src/HOL/Complex/NSComplex.thy

minimize imports

(*  Title:       NSComplex.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{*Nonstandard Complex Numbers*}

theory NSComplex
imports Complex "../Hyperreal/NSA"
begin

types hcomplex = "complex star"

abbreviation
  hcomplex_of_complex :: "complex => complex star" where
  "hcomplex_of_complex == star_of"

abbreviation
  hcmod :: "complex star => real star" where
  "hcmod == hnorm"


  (*--- real and Imaginary parts ---*)

definition
  hRe :: "hcomplex => hypreal" where
  "hRe = *f* Re"

definition
  hIm :: "hcomplex => hypreal" where
  "hIm = *f* Im"


  (*------ imaginary unit ----------*)

definition
  iii :: hcomplex where
  "iii = star_of ii"

  (*------- complex conjugate ------*)

definition
  hcnj :: "hcomplex => hcomplex" where
  "hcnj = *f* cnj"

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

definition
  hsgn :: "hcomplex => hcomplex" where
  "hsgn = *f* sgn"

definition
  harg :: "hcomplex => hypreal" where
  "harg = *f* arg"

definition
  (* abbreviation for (cos a + i sin a) *)
  hcis :: "hypreal => hcomplex" where
  "hcis = *f* cis"

  (*----- injection from hyperreals -----*)

definition
  hcomplex_of_hypreal :: "hypreal => hcomplex" where
  "hcomplex_of_hypreal = *f* complex_of_real"

definition
  (* abbreviation for r*(cos a + i sin a) *)
  hrcis :: "[hypreal, hypreal] => hcomplex" where
  "hrcis = *f2* rcis"

  (*------------ e ^ (x + iy) ------------*)
definition
  hexpi :: "hcomplex => hcomplex" where
  "hexpi = *f* expi"

definition
  HComplex :: "[hypreal,hypreal] => hcomplex" where
  "HComplex = *f2* Complex"
(*
definition
  hcpow :: "[hcomplex,hypnat] => hcomplex"  (infixr "hcpow" 80) where
  "(z::hcomplex) hcpow (n::hypnat) = ( *f2* op ^) z n"
*)
lemmas hcomplex_defs [transfer_unfold] =
  hRe_def hIm_def iii_def hcnj_def hsgn_def harg_def hcis_def
  hcomplex_of_hypreal_def hrcis_def hexpi_def HComplex_def

lemma Standard_hRe [simp]: "x \<in> Standard \<Longrightarrow> hRe x \<in> Standard"
by (simp add: hcomplex_defs)

lemma Standard_hIm [simp]: "x \<in> Standard \<Longrightarrow> hIm x \<in> Standard"
by (simp add: hcomplex_defs)

lemma Standard_iii [simp]: "iii \<in> Standard"
by (simp add: hcomplex_defs)

lemma Standard_hcnj [simp]: "x \<in> Standard \<Longrightarrow> hcnj x \<in> Standard"
by (simp add: hcomplex_defs)

lemma Standard_hsgn [simp]: "x \<in> Standard \<Longrightarrow> hsgn x \<in> Standard"
by (simp add: hcomplex_defs)

lemma Standard_harg [simp]: "x \<in> Standard \<Longrightarrow> harg x \<in> Standard"
by (simp add: hcomplex_defs)

lemma Standard_hcis [simp]: "r \<in> Standard \<Longrightarrow> hcis r \<in> Standard"
by (simp add: hcomplex_defs)

lemma Standard_hexpi [simp]: "x \<in> Standard \<Longrightarrow> hexpi x \<in> Standard"
by (simp add: hcomplex_defs)

lemma Standard_hrcis [simp]:
  "\<lbrakk>r \<in> Standard; s \<in> Standard\<rbrakk> \<Longrightarrow> hrcis r s \<in> Standard"
by (simp add: hcomplex_defs)

lemma Standard_HComplex [simp]:
  "\<lbrakk>r \<in> Standard; s \<in> Standard\<rbrakk> \<Longrightarrow> HComplex r s \<in> Standard"
by (simp add: hcomplex_defs)

lemma Standard_hcomplex_of_hypreal [simp]:
  "r \<in> Standard \<Longrightarrow> hcomplex_of_hypreal r \<in> Standard"
by (simp add: hcomplex_defs)

lemma hcmod_def: "hcmod = *f* cmod"
by (rule hnorm_def)


subsection{*Properties of Nonstandard Real and Imaginary Parts*}

lemma hcomplex_hRe_hIm_cancel_iff:
     "!!w z. (w=z) = (hRe(w) = hRe(z) & hIm(w) = hIm(z))"
by transfer (rule complex_Re_Im_cancel_iff)

lemma hcomplex_equality [intro?]:
  "!!z w. hRe z = hRe w ==> hIm z = hIm w ==> z = w"
by transfer (rule complex_equality)

lemma hcomplex_hRe_zero [simp]: "hRe 0 = 0"
by transfer (rule complex_Re_zero)

