(* Title: HOL/Complex.thy
Author: Jacques D. Fleuriot, 2001 University of Edinburgh
Author: Lawrence C Paulson, 2003/4
*)
section \<open>Complex Numbers: Rectangular and Polar Representations\<close>
theory Complex
imports Transcendental
begin
text \<open>
We use the \<^theory_text>\<open>codatatype\<close> command to define the type of complex numbers. This
allows us to use \<^theory_text>\<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?]: "Re x = Re y \<Longrightarrow> Im x = Im y \<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 standard (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 = x * inverse y" for x y :: complex
instance
by standard
(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)"
by (simp add: divide_complex_def 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 add: divide_simps)
lemma Complex_divide:
"(x / y) = Complex ((Re x * Re y + Im x * Im y) / ((Re y)\<^sup>2 + (Im y)\<^sup>2))
((Im x * Re y - Re x * Im y) / ((Re y)\<^sup>2 + (Im y)\<^sup>2))"
by (metis Im_divide Re_divide complex_surj)
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 R\<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)
lemma Re_prod_Reals: "(\<And>x. x \<in> A \<Longrightarrow> f x \<in> \<real>) \<Longrightarrow> Re (prod f A) = prod (\<lambda>x. Re (f x)) A"
proof (induction A rule: infinite_finite_induct)
case (insert x A)
hence "Re (prod f (insert x A)) = Re (f x) * Re (prod f A) - Im (f x) * Im (prod f A)"
by simp
also from insert.prems have "f x \<in> \<real>" by simp
hence "Im (f x) = 0" by (auto elim!: Reals_cases)
also have "Re (prod f A) = (\<Prod>x\<in>A. Re (f x))"
by (intro insert.IH insert.prems) auto
finally show ?case using insert.hyps by simp
qed auto
subsection \<open>The Complex Number $i$\<close>
primcorec imaginary_unit :: complex ("\<i>")
where
"Re \<i> = 0"
| "Im \<i> = 1"
lemma Complex_eq: "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]: "\<i> * \<i> = -1"
by (simp add: complex_eq_iff)
lemma power2_i [simp]: "\<i>\<^sup>2 = -1"
by (simp add: power2_eq_square)
lemma inverse_i [simp]: "inverse \<i> = - \<i>"
by (rule inverse_unique) simp
lemma divide_i [simp]: "x / \<i> = - \<i> * x"
by (simp add: divide_complex_def)
lemma complex_i_mult_minus [simp]: "\<i> * (\<i> * x) = - x"
by (simp add: mult.assoc [symmetric])
lemma complex_i_not_zero [simp]: "\<i> \<noteq> 0"
by (simp add: complex_eq_iff)
lemma complex_i_not_one [simp]: "\<i> \<noteq> 1"
by (simp add: complex_eq_iff)
lemma complex_i_not_numeral [simp]: "\<i> \<noteq> numeral w"
by (simp add: complex_eq_iff)
lemma complex_i_not_neg_numeral [simp]: "\<i> \<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_i_times [simp]: "Re (\<i> * z) = - Im z"
by simp
lemma Im_i_times [simp]: "Im (\<i> * z) = Re z"
by simp
lemma i_times_eq_iff: "\<i> * w = z \<longleftrightarrow> w = - (\<i> * z)"
by auto
lemma divide_numeral_i [simp]: "z / (numeral n * \<i>) = - (\<i> * z) / numeral n"
by (metis divide_divide_eq_left divide_i mult.commute mult_minus_right)
lemma imaginary_eq_real_iff [simp]:
assumes "y \<in> Reals" "x \<in> Reals"
shows "\<i> * y = x \<longleftrightarrow> x=0 \<and> y=0"
using assms
unfolding Reals_def
apply clarify
apply (rule iffI)
apply (metis Im_complex_of_real Im_i_times Re_complex_of_real mult_eq_0_iff of_real_0)
by simp
lemma real_eq_imaginary_iff [simp]:
assumes "y \<in> Reals" "x \<in> Reals"
shows "x = \<i> * y \<longleftrightarrow> x=0 \<and> y=0"
using assms imaginary_eq_real_iff by fastforce
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\<in>{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 \<i> = 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)\<^sup>2 = (Re z)\<^sup>2 + (Im z)\<^sup>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> \<bar>Re x\<bar> \<le> \<bar>Re y\<bar>"
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> \<bar>Im x\<bar> \<le> \<bar>Im y\<bar>"
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: "(\<And>x. x \<ge> 0 \<Longrightarrow> P x (x\<^sup>2)) \<Longrightarrow> P \<bar>x\<bar> (x\<^sup>2)"
