(*  Title:       Complex.thy

    Author:      Jacques D. Fleuriot

    Copyright:   2001 University of Edinburgh

    Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4

*)

header {* Complex Numbers: Rectangular and Polar Representations *}

theory Complex

imports Transcendental

begin

datatype complex = Complex real real

primrec

  Re :: "complex \<Rightarrow> real"

where

  Re: "Re (Complex x y) = x"

primrec

  Im :: "complex \<Rightarrow> real"

where

  Im: "Im (Complex x y) = y"

lemma complex_surj [simp]: "Complex (Re z) (Im z) = z"

  by (induct z) simp

lemma complex_equality [intro?]: "\<lbrakk>Re x = Re y; Im x = Im y\<rbrakk> \<Longrightarrow> x = y"

  by (induct x, induct y) simp

lemma expand_complex_eq: "x = y \<longleftrightarrow> Re x = Re y \<and> Im x = Im y"

  by (induct x, induct y) simp

lemmas complex_Re_Im_cancel_iff = expand_complex_eq

subsection {* Addition and Subtraction *}

instantiation complex :: ab_group_add

begin

definition

  complex_zero_def: "0 = Complex 0 0"

definition

  complex_add_def: "x + y = Complex (Re x + Re y) (Im x + Im y)"

definition

  complex_minus_def: "- x = Complex (- Re x) (- Im x)"

definition

  complex_diff_def: "x - (y\<Colon>complex) = x + - y"

lemma Complex_eq_0 [simp]: "Complex a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"

  by (simp add: complex_zero_def)

lemma complex_Re_zero [simp]: "Re 0 = 0"

  by (simp add: complex_zero_def)

lemma complex_Im_zero [simp]: "Im 0 = 0"

  by (simp add: complex_zero_def)

lemma complex_add [simp]:

  "Complex a b + Complex c d = Complex (a + c) (b + d)"

  by (simp add: complex_add_def)

lemma complex_Re_add [simp]: "Re (x + y) = Re x + Re y"

  by (simp add: complex_add_def)

lemma complex_Im_add [simp]: "Im (x + y) = Im x + Im y"

  by (simp add: complex_add_def)

lemma complex_minus [simp]:

  "- (Complex a b) = Complex (- a) (- b)"

  by (simp add: complex_minus_def)

lemma complex_Re_minus [simp]: "Re (- x) = - Re x"

  by (simp add: complex_minus_def)

lemma complex_Im_minus [simp]: "Im (- x) = - Im x"

  by (simp add: complex_minus_def)

lemma complex_diff [simp]:

  "Complex a b - Complex c d = Complex (a - c) (b - d)"

  by (simp add: complex_diff_def)

lemma complex_Re_diff [simp]: "Re (x - y) = Re x - Re y"

  by (simp add: complex_diff_def)

lemma complex_Im_diff [simp]: "Im (x - y) = Im x - Im y"

  by (simp add: complex_diff_def)

instance

  by intro_classes (simp_all add: complex_add_def complex_diff_def)

end

subsection {* Multiplication and Division *}

instantiation complex :: "{field, division_by_zero}"

begin

definition

  complex_one_def: "1 = Complex 1 0"

definition

  complex_mult_def: "x * y =

    Complex (Re x * Re y - Im x * Im y) (Re x * Im y + Im x * Re y)"

definition

  complex_inverse_def: "inverse x =

    Complex (Re x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)) (- Im x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>))"

definition

  complex_divide_def: "x / (y\<Colon>complex) = x * inverse y"

lemma Complex_eq_1 [simp]: "(Complex a b = 1) = (a = 1 \<and> b = 0)"

  by (simp add: complex_one_def)

lemma complex_Re_one [simp]: "Re 1 = 1"

  by (simp add: complex_one_def)

lemma complex_Im_one [simp]: "Im 1 = 0"

  by (simp add: complex_one_def)

lemma complex_mult [simp]:

  "Complex a b * Complex c d = Complex (a * c - b * d) (a * d + b * c)"

  by (simp add: complex_mult_def)

lemma complex_Re_mult [simp]: "Re (x * y) = Re x * Re y - Im x * Im y"

  by (simp add: complex_mult_def)

lemma complex_Im_mult [simp]: "Im (x * y) = Re x * Im y + Im x * Re y"

  by (simp add: complex_mult_def)

lemma complex_inverse [simp]:

  "inverse (Complex a b) = Complex (a / (a\<twosuperior> + b\<twosuperior>)) (- b / (a\<twosuperior> + b\<twosuperior>))"

  by (simp add: complex_inverse_def)

lemma complex_Re_inverse:

  "Re (inverse x) = Re x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)"

  by (simp add: complex_inverse_def)

lemma complex_Im_inverse:

  "Im (inverse x) = - Im x / ((Re x)\<twosuperior> + (Im x)\<twosuperior>)"

  by (simp add: complex_inverse_def)

instance

  by intro_classes (simp_all add: complex_mult_def

  right_distrib left_distrib right_diff_distrib left_diff_distrib

  complex_inverse_def complex_divide_def

  power2_eq_square add_divide_distrib [symmetric]

  expand_complex_eq)

end

subsection {* Exponentiation *}

instantiation complex :: recpower

begin

primrec power_complex where

  "z ^ 0     = (1\<Colon>complex)"

| "z ^ Suc n = (z\<Colon>complex) * z ^ n"

instance proof

qed simp_all

declare power_complex.simps [simp del]

end

subsection {* Numerals and Arithmetic *}

instantiation complex :: number_ring

begin

definition number_of_complex where

  complex_number_of_def: "number_of w = (of_int w \<Colon> complex)"

instance

  by intro_classes (simp only: complex_number_of_def)

end

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_number_of [simp]: "Re (number_of v) = number_of v"

unfolding number_of_eq by (rule complex_Re_of_int)

lemma complex_Im_number_of [simp]: "Im (number_of v) = 0"

unfolding number_of_eq by (rule complex_Im_of_int)

lemma Complex_eq_number_of [simp]:

  "(Complex a b = number_of w) = (a = number_of w \<and> b = 0)"

by (simp add: expand_complex_eq)

subsection {* Scalar Multiplication *}

instantiation complex :: real_field

begin

definition

  complex_scaleR_def: "scaleR r x = Complex (r * Re x) (r * Im x)"

lemma complex_scaleR [simp]:

  "scaleR r (Complex a b) = Complex (r * a) (r * b)"

  unfolding complex_scaleR_def by simp

lemma complex_Re_scaleR [simp]: "Re (scaleR r x) = r * Re x"

  unfolding complex_scaleR_def by simp

lemma complex_Im_scaleR [simp]: "Im (scaleR r x) = r * Im x"

  unfolding complex_scaleR_def by simp

instance

proof

  fix a b :: real and x y :: complex

  show "scaleR a (x + y) = scaleR a x + scaleR a y"

    by (simp add: expand_complex_eq right_distrib)

  show "scaleR (a + b) x = scaleR a x + scaleR b x"

    by (simp add: expand_complex_eq left_distrib)

  show "scaleR a (scaleR b x) = scaleR (a * b) x"

    by (simp add: expand_complex_eq mult_assoc)

  show "scaleR 1 x = x"

    by (simp add: expand_complex_eq)

  show "scaleR a x * y = scaleR a (x * y)"

    by (simp add: expand_complex_eq algebra_simps)

  show "x * scaleR a y = scaleR a (x * y)"

    by (simp add: expand_complex_eq algebra_simps)

qed

end

subsection{* Properties of Embedding from Reals *}

abbreviation

  complex_of_real :: "real \<Rightarrow> complex" where

    "complex_of_real \<equiv> of_real"

lemma complex_of_real_def: "complex_of_real r = Complex r 0"

by (simp add: of_real_def complex_scaleR_def)

lemma Re_complex_of_real [simp]: "Re (complex_of_real z) = z"

by (simp add: complex_of_real_def)

lemma Im_complex_of_real [simp]: "Im (complex_of_real z) = 0"

by (simp add: complex_of_real_def)

lemma Complex_add_complex_of_real [simp]:

     "Complex x y + complex_of_real r = Complex (x+r) y"

by (simp add: complex_of_real_def)

lemma complex_of_real_add_Complex [simp]:

     "complex_of_real r + Complex x y = Complex (r+x) y"

by (simp add: complex_of_real_def)

lemma Complex_mult_complex_of_real:

     "Complex x y * complex_of_real r = Complex (x*r) (y*r)"

by (simp add: complex_of_real_def)

lemma complex_of_real_mult_Complex:

     "complex_of_real r * Complex x y = Complex (r*x) (r*y)"

by (simp add: complex_of_real_def)

subsection {* Vector Norm *}

instantiation complex :: real_normed_field

begin

definition

  complex_norm_def: "norm z = sqrt ((Re z)\<twosuperior> + (Im z)\<twosuperior>)"

abbreviation

  cmod :: "complex \<Rightarrow> real" where

  "cmod \<equiv> norm"

definition

  complex_sgn_def: "sgn x = x /\<^sub>R cmod x"

lemmas cmod_def = complex_norm_def

lemma complex_norm [simp]: "cmod (Complex x y) = sqrt (x\<twosuperior> + y\<twosuperior>)"

  by (simp add: complex_norm_def)

instance

proof

  fix r :: real and x y :: complex

  show "0 \<le> norm x"

    by (induct x) simp

  show "(norm x = 0) = (x = 0)"

    by (induct x) simp

  show "norm (x + y) \<le> norm x + norm y"

    by (induct x, induct y)

       (simp add: real_sqrt_sum_squares_triangle_ineq)

  show "norm (scaleR r x) = \<bar>r\<bar> * norm x"

    by (induct x)

       (simp add: power_mult_distrib right_distrib [symmetric] real_sqrt_mult)

  show "norm (x * y) = norm x * norm y"

    by (induct x, induct y)

       (simp add: real_sqrt_mult [symmetric] power2_eq_square algebra_simps)

  show "sgn x = x /\<^sub>R cmod x" by(simp add: complex_sgn_def)

qed

end

lemma cmod_unit_one [simp]: "cmod (Complex (cos a) (sin a)) = 1"

by simp

lemma cmod_complex_polar [simp]:

     "cmod (complex_of_real r * Complex (cos a) (sin a)) = abs r"

by (simp add: norm_mult)

lemma complex_Re_le_cmod: "Re x \<le> cmod x"

unfolding complex_norm_def

by (rule real_sqrt_sum_squares_ge1)

lemma complex_mod_minus_le_complex_mod [simp]: "- cmod x \<le> cmod x"

by (rule order_trans [OF _ norm_ge_zero], simp)

lemma complex_mod_triangle_ineq2 [simp]: "cmod(b + a) - cmod b \<le> cmod a"

by (rule ord_le_eq_trans [OF norm_triangle_ineq2], simp)

lemmas real_sum_squared_expand = power2_sum [where 'a=real]

lemma abs_Re_le_cmod: "\<bar>Re x\<bar> \<le> cmod x"

by (cases x) simp

lemma abs_Im_le_cmod: "\<bar>Im x\<bar> \<le> cmod x"

by (cases x) simp

subsection {* Completeness of the Complexes *}

interpretation Re: bounded_linear "Re"

apply (unfold_locales, simp, simp)

apply (rule_tac x=1 in exI)

apply (simp add: complex_norm_def)

done

interpretation Im: bounded_linear "Im"

apply (unfold_locales, simp, simp)

apply (rule_tac x=1 in exI)

apply (simp add: complex_norm_def)

done

lemma LIMSEQ_Complex:

  "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. Complex (X n) (Y n)) ----> Complex a b"

apply (rule LIMSEQ_I)

apply (subgoal_tac "0 < r / sqrt 2")

apply (drule_tac r="r / sqrt 2" in LIMSEQ_D, safe)

apply (drule_tac r="r / sqrt 2" in LIMSEQ_D, safe)

apply (rename_tac M N, rule_tac x="max M N" in exI, safe)

apply (simp add: real_sqrt_sum_squares_less)

apply (simp add: divide_pos_pos)

done

instance complex :: banach

proof

  fix X :: "nat \<Rightarrow> complex"

  assume X: "Cauchy X"

  from Re.Cauchy [OF X] have 1: "(\<lambda>n. Re (X n)) ----> lim (\<lambda>n. Re (X n))"

    by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)

  from Im.Cauchy [OF X] have 2: "(\<lambda>n. Im (X n)) ----> lim (\<lambda>n. Im (X n))"

    by (simp add: Cauchy_convergent_iff convergent_LIMSEQ_iff)

  have "X ----> Complex (lim (\<lambda>n. Re (X n))) (lim (\<lambda>n. Im (X n)))"

    using LIMSEQ_Complex [OF 1 2] by simp

  thus "convergent X"

    by (rule convergentI)

qed

subsection {* The Complex Number @{term "\<i>"} *}

definition

  "ii" :: complex  ("\<i>") where

  i_def: "ii \<equiv> Complex 0 1"

lemma complex_Re_i [simp]: "Re ii = 0"

by (simp add: i_def)

lemma complex_Im_i [simp]: "Im ii = 1"

by (simp add: i_def)

lemma Complex_eq_i [simp]: "(Complex x y = ii) = (x = 0 \<and> y = 1)"

by (simp add: i_def)

lemma complex_i_not_zero [simp]: "ii \<noteq> 0"

by (simp add: expand_complex_eq)

lemma complex_i_not_one [simp]: "ii \<noteq> 1"

by (simp add: expand_complex_eq)

lemma complex_i_not_number_of [simp]: "ii \<noteq> number_of w"

by (simp add: expand_complex_eq)

lemma i_mult_Complex [simp]: "ii * Complex a b = Complex (- b) a"

by (simp add: expand_complex_eq)

lemma Complex_mult_i [simp]: "Complex a b * ii = Complex (- b) a"

by (simp add: expand_complex_eq)

lemma i_complex_of_real [simp]: "ii * complex_of_real r = Complex 0 r"

by (simp add: i_def complex_of_real_def)

lemma complex_of_real_i [simp]: "complex_of_real r * ii = Complex 0 r"

by (simp add: i_def complex_of_real_def)

lemma i_squared [simp]: "ii * ii = -1"

by (simp add: i_def)

lemma power2_i [simp]: "ii\<twosuperior> = -1"

by (simp add: power2_eq_square)

lemma inverse_i [simp]: "inverse ii = - ii"

by (rule inverse_unique, simp)

subsection {* Complex Conjugation *}

definition

  cnj :: "complex \<Rightarrow> complex" where

  "cnj z = Complex (Re z) (- Im z)"

lemma complex_cnj [simp]: "cnj (Complex a b) = Complex a (- b)"

by (simp add: cnj_def)

lemma complex_Re_cnj [simp]: "Re (cnj x) = Re x"

by (simp add: cnj_def)

lemma complex_Im_cnj [simp]: "Im (cnj x) = - Im x"

by (simp add: cnj_def)

lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)"

by (simp add: expand_complex_eq)

lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z"

by (simp add: cnj_def)

lemma complex_cnj_zero [simp]: "cnj 0 = 0"

by (simp add: expand_complex_eq)

lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)"

by (simp add: expand_complex_eq)

lemma complex_cnj_add: "cnj (x + y) = cnj x + cnj y"

by (simp add: expand_complex_eq)

lemma complex_cnj_diff: "cnj (x - y) = cnj x - cnj y"

by (simp add: expand_complex_eq)

lemma complex_cnj_minus: "cnj (- x) = - cnj x"

by (simp add: expand_complex_eq)

lemma complex_cnj_one [simp]: "cnj 1 = 1"

by (simp add: expand_complex_eq)

lemma complex_cnj_mult: "cnj (x * y) = cnj x * cnj y"

by (simp add: expand_complex_eq)

lemma complex_cnj_inverse: "cnj (inverse x) = inverse (cnj x)"

by (simp add: complex_inverse_def)

lemma complex_cnj_divide: "cnj (x / y) = cnj x / cnj y"

