
159   by (simp add: of_real_def)
161 lemma Im_complex_of_real [simp]: "Im (complex_of_real z) = 0"
162   by (simp add: of_real_def)
164 lemma Re_divide_numeral [simp]: "Re (z / numeral w) = Re z / numeral w"
165   by (simp add: Re_divide sqr_conv_mult)
167 lemma Im_divide_numeral [simp]: "Im (z / numeral w) = Im z / numeral w"
168   by (simp add: Im_divide sqr_conv_mult)
170 lemma Re_divide_of_nat [simp]: "Re (z / of_nat n) = Re z / of_nat n"
171   by (cases n) (simp_all add: Re_divide divide_simps power2_eq_square del: of_nat_Suc)
173 lemma Im_divide_of_nat [simp]: "Im (z / of_nat n) = Im z / of_nat n"
174   by (cases n) (simp_all add: Im_divide divide_simps power2_eq_square del: of_nat_Suc)
176 lemma of_real_Re [simp]:
177     "z \<in> \<real> \<Longrightarrow> of_real (Re z) = z"
178   by (auto simp: Reals_def)
180 lemma complex_Re_fact [simp]: "Re (fact n) = fact n"
181 proof -
182   have "(fact n :: complex) = of_real (fact n)" by simp
183   also have "Re \<dots> = fact n" by (subst Re_complex_of_real) simp_all
184   finally show ?thesis .
185 qed
187 lemma complex_Im_fact [simp]: "Im (fact n) = 0"
188   by (subst of_nat_fact [symmetric]) (simp only: complex_Im_of_nat)
191 subsection \<open>The Complex Number $i$\<close>
193 primcorec "ii" :: complex  ("\<i>") where
194   "Re ii = 0"
195 | "Im ii = 1"
197 lemma Complex_eq[simp]: "Complex a b = a + \<i> * b"
198   by (simp add: complex_eq_iff)
200 lemma complex_eq: "a = Re a + \<i> * Im a"
201   by (simp add: complex_eq_iff)
203 lemma fun_complex_eq: "f = (\<lambda>x. Re (f x) + \<i> * Im (f x))"
204   by (simp add: fun_eq_iff complex_eq)
206 lemma i_squared [simp]: "ii * ii = -1"
207   by (simp add: complex_eq_iff)
209 lemma power2_i [simp]: "ii\<^sup>2 = -1"
210   by (simp add: power2_eq_square)
212 lemma inverse_i [simp]: "inverse ii = - ii"
213   by (rule inverse_unique) simp
215 lemma divide_i [simp]: "x / ii = - ii * x"
216   by (simp add: divide_complex_def)
218 lemma complex_i_mult_minus [simp]: "ii * (ii * x) = - x"
219   by (simp add: mult.assoc [symmetric])
221 lemma complex_i_not_zero [simp]: "ii \<noteq> 0"
222   by (simp add: complex_eq_iff)
224 lemma complex_i_not_one [simp]: "ii \<noteq> 1"
225   by (simp add: complex_eq_iff)
227 lemma complex_i_not_numeral [simp]: "ii \<noteq> numeral w"
228   by (simp add: complex_eq_iff)
230 lemma complex_i_not_neg_numeral [simp]: "ii \<noteq> - numeral w"
231   by (simp add: complex_eq_iff)
233 lemma complex_split_polar: "\<exists>r a. z = complex_of_real r * (cos a + \<i> * sin a)"
234   by (simp add: complex_eq_iff polar_Ex)
236 lemma i_even_power [simp]: "\<i> ^ (n * 2) = (-1) ^ n"
237   by (metis mult.commute power2_i power_mult)
239 lemma Re_ii_times [simp]: "Re (ii*z) = - Im z"
240   by simp
242 lemma Im_ii_times [simp]: "Im (ii*z) = Re z"
243   by simp
245 lemma ii_times_eq_iff: "ii*w = z \<longleftrightarrow> w = -(ii*z)"
246   by auto
248 lemma divide_numeral_i [simp]: "z / (numeral n * ii) = -(ii*z) / numeral n"
249   by (metis divide_divide_eq_left divide_i mult.commute mult_minus_right)
251 subsection \<open>Vector Norm\<close>
253 instantiation complex :: real_normed_field
254 begin
256 definition "norm z = sqrt ((Re z)\<^sup>2 + (Im z)\<^sup>2)"
258 abbreviation cmod :: "complex \<Rightarrow> real"
259   where "cmod \<equiv> norm"
261 definition complex_sgn_def:
262   "sgn x = x /\<^sub>R cmod x"
264 definition dist_complex_def:
265   "dist x y = cmod (x - y)"
267 definition uniformity_complex_def [code del]:
268   "(uniformity :: (complex \<times> complex) filter) = (INF e:{0 <..}. principal {(x, y). dist x y < e})"
270 definition open_complex_def [code del]:
271   "open (U :: complex set) \<longleftrightarrow> (\<forall>x\<in>U. eventually (\<lambda>(x', y). x' = x \<longrightarrow> y \<in> U) uniformity)"
273 instance proof
274   fix r :: real and x y :: complex and S :: "complex set"
275   show "(norm x = 0) = (x = 0)"
276     by (simp add: norm_complex_def complex_eq_iff)
277   show "norm (x + y) \<le> norm x + norm y"
278     by (simp add: norm_complex_def complex_eq_iff real_sqrt_sum_squares_triangle_ineq)
279   show "norm (scaleR r x) = \<bar>r\<bar> * norm x"
280     by (simp add: norm_complex_def complex_eq_iff power_mult_distrib distrib_left [symmetric] real_sqrt_mult)
281   show "norm (x * y) = norm x * norm y"
282     by (simp add: norm_complex_def complex_eq_iff real_sqrt_mult [symmetric] power2_eq_square algebra_simps)
