(* Title: HOL/Complex.thy
Author: Jacques D. Fleuriot
Copyright: 2001 University of Edinburgh
Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4
*)
section \<open>Complex Numbers: Rectangular and Polar Representations\<close>
theory Complex
imports Transcendental
begin
text \<open>
We use the \<open>codatatype\<close> command to define the type of complex numbers. This allows us to use
\<open>primcorec\<close> to define complex functions by defining their real and imaginary result
separately.
\<close>
codatatype complex = Complex (Re: real) (Im: real)
lemma complex_surj: "Complex (Re z) (Im z) = z"
by (rule complex.collapse)
lemma complex_eqI [intro?]: "\<lbrakk>Re x = Re y; Im x = Im y\<rbrakk> \<Longrightarrow> x = y"
by (rule complex.expand) simp
lemma complex_eq_iff: "x = y \<longleftrightarrow> Re x = Re y \<and> Im x = Im y"
by (auto intro: complex.expand)
subsection \<open>Addition and Subtraction\<close>
instantiation complex :: ab_group_add
begin
primcorec zero_complex where
"Re 0 = 0"
| "Im 0 = 0"
primcorec plus_complex where
"Re (x + y) = Re x + Re y"
| "Im (x + y) = Im x + Im y"
primcorec uminus_complex where
"Re (- x) = - Re x"
| "Im (- x) = - Im x"
primcorec minus_complex where
"Re (x - y) = Re x - Re y"
| "Im (x - y) = Im x - Im y"
instance
by intro_classes (simp_all add: complex_eq_iff)
end
subsection \<open>Multiplication and Division\<close>
instantiation complex :: field
begin
primcorec one_complex where
"Re 1 = 1"
| "Im 1 = 0"
primcorec times_complex where
"Re (x * y) = Re x * Re y - Im x * Im y"
| "Im (x * y) = Re x * Im y + Im x * Re y"
primcorec inverse_complex where
"Re (inverse x) = Re x / ((Re x)\<^sup>2 + (Im x)\<^sup>2)"
| "Im (inverse x) = - Im x / ((Re x)\<^sup>2 + (Im x)\<^sup>2)"
definition "x div (y::complex) = x * inverse y"
instance
by intro_classes
(simp_all add: complex_eq_iff divide_complex_def
distrib_left distrib_right right_diff_distrib left_diff_distrib
power2_eq_square add_divide_distrib [symmetric])
end
lemma Re_divide: "Re (x / y) = (Re x * Re y + Im x * Im y) / ((Re y)\<^sup>2 + (Im y)\<^sup>2)"
unfolding divide_complex_def by (simp add: add_divide_distrib)
lemma Im_divide: "Im (x / y) = (Im x * Re y - Re x * Im y) / ((Re y)\<^sup>2 + (Im y)\<^sup>2)"
unfolding divide_complex_def times_complex.sel inverse_complex.sel
by (simp_all add: divide_simps)
lemma Re_power2: "Re (x ^ 2) = (Re x)^2 - (Im x)^2"
by (simp add: power2_eq_square)
lemma Im_power2: "Im (x ^ 2) = 2 * Re x * Im x"
by (simp add: power2_eq_square)
lemma Re_power_real [simp]: "Im x = 0 \<Longrightarrow> Re (x ^ n) = Re x ^ n "
by (induct n) simp_all
lemma Im_power_real [simp]: "Im x = 0 \<Longrightarrow> Im (x ^ n) = 0"
by (induct n) simp_all
subsection \<open>Scalar Multiplication\<close>
instantiation complex :: real_field
begin
primcorec scaleR_complex where
"Re (scaleR r x) = r * Re x"
| "Im (scaleR r x) = r * Im x"
instance
proof
fix a b :: real and x y :: complex
show "scaleR a (x + y) = scaleR a x + scaleR a y"
by (simp add: complex_eq_iff distrib_left)
show "scaleR (a + b) x = scaleR a x + scaleR b x"
by (simp add: complex_eq_iff distrib_right)
show "scaleR a (scaleR b x) = scaleR (a * b) x"
by (simp add: complex_eq_iff mult.assoc)
show "scaleR 1 x = x"
by (simp add: complex_eq_iff)
show "scaleR a x * y = scaleR a (x * y)"
by (simp add: complex_eq_iff algebra_simps)
show "x * scaleR a y = scaleR a (x * y)"
by (simp add: complex_eq_iff algebra_simps)
qed
end
subsection \<open>Numerals, Arithmetic, and Embedding from Reals\<close>
abbreviation complex_of_real :: "real \<Rightarrow> complex"
where "complex_of_real \<equiv> of_real"
declare [[coercion "of_real :: real \<Rightarrow> complex"]]
declare [[coercion "of_rat :: rat \<Rightarrow> complex"]]
declare [[coercion "of_int :: int \<Rightarrow> complex"]]
declare [[coercion "of_nat :: nat \<Rightarrow> complex"]]
lemma complex_Re_of_nat [simp]: "Re (of_nat n) = of_nat n"
by (induct n) simp_all
lemma complex_Im_of_nat [simp]: "Im (of_nat n) = 0"
by (induct n) simp_all
lemma complex_Re_of_int [simp]: "Re (of_int z) = of_int z"
by (cases z rule: int_diff_cases) simp
lemma complex_Im_of_int [simp]: "Im (of_int z) = 0"
by (cases z rule: int_diff_cases) simp
lemma complex_Re_numeral [simp]: "Re (numeral v) = numeral v"
using complex_Re_of_int [of "numeral v"] by simp
lemma complex_Im_numeral [simp]: "Im (numeral v) = 0"
using complex_Im_of_int [of "numeral v"] by simp
lemma Re_complex_of_real [simp]: "Re (complex_of_real z) = z"
by (simp add: of_real_def)
lemma Im_complex_of_real [simp]: "Im (complex_of_real z) = 0"
by (simp add: of_real_def)
lemma Re_divide_numeral [simp]: "Re (z / numeral w) = Re z / numeral w"
by (simp add: Re_divide sqr_conv_mult)
lemma Im_divide_numeral [simp]: "Im (z / numeral w) = Im z / numeral w"
by (simp add: Im_divide sqr_conv_mult)
lemma Re_divide_of_nat [simp]: "Re (z / of_nat n) = Re z / of_nat n"
by (cases n) (simp_all add: Re_divide divide_simps power2_eq_square del: of_nat_Suc)
lemma Im_divide_of_nat [simp]: "Im (z / of_nat n) = Im z / of_nat n"
by (cases n) (simp_all add: Im_divide divide_simps power2_eq_square del: of_nat_Suc)
lemma of_real_Re [simp]:
"z \<in> \<real> \<Longrightarrow> of_real (Re z) = z"
by (auto simp: Reals_def)
lemma complex_Re_fact [simp]: "Re (fact n) = fact n"
proof -
have "(fact n :: complex) = of_real (fact n)" by simp
also have "Re \<dots> = fact n" by (subst Re_complex_of_real) simp_all
finally show ?thesis .
