src/HOL/Complex/Complex.thy

remove redundant lemmas

(*  Title:       Complex.thy
    ID:      $Id$
    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 "../Hyperreal/Transcendental"
begin

datatype complex = Complex real real

instance complex :: "{zero, one, plus, times, minus, inverse, power}" ..

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

consts Re :: "complex => real"
primrec Re: "Re (Complex x y) = x"

consts Im :: "complex => real"
primrec Im: "Im (Complex x y) = y"

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

defs (overloaded)

  complex_zero_def:
  "0 == Complex 0 0"

  complex_one_def:
  "1 == Complex 1 0"

  i_def: "ii == Complex 0 1"

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

  complex_inverse_def:
   "inverse z ==
    Complex (Re z / ((Re z)\<twosuperior> + (Im z)\<twosuperior>)) (- Im z / ((Re z)\<twosuperior> + (Im z)\<twosuperior>))"

  complex_add_def:
    "z + w == Complex (Re z + Re w) (Im z + Im w)"

  complex_diff_def:
    "z - w == z + - (w::complex)"

  complex_mult_def:
    "z * w == Complex (Re z * Re w - Im z * Im w) (Re z * Im w + Im z * Re w)"

  complex_divide_def: "w / (z::complex) == w * inverse z"


lemma complex_equality [intro?]: "Re z = Re w ==> Im z = Im w ==> z = w"
  by (induct z, induct w) simp

lemma complex_Re_Im_cancel_iff: "(w=z) = (Re(w) = Re(z) & Im(w) = Im(z))"
by (induct w, induct z, simp)

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_zero_iff [simp]: "(Complex x y = 0) = (x = 0 \<and> y = 0)"
unfolding complex_zero_def by simp

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_Re_i [simp]: "Re(ii) = 0"
by (simp add: i_def)

lemma complex_Im_i [simp]: "Im(ii) = 1"
by (simp add: i_def)


subsection{*Unary Minus*}

lemma complex_minus [simp]: "- (Complex x y) = Complex (-x) (-y)"
by (simp add: complex_minus_def)

lemma complex_Re_minus [simp]: "Re (-z) = - Re z"
by (simp add: complex_minus_def)

lemma complex_Im_minus [simp]: "Im (-z) = - Im z"
by (simp add: complex_minus_def)


subsection{*Addition*}

lemma complex_add [simp]:
     "Complex x1 y1 + Complex x2 y2 = Complex (x1+x2) (y1+y2)"
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_add_commute: "(u::complex) + v = v + u"
by (simp add: complex_add_def add_commute)

lemma complex_add_assoc: "((u::complex) + v) + w = u + (v + w)"
by (simp add: complex_add_def add_assoc)

lemma complex_add_zero_left: "(0::complex) + z = z"
by (simp add: complex_add_def complex_zero_def)

lemma complex_add_zero_right: "z + (0::complex) = z"
by (simp add: complex_add_def complex_zero_def)

lemma complex_add_minus_left: "-z + z = (0::complex)"
by (simp add: complex_add_def complex_minus_def complex_zero_def)

lemma complex_diff:
      "Complex x1 y1 - Complex x2 y2 = Complex (x1-x2) (y1-y2)"
by (simp add: complex_add_def complex_minus_def 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)


subsection{*Multiplication*}

lemma complex_mult [simp]:
     "Complex x1 y1 * Complex x2 y2 = Complex (x1*x2 - y1*y2) (x1*y2 + y1*x2)"
by (simp add: complex_mult_def)

lemma complex_mult_commute: "(w::complex) * z = z * w"
by (simp add: complex_mult_def mult_commute add_commute)

lemma complex_mult_assoc: "((u::complex) * v) * w = u * (v * w)"
by (simp add: complex_mult_def mult_ac add_ac
              right_diff_distrib right_distrib left_diff_distrib left_distrib)

lemma complex_mult_one_left: "(1::complex) * z = z"
by (simp add: complex_mult_def complex_one_def)