283 qed (rule complex_sgn_def dist_complex_def open_complex_def uniformity_complex_def)+
285 end
287 declare uniformity_Abort[where 'a=complex, code]
289 lemma norm_ii [simp]: "norm ii = 1"
290   by (simp add: norm_complex_def)
292 lemma cmod_unit_one: "cmod (cos a + \<i> * sin a) = 1"
293   by (simp add: norm_complex_def)
295 lemma cmod_complex_polar: "cmod (r * (cos a + \<i> * sin a)) = \<bar>r\<bar>"
296   by (simp add: norm_mult cmod_unit_one)
298 lemma complex_Re_le_cmod: "Re x \<le> cmod x"
299   unfolding norm_complex_def
300   by (rule real_sqrt_sum_squares_ge1)
302 lemma complex_mod_minus_le_complex_mod: "- cmod x \<le> cmod x"
303   by (rule order_trans [OF _ norm_ge_zero]) simp
305 lemma complex_mod_triangle_ineq2: "cmod (b + a) - cmod b \<le> cmod a"
306   by (rule ord_le_eq_trans [OF norm_triangle_ineq2]) simp
308 lemma abs_Re_le_cmod: "\<bar>Re x\<bar> \<le> cmod x"
309   by (simp add: norm_complex_def)
311 lemma abs_Im_le_cmod: "\<bar>Im x\<bar> \<le> cmod x"
312   by (simp add: norm_complex_def)
314 lemma cmod_le: "cmod z \<le> \<bar>Re z\<bar> + \<bar>Im z\<bar>"
315   apply (subst complex_eq)
316   apply (rule order_trans)
317   apply (rule norm_triangle_ineq)
318   apply (simp add: norm_mult)
319   done
321 lemma cmod_eq_Re: "Im z = 0 \<Longrightarrow> cmod z = \<bar>Re z\<bar>"
322   by (simp add: norm_complex_def)
324 lemma cmod_eq_Im: "Re z = 0 \<Longrightarrow> cmod z = \<bar>Im z\<bar>"
325   by (simp add: norm_complex_def)
327 lemma cmod_power2: "cmod z ^ 2 = (Re z)^2 + (Im z)^2"
328   by (simp add: norm_complex_def)
330 lemma cmod_plus_Re_le_0_iff: "cmod z + Re z \<le> 0 \<longleftrightarrow> Re z = - cmod z"
331   using abs_Re_le_cmod[of z] by auto
333 lemma cmod_Re_le_iff: "Im x = Im y \<Longrightarrow> cmod x \<le> cmod y \<longleftrightarrow> abs (Re x) \<le> abs (Re y)"
334   by (metis add.commute add_le_cancel_left norm_complex_def real_sqrt_abs real_sqrt_le_iff)
336 lemma cmod_Im_le_iff: "Re x = Re y \<Longrightarrow> cmod x \<le> cmod y \<longleftrightarrow> abs (Im x) \<le> abs (Im y)"
337   by (metis add_le_cancel_left norm_complex_def real_sqrt_abs real_sqrt_le_iff)
339 lemma Im_eq_0: "\<bar>Re z\<bar> = cmod z \<Longrightarrow> Im z = 0"
340   by (subst (asm) power_eq_iff_eq_base[symmetric, where n=2])
341      (auto simp add: norm_complex_def)
343 lemma abs_sqrt_wlog:
344   fixes x::"'a::linordered_idom"
345   assumes "\<And>x::'a. x \<ge> 0 \<Longrightarrow> P x (x\<^sup>2)" shows "P \<bar>x\<bar> (x\<^sup>2)"
346 by (metis abs_ge_zero assms power2_abs)
348 lemma complex_abs_le_norm: "\<bar>Re z\<bar> + \<bar>Im z\<bar> \<le> sqrt 2 * norm z"
349   unfolding norm_complex_def
350   apply (rule abs_sqrt_wlog [where x="Re z"])
351   apply (rule abs_sqrt_wlog [where x="Im z"])
352   apply (rule power2_le_imp_le)
353   apply (simp_all add: power2_sum add.commute sum_squares_bound real_sqrt_mult [symmetric])
354   done
356 lemma complex_unit_circle: "z \<noteq> 0 \<Longrightarrow> (Re z / cmod z)\<^sup>2 + (Im z / cmod z)\<^sup>2 = 1"
357   by (simp add: norm_complex_def divide_simps complex_eq_iff)
360 text \<open>Properties of complex signum.\<close>
362 lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)"
363   by (simp add: sgn_div_norm divide_inverse scaleR_conv_of_real mult.commute)
365 lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z"
366   by (simp add: complex_sgn_def divide_inverse)
368 lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z"
369   by (simp add: complex_sgn_def divide_inverse)
372 subsection \<open>Completeness of the Complexes\<close>
374 lemma bounded_linear_Re: "bounded_linear Re"
375   by (rule bounded_linear_intro [where K=1], simp_all add: norm_complex_def)
377 lemma bounded_linear_Im: "bounded_linear Im"
378   by (rule bounded_linear_intro [where K=1], simp_all add: norm_complex_def)
380 lemmas Cauchy_Re = bounded_linear.Cauchy [OF bounded_linear_Re]
381 lemmas Cauchy_Im = bounded_linear.Cauchy [OF bounded_linear_Im]
382 lemmas tendsto_Re [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_Re]
383 lemmas tendsto_Im [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_Im]
384 lemmas isCont_Re [simp] = bounded_linear.isCont [OF bounded_linear_Re]
385 lemmas isCont_Im [simp] = bounded_linear.isCont [OF bounded_linear_Im]
386 lemmas continuous_Re [simp] = bounded_linear.continuous [OF bounded_linear_Re]
387 lemmas continuous_Im [simp] = bounded_linear.continuous [OF bounded_linear_Im]
388 lemmas continuous_on_Re [continuous_intros] = bounded_linear.continuous_on[OF bounded_linear_Re]