lemma hcomplex_hIm_zero [simp]: "hIm 0 = 0"
by transfer (rule complex_Im_zero)

lemma hcomplex_hRe_one [simp]: "hRe 1 = 1"
by transfer (rule complex_Re_one)

lemma hcomplex_hIm_one [simp]: "hIm 1 = 0"
by transfer (rule complex_Im_one)


subsection{*Addition for Nonstandard Complex Numbers*}

lemma hRe_add: "!!x y. hRe(x + y) = hRe(x) + hRe(y)"
by transfer (rule complex_Re_add)

lemma hIm_add: "!!x y. hIm(x + y) = hIm(x) + hIm(y)"
by transfer (rule complex_Im_add)

subsection{*More Minus Laws*}

lemma hRe_minus: "!!z. hRe(-z) = - hRe(z)"
by transfer (rule complex_Re_minus)

lemma hIm_minus: "!!z. hIm(-z) = - hIm(z)"
by transfer (rule complex_Im_minus)

lemma hcomplex_add_minus_eq_minus:
      "x + y = (0::hcomplex) ==> x = -y"
apply (drule OrderedGroup.equals_zero_I)
apply (simp add: minus_equation_iff [of x y])
done

lemma hcomplex_i_mult_eq [simp]: "iii * iii = - 1"
by transfer (rule i_mult_eq2)

lemma hcomplex_i_mult_left [simp]: "!!z. iii * (iii * z) = -z"
by transfer (rule complex_i_mult_minus)

lemma hcomplex_i_not_zero [simp]: "iii \<noteq> 0"
by transfer (rule complex_i_not_zero)


subsection{*More Multiplication Laws*}

lemma hcomplex_mult_minus_one: "- 1 * (z::hcomplex) = -z"
by simp

lemma hcomplex_mult_minus_one_right: "(z::hcomplex) * - 1 = -z"
by simp

lemma hcomplex_mult_left_cancel:
     "(c::hcomplex) \<noteq> (0::hcomplex) ==> (c*a=c*b) = (a=b)"
by simp

lemma hcomplex_mult_right_cancel:
     "(c::hcomplex) \<noteq> (0::hcomplex) ==> (a*c=b*c) = (a=b)"
by simp


subsection{*Subraction and Division*}

lemma hcomplex_diff_eq_eq [simp]: "((x::hcomplex) - y = z) = (x = z + y)"
by (rule OrderedGroup.diff_eq_eq)

lemma hcomplex_add_divide_distrib: "(x+y)/(z::hcomplex) = x/z + y/z"
by (rule Ring_and_Field.add_divide_distrib)


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

lemma hcomplex_of_hypreal_cancel_iff [iff]:
     "!!x y. (hcomplex_of_hypreal x = hcomplex_of_hypreal y) = (x = y)"
by transfer (rule of_real_eq_iff)

lemma hcomplex_of_hypreal_one [simp]: "hcomplex_of_hypreal 1 = 1"
by transfer (rule of_real_1)

lemma hcomplex_of_hypreal_zero [simp]: "hcomplex_of_hypreal 0 = 0"
by transfer (rule of_real_0)

lemma hcomplex_of_hypreal_minus [simp]:
     "!!x. hcomplex_of_hypreal(-x) = - hcomplex_of_hypreal x"
by transfer (rule of_real_minus)

lemma hcomplex_of_hypreal_inverse [simp]:
     "!!x. hcomplex_of_hypreal(inverse x) = inverse(hcomplex_of_hypreal x)"
by transfer (rule of_real_inverse)

lemma hcomplex_of_hypreal_add [simp]:
     "!!x y. hcomplex_of_hypreal (x + y) =
      hcomplex_of_hypreal x + hcomplex_of_hypreal y"
by transfer (rule of_real_add)

lemma hcomplex_of_hypreal_diff [simp]:
     "!!x y. hcomplex_of_hypreal (x - y) =
      hcomplex_of_hypreal x - hcomplex_of_hypreal y "
by transfer (rule of_real_diff)

lemma hcomplex_of_hypreal_mult [simp]:
     "!!x y. hcomplex_of_hypreal (x * y) =
      hcomplex_of_hypreal x * hcomplex_of_hypreal y"
by transfer (rule of_real_mult)

lemma hcomplex_of_hypreal_divide [simp]:
     "!!x y. hcomplex_of_hypreal(x/y) =
      hcomplex_of_hypreal x / hcomplex_of_hypreal y"
by transfer (rule of_real_divide)

lemma hRe_hcomplex_of_hypreal [simp]: "!!z. hRe(hcomplex_of_hypreal z) = z"
by transfer (rule Re_complex_of_real)

lemma hIm_hcomplex_of_hypreal [simp]: "!!z. hIm(hcomplex_of_hypreal z) = 0"
by transfer (rule Im_complex_of_real)