qed
lemma complex_Im_fact [simp]: "Im (fact n) = 0"
by (subst of_nat_fact [symmetric]) (simp only: complex_Im_of_nat)
subsection \<open>The Complex Number $i$\<close>
primcorec "ii" :: complex ("\<i>") where
"Re ii = 0"
| "Im ii = 1"
lemma Complex_eq[simp]: "Complex a b = a + \<i> * b"
by (simp add: complex_eq_iff)
lemma complex_eq: "a = Re a + \<i> * Im a"
by (simp add: complex_eq_iff)
lemma fun_complex_eq: "f = (\<lambda>x. Re (f x) + \<i> * Im (f x))"
by (simp add: fun_eq_iff complex_eq)
lemma i_squared [simp]: "ii * ii = -1"
by (simp add: complex_eq_iff)
lemma power2_i [simp]: "ii\<^sup>2 = -1"
by (simp add: power2_eq_square)
lemma inverse_i [simp]: "inverse ii = - ii"
by (rule inverse_unique) simp
lemma divide_i [simp]: "x / ii = - ii * x"
by (simp add: divide_complex_def)
lemma complex_i_mult_minus [simp]: "ii * (ii * x) = - x"
by (simp add: mult.assoc [symmetric])
lemma complex_i_not_zero [simp]: "ii \<noteq> 0"
by (simp add: complex_eq_iff)
lemma complex_i_not_one [simp]: "ii \<noteq> 1"
by (simp add: complex_eq_iff)
lemma complex_i_not_numeral [simp]: "ii \<noteq> numeral w"
by (simp add: complex_eq_iff)
lemma complex_i_not_neg_numeral [simp]: "ii \<noteq> - numeral w"
by (simp add: complex_eq_iff)
lemma complex_split_polar: "\<exists>r a. z = complex_of_real r * (cos a + \<i> * sin a)"
by (simp add: complex_eq_iff polar_Ex)
lemma i_even_power [simp]: "\<i> ^ (n * 2) = (-1) ^ n"
by (metis mult.commute power2_i power_mult)
lemma Re_ii_times [simp]: "Re (ii*z) = - Im z"
by simp
lemma Im_ii_times [simp]: "Im (ii*z) = Re z"
by simp
lemma ii_times_eq_iff: "ii*w = z \<longleftrightarrow> w = -(ii*z)"
by auto
lemma divide_numeral_i [simp]: "z / (numeral n * ii) = -(ii*z) / numeral n"
by (metis divide_divide_eq_left divide_i mult.commute mult_minus_right)
subsection \<open>Vector Norm\<close>
instantiation complex :: real_normed_field
begin
definition "norm z = sqrt ((Re z)\<^sup>2 + (Im z)\<^sup>2)"
abbreviation cmod :: "complex \<Rightarrow> real"
where "cmod \<equiv> norm"
definition complex_sgn_def:
"sgn x = x /\<^sub>R cmod x"
definition dist_complex_def:
"dist x y = cmod (x - y)"
definition uniformity_complex_def [code del]:
"(uniformity :: (complex \<times> complex) filter) = (INF e:{0 <..}. principal {(x, y). dist x y < e})"
definition open_complex_def [code del]:
"open (U :: complex set) \<longleftrightarrow> (\<forall>x\<in>U. eventually (\<lambda>(x', y). x' = x \<longrightarrow> y \<in> U) uniformity)"
instance proof
fix r :: real and x y :: complex and S :: "complex set"
show "(norm x = 0) = (x = 0)"
by (simp add: norm_complex_def complex_eq_iff)
show "norm (x + y) \<le> norm x + norm y"
by (simp add: norm_complex_def complex_eq_iff real_sqrt_sum_squares_triangle_ineq)
show "norm (scaleR r x) = \<bar>r\<bar> * norm x"
by (simp add: norm_complex_def complex_eq_iff power_mult_distrib distrib_left [symmetric] real_sqrt_mult)
show "norm (x * y) = norm x * norm y"
by (simp add: norm_complex_def complex_eq_iff real_sqrt_mult [symmetric] power2_eq_square algebra_simps)
qed (rule complex_sgn_def dist_complex_def open_complex_def uniformity_complex_def)+
end
declare uniformity_Abort[where 'a=complex, code]
lemma norm_ii [simp]: "norm ii = 1"
by (simp add: norm_complex_def)
lemma cmod_unit_one: "cmod (cos a + \<i> * sin a) = 1"
by (simp add: norm_complex_def)
lemma cmod_complex_polar: "cmod (r * (cos a + \<i> * sin a)) = \<bar>r\<bar>"
by (simp add: norm_mult cmod_unit_one)
lemma complex_Re_le_cmod: "Re x \<le> cmod x"
unfolding norm_complex_def
by (rule real_sqrt_sum_squares_ge1)
lemma complex_mod_minus_le_complex_mod: "- cmod x \<le> cmod x"
by (rule order_trans [OF _ norm_ge_zero]) simp
lemma complex_mod_triangle_ineq2: "cmod (b + a) - cmod b \<le> cmod a"
by (rule ord_le_eq_trans [OF norm_triangle_ineq2]) simp
lemma abs_Re_le_cmod: "\<bar>Re x\<bar> \<le> cmod x"
by (simp add: norm_complex_def)
lemma abs_Im_le_cmod: "\<bar>Im x\<bar> \<le> cmod x"
by (simp add: norm_complex_def)
lemma cmod_le: "cmod z \<le> \<bar>Re z\<bar> + \<bar>Im z\<bar>"
apply (subst complex_eq)
apply (rule order_trans)
apply (rule norm_triangle_ineq)
apply (simp add: norm_mult)
done