389 lemmas continuous_on_Im [continuous_intros] = bounded_linear.continuous_on[OF bounded_linear_Im]
390 lemmas has_derivative_Re [derivative_intros] = bounded_linear.has_derivative[OF bounded_linear_Re]
391 lemmas has_derivative_Im [derivative_intros] = bounded_linear.has_derivative[OF bounded_linear_Im]
392 lemmas sums_Re = bounded_linear.sums [OF bounded_linear_Re]
393 lemmas sums_Im = bounded_linear.sums [OF bounded_linear_Im]
395 lemma tendsto_Complex [tendsto_intros]:
396   "(f \<longlongrightarrow> a) F \<Longrightarrow> (g \<longlongrightarrow> b) F \<Longrightarrow> ((\<lambda>x. Complex (f x) (g x)) \<longlongrightarrow> Complex a b) F"
397   by (auto intro!: tendsto_intros)
399 lemma tendsto_complex_iff:
400   "(f \<longlongrightarrow> x) F \<longleftrightarrow> (((\<lambda>x. Re (f x)) \<longlongrightarrow> Re x) F \<and> ((\<lambda>x. Im (f x)) \<longlongrightarrow> Im x) F)"
401 proof safe
402   assume "((\<lambda>x. Re (f x)) \<longlongrightarrow> Re x) F" "((\<lambda>x. Im (f x)) \<longlongrightarrow> Im x) F"
403   from tendsto_Complex[OF this] show "(f \<longlongrightarrow> x) F"
404     unfolding complex.collapse .
405 qed (auto intro: tendsto_intros)
407 lemma continuous_complex_iff: "continuous F f \<longleftrightarrow>
408     continuous F (\<lambda>x. Re (f x)) \<and> continuous F (\<lambda>x. Im (f x))"
409   unfolding continuous_def tendsto_complex_iff ..
411 lemma has_vector_derivative_complex_iff: "(f has_vector_derivative x) F \<longleftrightarrow>
412     ((\<lambda>x. Re (f x)) has_field_derivative (Re x)) F \<and>
413     ((\<lambda>x. Im (f x)) has_field_derivative (Im x)) F"
414   unfolding has_vector_derivative_def has_field_derivative_def has_derivative_def tendsto_complex_iff
415   by (simp add: field_simps bounded_linear_scaleR_left bounded_linear_mult_right)
417 lemma has_field_derivative_Re[derivative_intros]:
418   "(f has_vector_derivative D) F \<Longrightarrow> ((\<lambda>x. Re (f x)) has_field_derivative (Re D)) F"
419   unfolding has_vector_derivative_complex_iff by safe
421 lemma has_field_derivative_Im[derivative_intros]:
422   "(f has_vector_derivative D) F \<Longrightarrow> ((\<lambda>x. Im (f x)) has_field_derivative (Im D)) F"
423   unfolding has_vector_derivative_complex_iff by safe
425 instance complex :: banach
426 proof
427   fix X :: "nat \<Rightarrow> complex"
428   assume X: "Cauchy X"
429   then have "(\<lambda>n. Complex (Re (X n)) (Im (X n))) \<longlonglongrightarrow> Complex (lim (\<lambda>n. Re (X n))) (lim (\<lambda>n. Im (X n)))"
430     by (intro tendsto_Complex convergent_LIMSEQ_iff[THEN iffD1] Cauchy_convergent_iff[THEN iffD1] Cauchy_Re Cauchy_Im)
431   then show "convergent X"
432     unfolding complex.collapse by (rule convergentI)
433 qed
435 declare
436   DERIV_power[where 'a=complex, unfolded of_nat_def[symmetric], derivative_intros]
438 subsection \<open>Complex Conjugation\<close>
440 primcorec cnj :: "complex \<Rightarrow> complex" where
441   "Re (cnj z) = Re z"
442 | "Im (cnj z) = - Im z"
444 lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)"
445   by (simp add: complex_eq_iff)
447 lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z"
448   by (simp add: complex_eq_iff)
450 lemma complex_cnj_zero [simp]: "cnj 0 = 0"
451   by (simp add: complex_eq_iff)
453 lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)"
454   by (simp add: complex_eq_iff)
456 lemma complex_cnj_add [simp]: "cnj (x + y) = cnj x + cnj y"
457   by (simp add: complex_eq_iff)
459 lemma cnj_setsum [simp]: "cnj (setsum f s) = (\<Sum>x\<in>s. cnj (f x))"
460   by (induct s rule: infinite_finite_induct) auto
462 lemma complex_cnj_diff [simp]: "cnj (x - y) = cnj x - cnj y"
463   by (simp add: complex_eq_iff)
465 lemma complex_cnj_minus [simp]: "cnj (- x) = - cnj x"
466   by (simp add: complex_eq_iff)
468 lemma complex_cnj_one [simp]: "cnj 1 = 1"
469   by (simp add: complex_eq_iff)
471 lemma complex_cnj_mult [simp]: "cnj (x * y) = cnj x * cnj y"
472   by (simp add: complex_eq_iff)
474 lemma cnj_setprod [simp]: "cnj (setprod f s) = (\<Prod>x\<in>s. cnj (f x))"
475   by (induct s rule: infinite_finite_induct) auto
477 lemma complex_cnj_inverse [simp]: "cnj (inverse x) = inverse (cnj x)"
478   by (simp add: complex_eq_iff)
480 lemma complex_cnj_divide [simp]: "cnj (x / y) = cnj x / cnj y"
481   by (simp add: divide_complex_def)
483 lemma complex_cnj_power [simp]: "cnj (x ^ n) = cnj x ^ n"
484   by (induct n) simp_all