lemma hcomplex_of_hypreal_zero_iff [simp]:
  "\<And>x. (hcomplex_of_hypreal x = 0) = (x = 0)"
by transfer (rule of_real_eq_0_iff)

lemma hcomplex_of_hypreal_epsilon_not_zero [simp]:
     "hcomplex_of_hypreal epsilon \<noteq> 0"
by (simp add: hypreal_epsilon_not_zero)

subsection{*HComplex theorems*}

lemma hRe_HComplex [simp]: "!!x y. hRe (HComplex x y) = x"
by transfer (rule Re)

lemma hIm_HComplex [simp]: "!!x y. hIm (HComplex x y) = y"
by transfer (rule Im)

lemma hcomplex_surj [simp]: "!!z. HComplex (hRe z) (hIm z) = z"
by transfer (rule complex_surj)

lemma hcomplex_induct [case_names rect(*, induct type: hcomplex*)]:
     "(\<And>x y. P (HComplex x y)) ==> P z"
by (rule hcomplex_surj [THEN subst], blast)


subsection{*Modulus (Absolute Value) of Nonstandard Complex Number*}

lemma hcmod_zero [simp]: "hcmod(0) = 0"
by (rule hnorm_zero)

lemma hcmod_one [simp]: "hcmod(1) = 1"
by (rule hnorm_one)

lemma hcmod_hcomplex_of_hypreal [simp]:
  "!!x. hcmod(hcomplex_of_hypreal x) = abs x"
by transfer (rule norm_of_real)

lemma hcomplex_of_hypreal_abs:
     "hcomplex_of_hypreal (abs x) =
      hcomplex_of_hypreal(hcmod(hcomplex_of_hypreal x))"
by simp

lemma HComplex_inject [simp]:
  "!!x y x' y'. HComplex x y = HComplex x' y' = (x=x' & y=y')"
by transfer (rule complex.inject)

lemma HComplex_add [simp]:
  "!!x1 y1 x2 y2. HComplex x1 y1 + HComplex x2 y2 = HComplex (x1+x2) (y1+y2)"
by transfer (rule complex_add)

lemma HComplex_minus [simp]: "!!x y. - HComplex x y = HComplex (-x) (-y)"
by transfer (rule complex_minus)

lemma HComplex_diff [simp]:
  "!!x1 y1 x2 y2. HComplex x1 y1 - HComplex x2 y2 = HComplex (x1-x2) (y1-y2)"
by transfer (rule complex_diff)

lemma HComplex_mult [simp]:
  "!!x1 y1 x2 y2. HComplex x1 y1 * HComplex x2 y2 =
   HComplex (x1*x2 - y1*y2) (x1*y2 + y1*x2)"
by transfer (rule complex_mult)

(*HComplex_inverse is proved below*)

lemma hcomplex_of_hypreal_eq: "!!r. hcomplex_of_hypreal r = HComplex r 0"
by transfer (rule complex_of_real_def)

lemma HComplex_add_hcomplex_of_hypreal [simp]:
     "!!x y r. HComplex x y + hcomplex_of_hypreal r = HComplex (x+r) y"
by transfer (rule Complex_add_complex_of_real)

lemma hcomplex_of_hypreal_add_HComplex [simp]:
     "!!r x y. hcomplex_of_hypreal r + HComplex x y = HComplex (r+x) y"
by transfer (rule complex_of_real_add_Complex)

lemma HComplex_mult_hcomplex_of_hypreal:
     "!!x y r. HComplex x y * hcomplex_of_hypreal r = HComplex (x*r) (y*r)"
by transfer (rule Complex_mult_complex_of_real)

lemma hcomplex_of_hypreal_mult_HComplex:
     "!!r x y. hcomplex_of_hypreal r * HComplex x y = HComplex (r*x) (r*y)"
by transfer (rule complex_of_real_mult_Complex)

lemma i_hcomplex_of_hypreal [simp]:
     "!!r. iii * hcomplex_of_hypreal r = HComplex 0 r"
by transfer (rule i_complex_of_real)

lemma hcomplex_of_hypreal_i [simp]:
     "!!r. hcomplex_of_hypreal r * iii = HComplex 0 r"
by transfer (rule complex_of_real_i)


subsection{*Conjugation*}

lemma hcomplex_hcnj_cancel_iff [iff]: "!!x y. (hcnj x = hcnj y) = (x = y)"
by transfer (rule complex_cnj_cancel_iff)

lemma hcomplex_hcnj_hcnj [simp]: "!!z. hcnj (hcnj z) = z"
by transfer (rule complex_cnj_cnj)

lemma hcomplex_hcnj_hcomplex_of_hypreal [simp]:
     "!!x. hcnj (hcomplex_of_hypreal x) = hcomplex_of_hypreal x"
by transfer (rule complex_cnj_complex_of_real)

lemma hcomplex_hmod_hcnj [simp]: "!!z. hcmod (hcnj z) = hcmod z"
by transfer (rule complex_mod_cnj)