for x::"'a::linordered_idom"
by (metis abs_ge_zero 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>Absolute value\<close>
instantiation complex :: field_abs_sgn
begin
definition abs_complex :: "complex \<Rightarrow> complex"
where "abs_complex = of_real \<circ> norm"
instance
apply standard
apply (auto simp add: abs_complex_def complex_sgn_def norm_mult)
apply (auto simp add: scaleR_conv_of_real field_simps)
done
end
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"
unfolding Complex_eq 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))"
by (simp only: continuous_def tendsto_complex_iff)
lemma continuous_on_of_real_o_iff [simp]:
"continuous_on S (\<lambda>x. complex_of_real (g x)) = continuous_on S g"
using continuous_on_Re continuous_on_of_real by fastforce
lemma continuous_on_of_real_id [simp]:
"continuous_on S (of_real :: real \<Rightarrow> 'a::real_normed_algebra_1)"
by (rule continuous_on_of_real [OF continuous_on_id])
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"
by (simp add: has_vector_derivative_def has_field_derivative_def has_derivative_def
tendsto_complex_iff 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 \<longleftrightarrow> 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 \<longleftrightarrow> z = 0"
by (simp add: complex_eq_iff)
lemma complex_cnj_one_iff [simp]: "cnj z = 1 \<longleftrightarrow> z = 1"
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_sum [simp]: "cnj (sum 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_prod [simp]: "cnj (prod 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 \<i> = - \<i>"
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) * \<i>"
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 cnj_add_mult_eq_Re: "z * cnj w + cnj z * w = 2 * Re (z * cnj w)"
by (rule complex_eqI) auto
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 (induct 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]
and isCont_cnj [simp] = bounded_linear.isCont [OF bounded_linear_cnj]
and continuous_cnj [simp, continuous_intros] = bounded_linear.continuous [OF bounded_linear_cnj]
and continuous_on_cnj [simp, continuous_intros] = bounded_linear.continuous_on [OF bounded_linear_cnj]
and 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_sum [symmetric] del: cnj_sum)
lemma differentiable_cnj_iff:
"(\<lambda>z. cnj (f z)) differentiable at x within A \<longleftrightarrow> f differentiable at x within A"
proof
assume "(\<lambda>z. cnj (f z)) differentiable at x within A"
then obtain D where "((\<lambda>z. cnj (f z)) has_derivative D) (at x within A)"
by (auto simp: differentiable_def)
from has_derivative_cnj[OF this] show "f differentiable at x within A"
by (auto simp: differentiable_def)
next
assume "f differentiable at x within A"
then obtain D where "(f has_derivative D) (at x within A)"
by (auto simp: differentiable_def)
from has_derivative_cnj[OF this] show "(\<lambda>z. cnj (f z)) differentiable at x within A"
by (auto simp: differentiable_def)
qed
lemma has_vector_derivative_cnj [derivative_intros]:
assumes "(f has_vector_derivative f') (at z within A)"
shows "((\<lambda>z. cnj (f z)) has_vector_derivative cnj f') (at z within A)"
using assms by (auto simp: has_vector_derivative_complex_iff intro: derivative_intros)
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)\<^sup>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 "b = 0")
case True
then show ?thesis by auto
next
case False
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 [simp]: "r \<in> \<real> \<Longrightarrow> Re (z / r) = Re z / Re r"
by (metis Re_divide_of_real of_real_Re)
lemma Im_divide_Reals [simp]: "r \<in> \<real> \<Longrightarrow> Im (z / r) = Im z / Re r"
by (metis Im_divide_of_real of_real_Re)
lemma Re_sum[simp]: "Re (sum f s) = (\<Sum>x\<in>s. Re (f x))"
by (induct s rule: infinite_finite_induct) auto
lemma Im_sum[simp]: "Im (sum 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_sum Re_sum ..