486 lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n"
487   by (simp add: complex_eq_iff)
489 lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z"
490   by (simp add: complex_eq_iff)
492 lemma complex_cnj_numeral [simp]: "cnj (numeral w) = numeral w"
493   by (simp add: complex_eq_iff)
495 lemma complex_cnj_neg_numeral [simp]: "cnj (- numeral w) = - numeral w"
496   by (simp add: complex_eq_iff)
498 lemma complex_cnj_scaleR [simp]: "cnj (scaleR r x) = scaleR r (cnj x)"
499   by (simp add: complex_eq_iff)
501 lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z"
502   by (simp add: norm_complex_def)
504 lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x"
505   by (simp add: complex_eq_iff)
507 lemma complex_cnj_i [simp]: "cnj ii = - ii"
508   by (simp add: complex_eq_iff)
510 lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)"
511   by (simp add: complex_eq_iff)
513 lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im z) * ii"
514   by (simp add: complex_eq_iff)
516 lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\<^sup>2 + (Im z)\<^sup>2)"
517   by (simp add: complex_eq_iff power2_eq_square)
519 lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<^sup>2"
520   by (simp add: norm_mult power2_eq_square)
522 lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))"
523   by (simp add: norm_complex_def power2_eq_square)
525 lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0"
526   by simp
528 lemma complex_cnj_fact [simp]: "cnj (fact n) = fact n"
529   by (subst of_nat_fact [symmetric], subst complex_cnj_of_nat) simp
531 lemma complex_cnj_pochhammer [simp]: "cnj (pochhammer z n) = pochhammer (cnj z) n"
532   by (induction n arbitrary: z) (simp_all add: pochhammer_rec)
534 lemma bounded_linear_cnj: "bounded_linear cnj"
535   using complex_cnj_add complex_cnj_scaleR
536   by (rule bounded_linear_intro [where K=1], simp)
538 lemmas tendsto_cnj [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_cnj]
539 lemmas isCont_cnj [simp] = bounded_linear.isCont [OF bounded_linear_cnj]
540 lemmas continuous_cnj [simp, continuous_intros] = bounded_linear.continuous [OF bounded_linear_cnj]
541 lemmas continuous_on_cnj [simp, continuous_intros] = bounded_linear.continuous_on [OF bounded_linear_cnj]
542 lemmas has_derivative_cnj [simp, derivative_intros] = bounded_linear.has_derivative [OF bounded_linear_cnj]
544 lemma lim_cnj: "((\<lambda>x. cnj(f x)) \<longlongrightarrow> cnj l) F \<longleftrightarrow> (f \<longlongrightarrow> l) F"
545   by (simp add: tendsto_iff dist_complex_def complex_cnj_diff [symmetric] del: complex_cnj_diff)
547 lemma sums_cnj: "((\<lambda>x. cnj(f x)) sums cnj l) \<longleftrightarrow> (f sums l)"
548   by (simp add: sums_def lim_cnj cnj_setsum [symmetric] del: cnj_setsum)
551 subsection\<open>Basic Lemmas\<close>
553 lemma complex_eq_0: "z=0 \<longleftrightarrow> (Re z)\<^sup>2 + (Im z)\<^sup>2 = 0"
554   by (metis zero_complex.sel complex_eqI sum_power2_eq_zero_iff)
556 lemma complex_neq_0: "z\<noteq>0 \<longleftrightarrow> (Re z)\<^sup>2 + (Im z)\<^sup>2 > 0"
557   by (metis complex_eq_0 less_numeral_extra(3) sum_power2_gt_zero_iff)
559 lemma complex_norm_square: "of_real ((norm z)\<^sup>2) = z * cnj z"
560 by (cases z)
561    (auto simp: complex_eq_iff norm_complex_def power2_eq_square[symmetric] of_real_power[symmetric]
562          simp del: of_real_power)
564 lemma complex_div_cnj: "a / b = (a * cnj b) / (norm b)^2"
565   using complex_norm_square by auto
567 lemma Re_complex_div_eq_0: "Re (a / b) = 0 \<longleftrightarrow> Re (a * cnj b) = 0"
568   by (auto simp add: Re_divide)
570 lemma Im_complex_div_eq_0: "Im (a / b) = 0 \<longleftrightarrow> Im (a * cnj b) = 0"
571   by (auto simp add: Im_divide)
573 lemma complex_div_gt_0:
574   "(Re (a / b) > 0 \<longleftrightarrow> Re (a * cnj b) > 0) \<and> (Im (a / b) > 0 \<longleftrightarrow> Im (a * cnj b) > 0)"
575 proof cases
576   assume "b = 0" then show ?thesis by auto
577 next
578   assume "b \<noteq> 0"
579   then have "0 < (Re b)\<^sup>2 + (Im b)\<^sup>2"
580     by (simp add: complex_eq_iff sum_power2_gt_zero_iff)
581   then show ?thesis
582     by (simp add: Re_divide Im_divide zero_less_divide_iff)
583 qed
585 lemma Re_complex_div_gt_0: "Re (a / b) > 0 \<longleftrightarrow> Re (a * cnj b) > 0"
586   and Im_complex_div_gt_0: "Im (a / b) > 0 \<longleftrightarrow> Im (a * cnj b) > 0"
587   using complex_div_gt_0 by auto
589 lemma Re_complex_div_ge_0: "Re(a / b) \<ge> 0 \<longleftrightarrow> Re(a * cnj b) \<ge> 0"