lemma complex_mult_one_right: "z * (1::complex) = z"
by (simp add: complex_mult_def complex_one_def)


subsection{*Inverse*}

lemma complex_inverse [simp]:
     "inverse (Complex x y) = Complex (x/(x ^ 2 + y ^ 2)) (-y/(x ^ 2 + y ^ 2))"
by (simp add: complex_inverse_def)

lemma complex_mult_inv_left: "z \<noteq> (0::complex) ==> inverse(z) * z = 1"
apply (induct z)
apply (rename_tac x y)
apply (auto simp add:
             complex_one_def complex_zero_def add_divide_distrib [symmetric] 
             power2_eq_square mult_ac)
apply (simp_all add: real_sum_squares_not_zero real_sum_squares_not_zero2) 
done


subsection {* The field of complex numbers *}

instance complex :: field
proof
  fix z u v w :: complex
  show "(u + v) + w = u + (v + w)"
    by (rule complex_add_assoc)
  show "z + w = w + z"
    by (rule complex_add_commute)
  show "0 + z = z"
    by (rule complex_add_zero_left)
  show "-z + z = 0"
    by (rule complex_add_minus_left)
  show "z - w = z + -w"
    by (simp add: complex_diff_def)
  show "(u * v) * w = u * (v * w)"
    by (rule complex_mult_assoc)
  show "z * w = w * z"
    by (rule complex_mult_commute)
  show "1 * z = z"
    by (rule complex_mult_one_left)
  show "0 \<noteq> (1::complex)"
    by (simp add: complex_zero_def complex_one_def)
  show "(u + v) * w = u * w + v * w"
    by (simp add: complex_mult_def complex_add_def left_distrib 
                  diff_minus add_ac)
  show "z / w = z * inverse w"
    by (simp add: complex_divide_def)
  assume "w \<noteq> 0"
  thus "inverse w * w = 1"
    by (simp add: complex_mult_inv_left)
qed

instance complex :: division_by_zero
proof
  show "inverse 0 = (0::complex)"
    by (simp add: complex_inverse_def complex_zero_def)
qed


subsection{*The real algebra of complex numbers*}

instance complex :: scaleR ..

defs (overloaded)
  complex_scaleR_def: "r *# x == Complex r 0 * x"

instance complex :: real_field
proof
  fix a b :: real
  fix x y :: complex
  show "a *# (x + y) = a *# x + a *# y"
    by (simp add: complex_scaleR_def right_distrib)
  show "(a + b) *# x = a *# x + b *# x"
    by (simp add: complex_scaleR_def left_distrib [symmetric])
  show "a *# b *# x = (a * b) *# x"
    by (simp add: complex_scaleR_def mult_assoc [symmetric])
  show "1 *# x = x"
    by (simp add: complex_scaleR_def complex_one_def [symmetric])
  show "a *# x * y = a *# (x * y)"
    by (simp add: complex_scaleR_def mult_assoc)
  show "x * a *# y = a *# (x * y)"
    by (simp add: complex_scaleR_def mult_left_commute)
qed


subsection{*Embedding Properties for @{term complex_of_real} Map*}

abbreviation
  complex_of_real :: "real => complex" where
  "complex_of_real == 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: i_def 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: i_def complex_of_real_def)

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)


subsection{*The Functions @{term Re} and @{term Im}*}

lemma complex_Re_mult_eq: "Re (w * z) = Re w * Re z - Im w * Im z"
by (induct z, induct w, simp)

lemma complex_Im_mult_eq: "Im (w * z) = Re w * Im z + Im w * Re z"
by (induct z, induct w, simp)

lemma Re_i_times [simp]: "Re(ii * z) = - Im z"
by (simp add: complex_Re_mult_eq)

lemma Re_times_i [simp]: "Re(z * ii) = - Im z"
by (simp add: complex_Re_mult_eq)

lemma Im_i_times [simp]: "Im(ii * z) = Re z"
by (simp add: complex_Im_mult_eq)