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 Re_Reals_divide: "r \<in> \<real> \<Longrightarrow> Re (r / z) = Re r * Re z / (norm z)\<^sup>2"
by (simp add: Re_divide complex_is_Real_iff cmod_power2)
lemma Im_Reals_divide: "r \<in> \<real> \<Longrightarrow> Im (r / z) = -Re r * Im z / (norm z)\<^sup>2"
by (simp add: Im_divide complex_is_Real_iff cmod_power2)
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:
assumes "norm z = 1"
obtains t where "0 \<le> t" "t < 2 * pi" "z = Complex (cos t) (sin t)"
by (metis cmod_power2 one_power2 complex_surj sincos_total_2pi [of "Re z" "Im z"] assms)
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_2pi [simp]: "cis (2 * pi) = 1"
by (simp add: cis.ctr complex_eq_iff)
lemma cis_neq_zero [simp]: "cis a \<noteq> 0"
by (metis norm_cis norm_zero zero_neq_one)
lemma cis_cnj: "cnj (cis t) = cis (-t)"
by (simp add: complex_eq_iff)
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: 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 [simp]: "cis pi = -1"
by (simp add: complex_eq_iff)
lemma cis_pi_half[simp]: "cis (pi / 2) = \<i>"
by (simp add: cis.ctr complex_eq_iff)
lemma cis_minus_pi_half[simp]: "cis (-(pi / 2)) = -\<i>"
by (simp add: cis.ctr complex_eq_iff)
lemma cis_multiple_2pi[simp]: "n \<in> \<int> \<Longrightarrow> cis (2 * pi * n) = 1"
by (auto elim!: Ints_cases simp: cis.ctr one_complex.ctr)
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 exp_Reals_eq:
assumes "z \<in> \<real>"
shows "exp z = of_real (exp (Re z))"
using assms by (auto elim!: Reals_cases simp: exp_of_real)
lemma cis_conv_exp: "cis b = exp (\<i> * b)"
proof -
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)"
for n :: nat
proof -
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 add_nonneg_eq_0_iff)
then show ?thesis
by (simp add: field_simps)
qed
then show ?thesis
using sin_converges [of b] cos_converges [of b]
by (auto simp add: Complex_eq 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 add: Complex_eq)
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 Complex_eq)
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 = "\<i> * complex_of_real a" in exI)
apply auto
done
lemma exp_pi_i [simp]: "exp (of_real pi * \<i>) = -1"
by (metis cis_conv_exp cis_pi mult.commute)
lemma exp_pi_i' [simp]: "exp (\<i> * of_real pi) = -1"
using cis_conv_exp cis_pi by auto
lemma exp_two_pi_i [simp]: "exp (2 * of_real pi * \<i>) = 1"
by (simp add: exp_eq_polar complex_eq_iff)
lemma exp_two_pi_i' [simp]: "exp (\<i> * (of_real pi * 2)) = 1"
by (metis exp_two_pi_i mult.commute)
lemma continuous_on_cis [continuous_intros]:
"continuous_on A f \<Longrightarrow> continuous_on A (\<lambda>x. cis (f x))"
by (auto simp: cis_conv_exp intro!: continuous_intros)
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
define d where "d = 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)
then have cos: "cos d = 1"
by (simp add: d_def cos_diff)
moreover from cos have "sin d = 0"
by (rule cos_one_sin_zero)
ultimately have "d = 0"
by (auto simp: sin_zero_iff elim!: evenE dest!: less_2_cases)
then show "a = x"
by (simp add: d_def)
qed (simp add: assms del: Re_sgn Im_sgn)
with \<open>z \<noteq> 0\<close> show "arg z = x"
by (simp add: arg_def)
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
define b where "b = (if 0 < r then a else a + pi)"
have b: "sgn z = cis b"
using \<open>r \<noteq> 0\<close> by (simp add: z b_def rcis_def of_real_def sgn_scaleR sgn_if complex_eq_iff)