590   by (metis le_less Re_complex_div_eq_0 Re_complex_div_gt_0)
592 lemma Im_complex_div_ge_0: "Im(a / b) \<ge> 0 \<longleftrightarrow> Im(a * cnj b) \<ge> 0"
593   by (metis Im_complex_div_eq_0 Im_complex_div_gt_0 le_less)
595 lemma Re_complex_div_lt_0: "Re(a / b) < 0 \<longleftrightarrow> Re(a * cnj b) < 0"
596   by (metis less_asym neq_iff Re_complex_div_eq_0 Re_complex_div_gt_0)
598 lemma Im_complex_div_lt_0: "Im(a / b) < 0 \<longleftrightarrow> Im(a * cnj b) < 0"
599   by (metis Im_complex_div_eq_0 Im_complex_div_gt_0 less_asym neq_iff)
601 lemma Re_complex_div_le_0: "Re(a / b) \<le> 0 \<longleftrightarrow> Re(a * cnj b) \<le> 0"
602   by (metis not_le Re_complex_div_gt_0)
604 lemma Im_complex_div_le_0: "Im(a / b) \<le> 0 \<longleftrightarrow> Im(a * cnj b) \<le> 0"
605   by (metis Im_complex_div_gt_0 not_le)
607 lemma Re_divide_of_real [simp]: "Re (z / of_real r) = Re z / r"
608   by (simp add: Re_divide power2_eq_square)
610 lemma Im_divide_of_real [simp]: "Im (z / of_real r) = Im z / r"
611   by (simp add: Im_divide power2_eq_square)
613 lemma Re_divide_Reals: "r \<in> Reals \<Longrightarrow> Re (z / r) = Re z / Re r"
614   by (metis Re_divide_of_real of_real_Re)
616 lemma Im_divide_Reals: "r \<in> Reals \<Longrightarrow> Im (z / r) = Im z / Re r"
617   by (metis Im_divide_of_real of_real_Re)
619 lemma Re_setsum[simp]: "Re (setsum f s) = (\<Sum>x\<in>s. Re (f x))"
620   by (induct s rule: infinite_finite_induct) auto
622 lemma Im_setsum[simp]: "Im (setsum f s) = (\<Sum>x\<in>s. Im(f x))"
623   by (induct s rule: infinite_finite_induct) auto
625 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)"
626   unfolding sums_def tendsto_complex_iff Im_setsum Re_setsum ..
628 lemma summable_complex_iff: "summable f \<longleftrightarrow> summable (\<lambda>x. Re (f x)) \<and>  summable (\<lambda>x. Im (f x))"
629   unfolding summable_def sums_complex_iff[abs_def] by (metis complex.sel)
631 lemma summable_complex_of_real [simp]: "summable (\<lambda>n. complex_of_real (f n)) \<longleftrightarrow> summable f"
632   unfolding summable_complex_iff by simp
634 lemma summable_Re: "summable f \<Longrightarrow> summable (\<lambda>x. Re (f x))"
635   unfolding summable_complex_iff by blast
637 lemma summable_Im: "summable f \<Longrightarrow> summable (\<lambda>x. Im (f x))"
638   unfolding summable_complex_iff by blast
640 lemma complex_is_Nat_iff: "z \<in> \<nat> \<longleftrightarrow> Im z = 0 \<and> (\<exists>i. Re z = of_nat i)"
641   by (auto simp: Nats_def complex_eq_iff)
643 lemma complex_is_Int_iff: "z \<in> \<int> \<longleftrightarrow> Im z = 0 \<and> (\<exists>i. Re z = of_int i)"
644   by (auto simp: Ints_def complex_eq_iff)
646 lemma complex_is_Real_iff: "z \<in> \<real> \<longleftrightarrow> Im z = 0"
647   by (auto simp: Reals_def complex_eq_iff)
649 lemma Reals_cnj_iff: "z \<in> \<real> \<longleftrightarrow> cnj z = z"
650   by (auto simp: complex_is_Real_iff complex_eq_iff)
652 lemma in_Reals_norm: "z \<in> \<real> \<Longrightarrow> norm z = \<bar>Re z\<bar>"
653   by (simp add: complex_is_Real_iff norm_complex_def)
655 lemma series_comparison_complex:
656   fixes f:: "nat \<Rightarrow> 'a::banach"
657   assumes sg: "summable g"
658      and "\<And>n. g n \<in> \<real>" "\<And>n. Re (g n) \<ge> 0"
659      and fg: "\<And>n. n \<ge> N \<Longrightarrow> norm(f n) \<le> norm(g n)"
660   shows "summable f"
661 proof -
662   have g: "\<And>n. cmod (g n) = Re (g n)" using assms
663     by (metis abs_of_nonneg in_Reals_norm)
664   show ?thesis
665     apply (rule summable_comparison_test' [where g = "\<lambda>n. norm (g n)" and N=N])
666     using sg
667     apply (auto simp: summable_def)
668     apply (rule_tac x="Re s" in exI)
669     apply (auto simp: g sums_Re)
670     apply (metis fg g)
671     done
672 qed
674 subsection\<open>Polar Form for Complex Numbers\<close>
676 lemma complex_unimodular_polar: "(norm z = 1) \<Longrightarrow> \<exists>x. z = Complex (cos x) (sin x)"
677   using sincos_total_2pi [of "Re z" "Im z"]
678   by auto (metis cmod_power2 complex_eq power_one)
680 subsubsection \<open>$\cos \theta + i \sin \theta$\<close>
682 primcorec cis :: "real \<Rightarrow> complex" where
683   "Re (cis a) = cos a"
684 | "Im (cis a) = sin a"
686 lemma cis_zero [simp]: "cis 0 = 1"
687   by (simp add: complex_eq_iff)
689 lemma norm_cis [simp]: "norm (cis a) = 1"
690   by (simp add: norm_complex_def)
692 lemma sgn_cis [simp]: "sgn (cis a) = cis a"