lemma cmod_eq_Re: "Im z = 0 \<Longrightarrow> cmod z = \<bar>Re z\<bar>"
by (simp add: norm_complex_def)
lemma cmod_eq_Im: "Re z = 0 \<Longrightarrow> cmod z = \<bar>Im z\<bar>"
by (simp add: norm_complex_def)
lemma cmod_power2: "cmod z ^ 2 = (Re z)^2 + (Im z)^2"
by (simp add: norm_complex_def)
lemma cmod_plus_Re_le_0_iff: "cmod z + Re z \<le> 0 \<longleftrightarrow> Re z = - cmod z"
using abs_Re_le_cmod[of z] by auto
lemma cmod_Re_le_iff: "Im x = Im y \<Longrightarrow> cmod x \<le> cmod y \<longleftrightarrow> abs (Re x) \<le> abs (Re y)"
by (metis add.commute add_le_cancel_left norm_complex_def real_sqrt_abs real_sqrt_le_iff)
lemma cmod_Im_le_iff: "Re x = Re y \<Longrightarrow> cmod x \<le> cmod y \<longleftrightarrow> abs (Im x) \<le> abs (Im y)"
by (metis add_le_cancel_left norm_complex_def real_sqrt_abs real_sqrt_le_iff)
lemma Im_eq_0: "\<bar>Re z\<bar> = cmod z \<Longrightarrow> Im z = 0"
by (subst (asm) power_eq_iff_eq_base[symmetric, where n=2])
(auto simp add: norm_complex_def)
lemma abs_sqrt_wlog:
fixes x::"'a::linordered_idom"
assumes "\<And>x::'a. x \<ge> 0 \<Longrightarrow> P x (x\<^sup>2)" shows "P \<bar>x\<bar> (x\<^sup>2)"
by (metis abs_ge_zero assms power2_abs)
lemma complex_abs_le_norm: "\<bar>Re z\<bar> + \<bar>Im z\<bar> \<le> sqrt 2 * norm z"
unfolding norm_complex_def
apply (rule abs_sqrt_wlog [where x="Re z"])
apply (rule abs_sqrt_wlog [where x="Im z"])
apply (rule power2_le_imp_le)
apply (simp_all add: power2_sum add.commute sum_squares_bound real_sqrt_mult [symmetric])
done
lemma complex_unit_circle: "z \<noteq> 0 \<Longrightarrow> (Re z / cmod z)\<^sup>2 + (Im z / cmod z)\<^sup>2 = 1"
by (simp add: norm_complex_def divide_simps complex_eq_iff)
text \<open>Properties of complex signum.\<close>
lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)"
by (simp add: sgn_div_norm divide_inverse scaleR_conv_of_real mult.commute)
lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z"
by (simp add: complex_sgn_def divide_inverse)
lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z"
by (simp add: complex_sgn_def divide_inverse)
subsection \<open>Completeness of the Complexes\<close>
lemma bounded_linear_Re: "bounded_linear Re"
by (rule bounded_linear_intro [where K=1], simp_all add: norm_complex_def)
lemma bounded_linear_Im: "bounded_linear Im"
by (rule bounded_linear_intro [where K=1], simp_all add: norm_complex_def)
lemmas Cauchy_Re = bounded_linear.Cauchy [OF bounded_linear_Re]
lemmas Cauchy_Im = bounded_linear.Cauchy [OF bounded_linear_Im]
lemmas tendsto_Re [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_Re]
lemmas tendsto_Im [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_Im]
lemmas isCont_Re [simp] = bounded_linear.isCont [OF bounded_linear_Re]
lemmas isCont_Im [simp] = bounded_linear.isCont [OF bounded_linear_Im]
lemmas continuous_Re [simp] = bounded_linear.continuous [OF bounded_linear_Re]
lemmas continuous_Im [simp] = bounded_linear.continuous [OF bounded_linear_Im]
lemmas continuous_on_Re [continuous_intros] = bounded_linear.continuous_on[OF bounded_linear_Re]
lemmas continuous_on_Im [continuous_intros] = bounded_linear.continuous_on[OF bounded_linear_Im]
lemmas has_derivative_Re [derivative_intros] = bounded_linear.has_derivative[OF bounded_linear_Re]
lemmas has_derivative_Im [derivative_intros] = bounded_linear.has_derivative[OF bounded_linear_Im]
lemmas sums_Re = bounded_linear.sums [OF bounded_linear_Re]
lemmas sums_Im = bounded_linear.sums [OF bounded_linear_Im]
lemma tendsto_Complex [tendsto_intros]:
"(f \<longlongrightarrow> a) F \<Longrightarrow> (g \<longlongrightarrow> b) F \<Longrightarrow> ((\<lambda>x. Complex (f x) (g x)) \<longlongrightarrow> Complex a b) F"
by (auto intro!: tendsto_intros)
lemma tendsto_complex_iff:
"(f \<longlongrightarrow> x) F \<longleftrightarrow> (((\<lambda>x. Re (f x)) \<longlongrightarrow> Re x) F \<and> ((\<lambda>x. Im (f x)) \<longlongrightarrow> Im x) F)"
proof safe
assume "((\<lambda>x. Re (f x)) \<longlongrightarrow> Re x) F" "((\<lambda>x. Im (f x)) \<longlongrightarrow> Im x) F"
from tendsto_Complex[OF this] show "(f \<longlongrightarrow> x) F"
unfolding complex.collapse .