lemma hcomplex_hcnj_minus: "!!z. hcnj (-z) = - hcnj z"
by transfer (rule complex_cnj_minus)

lemma hcomplex_hcnj_inverse: "!!z. hcnj(inverse z) = inverse(hcnj z)"
by transfer (rule complex_cnj_inverse)

lemma hcomplex_hcnj_add: "!!w z. hcnj(w + z) = hcnj(w) + hcnj(z)"
by transfer (rule complex_cnj_add)

lemma hcomplex_hcnj_diff: "!!w z. hcnj(w - z) = hcnj(w) - hcnj(z)"
by transfer (rule complex_cnj_diff)

lemma hcomplex_hcnj_mult: "!!w z. hcnj(w * z) = hcnj(w) * hcnj(z)"
by transfer (rule complex_cnj_mult)

lemma hcomplex_hcnj_divide: "!!w z. hcnj(w / z) = (hcnj w)/(hcnj z)"
by transfer (rule complex_cnj_divide)

lemma hcnj_one [simp]: "hcnj 1 = 1"
by transfer (rule complex_cnj_one)

lemma hcomplex_hcnj_zero [simp]: "hcnj 0 = 0"
by transfer (rule complex_cnj_zero)

lemma hcomplex_hcnj_zero_iff [iff]: "!!z. (hcnj z = 0) = (z = 0)"
by transfer (rule complex_cnj_zero_iff)

lemma hcomplex_mult_hcnj:
     "!!z. z * hcnj z = hcomplex_of_hypreal (hRe(z) ^ 2 + hIm(z) ^ 2)"
by transfer (rule complex_mult_cnj)


subsection{*More Theorems about the Function @{term hcmod}*}

lemma hcomplex_hcmod_eq_zero_cancel [simp]: "!!x. (hcmod x = 0) = (x = 0)"
by transfer (rule complex_mod_eq_zero_cancel)

lemma hcmod_hcomplex_of_hypreal_of_nat [simp]:
     "hcmod (hcomplex_of_hypreal(hypreal_of_nat n)) = hypreal_of_nat n"
by simp

lemma hcmod_hcomplex_of_hypreal_of_hypnat [simp]:
     "hcmod (hcomplex_of_hypreal(hypreal_of_hypnat n)) = hypreal_of_hypnat n"
by simp

lemma hcmod_minus [simp]: "!!x. hcmod (-x) = hcmod(x)"
by transfer (rule complex_mod_minus)

lemma hcmod_mult_hcnj: "!!z. hcmod(z * hcnj(z)) = hcmod(z) ^ 2"
by transfer (rule complex_mod_mult_cnj)

lemma hcmod_ge_zero [simp]: "!!x. (0::hypreal) \<le> hcmod x"
by transfer (rule complex_mod_ge_zero)

lemma hrabs_hcmod_cancel [simp]: "abs(hcmod x) = hcmod x"
by (simp add: abs_if linorder_not_less)

lemma hcmod_mult: "!!x y. hcmod(x*y) = hcmod(x) * hcmod(y)"
by transfer (rule complex_mod_mult)

lemma hcmod_add_squared_eq:
  "!!x y. hcmod(x + y) ^ 2 = hcmod(x) ^ 2 + hcmod(y) ^ 2 + 2 * hRe(x * hcnj y)"
by transfer (rule complex_mod_add_squared_eq)

lemma hcomplex_hRe_mult_hcnj_le_hcmod [simp]:
  "!!x y. hRe(x * hcnj y) \<le> hcmod(x * hcnj y)"
by transfer (rule complex_Re_mult_cnj_le_cmod)

lemma hcomplex_hRe_mult_hcnj_le_hcmod2 [simp]:
  "!!x y. hRe(x * hcnj y) \<le> hcmod(x * y)"
by transfer (rule complex_Re_mult_cnj_le_cmod2)

lemma hcmod_triangle_squared [simp]:
  "!!x y. hcmod (x + y) ^ 2 \<le> (hcmod(x) + hcmod(y)) ^ 2"
by transfer (rule complex_mod_triangle_squared)

lemma hcmod_triangle_ineq [simp]:
  "!!x y. hcmod (x + y) \<le> hcmod(x) + hcmod(y)"
by transfer (rule complex_mod_triangle_ineq)

lemma hcmod_triangle_ineq2 [simp]:
  "!!a b. hcmod(b + a) - hcmod b \<le> hcmod a"
by transfer (rule complex_mod_triangle_ineq2)

lemma hcmod_diff_commute: "!!x y. hcmod (x - y) = hcmod (y - x)"
by transfer (rule complex_mod_diff_commute)

lemma hcmod_add_less:
  "!!x y r s. [| hcmod x < r; hcmod y < s |] ==> hcmod (x + y) < r + s"
by transfer (rule complex_mod_add_less)

lemma hcmod_mult_less:
  "!!x y r s. [| hcmod x < r; hcmod y < s |] ==> hcmod (x * y) < r * s"
by transfer (rule complex_mod_mult_less)