693   by (simp add: sgn_div_norm)
695 lemma cis_neq_zero [simp]: "cis a \<noteq> 0"
696   by (metis norm_cis norm_zero zero_neq_one)
698 lemma cis_mult: "cis a * cis b = cis (a + b)"
701 lemma DeMoivre: "(cis a) ^ n = cis (real n * a)"
702   by (induct n, simp_all add: of_nat_Suc algebra_simps cis_mult)
704 lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)"
705   by (simp add: complex_eq_iff)
707 lemma cis_divide: "cis a / cis b = cis (a - b)"
708   by (simp add: divide_complex_def cis_mult)
710 lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)"
711   by (auto simp add: DeMoivre)
713 lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)"
714   by (auto simp add: DeMoivre)
716 lemma cis_pi: "cis pi = -1"
717   by (simp add: complex_eq_iff)
719 subsubsection \<open>$r(\cos \theta + i \sin \theta)$\<close>
721 definition rcis :: "real \<Rightarrow> real \<Rightarrow> complex" where
722   "rcis r a = complex_of_real r * cis a"
724 lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a"
725   by (simp add: rcis_def)
727 lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a"
728   by (simp add: rcis_def)
730 lemma rcis_Ex: "\<exists>r a. z = rcis r a"
731   by (simp add: complex_eq_iff polar_Ex)
733 lemma complex_mod_rcis [simp]: "cmod (rcis r a) = \<bar>r\<bar>"
734   by (simp add: rcis_def norm_mult)
736 lemma cis_rcis_eq: "cis a = rcis 1 a"
737   by (simp add: rcis_def)
739 lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)"
740   by (simp add: rcis_def cis_mult)
742 lemma rcis_zero_mod [simp]: "rcis 0 a = 0"
743   by (simp add: rcis_def)
745 lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r"
746   by (simp add: rcis_def)
748 lemma rcis_eq_zero_iff [simp]: "rcis r a = 0 \<longleftrightarrow> r = 0"
749   by (simp add: rcis_def)
751 lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)"
752   by (simp add: rcis_def power_mult_distrib DeMoivre)
754 lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)"
755   by (simp add: divide_inverse rcis_def)
757 lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)"
758   by (simp add: rcis_def cis_divide [symmetric])
760 subsubsection \<open>Complex exponential\<close>
762 lemma cis_conv_exp: "cis b = exp (\<i> * b)"
763 proof -
764   { fix n :: nat
765     have "\<i> ^ n = fact n *\<^sub>R (cos_coeff n + \<i> * sin_coeff n)"
766       by (induct n)
767          (simp_all add: sin_coeff_Suc cos_coeff_Suc complex_eq_iff Re_divide Im_divide field_simps
768                         power2_eq_square of_nat_Suc add_nonneg_eq_0_iff)
769     then have "(\<i> * complex_of_real b) ^ n /\<^sub>R fact n =
770         of_real (cos_coeff n * b^n) + \<i> * of_real (sin_coeff n * b^n)"
771       by (simp add: field_simps) }
772   then show ?thesis using sin_converges [of b] cos_converges [of b]
773     by (auto simp add: cis.ctr exp_def simp del: of_real_mult
774              intro!: sums_unique sums_add sums_mult sums_of_real)
775 qed
777 lemma exp_eq_polar: "exp z = exp (Re z) * cis (Im z)"
778   unfolding cis_conv_exp exp_of_real [symmetric] mult_exp_exp by (cases z) simp
780 lemma Re_exp: "Re (exp z) = exp (Re z) * cos (Im z)"
781   unfolding exp_eq_polar by simp
783 lemma Im_exp: "Im (exp z) = exp (Re z) * sin (Im z)"
784   unfolding exp_eq_polar by simp
786 lemma norm_cos_sin [simp]: "norm (Complex (cos t) (sin t)) = 1"
787   by (simp add: norm_complex_def)
789 lemma norm_exp_eq_Re [simp]: "norm (exp z) = exp (Re z)"
790   by (simp add: cis.code cmod_complex_polar exp_eq_polar)
792 lemma complex_exp_exists: "\<exists>a r. z = complex_of_real r * exp a"
793   apply (insert rcis_Ex [of z])
794   apply (auto simp add: exp_eq_polar rcis_def mult.assoc [symmetric])
795   apply (rule_tac x = "ii * complex_of_real a" in exI, auto)
796   done
798 lemma exp_pi_i [simp]: "exp(of_real pi * ii) = -1"
799   by (metis cis_conv_exp cis_pi mult.commute)
801 lemma exp_two_pi_i [simp]: "exp(2 * of_real pi * ii) = 1"
802   by (simp add: exp_eq_polar complex_eq_iff)
804 subsubsection \<open>Complex argument\<close>
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))"
809 lemma arg_zero: "arg 0 = 0"
810   by (simp add: arg_def)
812 lemma arg_unique:
813   assumes "sgn z = cis x" and "-pi < x" and "x \<le> pi"
814   shows "arg z = x"
815 proof -
816   from assms have "z \<noteq> 0" by auto
817   have "(SOME a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi) = x"
818   proof
819     fix a def d \<equiv> "a - x"