qed (auto intro: tendsto_intros)
lemma continuous_complex_iff: "continuous F f \<longleftrightarrow>
continuous F (\<lambda>x. Re (f x)) \<and> continuous F (\<lambda>x. Im (f x))"
unfolding continuous_def tendsto_complex_iff ..
lemma has_vector_derivative_complex_iff: "(f has_vector_derivative x) F \<longleftrightarrow>
((\<lambda>x. Re (f x)) has_field_derivative (Re x)) F \<and>
((\<lambda>x. Im (f x)) has_field_derivative (Im x)) F"
unfolding has_vector_derivative_def has_field_derivative_def has_derivative_def tendsto_complex_iff
by (simp add: field_simps bounded_linear_scaleR_left bounded_linear_mult_right)
lemma has_field_derivative_Re[derivative_intros]:
"(f has_vector_derivative D) F \<Longrightarrow> ((\<lambda>x. Re (f x)) has_field_derivative (Re D)) F"
unfolding has_vector_derivative_complex_iff by safe
lemma has_field_derivative_Im[derivative_intros]:
"(f has_vector_derivative D) F \<Longrightarrow> ((\<lambda>x. Im (f x)) has_field_derivative (Im D)) F"
unfolding has_vector_derivative_complex_iff by safe
instance complex :: banach
proof
fix X :: "nat \<Rightarrow> complex"
assume X: "Cauchy X"
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)))"
by (intro tendsto_Complex convergent_LIMSEQ_iff[THEN iffD1] Cauchy_convergent_iff[THEN iffD1] Cauchy_Re Cauchy_Im)
then show "convergent X"
unfolding complex.collapse by (rule convergentI)
qed
declare
DERIV_power[where 'a=complex, unfolded of_nat_def[symmetric], derivative_intros]
subsection \<open>Complex Conjugation\<close>
primcorec cnj :: "complex \<Rightarrow> complex" where
"Re (cnj z) = Re z"
| "Im (cnj z) = - Im z"
lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)"
by (simp add: complex_eq_iff)
lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z"
by (simp add: complex_eq_iff)
lemma complex_cnj_zero [simp]: "cnj 0 = 0"
by (simp add: complex_eq_iff)
lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)"
by (simp add: complex_eq_iff)
lemma complex_cnj_add [simp]: "cnj (x + y) = cnj x + cnj y"
by (simp add: complex_eq_iff)
lemma cnj_setsum [simp]: "cnj (setsum f s) = (\<Sum>x\<in>s. cnj (f x))"
by (induct s rule: infinite_finite_induct) auto
lemma complex_cnj_diff [simp]: "cnj (x - y) = cnj x - cnj y"
by (simp add: complex_eq_iff)
lemma complex_cnj_minus [simp]: "cnj (- x) = - cnj x"
by (simp add: complex_eq_iff)
lemma complex_cnj_one [simp]: "cnj 1 = 1"
by (simp add: complex_eq_iff)
lemma complex_cnj_mult [simp]: "cnj (x * y) = cnj x * cnj y"
by (simp add: complex_eq_iff)
lemma cnj_setprod [simp]: "cnj (setprod f s) = (\<Prod>x\<in>s. cnj (f x))"
by (induct s rule: infinite_finite_induct) auto
lemma complex_cnj_inverse [simp]: "cnj (inverse x) = inverse (cnj x)"
by (simp add: complex_eq_iff)
lemma complex_cnj_divide [simp]: "cnj (x / y) = cnj x / cnj y"
by (simp add: divide_complex_def)
lemma complex_cnj_power [simp]: "cnj (x ^ n) = cnj x ^ n"
by (induct n) simp_all
lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n"
by (simp add: complex_eq_iff)
lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z"
by (simp add: complex_eq_iff)
lemma complex_cnj_numeral [simp]: "cnj (numeral w) = numeral w"
by (simp add: complex_eq_iff)
lemma complex_cnj_neg_numeral [simp]: "cnj (- numeral w) = - numeral w"
by (simp add: complex_eq_iff)
lemma complex_cnj_scaleR [simp]: "cnj (scaleR r x) = scaleR r (cnj x)"
by (simp add: complex_eq_iff)
lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z"
by (simp add: norm_complex_def)
lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x"
by (simp add: complex_eq_iff)
lemma complex_cnj_i [simp]: "cnj ii = - ii"
by (simp add: complex_eq_iff)
lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)"
by (simp add: complex_eq_iff)
lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im z) * ii"
by (simp add: complex_eq_iff)
lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\<^sup>2 + (Im z)\<^sup>2)"
by (simp add: complex_eq_iff power2_eq_square)
lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<^sup>2"
by (simp add: norm_mult power2_eq_square)
lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))"
by (simp add: norm_complex_def power2_eq_square)
lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0"
by simp
lemma complex_cnj_fact [simp]: "cnj (fact n) = fact n"
by (subst of_nat_fact [symmetric], subst complex_cnj_of_nat) simp