have cis_2pi_nat: "cis (2 * pi * real_of_nat n) = 1" for n
by (induct n) (simp_all add: distrib_left cis_mult [symmetric] complex_eq_iff)
have cis_2pi_int: "cis (2 * pi * real_of_int x) = 1" for x
by (cases x rule: int_diff_cases)
(simp add: right_diff_distrib cis_divide [symmetric] cis_2pi_nat)
define c where "c = b - 2 * pi * of_int \<lceil>(b - pi) / (2 * pi)\<rceil>"
have "sgn z = cis c"
by (simp add: b c_def 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>Complex n-th roots\<close>
lemma bij_betw_roots_unity:
assumes "n > 0"
shows "bij_betw (\<lambda>k. cis (2 * pi * real k / real n)) {..<n} {z. z ^ n = 1}"
(is "bij_betw ?f _ _")
unfolding bij_betw_def
proof (intro conjI)
show inj: "inj_on ?f {..<n}" unfolding inj_on_def
proof (safe, goal_cases)
case (1 k l)
hence kl: "k < n" "l < n" by simp_all
from 1 have "1 = ?f k / ?f l" by simp
also have "\<dots> = cis (2*pi*(real k - real l)/n)"
using assms by (simp add: field_simps cis_divide)
finally have "cos (2*pi*(real k - real l) / n) = 1"
by (simp add: complex_eq_iff)
then obtain m :: int where "2 * pi * (real k - real l) / real n = real_of_int m * 2 * pi"
by (subst (asm) cos_one_2pi_int) blast
hence "real_of_int (int k - int l) = real_of_int (m * int n)"
unfolding of_int_diff of_int_mult using assms by (simp add: divide_simps)
also note of_int_eq_iff
finally have *: "abs m * n = abs (int k - int l)" by (simp add: abs_mult)
also have "\<dots> < int n" using kl by linarith
finally have "m = 0" using assms by simp
with * show "k = l" by simp
qed
have subset: "?f ` {..<n} \<subseteq> {z. z ^ n = 1}"
proof safe
fix k :: nat
have "cis (2 * pi * real k / real n) ^ n = cis (2 * pi) ^ k"
using assms by (simp add: DeMoivre mult_ac)
also have "cis (2 * pi) = 1" by (simp add: complex_eq_iff)
finally show "?f k ^ n = 1" by simp
qed
have "n = card {..<n}" by simp
also from assms and subset have "\<dots> \<le> card {z::complex. z ^ n = 1}"
by (intro card_inj_on_le[OF inj]) (auto simp: finite_roots_unity)
finally have card: "card {z::complex. z ^ n = 1} = n"
using assms by (intro antisym card_roots_unity) auto
have "card (?f ` {..<n}) = card {z::complex. z ^ n = 1}"
using card inj by (subst card_image) auto
with subset and assms show "?f ` {..<n} = {z::complex. z ^ n = 1}"
by (intro card_subset_eq finite_roots_unity) auto
qed
lemma card_roots_unity_eq:
assumes "n > 0"
shows "card {z::complex. z ^ n = 1} = n"
using bij_betw_same_card [OF bij_betw_roots_unity [OF assms]] by simp
lemma bij_betw_nth_root_unity:
fixes c :: complex and n :: nat
assumes c: "c \<noteq> 0" and n: "n > 0"
defines "c' \<equiv> root n (norm c) * cis (arg c / n)"
shows "bij_betw (\<lambda>z. c' * z) {z. z ^ n = 1} {z. z ^ n = c}"
proof -
have "c' ^ n = of_real (root n (norm c) ^ n) * cis (arg c)"
unfolding of_real_power using n by (simp add: c'_def power_mult_distrib DeMoivre)
also from n have "root n (norm c) ^ n = norm c" by simp
also from c have "of_real \<dots> * cis (arg c) = c" by (simp add: cis_arg Complex.sgn_eq)
finally have [simp]: "c' ^ n = c" .
show ?thesis unfolding bij_betw_def inj_on_def
proof safe
fix z :: complex assume "z ^ n = 1"
hence "(c' * z) ^ n = c' ^ n" by (simp add: power_mult_distrib)
also have "c' ^ n = of_real (root n (norm c) ^ n) * cis (arg c)"
unfolding of_real_power using n by (simp add: c'_def power_mult_distrib DeMoivre)
also from n have "root n (norm c) ^ n = norm c" by simp
also from c have "\<dots> * cis (arg c) = c" by (simp add: cis_arg Complex.sgn_eq)
finally show "(c' * z) ^ n = c" .