820     assume a: "sgn z = cis a \<and> - pi < a \<and> a \<le> pi"
821     from a assms have "- (2*pi) < d \<and> d < 2*pi"
822       unfolding d_def by simp
823     moreover from a assms have "cos a = cos x" and "sin a = sin x"
824       by (simp_all add: complex_eq_iff)
825     hence cos: "cos d = 1" unfolding d_def cos_diff by simp
826     moreover from cos have "sin d = 0" by (rule cos_one_sin_zero)
827     ultimately have "d = 0"
828       unfolding sin_zero_iff
829       by (auto elim!: evenE dest!: less_2_cases)
830     thus "a = x" unfolding d_def by simp
831   qed (simp add: assms del: Re_sgn Im_sgn)
832   with \<open>z \<noteq> 0\<close> show "arg z = x"
833     unfolding arg_def by simp
834 qed
836 lemma arg_correct:
837   assumes "z \<noteq> 0" shows "sgn z = cis (arg z) \<and> -pi < arg z \<and> arg z \<le> pi"
838 proof (simp add: arg_def assms, rule someI_ex)
839   obtain r a where z: "z = rcis r a" using rcis_Ex by fast
840   with assms have "r \<noteq> 0" by auto
841   def b \<equiv> "if 0 < r then a else a + pi"
842   have b: "sgn z = cis b"
843     unfolding z b_def rcis_def using \<open>r \<noteq> 0\<close>
844     by (simp add: of_real_def sgn_scaleR sgn_if complex_eq_iff)
845   have cis_2pi_nat: "\<And>n. cis (2 * pi * real_of_nat n) = 1"
846     by (induct_tac n) (simp_all add: distrib_left cis_mult [symmetric] complex_eq_iff)
847   have cis_2pi_int: "\<And>x. cis (2 * pi * real_of_int x) = 1"
848     by (case_tac x rule: int_diff_cases)
849        (simp add: right_diff_distrib cis_divide [symmetric] cis_2pi_nat)
850   def c \<equiv> "b - 2*pi * of_int \<lceil>(b - pi) / (2*pi)\<rceil>"
851   have "sgn z = cis c"
852     unfolding b c_def
853     by (simp add: cis_divide [symmetric] cis_2pi_int)
854   moreover have "- pi < c \<and> c \<le> pi"
855     using ceiling_correct [of "(b - pi) / (2*pi)"]
856     by (simp add: c_def less_divide_eq divide_le_eq algebra_simps del: le_of_int_ceiling)
857   ultimately show "\<exists>a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi" by fast
858 qed
860 lemma arg_bounded: "- pi < arg z \<and> arg z \<le> pi"
861   by (cases "z = 0") (simp_all add: arg_zero arg_correct)
863 lemma cis_arg: "z \<noteq> 0 \<Longrightarrow> cis (arg z) = sgn z"
864   by (simp add: arg_correct)
866 lemma rcis_cmod_arg: "rcis (cmod z) (arg z) = z"
867   by (cases "z = 0") (simp_all add: rcis_def cis_arg sgn_div_norm of_real_def)
869 lemma cos_arg_i_mult_zero [simp]: "y \<noteq> 0 \<Longrightarrow> Re y = 0 \<Longrightarrow> cos (arg y) = 0"
870   using cis_arg [of y] by (simp add: complex_eq_iff)
872 subsection \<open>Square root of complex numbers\<close>
874 primcorec csqrt :: "complex \<Rightarrow> complex" where
875   "Re (csqrt z) = sqrt ((cmod z + Re z) / 2)"
876 | "Im (csqrt z) = (if Im z = 0 then 1 else sgn (Im z)) * sqrt ((cmod z - Re z) / 2)"
878 lemma csqrt_of_real_nonneg [simp]: "Im x = 0 \<Longrightarrow> Re x \<ge> 0 \<Longrightarrow> csqrt x = sqrt (Re x)"
879   by (simp add: complex_eq_iff norm_complex_def)
881 lemma csqrt_of_real_nonpos [simp]: "Im x = 0 \<Longrightarrow> Re x \<le> 0 \<Longrightarrow> csqrt x = \<i> * sqrt \<bar>Re x\<bar>"
882   by (simp add: complex_eq_iff norm_complex_def)
884 lemma of_real_sqrt: "x \<ge> 0 \<Longrightarrow> of_real (sqrt x) = csqrt (of_real x)"
885   by (simp add: complex_eq_iff norm_complex_def)
887 lemma csqrt_0 [simp]: "csqrt 0 = 0"
888   by simp
890 lemma csqrt_1 [simp]: "csqrt 1 = 1"
891   by simp
893 lemma csqrt_ii [simp]: "csqrt \<i> = (1 + \<i>) / sqrt 2"
894   by (simp add: complex_eq_iff Re_divide Im_divide real_sqrt_divide real_div_sqrt)
896 lemma power2_csqrt[simp,algebra]: "(csqrt z)\<^sup>2 = z"
897 proof cases
898   assume "Im z = 0" then show ?thesis
899     using real_sqrt_pow2[of "Re z"] real_sqrt_pow2[of "- Re z"]
900     by (cases "0::real" "Re z" rule: linorder_cases)
901        (simp_all add: complex_eq_iff Re_power2 Im_power2 power2_eq_square cmod_eq_Re)
902 next
903   assume "Im z \<noteq> 0"
904   moreover
905   have "cmod z * cmod z - Re z * Re z = Im z * Im z"
906     by (simp add: norm_complex_def power2_eq_square)
907   moreover
908   have "\<bar>Re z\<bar> \<le> cmod z"
909     by (simp add: norm_complex_def)
910   ultimately show ?thesis
911     by (simp add: Re_power2 Im_power2 complex_eq_iff real_sgn_eq
912                   field_simps real_sqrt_mult[symmetric] real_sqrt_divide)
913 qed
915 lemma csqrt_eq_0 [simp]: "csqrt z = 0 \<longleftrightarrow> z = 0"
916   by auto (metis power2_csqrt power_eq_0_iff)