by (simp add: complex_divide_def complex_cnj_mult complex_cnj_inverse)

lemma complex_cnj_power: "cnj (x ^ n) = cnj x ^ n"

by (induct n, simp_all add: complex_cnj_mult)

lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n"

by (simp add: expand_complex_eq)

lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z"

by (simp add: expand_complex_eq)

lemma complex_cnj_number_of [simp]: "cnj (number_of w) = number_of w"

by (simp add: expand_complex_eq)

lemma complex_cnj_scaleR: "cnj (scaleR r x) = scaleR r (cnj x)"

by (simp add: expand_complex_eq)

lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z"

by (simp add: complex_norm_def)

lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x"

by (simp add: expand_complex_eq)

lemma complex_cnj_i [simp]: "cnj ii = - ii"

by (simp add: expand_complex_eq)

lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)"

by (simp add: expand_complex_eq)

lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im z) * ii"

by (simp add: expand_complex_eq)

lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\<twosuperior> + (Im z)\<twosuperior>)"

by (simp add: expand_complex_eq power2_eq_square)

lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<twosuperior>"

by (simp add: norm_mult power2_eq_square)

interpretation cnj: bounded_linear "cnj"

apply (unfold_locales)

apply (rule complex_cnj_add)

apply (rule complex_cnj_scaleR)

apply (rule_tac x=1 in exI, simp)

done

subsection{*The Functions @{term sgn} and @{term arg}*}

text {*------------ Argand -------------*}

definition

  arg :: "complex => real" where

  "arg z = (SOME a. Re(sgn z) = cos a & Im(sgn z) = sin a & -pi < a & a \<le> pi)"

lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)"

by (simp add: complex_sgn_def divide_inverse scaleR_conv_of_real mult_commute)

lemma i_mult_eq: "ii * ii = complex_of_real (-1)"

by (simp add: i_def complex_of_real_def)

lemma i_mult_eq2 [simp]: "ii * ii = -(1::complex)"

by (simp add: i_def complex_one_def)

lemma complex_eq_cancel_iff2 [simp]:

     "(Complex x y = complex_of_real xa) = (x = xa & y = 0)"

by (simp add: complex_of_real_def)

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)

lemma complex_inverse_complex_split:

     "inverse(complex_of_real x + ii * complex_of_real y) =

      complex_of_real(x/(x ^ 2 + y ^ 2)) -

      ii * complex_of_real(y/(x ^ 2 + y ^ 2))"

by (simp add: complex_of_real_def i_def diff_minus divide_inverse)

(*----------------------------------------------------------------------------*)

(* Many of the theorems below need to be moved elsewhere e.g. Transc. Also *)

(* many of the theorems are not used - so should they be kept?                *)

(*----------------------------------------------------------------------------*)

lemma cos_arg_i_mult_zero_pos:

   "0 < y ==> cos (arg(Complex 0 y)) = 0"

apply (simp add: arg_def abs_if)

apply (rule_tac a = "pi/2" in someI2, auto)

apply (rule order_less_trans [of _ 0], auto)

done

lemma cos_arg_i_mult_zero_neg:

   "y < 0 ==> cos (arg(Complex 0 y)) = 0"

apply (simp add: arg_def abs_if)

apply (rule_tac a = "- pi/2" in someI2, auto)

apply (rule order_trans [of _ 0], auto)

done

lemma cos_arg_i_mult_zero [simp]:

     "y \<noteq> 0 ==> cos (arg(Complex 0 y)) = 0"

by (auto simp add: linorder_neq_iff cos_arg_i_mult_zero_pos cos_arg_i_mult_zero_neg)

subsection{*Finally! Polar Form for Complex Numbers*}

definition

  (* abbreviation for (cos a + i sin a) *)

  cis :: "real => complex" where

  "cis a = Complex (cos a) (sin a)"

definition

  (* abbreviation for r*(cos a + i sin a) *)

  rcis :: "[real, real] => complex" where

  "rcis r a = complex_of_real r * cis a"

definition

  (* e ^ (x + iy) *)

  expi :: "complex => complex" where

  "expi z = complex_of_real(exp (Re z)) * cis (Im z)"

lemma complex_split_polar:

     "\<exists>r a. z = complex_of_real r * (Complex (cos a) (sin a))"

apply (induct z)

apply (auto simp add: polar_Ex complex_of_real_mult_Complex)

done

lemma rcis_Ex: "\<exists>r a. z = rcis r a"

apply (induct z)

apply (simp add: rcis_def cis_def polar_Ex complex_of_real_mult_Complex)

done

lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a"

by (simp add: rcis_def cis_def)

lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a"

by (simp add: rcis_def cis_def)

lemma sin_cos_squared_add2_mult: "(r * cos a)\<twosuperior> + (r * sin a)\<twosuperior> = r\<twosuperior>"

proof -

  have "(r * cos a)\<twosuperior> + (r * sin a)\<twosuperior> = r\<twosuperior> * ((cos a)\<twosuperior> + (sin a)\<twosuperior>)"

    by (simp only: power_mult_distrib right_distrib)

  thus ?thesis by simp

qed

lemma complex_mod_rcis [simp]: "cmod(rcis r a) = abs r"

by (simp add: rcis_def cis_def sin_cos_squared_add2_mult)

lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))"

by (simp add: cmod_def power2_eq_square)

lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0"

by simp

(*---------------------------------------------------------------------------*)

(*  (r1 * cis a) * (r2 * cis b) = r1 * r2 * cis (a + b)                      *)

(*---------------------------------------------------------------------------*)

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_def cos_add sin_add right_distrib right_diff_distrib

              complex_of_real_def)

lemma cis_mult: "cis a * cis b = cis (a + b)"

by (simp add: cis_rcis_eq rcis_mult)

lemma cis_zero [simp]: "cis 0 = 1"

by (simp add: cis_def complex_one_def)

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 complex_of_real_minus_one:

   "complex_of_real (-(1::real)) = -(1::complex)"

by (simp add: complex_of_real_def complex_one_def)

lemma complex_i_mult_minus [simp]: "ii * (ii * x) = - x"

by (simp add: mult_assoc [symmetric])

lemma cis_real_of_nat_Suc_mult:

   "cis (real (Suc n) * a) = cis a * cis (real n * a)"

by (simp add: cis_def real_of_nat_Suc left_distrib cos_add sin_add right_distrib)

lemma DeMoivre: "(cis a) ^ n = cis (real n * a)"

apply (induct_tac "n")

apply (auto simp add: cis_real_of_nat_Suc_mult)

done

lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)"

by (simp add: rcis_def power_mult_distrib DeMoivre)

lemma cis_inverse [simp]: "inverse(cis a) = cis (-a)"

by (simp add: cis_def complex_inverse_complex_split diff_minus)

lemma rcis_inverse: "inverse(rcis r a) = rcis (1/r) (-a)"

by (simp add: divide_inverse rcis_def)

lemma cis_divide: "cis a / cis b = cis (a - b)"

by (simp add: complex_divide_def cis_mult real_diff_def)

lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1/r2) (a - b)"

apply (simp add: complex_divide_def)

apply (case_tac "r2=0", simp)

apply (simp add: rcis_inverse rcis_mult real_diff_def)

done

lemma Re_cis [simp]: "Re(cis a) = cos a"

by (simp add: cis_def)

lemma Im_cis [simp]: "Im(cis a) = sin a"

by (simp add: cis_def)

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 expi_add: "expi(a + b) = expi(a) * expi(b)"

by (simp add: expi_def exp_add cis_mult [symmetric] mult_ac)

lemma expi_zero [simp]: "expi (0::complex) = 1"

by (simp add: expi_def)

lemma complex_expi_Ex: "\<exists>a r. z = complex_of_real r * expi a"

apply (insert rcis_Ex [of z])

apply (auto simp add: expi_def rcis_def mult_assoc [symmetric])

apply (rule_tac x = "ii * complex_of_real a" in exI, auto)

done

lemma expi_two_pi_i [simp]: "expi((2::complex) * complex_of_real pi * ii) = 1"

by (simp add: expi_def cis_def)

end