next
fix z assume z: "c = z ^ n"
define z' where "z' = z / c'"
from c and n have "c' \<noteq> 0" by (auto simp: c'_def)
with n c have "z = c' * z'" and "z' ^ n = 1"
by (auto simp: z'_def power_divide z)
thus "z \<in> (\<lambda>z. c' * z) ` {z. z ^ n = 1}" by blast
qed (insert c n, auto simp: c'_def)
qed
lemma finite_nth_roots [intro]:
assumes "n > 0"
shows "finite {z::complex. z ^ n = c}"
proof (cases "c = 0")
case True
with assms have "{z::complex. z ^ n = c} = {0}" by auto
thus ?thesis by simp
next
case False
from assms have "finite {z::complex. z ^ n = 1}" by (intro finite_roots_unity) simp_all
also have "?this \<longleftrightarrow> ?thesis"
by (rule bij_betw_finite, rule bij_betw_nth_root_unity) fact+
finally show ?thesis .
qed
lemma card_nth_roots:
assumes "c \<noteq> 0" "n > 0"
shows "card {z::complex. z ^ n = c} = n"
proof -
have "card {z. z ^ n = c} = card {z::complex. z ^ n = 1}"
by (rule sym, rule bij_betw_same_card, rule bij_betw_nth_root_unity) fact+
also have "\<dots> = n" by (rule card_roots_unity_eq) fact+
finally show ?thesis .
qed
lemma sum_roots_unity:
assumes "n > 1"
shows "\<Sum>{z::complex. z ^ n = 1} = 0"
proof -
define \<omega> where "\<omega> = cis (2 * pi / real n)"
have [simp]: "\<omega> \<noteq> 1"
proof
assume "\<omega> = 1"
with assms obtain k :: int where "2 * pi / real n = 2 * pi * of_int k"
by (auto simp: \<omega>_def complex_eq_iff cos_one_2pi_int)
with assms have "real n * of_int k = 1" by (simp add: field_simps)
also have "real n * of_int k = of_int (int n * k)" by simp
also have "1 = (of_int 1 :: real)" by simp
also note of_int_eq_iff
finally show False using assms by (auto simp: zmult_eq_1_iff)
qed
have "(\<Sum>z | z ^ n = 1. z :: complex) = (\<Sum>k<n. cis (2 * pi * real k / real n))"
using assms by (intro sum.reindex_bij_betw [symmetric] bij_betw_roots_unity) auto
also have "\<dots> = (\<Sum>k<n. \<omega> ^ k)"
by (intro sum.cong refl) (auto simp: \<omega>_def DeMoivre mult_ac)
also have "\<dots> = (\<omega> ^ n - 1) / (\<omega> - 1)"
by (subst geometric_sum) auto
also have "\<omega> ^ n - 1 = cis (2 * pi) - 1" using assms by (auto simp: \<omega>_def DeMoivre)
also have "\<dots> = 0" by (simp add: complex_eq_iff)
finally show ?thesis by simp
qed
lemma sum_nth_roots:
assumes "n > 1"
shows "\<Sum>{z::complex. z ^ n = c} = 0"
proof (cases "c = 0")
case True
with assms have "{z::complex. z ^ n = c} = {0}" by auto
also have "\<Sum>\<dots> = 0" by simp
finally show ?thesis .
next
case False
define c' where "c' = root n (norm c) * cis (arg c / n)"
from False and assms have "(\<Sum>{z. z ^ n = c}) = (\<Sum>z | z ^ n = 1. c' * z)"
by (subst sum.reindex_bij_betw [OF bij_betw_nth_root_unity, symmetric])
(auto simp: sum_distrib_left finite_roots_unity c'_def)
also from assms have "\<dots> = 0"
by (simp add: sum_distrib_left [symmetric] sum_roots_unity)
finally show ?thesis .
qed
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 "Im z = 0")
case True
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
case False
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"
by (simp add: power2_eq_iff[symmetric])
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\<^sup>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 cmod_def = norm_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 = \<i> \<longleftrightarrow> x = 0 \<and> y = 1"
and i_mult_Complex: "\<i> * Complex a b = Complex (- b) a"
and Complex_mult_i: "Complex a b * \<i> = Complex (- b) a"
and i_complex_of_real: "\<i> * complex_of_real r = Complex 0 r"
and complex_of_real_i: "complex_of_real r * \<i> = 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 \<and> y = 0)"
and complex_cnj: "cnj (Complex a b) = Complex a (- b)"
and Complex_sum': "sum (\<lambda>x. Complex (f x) 0) s = Complex (sum f s) 0"
and Complex_sum: "Complex (sum f s) 0 = sum (\<lambda>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)
lemma Complex_in_Reals: "Complex x 0 \<in> \<real>"
by (metis Reals_of_real complex_of_real_def)
end