lemma complex_cnj_pochhammer [simp]: "cnj (pochhammer z n) = pochhammer (cnj z) n"
by (induction n arbitrary: z) (simp_all add: pochhammer_rec)
lemma bounded_linear_cnj: "bounded_linear cnj"
using complex_cnj_add complex_cnj_scaleR
by (rule bounded_linear_intro [where K=1], simp)
lemmas tendsto_cnj [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_cnj]
lemmas isCont_cnj [simp] = bounded_linear.isCont [OF bounded_linear_cnj]
lemmas continuous_cnj [simp, continuous_intros] = bounded_linear.continuous [OF bounded_linear_cnj]
lemmas continuous_on_cnj [simp, continuous_intros] = bounded_linear.continuous_on [OF bounded_linear_cnj]
lemmas has_derivative_cnj [simp, derivative_intros] = bounded_linear.has_derivative [OF bounded_linear_cnj]
lemma lim_cnj: "((\<lambda>x. cnj(f x)) \<longlongrightarrow> cnj l) F \<longleftrightarrow> (f \<longlongrightarrow> l) F"
by (simp add: tendsto_iff dist_complex_def complex_cnj_diff [symmetric] del: complex_cnj_diff)
lemma sums_cnj: "((\<lambda>x. cnj(f x)) sums cnj l) \<longleftrightarrow> (f sums l)"
by (simp add: sums_def lim_cnj cnj_setsum [symmetric] del: cnj_setsum)
subsection\<open>Basic Lemmas\<close>
lemma complex_eq_0: "z=0 \<longleftrightarrow> (Re z)\<^sup>2 + (Im z)\<^sup>2 = 0"
by (metis zero_complex.sel complex_eqI sum_power2_eq_zero_iff)
lemma complex_neq_0: "z\<noteq>0 \<longleftrightarrow> (Re z)\<^sup>2 + (Im z)\<^sup>2 > 0"
by (metis complex_eq_0 less_numeral_extra(3) sum_power2_gt_zero_iff)
lemma complex_norm_square: "of_real ((norm z)\<^sup>2) = z * cnj z"
by (cases z)
(auto simp: complex_eq_iff norm_complex_def power2_eq_square[symmetric] of_real_power[symmetric]
simp del: of_real_power)
lemma complex_div_cnj: "a / b = (a * cnj b) / (norm b)^2"
using complex_norm_square by auto
lemma Re_complex_div_eq_0: "Re (a / b) = 0 \<longleftrightarrow> Re (a * cnj b) = 0"
by (auto simp add: Re_divide)
lemma Im_complex_div_eq_0: "Im (a / b) = 0 \<longleftrightarrow> Im (a * cnj b) = 0"
by (auto simp add: Im_divide)
lemma complex_div_gt_0:
"(Re (a / b) > 0 \<longleftrightarrow> Re (a * cnj b) > 0) \<and> (Im (a / b) > 0 \<longleftrightarrow> Im (a * cnj b) > 0)"
proof cases
assume "b = 0" then show ?thesis by auto
next
assume "b \<noteq> 0"
then have "0 < (Re b)\<^sup>2 + (Im b)\<^sup>2"
by (simp add: complex_eq_iff sum_power2_gt_zero_iff)
then show ?thesis
by (simp add: Re_divide Im_divide zero_less_divide_iff)
qed
lemma Re_complex_div_gt_0: "Re (a / b) > 0 \<longleftrightarrow> Re (a * cnj b) > 0"
and Im_complex_div_gt_0: "Im (a / b) > 0 \<longleftrightarrow> Im (a * cnj b) > 0"
using complex_div_gt_0 by auto
lemma Re_complex_div_ge_0: "Re(a / b) \<ge> 0 \<longleftrightarrow> Re(a * cnj b) \<ge> 0"
by (metis le_less Re_complex_div_eq_0 Re_complex_div_gt_0)
lemma Im_complex_div_ge_0: "Im(a / b) \<ge> 0 \<longleftrightarrow> Im(a * cnj b) \<ge> 0"
by (metis Im_complex_div_eq_0 Im_complex_div_gt_0 le_less)
lemma Re_complex_div_lt_0: "Re(a / b) < 0 \<longleftrightarrow> Re(a * cnj b) < 0"
by (metis less_asym neq_iff Re_complex_div_eq_0 Re_complex_div_gt_0)
lemma Im_complex_div_lt_0: "Im(a / b) < 0 \<longleftrightarrow> Im(a * cnj b) < 0"
by (metis Im_complex_div_eq_0 Im_complex_div_gt_0 less_asym neq_iff)
lemma Re_complex_div_le_0: "Re(a / b) \<le> 0 \<longleftrightarrow> Re(a * cnj b) \<le> 0"
by (metis not_le Re_complex_div_gt_0)
lemma Im_complex_div_le_0: "Im(a / b) \<le> 0 \<longleftrightarrow> Im(a * cnj b) \<le> 0"
by (metis Im_complex_div_gt_0 not_le)
lemma Re_divide_of_real [simp]: "Re (z / of_real r) = Re z / r"
by (simp add: Re_divide power2_eq_square)
lemma Im_divide_of_real [simp]: "Im (z / of_real r) = Im z / r"
by (simp add: Im_divide power2_eq_square)
lemma Re_divide_Reals: "r \<in> Reals \<Longrightarrow> Re (z / r) = Re z / Re r"
by (metis Re_divide_of_real of_real_Re)
lemma Im_divide_Reals: "r \<in> Reals \<Longrightarrow> Im (z / r) = Im z / Re r"
by (metis Im_divide_of_real of_real_Re)
lemma Re_setsum[simp]: "Re (setsum f s) = (\<Sum>x\<in>s. Re (f x))"
by (induct s rule: infinite_finite_induct) auto
lemma Im_setsum[simp]: "Im (setsum f s) = (\<Sum>x\<in>s. Im(f x))"
by (induct s rule: infinite_finite_induct) auto
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)"
unfolding sums_def tendsto_complex_iff Im_setsum Re_setsum ..