lemma Im_times_i [simp]: "Im(z * ii) = Re z"
by (simp add: complex_Im_mult_eq)

lemma complex_Re_mult: "[| Im w = 0; Im z = 0 |] ==> Re(w * z) = Re(w) * Re(z)"
by (simp add: complex_Re_mult_eq)

lemma complex_Re_mult_complex_of_real [simp]:
     "Re (z * complex_of_real c) = Re(z) * c"
by (simp add: complex_Re_mult_eq)

lemma complex_Im_mult_complex_of_real [simp]:
     "Im (z * complex_of_real c) = Im(z) * c"
by (simp add: complex_Im_mult_eq)

lemma complex_Re_mult_complex_of_real2 [simp]:
     "Re (complex_of_real c * z) = c * Re(z)"
by (simp add: complex_Re_mult_eq)

lemma complex_Im_mult_complex_of_real2 [simp]:
     "Im (complex_of_real c * z) = c * Im(z)"
by (simp add: complex_Im_mult_eq)


subsection{*Conjugation is an Automorphism*}

definition
  cnj :: "complex => complex" where
  "cnj z = Complex (Re z) (-Im z)"

lemma complex_cnj: "cnj (Complex x y) = Complex x (-y)"
by (simp add: cnj_def)

lemma complex_cnj_cancel_iff [simp]: "(cnj x = cnj y) = (x = y)"
by (simp add: cnj_def complex_Re_Im_cancel_iff)

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

lemma complex_cnj_complex_of_real [simp]:
     "cnj (complex_of_real x) = complex_of_real x"
by (simp add: complex_of_real_def complex_cnj)

lemma complex_cnj_minus: "cnj (-z) = - cnj z"
by (simp add: cnj_def)

lemma complex_cnj_inverse: "cnj(inverse z) = inverse(cnj z)"
by (induct z, simp add: complex_cnj power2_eq_square)

lemma complex_cnj_add: "cnj(w + z) = cnj(w) + cnj(z)"
by (induct w, induct z, simp add: complex_cnj)

lemma complex_cnj_diff: "cnj(w - z) = cnj(w) - cnj(z)"
by (simp add: diff_minus complex_cnj_add complex_cnj_minus)

lemma complex_cnj_mult: "cnj(w * z) = cnj(w) * cnj(z)"
by (induct w, induct z, simp add: complex_cnj)

lemma complex_cnj_divide: "cnj(w / z) = (cnj w)/(cnj z)"
by (simp add: complex_divide_def complex_cnj_mult complex_cnj_inverse)

lemma complex_cnj_one [simp]: "cnj 1 = 1"
by (simp add: cnj_def complex_one_def)

lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re(z))"
by (induct z, simp add: complex_cnj complex_of_real_def)

lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im(z)) * ii"
apply (induct z)
apply (simp add: complex_add complex_cnj complex_of_real_def diff_minus
                 complex_minus i_def complex_mult)
done

lemma complex_cnj_zero [simp]: "cnj 0 = 0"
by (simp add: cnj_def complex_zero_def)

lemma complex_cnj_zero_iff [iff]: "(cnj z = 0) = (z = 0)"
by (induct z, simp add: complex_zero_def complex_cnj)

lemma complex_mult_cnj: "z * cnj z = complex_of_real (Re(z) ^ 2 + Im(z) ^ 2)"
by (induct z, simp add: complex_cnj complex_of_real_def power2_eq_square)


subsection{*Modulus*}

instance complex :: norm
  complex_norm_def: "norm z \<equiv> sqrt ((Re z)\<twosuperior> + (Im z)\<twosuperior>)" ..