lemma hcmod_diff_ineq [simp]: "!!a b. hcmod(a) - hcmod(b) \<le> hcmod(a + b)"
by transfer (rule complex_mod_diff_ineq)


subsection{*A Few Nonlinear Theorems*}

lemma hcomplex_of_hypreal_hyperpow:
     "!!x n. hcomplex_of_hypreal (x pow n) = (hcomplex_of_hypreal x) pow n"
by transfer (rule complex_of_real_pow)

lemma hcmod_hcpow: "!!x n. hcmod(x pow n) = hcmod(x) pow n"
by transfer (rule complex_mod_complexpow)

lemma hcmod_hcomplex_inverse: "!!x. hcmod(inverse x) = inverse(hcmod x)"
by transfer (rule complex_mod_inverse)

lemma hcmod_divide: "!!x y. hcmod(x/y) = hcmod(x)/(hcmod y)"
by transfer (rule norm_divide)


subsection{*Exponentiation*}

lemma hcomplexpow_0 [simp]:   "z ^ 0       = (1::hcomplex)"
by (rule power_0)

lemma hcomplexpow_Suc [simp]: "z ^ (Suc n) = (z::hcomplex) * (z ^ n)"
by (rule power_Suc)

lemma hcomplexpow_i_squared [simp]: "iii ^ 2 = - 1"
by transfer (rule complexpow_i_squared)

lemma hcomplex_of_hypreal_pow:
     "!!x. hcomplex_of_hypreal (x ^ n) = (hcomplex_of_hypreal x) ^ n"
by transfer (rule of_real_power)

lemma hcomplex_hcnj_pow: "!!z. hcnj(z ^ n) = hcnj(z) ^ n"
by transfer (rule complex_cnj_pow)

lemma hcmod_hcomplexpow: "!!x. hcmod(x ^ n) = hcmod(x) ^ n"
by transfer (rule norm_power)

lemma hcpow_minus:
     "!!x n. (-x::hcomplex) pow n =
      (if ( *p* even) n then (x pow n) else -(x pow n))"
by transfer (rule neg_power_if)

lemma hcpow_mult:
  "!!r s n. ((r::hcomplex) * s) pow n = (r pow n) * (s pow n)"
by transfer (rule power_mult_distrib)

lemma hcpow_zero2 [simp]:
  "\<And>n. 0 pow (hSuc n) = (0::'a::{recpower,semiring_0} star)"
by transfer (rule power_0_Suc)

lemma hcpow_not_zero [simp,intro]:
  "!!r n. r \<noteq> 0 ==> r pow n \<noteq> (0::hcomplex)"
by (rule hyperpow_not_zero)

lemma hcpow_zero_zero: "r pow n = (0::hcomplex) ==> r = 0"
by (blast intro: ccontr dest: hcpow_not_zero)

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

lemma hsgn_zero [simp]: "hsgn 0 = 0"
by transfer (rule sgn_zero)

lemma hsgn_one [simp]: "hsgn 1 = 1"
by transfer (rule sgn_one)

lemma hsgn_minus: "!!z. hsgn (-z) = - hsgn(z)"
by transfer (rule sgn_minus)

lemma hsgn_eq: "!!z. hsgn z = z / hcomplex_of_hypreal (hcmod z)"
by transfer (rule sgn_eq)

lemma hcmod_i: "!!x y. hcmod (HComplex x y) = ( *f* sqrt) (x ^ 2 + y ^ 2)"
by transfer (rule complex_mod)

lemma hcomplex_eq_cancel_iff1 [simp]:
     "(hcomplex_of_hypreal xa = HComplex x y) = (xa = x & y = 0)"
by (simp add: hcomplex_of_hypreal_eq)

lemma hcomplex_eq_cancel_iff2 [simp]:
     "(HComplex x y = hcomplex_of_hypreal xa) = (x = xa & y = 0)"
by (simp add: hcomplex_of_hypreal_eq)

lemma HComplex_eq_0 [simp]: "!!x y. (HComplex x y = 0) = (x = 0 & y = 0)"
by transfer (rule Complex_eq_0)

lemma HComplex_eq_1 [simp]: "!!x y. (HComplex x y = 1) = (x = 1 & y = 0)"
by transfer (rule Complex_eq_1)

lemma i_eq_HComplex_0_1: "iii = HComplex 0 1"
by transfer (rule i_def [THEN meta_eq_to_obj_eq])

lemma HComplex_eq_i [simp]: "!!x y. (HComplex x y = iii) = (x = 0 & y = 1)"
by transfer (rule Complex_eq_i)

lemma hRe_hsgn [simp]: "!!z. hRe(hsgn z) = hRe(z)/hcmod z"
by transfer (rule Re_sgn)

lemma hIm_hsgn [simp]: "!!z. hIm(hsgn z) = hIm(z)/hcmod z"
by transfer (rule Im_sgn)