lemma summable_complex_iff: "summable f \<longleftrightarrow> summable (\<lambda>x. Re (f x)) \<and> summable (\<lambda>x. Im (f x))"
unfolding summable_def sums_complex_iff[abs_def] by (metis complex.sel)
lemma summable_complex_of_real [simp]: "summable (\<lambda>n. complex_of_real (f n)) \<longleftrightarrow> summable f"
unfolding summable_complex_iff by simp
lemma summable_Re: "summable f \<Longrightarrow> summable (\<lambda>x. Re (f x))"
unfolding summable_complex_iff by blast
lemma summable_Im: "summable f \<Longrightarrow> summable (\<lambda>x. Im (f x))"
unfolding summable_complex_iff by blast
lemma complex_is_Nat_iff: "z \<in> \<nat> \<longleftrightarrow> Im z = 0 \<and> (\<exists>i. Re z = of_nat i)"
by (auto simp: Nats_def complex_eq_iff)
lemma complex_is_Int_iff: "z \<in> \<int> \<longleftrightarrow> Im z = 0 \<and> (\<exists>i. Re z = of_int i)"
by (auto simp: Ints_def complex_eq_iff)
lemma complex_is_Real_iff: "z \<in> \<real> \<longleftrightarrow> Im z = 0"
by (auto simp: Reals_def complex_eq_iff)
lemma Reals_cnj_iff: "z \<in> \<real> \<longleftrightarrow> cnj z = z"
by (auto simp: complex_is_Real_iff complex_eq_iff)
lemma in_Reals_norm: "z \<in> \<real> \<Longrightarrow> norm z = \<bar>Re z\<bar>"
by (simp add: complex_is_Real_iff norm_complex_def)
lemma series_comparison_complex:
fixes f:: "nat \<Rightarrow> 'a::banach"
assumes sg: "summable g"
and "\<And>n. g n \<in> \<real>" "\<And>n. Re (g n) \<ge> 0"
and fg: "\<And>n. n \<ge> N \<Longrightarrow> norm(f n) \<le> norm(g n)"
shows "summable f"
proof -
have g: "\<And>n. cmod (g n) = Re (g n)" using assms
by (metis abs_of_nonneg in_Reals_norm)
show ?thesis
apply (rule summable_comparison_test' [where g = "\<lambda>n. norm (g n)" and N=N])
using sg
apply (auto simp: summable_def)
apply (rule_tac x="Re s" in exI)
apply (auto simp: g sums_Re)
apply (metis fg g)
done
qed
subsection\<open>Polar Form for Complex Numbers\<close>
lemma complex_unimodular_polar: "(norm z = 1) \<Longrightarrow> \<exists>x. z = Complex (cos x) (sin x)"
using sincos_total_2pi [of "Re z" "Im z"]
by auto (metis cmod_power2 complex_eq power_one)
subsubsection \<open>$\cos \theta + i \sin \theta$\<close>
primcorec cis :: "real \<Rightarrow> complex" where
"Re (cis a) = cos a"
| "Im (cis a) = sin a"
lemma cis_zero [simp]: "cis 0 = 1"
by (simp add: complex_eq_iff)
lemma norm_cis [simp]: "norm (cis a) = 1"
by (simp add: norm_complex_def)
lemma sgn_cis [simp]: "sgn (cis a) = cis a"
by (simp add: sgn_div_norm)
lemma cis_neq_zero [simp]: "cis a \<noteq> 0"
by (metis norm_cis norm_zero zero_neq_one)
lemma cis_mult: "cis a * cis b = cis (a + b)"
by (simp add: complex_eq_iff cos_add sin_add)
lemma DeMoivre: "(cis a) ^ n = cis (real n * a)"
by (induct n, simp_all add: of_nat_Suc algebra_simps cis_mult)
lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)"
by (simp add: complex_eq_iff)
lemma cis_divide: "cis a / cis b = cis (a - b)"
by (simp add: divide_complex_def cis_mult)
lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re(cis a ^ n)"
by (auto simp add: DeMoivre)
lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im(cis a ^ n)"
by (auto simp add: DeMoivre)
lemma cis_pi: "cis pi = -1"
by (simp add: complex_eq_iff)
subsubsection \<open>$r(\cos \theta + i \sin \theta)$\<close>
definition rcis :: "real \<Rightarrow> real \<Rightarrow> complex" where
"rcis r a = complex_of_real r * cis a"
lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a"
by (simp add: rcis_def)
lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a"
by (simp add: rcis_def)
lemma rcis_Ex: "\<exists>r a. z = rcis r a"
by (simp add: complex_eq_iff polar_Ex)
lemma complex_mod_rcis [simp]: "cmod (rcis r a) = \<bar>r\<bar>"
by (simp add: rcis_def norm_mult)
lemma cis_rcis_eq: "cis a = rcis 1 a"
by (simp add: rcis_def)
lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1*r2) (a + b)"
by (simp add: rcis_def cis_mult)
lemma rcis_zero_mod [simp]: "rcis 0 a = 0"
by (simp add: rcis_def)
lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r"
by (simp add: rcis_def)
lemma rcis_eq_zero_iff [simp]: "rcis r a = 0 \<longleftrightarrow> r = 0"
by (simp add: rcis_def)
lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)"
by (simp add: rcis_def power_mult_distrib DeMoivre)
lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)"
by (simp add: divide_inverse rcis_def)
lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)"
by (simp add: rcis_def cis_divide [symmetric])
subsubsection \<open>Complex exponential\<close>
lemma cis_conv_exp: "cis b = exp (\<i> * b)"
proof -
{ fix n :: nat
have "\<i> ^ n = fact n *\<^sub>R (cos_coeff n + \<i> * sin_coeff n)"
by (induct n)
(simp_all add: sin_coeff_Suc cos_coeff_Suc complex_eq_iff Re_divide Im_divide field_simps
power2_eq_square of_nat_Suc add_nonneg_eq_0_iff)
then have "(\<i> * complex_of_real b) ^ n /\<^sub>R fact n =