abbreviation
  cmod :: "complex \<Rightarrow> real" where
  "cmod \<equiv> norm"

lemmas cmod_def = complex_norm_def

lemma complex_mod [simp]: "cmod (Complex x y) = sqrt (x\<twosuperior> + y\<twosuperior>)"
by (simp add: cmod_def)

lemma complex_mod_triangle_ineq [simp]: "cmod (x + y) \<le> cmod x + cmod y"
apply (simp add: cmod_def)
apply (rule real_sqrt_sum_squares_triangle_ineq)
done

lemma complex_mod_mult: "cmod (x * y) = cmod x * cmod y"
apply (induct x, induct y)
apply (simp add: real_sqrt_mult_distrib [symmetric])
apply (rule_tac f=sqrt in arg_cong)
apply (simp add: power2_sum power2_diff power_mult_distrib ring_distrib)
done

lemma complex_mod_complex_of_real: "cmod (complex_of_real x) = \<bar>x\<bar>"
by (simp add: complex_of_real_def)

lemma complex_norm_scaleR:
  "norm (scaleR a x) = \<bar>a\<bar> * norm (x::complex)"
unfolding scaleR_conv_of_real
by (simp only: complex_mod_mult complex_mod_complex_of_real)

instance complex :: real_normed_field
proof
  fix r :: real
  fix x y :: complex
  show "0 \<le> cmod x"
    by (induct x) simp
  show "(cmod x = 0) = (x = 0)"
    by (induct x) simp
  show "cmod (x + y) \<le> cmod x + cmod y"
    by (rule complex_mod_triangle_ineq)
  show "cmod (scaleR r x) = \<bar>r\<bar> * cmod x"
    by (rule complex_norm_scaleR)
  show "cmod (x * y) = cmod x * cmod y"
    by (rule complex_mod_mult)
qed

lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z"
by (induct z, simp add: complex_cnj)

lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<twosuperior>"
by (simp add: complex_mod_mult power2_eq_square)

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"
apply (simp only: cmod_unit_one complex_mod_mult)
apply (simp add: complex_mod_complex_of_real)
done

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]


subsection{*Exponentiation*}

primrec
     complexpow_0:   "z ^ 0       = 1"
     complexpow_Suc: "z ^ (Suc n) = (z::complex) * (z ^ n)"


instance complex :: recpower
proof
  fix z :: complex
  fix n :: nat
  show "z^0 = 1" by simp
  show "z^(Suc n) = z * (z^n)" by simp
qed

lemma complex_cnj_pow: "cnj(z ^ n) = cnj(z) ^ n"
apply (induct_tac "n")
apply (auto simp add: complex_cnj_mult)
done

lemma complexpow_i_squared [simp]: "ii ^ 2 = -(1::complex)"
by (simp add: i_def complex_one_def numeral_2_eq_2)

lemma complex_i_not_zero [simp]: "ii \<noteq> 0"
by (simp add: i_def complex_zero_def)


subsection{*The Function @{term sgn}*}

definition
  (*------------ Argand -------------*)

  sgn :: "complex => complex" where
  "sgn z = z / complex_of_real(cmod z)"

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_zero [simp]: "sgn 0 = 0"
by (simp add: sgn_def)

lemma sgn_one [simp]: "sgn 1 = 1"
by (simp add: sgn_def)

lemma sgn_minus: "sgn (-z) = - sgn(z)"
by (simp add: sgn_def)

lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)"
by (simp add: sgn_def)

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 Complex_eq_0 [simp]: "(Complex x y = 0) = (x = 0 & y = 0)"
by (simp add: complex_zero_def)

lemma Complex_eq_1 [simp]: "(Complex x y = 1) = (x = 1 & y = 0)"
by (simp add: complex_one_def)

lemma Complex_eq_i [simp]: "(Complex x y = ii) = (x = 0 & y = 1)"
by (simp add: i_def)



lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z"
proof (induct z)
  case (Complex x y)
    have "sqrt (x\<twosuperior> + y\<twosuperior>) * inverse (x\<twosuperior> + y\<twosuperior>) = inverse (sqrt (x\<twosuperior> + y\<twosuperior>))"
      by (simp add: divide_inverse [symmetric] sqrt_divide_self_eq)
    thus "Re (sgn (Complex x y)) = Re (Complex x y) /cmod (Complex x y)"
       by (simp add: sgn_def complex_of_real_def divide_inverse)
qed


lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z"
proof (induct z)
  case (Complex x y)
    have "sqrt (x\<twosuperior> + y\<twosuperior>) * inverse (x\<twosuperior> + y\<twosuperior>) = inverse (sqrt (x\<twosuperior> + y\<twosuperior>))"
      by (simp add: divide_inverse [symmetric] sqrt_divide_self_eq)
    thus "Im (sgn (Complex x y)) = Im (Complex x y) /cmod (Complex x y)"
       by (simp add: sgn_def complex_of_real_def divide_inverse)
qed