918 lemma csqrt_eq_1 [simp]: "csqrt z = 1 \<longleftrightarrow> z = 1"
919   by auto (metis power2_csqrt power2_eq_1_iff)
921 lemma csqrt_principal: "0 < Re (csqrt z) \<or> Re (csqrt z) = 0 \<and> 0 \<le> Im (csqrt z)"
922   by (auto simp add: not_less cmod_plus_Re_le_0_iff Im_eq_0)
924 lemma Re_csqrt: "0 \<le> Re (csqrt z)"
925   by (metis csqrt_principal le_less)
927 lemma csqrt_square:
928   assumes "0 < Re b \<or> (Re b = 0 \<and> 0 \<le> Im b)"
929   shows "csqrt (b^2) = b"
930 proof -
931   have "csqrt (b^2) = b \<or> csqrt (b^2) = - b"
932     unfolding power2_eq_iff[symmetric] by (simp add: power2_csqrt)
933   moreover have "csqrt (b^2) \<noteq> -b \<or> b = 0"
934     using csqrt_principal[of "b ^ 2"] assms by (intro disjCI notI) (auto simp: complex_eq_iff)
935   ultimately show ?thesis
936     by auto
937 qed
939 lemma csqrt_unique:
940     "w^2 = z \<Longrightarrow> (0 < Re w \<or> Re w = 0 \<and> 0 \<le> Im w) \<Longrightarrow> csqrt z = w"
941   by (auto simp: csqrt_square)
943 lemma csqrt_minus [simp]:
944   assumes "Im x < 0 \<or> (Im x = 0 \<and> 0 \<le> Re x)"
945   shows "csqrt (- x) = \<i> * csqrt x"
946 proof -
947   have "csqrt ((\<i> * csqrt x)^2) = \<i> * csqrt x"
948   proof (rule csqrt_square)
949     have "Im (csqrt x) \<le> 0"
950       using assms by (auto simp add: cmod_eq_Re mult_le_0_iff field_simps complex_Re_le_cmod)
951     then show "0 < Re (\<i> * csqrt x) \<or> Re (\<i> * csqrt x) = 0 \<and> 0 \<le> Im (\<i> * csqrt x)"
952       by (auto simp add: Re_csqrt simp del: csqrt.simps)
953   qed
954   also have "(\<i> * csqrt x)^2 = - x"
955     by (simp add: power_mult_distrib)
956   finally show ?thesis .
957 qed
959 text \<open>Legacy theorem names\<close>
961 lemmas expand_complex_eq = complex_eq_iff
962 lemmas complex_Re_Im_cancel_iff = complex_eq_iff
963 lemmas complex_equality = complex_eqI
964 lemmas cmod_def = norm_complex_def
965 lemmas complex_norm_def = norm_complex_def
966 lemmas complex_divide_def = divide_complex_def
968 lemma legacy_Complex_simps:
969   shows Complex_eq_0: "Complex a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
970     and complex_add: "Complex a b + Complex c d = Complex (a + c) (b + d)"
971     and complex_minus: "- (Complex a b) = Complex (- a) (- b)"
972     and complex_diff: "Complex a b - Complex c d = Complex (a - c) (b - d)"
973     and Complex_eq_1: "Complex a b = 1 \<longleftrightarrow> a = 1 \<and> b = 0"
974     and Complex_eq_neg_1: "Complex a b = - 1 \<longleftrightarrow> a = - 1 \<and> b = 0"
975     and complex_mult: "Complex a b * Complex c d = Complex (a * c - b * d) (a * d + b * c)"
976     and complex_inverse: "inverse (Complex a b) = Complex (a / (a\<^sup>2 + b\<^sup>2)) (- b / (a\<^sup>2 + b\<^sup>2))"
977     and Complex_eq_numeral: "Complex a b = numeral w \<longleftrightarrow> a = numeral w \<and> b = 0"
978     and Complex_eq_neg_numeral: "Complex a b = - numeral w \<longleftrightarrow> a = - numeral w \<and> b = 0"
979     and complex_scaleR: "scaleR r (Complex a b) = Complex (r * a) (r * b)"
980     and Complex_eq_i: "(Complex x y = ii) = (x = 0 \<and> y = 1)"
981     and i_mult_Complex: "ii * Complex a b = Complex (- b) a"
982     and Complex_mult_i: "Complex a b * ii = Complex (- b) a"
983     and i_complex_of_real: "ii * complex_of_real r = Complex 0 r"
984     and complex_of_real_i: "complex_of_real r * ii = Complex 0 r"
985     and Complex_add_complex_of_real: "Complex x y + complex_of_real r = Complex (x+r) y"
986     and complex_of_real_add_Complex: "complex_of_real r + Complex x y = Complex (r+x) y"
987     and Complex_mult_complex_of_real: "Complex x y * complex_of_real r = Complex (x*r) (y*r)"
988     and complex_of_real_mult_Complex: "complex_of_real r * Complex x y = Complex (r*x) (r*y)"
989     and complex_eq_cancel_iff2: "(Complex x y = complex_of_real xa) = (x = xa & y = 0)"
990     and complex_cn: "cnj (Complex a b) = Complex a (- b)"
991     and Complex_setsum': "setsum (%x. Complex (f x) 0) s = Complex (setsum f s) 0"
992     and Complex_setsum: "Complex (setsum f s) 0 = setsum (%x. Complex (f x) 0) s"
993     and complex_of_real_def: "complex_of_real r = Complex r 0"
994     and complex_norm: "cmod (Complex x y) = sqrt (x\<^sup>2 + y\<^sup>2)"
995   by (simp_all add: norm_complex_def field_simps complex_eq_iff Re_divide Im_divide del: Complex_eq)
997 lemma Complex_in_Reals: "Complex x 0 \<in> \<real>"
998   by (metis Reals_of_real complex_of_real_def)
1000 end