of_real (cos_coeff n * b^n) + \<i> * of_real (sin_coeff n * b^n)"
by (simp add: field_simps) }
then show ?thesis using sin_converges [of b] cos_converges [of b]
by (auto simp add: cis.ctr exp_def simp del: of_real_mult
intro!: sums_unique sums_add sums_mult sums_of_real)
qed
lemma exp_eq_polar: "exp z = exp (Re z) * cis (Im z)"
unfolding cis_conv_exp exp_of_real [symmetric] mult_exp_exp by (cases z) simp
lemma Re_exp: "Re (exp z) = exp (Re z) * cos (Im z)"
unfolding exp_eq_polar by simp
lemma Im_exp: "Im (exp z) = exp (Re z) * sin (Im z)"
unfolding exp_eq_polar by simp
lemma norm_cos_sin [simp]: "norm (Complex (cos t) (sin t)) = 1"
by (simp add: norm_complex_def)
lemma norm_exp_eq_Re [simp]: "norm (exp z) = exp (Re z)"
by (simp add: cis.code cmod_complex_polar exp_eq_polar)
lemma complex_exp_exists: "\<exists>a r. z = complex_of_real r * exp a"
apply (insert rcis_Ex [of z])
apply (auto simp add: exp_eq_polar rcis_def mult.assoc [symmetric])
apply (rule_tac x = "ii * complex_of_real a" in exI, auto)
done
lemma exp_pi_i [simp]: "exp(of_real pi * ii) = -1"
by (metis cis_conv_exp cis_pi mult.commute)
lemma exp_two_pi_i [simp]: "exp(2 * of_real pi * ii) = 1"
by (simp add: exp_eq_polar complex_eq_iff)
subsubsection \<open>Complex argument\<close>
definition arg :: "complex \<Rightarrow> real" where
"arg z = (if z = 0 then 0 else (SOME a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi))"
lemma arg_zero: "arg 0 = 0"
by (simp add: arg_def)
lemma arg_unique:
assumes "sgn z = cis x" and "-pi < x" and "x \<le> pi"
shows "arg z = x"
proof -
from assms have "z \<noteq> 0" by auto
have "(SOME a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi) = x"
proof
fix a def d \<equiv> "a - x"
assume a: "sgn z = cis a \<and> - pi < a \<and> a \<le> pi"
from a assms have "- (2*pi) < d \<and> d < 2*pi"
unfolding d_def by simp
moreover from a assms have "cos a = cos x" and "sin a = sin x"
by (simp_all add: complex_eq_iff)
hence cos: "cos d = 1" unfolding d_def cos_diff by simp
moreover from cos have "sin d = 0" by (rule cos_one_sin_zero)
ultimately have "d = 0"
unfolding sin_zero_iff
by (auto elim!: evenE dest!: less_2_cases)
thus "a = x" unfolding d_def by simp
qed (simp add: assms del: Re_sgn Im_sgn)
with \<open>z \<noteq> 0\<close> show "arg z = x"
unfolding arg_def by simp
qed
lemma arg_correct:
assumes "z \<noteq> 0" shows "sgn z = cis (arg z) \<and> -pi < arg z \<and> arg z \<le> pi"
proof (simp add: arg_def assms, rule someI_ex)
obtain r a where z: "z = rcis r a" using rcis_Ex by fast
with assms have "r \<noteq> 0" by auto
def b \<equiv> "if 0 < r then a else a + pi"
have b: "sgn z = cis b"
unfolding z b_def rcis_def using \<open>r \<noteq> 0\<close>
by (simp add: of_real_def sgn_scaleR sgn_if complex_eq_iff)
have cis_2pi_nat: "\<And>n. cis (2 * pi * real_of_nat n) = 1"
by (induct_tac n) (simp_all add: distrib_left cis_mult [symmetric] complex_eq_iff)
have cis_2pi_int: "\<And>x. cis (2 * pi * real_of_int x) = 1"
by (case_tac x rule: int_diff_cases)
(simp add: right_diff_distrib cis_divide [symmetric] cis_2pi_nat)
def c \<equiv> "b - 2*pi * of_int \<lceil>(b - pi) / (2*pi)\<rceil>"
have "sgn z = cis c"
unfolding b c_def
by (simp add: cis_divide [symmetric] cis_2pi_int)
moreover have "- pi < c \<and> c \<le> pi"
using ceiling_correct [of "(b - pi) / (2*pi)"]
by (simp add: c_def less_divide_eq divide_le_eq algebra_simps del: le_of_int_ceiling)
ultimately show "\<exists>a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi" by fast
qed
lemma arg_bounded: "- pi < arg z \<and> arg z \<le> pi"
by (cases "z = 0") (simp_all add: arg_zero arg_correct)
lemma cis_arg: "z \<noteq> 0 \<Longrightarrow> cis (arg z) = sgn z"
by (simp add: arg_correct)
lemma rcis_cmod_arg: "rcis (cmod z) (arg z) = z"
by (cases "z = 0") (simp_all add: rcis_def cis_arg sgn_div_norm of_real_def)
lemma cos_arg_i_mult_zero [simp]: "y \<noteq> 0 \<Longrightarrow> Re y = 0 \<Longrightarrow> cos (arg y) = 0"
using cis_arg [of y] by (simp add: complex_eq_iff)
subsection \<open>Square root of complex numbers\<close>
primcorec csqrt :: "complex \<Rightarrow> complex" where
"Re (csqrt z) = sqrt ((cmod z + Re z) / 2)"
| "Im (csqrt z) = (if Im z = 0 then 1 else sgn (Im z)) * sqrt ((cmod z - Re z) / 2)"
lemma csqrt_of_real_nonneg [simp]: "Im x = 0 \<Longrightarrow> Re x \<ge> 0 \<Longrightarrow> csqrt x = sqrt (Re x)"
by (simp add: complex_eq_iff norm_complex_def)
lemma csqrt_of_real_nonpos [simp]: "Im x = 0 \<Longrightarrow> Re x \<le> 0 \<Longrightarrow> csqrt x = \<i> * sqrt \<bar>Re x\<bar>"
by (simp add: complex_eq_iff norm_complex_def)
lemma of_real_sqrt: "x \<ge> 0 \<Longrightarrow> of_real (sqrt x) = csqrt (of_real x)"
by (simp add: complex_eq_iff norm_complex_def)
lemma csqrt_0 [simp]: "csqrt 0 = 0"
by simp
lemma csqrt_1 [simp]: "csqrt 1 = 1"
by simp
lemma csqrt_ii [simp]: "csqrt \<i> = (1 + \<i>) / sqrt 2"