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))"
apply (simp add: cmod_def)
apply (rule real_sqrt_eq_iff [THEN iffD2])
apply (auto simp add: complex_mult_cnj
            simp del: of_real_add)
done

lemma complex_Re_cnj [simp]: "Re(cnj z) = Re z"
by (induct z, simp add: complex_cnj)

lemma complex_Im_cnj [simp]: "Im(cnj z) = - Im z"
by (induct z, simp add: complex_cnj)

lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0"
by (induct z, simp add: complex_cnj complex_mult)


(*---------------------------------------------------------------------------*)
(*  (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: complex_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 complex_mult_assoc [symmetric])
apply (rule_tac x = "ii * complex_of_real a" in exI, auto)
done


subsection{*Numerals and Arithmetic*}

instance complex :: number ..

defs (overloaded)
  complex_number_of_def: "(number_of w :: complex) == of_int w"
    --{*the type constraint is essential!*}

instance complex :: number_ring
by (intro_classes, simp add: complex_number_of_def)

lemma complex_number_of: "complex_of_real (number_of w) = number_of w"
by (rule of_real_number_of_eq)

lemma complex_number_of_cnj [simp]: "cnj(number_of v :: complex) = number_of v"
by (simp only: complex_number_of [symmetric] complex_cnj_complex_of_real)

lemma complex_number_of_cmod: 
      "cmod(number_of v :: complex) = abs (number_of v :: real)"
by (simp only: complex_number_of [symmetric] complex_mod_complex_of_real)

lemma complex_number_of_Re [simp]: "Re(number_of v :: complex) = number_of v"
by (simp only: complex_number_of [symmetric] Re_complex_of_real)

lemma complex_number_of_Im [simp]: "Im(number_of v :: complex) = 0"
by (simp only: complex_number_of [symmetric] Im_complex_of_real)

lemma expi_two_pi_i [simp]: "expi((2::complex) * complex_of_real pi * ii) = 1"
by (simp add: expi_def complex_Re_mult_eq complex_Im_mult_eq cis_def)


(*examples:
print_depth 22
set timing;
set trace_simp;
fun test s = (Goal s, by (Simp_tac 1)); 

test "23 * ii + 45 * ii= (x::complex)";

test "5 * ii + 12 - 45 * ii= (x::complex)";
test "5 * ii + 40 - 12 * ii + 9 = (x::complex) + 89 * ii";
test "5 * ii + 40 - 12 * ii + 9 - 78 = (x::complex) + 89 * ii";

test "l + 10 * ii + 90 + 3*l +  9 + 45 * ii= (x::complex)";
test "87 + 10 * ii + 90 + 3*7 +  9 + 45 * ii= (x::complex)";


fun test s = (Goal s; by (Asm_simp_tac 1)); 

test "x*k = k*(y::complex)";
test "k = k*(y::complex)"; 
test "a*(b*c) = (b::complex)";
test "a*(b*c) = d*(b::complex)*(x*a)";


test "(x*k) / (k*(y::complex)) = (uu::complex)";
test "(k) / (k*(y::complex)) = (uu::complex)"; 
test "(a*(b*c)) / ((b::complex)) = (uu::complex)";
test "(a*(b*c)) / (d*(b::complex)*(x*a)) = (uu::complex)";

FIXME: what do we do about this?
test "a*(b*c)/(y*z) = d*(b::complex)*(x*a)/z";
*)

end