(*????move to RealDef????*)
lemma real_two_squares_add_zero_iff [simp]: "(x*x + y*y = 0) = ((x::real) = 0 & y = 0)"
by (auto intro: real_sum_squares_cancel iff: real_add_eq_0_iff)

lemma hcomplex_inverse_complex_split:
     "!!x y. inverse(hcomplex_of_hypreal x + iii * hcomplex_of_hypreal y) =
      hcomplex_of_hypreal(x/(x ^ 2 + y ^ 2)) -
      iii * hcomplex_of_hypreal(y/(x ^ 2 + y ^ 2))"
by transfer (rule complex_inverse_complex_split)

lemma HComplex_inverse:
     "!!x y. inverse (HComplex x y) =
      HComplex (x/(x ^ 2 + y ^ 2)) (-y/(x ^ 2 + y ^ 2))"
by transfer (rule complex_inverse)

lemma hRe_mult_i_eq[simp]:
    "!!y. hRe (iii * hcomplex_of_hypreal y) = 0"
by transfer simp

lemma hIm_mult_i_eq [simp]:
    "!!y. hIm (iii * hcomplex_of_hypreal y) = y"
by transfer simp

lemma hcmod_mult_i [simp]: "!!y. hcmod (iii * hcomplex_of_hypreal y) = abs y"
by transfer simp

lemma hcmod_mult_i2 [simp]: "!!y. hcmod (hcomplex_of_hypreal y * iii) = abs y"
by transfer simp

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

lemma cos_harg_i_mult_zero_pos:
     "!!y. 0 < y ==> ( *f* cos) (harg(HComplex 0 y)) = 0"
by transfer (rule cos_arg_i_mult_zero_pos)

lemma cos_harg_i_mult_zero_neg:
     "!!y. y < 0 ==> ( *f* cos) (harg(HComplex 0 y)) = 0"
by transfer (rule cos_arg_i_mult_zero_neg)

lemma cos_harg_i_mult_zero [simp]:
     "!!y. y \<noteq> 0 ==> ( *f* cos) (harg(HComplex 0 y)) = 0"
by transfer (rule cos_arg_i_mult_zero)

lemma hcomplex_of_hypreal_zero_iff [simp]:
     "!!y. (hcomplex_of_hypreal y = 0) = (y = 0)"
by transfer (rule of_real_eq_0_iff)


subsection{*Polar Form for Nonstandard Complex Numbers*}

lemma complex_split_polar2:
     "\<forall>n. \<exists>r a. (z n) =  complex_of_real r * (Complex (cos a) (sin a))"
by (blast intro: complex_split_polar)

lemma hcomplex_split_polar:
  "!!z. \<exists>r a. z = hcomplex_of_hypreal r * (HComplex(( *f* cos) a)(( *f* sin) a))"
by transfer (rule complex_split_polar)

lemma hcis_eq:
   "!!a. hcis a =
    (hcomplex_of_hypreal(( *f* cos) a) +
    iii * hcomplex_of_hypreal(( *f* sin) a))"
by transfer (simp add: cis_def)

lemma hrcis_Ex: "!!z. \<exists>r a. z = hrcis r a"
by transfer (rule rcis_Ex)

lemma hRe_hcomplex_polar [simp]:
  "!!r a. hRe (hcomplex_of_hypreal r * HComplex (( *f* cos) a) (( *f* sin) a)) = 
      r * ( *f* cos) a"
by transfer simp

lemma hRe_hrcis [simp]: "!!r a. hRe(hrcis r a) = r * ( *f* cos) a"
by transfer (rule Re_rcis)

lemma hIm_hcomplex_polar [simp]:
  "!!r a. hIm (hcomplex_of_hypreal r * HComplex (( *f* cos) a) (( *f* sin) a)) = 
      r * ( *f* sin) a"
by transfer simp

lemma hIm_hrcis [simp]: "!!r a. hIm(hrcis r a) = r * ( *f* sin) a"
by transfer (rule Im_rcis)

lemma hcmod_unit_one [simp]:
     "!!a. hcmod (HComplex (( *f* cos) a) (( *f* sin) a)) = 1"
by transfer (rule cmod_unit_one)

lemma hcmod_complex_polar [simp]:
  "!!r a. hcmod (hcomplex_of_hypreal r * HComplex (( *f* cos) a) (( *f* sin) a)) =
      abs r"
by transfer (rule cmod_complex_polar)

lemma hcmod_hrcis [simp]: "!!r a. hcmod(hrcis r a) = abs r"
by transfer (rule complex_mod_rcis)