by (simp add: complex_eq_iff Re_divide Im_divide real_sqrt_divide real_div_sqrt)
lemma power2_csqrt[simp,algebra]: "(csqrt z)\<^sup>2 = z"
proof cases
assume "Im z = 0" then show ?thesis
using real_sqrt_pow2[of "Re z"] real_sqrt_pow2[of "- Re z"]
by (cases "0::real" "Re z" rule: linorder_cases)
(simp_all add: complex_eq_iff Re_power2 Im_power2 power2_eq_square cmod_eq_Re)
next
assume "Im z \<noteq> 0"
moreover
have "cmod z * cmod z - Re z * Re z = Im z * Im z"
by (simp add: norm_complex_def power2_eq_square)
moreover
have "\<bar>Re z\<bar> \<le> cmod z"
by (simp add: norm_complex_def)
ultimately show ?thesis
by (simp add: Re_power2 Im_power2 complex_eq_iff real_sgn_eq
field_simps real_sqrt_mult[symmetric] real_sqrt_divide)
qed
lemma csqrt_eq_0 [simp]: "csqrt z = 0 \<longleftrightarrow> z = 0"
by auto (metis power2_csqrt power_eq_0_iff)
lemma csqrt_eq_1 [simp]: "csqrt z = 1 \<longleftrightarrow> z = 1"
by auto (metis power2_csqrt power2_eq_1_iff)
lemma csqrt_principal: "0 < Re (csqrt z) \<or> Re (csqrt z) = 0 \<and> 0 \<le> Im (csqrt z)"
by (auto simp add: not_less cmod_plus_Re_le_0_iff Im_eq_0)
lemma Re_csqrt: "0 \<le> Re (csqrt z)"
by (metis csqrt_principal le_less)
lemma csqrt_square:
assumes "0 < Re b \<or> (Re b = 0 \<and> 0 \<le> Im b)"
shows "csqrt (b^2) = b"
proof -
have "csqrt (b^2) = b \<or> csqrt (b^2) = - b"
unfolding power2_eq_iff[symmetric] by (simp add: power2_csqrt)
moreover have "csqrt (b^2) \<noteq> -b \<or> b = 0"
using csqrt_principal[of "b ^ 2"] assms by (intro disjCI notI) (auto simp: complex_eq_iff)
ultimately show ?thesis
by auto
qed
lemma csqrt_unique:
"w^2 = z \<Longrightarrow> (0 < Re w \<or> Re w = 0 \<and> 0 \<le> Im w) \<Longrightarrow> csqrt z = w"
by (auto simp: csqrt_square)
lemma csqrt_minus [simp]:
assumes "Im x < 0 \<or> (Im x = 0 \<and> 0 \<le> Re x)"
shows "csqrt (- x) = \<i> * csqrt x"
proof -
have "csqrt ((\<i> * csqrt x)^2) = \<i> * csqrt x"
proof (rule csqrt_square)
have "Im (csqrt x) \<le> 0"
using assms by (auto simp add: cmod_eq_Re mult_le_0_iff field_simps complex_Re_le_cmod)
then show "0 < Re (\<i> * csqrt x) \<or> Re (\<i> * csqrt x) = 0 \<and> 0 \<le> Im (\<i> * csqrt x)"
by (auto simp add: Re_csqrt simp del: csqrt.simps)
qed
also have "(\<i> * csqrt x)^2 = - x"
by (simp add: power_mult_distrib)
finally show ?thesis .
qed
text \<open>Legacy theorem names\<close>
lemmas expand_complex_eq = complex_eq_iff
lemmas complex_Re_Im_cancel_iff = complex_eq_iff
lemmas complex_equality = complex_eqI
lemmas cmod_def = norm_complex_def
lemmas complex_norm_def = norm_complex_def
lemmas complex_divide_def = divide_complex_def
lemma legacy_Complex_simps:
shows Complex_eq_0: "Complex a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
and complex_add: "Complex a b + Complex c d = Complex (a + c) (b + d)"
and complex_minus: "- (Complex a b) = Complex (- a) (- b)"
and complex_diff: "Complex a b - Complex c d = Complex (a - c) (b - d)"
and Complex_eq_1: "Complex a b = 1 \<longleftrightarrow> a = 1 \<and> b = 0"
and Complex_eq_neg_1: "Complex a b = - 1 \<longleftrightarrow> a = - 1 \<and> b = 0"
and complex_mult: "Complex a b * Complex c d = Complex (a * c - b * d) (a * d + b * c)"
and complex_inverse: "inverse (Complex a b) = Complex (a / (a\<^sup>2 + b\<^sup>2)) (- b / (a\<^sup>2 + b\<^sup>2))"
and Complex_eq_numeral: "Complex a b = numeral w \<longleftrightarrow> a = numeral w \<and> b = 0"
and Complex_eq_neg_numeral: "Complex a b = - numeral w \<longleftrightarrow> a = - numeral w \<and> b = 0"
and complex_scaleR: "scaleR r (Complex a b) = Complex (r * a) (r * b)"
and Complex_eq_i: "(Complex x y = ii) = (x = 0 \<and> y = 1)"
and i_mult_Complex: "ii * Complex a b = Complex (- b) a"
and Complex_mult_i: "Complex a b * ii = Complex (- b) a"
and i_complex_of_real: "ii * complex_of_real r = Complex 0 r"
and complex_of_real_i: "complex_of_real r * ii = Complex 0 r"
and Complex_add_complex_of_real: "Complex x y + complex_of_real r = Complex (x+r) y"
and complex_of_real_add_Complex: "complex_of_real r + Complex x y = Complex (r+x) y"
and Complex_mult_complex_of_real: "Complex x y * complex_of_real r = Complex (x*r) (y*r)"
and complex_of_real_mult_Complex: "complex_of_real r * Complex x y = Complex (r*x) (r*y)"
and complex_eq_cancel_iff2: "(Complex x y = complex_of_real xa) = (x = xa & y = 0)"
and complex_cn: "cnj (Complex a b) = Complex a (- b)"
and Complex_setsum': "setsum (%x. Complex (f x) 0) s = Complex (setsum f s) 0"
and Complex_setsum: "Complex (setsum f s) 0 = setsum (%x. Complex (f x) 0) s"
and complex_of_real_def: "complex_of_real r = Complex r 0"
and complex_norm: "cmod (Complex x y) = sqrt (x\<^sup>2 + y\<^sup>2)"
by (simp_all add: norm_complex_def field_simps complex_eq_iff Re_divide Im_divide del: Complex_eq)
lemma Complex_in_Reals: "Complex x 0 \<in> \<real>"
by (metis Reals_of_real complex_of_real_def)
end