(*---------------------------------------------------------------------------*)
(*  (r1 * hrcis a) * (r2 * hrcis b) = r1 * r2 * hrcis (a + b)                *)
(*---------------------------------------------------------------------------*)

lemma hcis_hrcis_eq: "!!a. hcis a = hrcis 1 a"
by transfer (rule cis_rcis_eq)
declare hcis_hrcis_eq [symmetric, simp]

lemma hrcis_mult:
  "!!a b r1 r2. hrcis r1 a * hrcis r2 b = hrcis (r1*r2) (a + b)"
by transfer (rule rcis_mult)

lemma hcis_mult: "!!a b. hcis a * hcis b = hcis (a + b)"
by transfer (rule cis_mult)

lemma hcis_zero [simp]: "hcis 0 = 1"
by transfer (rule cis_zero)

lemma hrcis_zero_mod [simp]: "!!a. hrcis 0 a = 0"
by transfer (rule rcis_zero_mod)

lemma hrcis_zero_arg [simp]: "!!r. hrcis r 0 = hcomplex_of_hypreal r"
by transfer (rule rcis_zero_arg)

lemma hcomplex_i_mult_minus [simp]: "!!x. iii * (iii * x) = - x"
by transfer (rule complex_i_mult_minus)

lemma hcomplex_i_mult_minus2 [simp]: "iii * iii * x = - x"
by simp

lemma hcis_hypreal_of_nat_Suc_mult:
   "!!a. hcis (hypreal_of_nat (Suc n) * a) =
     hcis a * hcis (hypreal_of_nat n * a)"
apply transfer
apply (fold real_of_nat_def)
apply (rule cis_real_of_nat_Suc_mult)
done

lemma NSDeMoivre: "!!a. (hcis a) ^ n = hcis (hypreal_of_nat n * a)"
apply transfer
apply (fold real_of_nat_def)
apply (rule DeMoivre)
done

lemma hcis_hypreal_of_hypnat_Suc_mult:
     "!! a n. hcis (hypreal_of_hypnat (n + 1) * a) =
      hcis a * hcis (hypreal_of_hypnat n * a)"
by transfer (fold real_of_nat_def, simp add: cis_real_of_nat_Suc_mult)

lemma NSDeMoivre_ext:
  "!!a n. (hcis a) pow n = hcis (hypreal_of_hypnat n * a)"
by transfer (fold real_of_nat_def, rule DeMoivre)

lemma NSDeMoivre2:
  "!!a r. (hrcis r a) ^ n = hrcis (r ^ n) (hypreal_of_nat n * a)"
by transfer (fold real_of_nat_def, rule DeMoivre2)

lemma DeMoivre2_ext:
  "!! a r n. (hrcis r a) pow n = hrcis (r pow n) (hypreal_of_hypnat n * a)"
by transfer (fold real_of_nat_def, rule DeMoivre2)

lemma hcis_inverse [simp]: "!!a. inverse(hcis a) = hcis (-a)"
by transfer (rule cis_inverse)

lemma hrcis_inverse: "!!a r. inverse(hrcis r a) = hrcis (inverse r) (-a)"
by transfer (simp add: rcis_inverse inverse_eq_divide [symmetric])

lemma hRe_hcis [simp]: "!!a. hRe(hcis a) = ( *f* cos) a"
by transfer (rule Re_cis)

lemma hIm_hcis [simp]: "!!a. hIm(hcis a) = ( *f* sin) a"
by transfer (rule Im_cis)

lemma cos_n_hRe_hcis_pow_n: "( *f* cos) (hypreal_of_nat n * a) = hRe(hcis a ^ n)"
by (simp add: NSDeMoivre)

lemma sin_n_hIm_hcis_pow_n: "( *f* sin) (hypreal_of_nat n * a) = hIm(hcis a ^ n)"
by (simp add: NSDeMoivre)

lemma cos_n_hRe_hcis_hcpow_n: "( *f* cos) (hypreal_of_hypnat n * a) = hRe(hcis a pow n)"
by (simp add: NSDeMoivre_ext)

lemma sin_n_hIm_hcis_hcpow_n: "( *f* sin) (hypreal_of_hypnat n * a) = hIm(hcis a pow n)"
by (simp add: NSDeMoivre_ext)

lemma hexpi_add: "!!a b. hexpi(a + b) = hexpi(a) * hexpi(b)"
by transfer (rule expi_add)


subsection{*@{term hcomplex_of_complex}: the Injection from
  type @{typ complex} to to @{typ hcomplex}*}

lemma inj_hcomplex_of_complex: "inj(hcomplex_of_complex)"
by (rule inj_onI, simp)

lemma hcomplex_of_complex_i: "iii = hcomplex_of_complex ii"
by (rule iii_def)

lemma hRe_hcomplex_of_complex:
   "hRe (hcomplex_of_complex z) = hypreal_of_real (Re z)"
by transfer (rule refl)

lemma hIm_hcomplex_of_complex:
   "hIm (hcomplex_of_complex z) = hypreal_of_real (Im z)"
by transfer (rule refl)

lemma hcmod_hcomplex_of_complex:
     "hcmod (hcomplex_of_complex x) = hypreal_of_real (cmod x)"
by transfer (rule refl)


subsection{*Numerals and Arithmetic*}

lemma hcomplex_number_of_def: "(number_of w :: hcomplex) == of_int w"
by transfer (rule number_of_eq [THEN eq_reflection])

lemma hcomplex_of_hypreal_eq_hcomplex_of_complex: 
     "hcomplex_of_hypreal (hypreal_of_real x) =  
      hcomplex_of_complex (complex_of_real x)"
by transfer (rule refl)

lemma hcomplex_hypreal_number_of: 
  "hcomplex_of_complex (number_of w) = hcomplex_of_hypreal(number_of w)"
by transfer (rule complex_number_of [symmetric])

text{*This theorem is necessary because theorems such as
   @{text iszero_number_of_0} only hold for ordered rings. They cannot
   be generalized to fields in general because they fail for finite fields.
   They work for type complex because the reals can be embedded in them.*}
lemma iszero_hcomplex_number_of [simp]:
     "iszero (number_of w :: hcomplex) = iszero (number_of w :: real)"
by (transfer iszero_def, simp)


(*
Goal "z + hcnj z =  
      hcomplex_of_hypreal (2 * hRe(z))"
by (res_inst_tac [("z","z")] eq_Abs_star 1);
by (auto_tac (claset(),HOL_ss addsimps [hRe,hcnj,star_n_add,
    hypreal_mult,hcomplex_of_hypreal,complex_add_cnj]));
qed "star_n_add_hcnj";

Goal "z - hcnj z = \
\     hcomplex_of_hypreal (hypreal_of_real #2 * hIm(z)) * iii";
by (res_inst_tac [("z","z")] eq_Abs_star 1);
by (auto_tac (claset(),simpset() addsimps [hIm,hcnj,hcomplex_diff,
    hypreal_of_real_def,hypreal_mult,hcomplex_of_hypreal,
    complex_diff_cnj,iii_def,star_n_mult]));
qed "hcomplex_diff_hcnj";
*)


(*** Real and imaginary stuff ***)

(*Convert???
Goalw [hcomplex_number_of_def] 
  "((number_of xa :: hcomplex) + iii * number_of ya =  
        number_of xb + iii * number_of yb) =  
   (((number_of xa :: hcomplex) = number_of xb) &  
    ((number_of ya :: hcomplex) = number_of yb))"
by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iff,
     hcomplex_hypreal_number_of]));
qed "hcomplex_number_of_eq_cancel_iff";
Addsimps [hcomplex_number_of_eq_cancel_iff];

Goalw [hcomplex_number_of_def] 
  "((number_of xa :: hcomplex) + number_of ya * iii = \
\       number_of xb + number_of yb * iii) = \
\  (((number_of xa :: hcomplex) = number_of xb) & \
\   ((number_of ya :: hcomplex) = number_of yb))";
by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iffA,
    hcomplex_hypreal_number_of]));
qed "hcomplex_number_of_eq_cancel_iffA";
Addsimps [hcomplex_number_of_eq_cancel_iffA];

Goalw [hcomplex_number_of_def] 
  "((number_of xa :: hcomplex) + number_of ya * iii = \
\       number_of xb + iii * number_of yb) = \
\  (((number_of xa :: hcomplex) = number_of xb) & \
\   ((number_of ya :: hcomplex) = number_of yb))";
by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iffB,
    hcomplex_hypreal_number_of]));
qed "hcomplex_number_of_eq_cancel_iffB";
Addsimps [hcomplex_number_of_eq_cancel_iffB];

Goalw [hcomplex_number_of_def] 
  "((number_of xa :: hcomplex) + iii * number_of ya = \
\       number_of xb + number_of yb * iii) = \
\  (((number_of xa :: hcomplex) = number_of xb) & \
\   ((number_of ya :: hcomplex) = number_of yb))";
by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iffC,
     hcomplex_hypreal_number_of]));
qed "hcomplex_number_of_eq_cancel_iffC";
Addsimps [hcomplex_number_of_eq_cancel_iffC];

Goalw [hcomplex_number_of_def] 
  "((number_of xa :: hcomplex) + iii * number_of ya = \
\       number_of xb) = \
\  (((number_of xa :: hcomplex) = number_of xb) & \
\   ((number_of ya :: hcomplex) = 0))";
by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iff2,
    hcomplex_hypreal_number_of,hcomplex_of_hypreal_zero_iff]));
qed "hcomplex_number_of_eq_cancel_iff2";
Addsimps [hcomplex_number_of_eq_cancel_iff2];

Goalw [hcomplex_number_of_def] 
  "((number_of xa :: hcomplex) + number_of ya * iii = \
\       number_of xb) = \
\  (((number_of xa :: hcomplex) = number_of xb) & \
\   ((number_of ya :: hcomplex) = 0))";
by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iff2a,
    hcomplex_hypreal_number_of,hcomplex_of_hypreal_zero_iff]));
qed "hcomplex_number_of_eq_cancel_iff2a";
Addsimps [hcomplex_number_of_eq_cancel_iff2a];

Goalw [hcomplex_number_of_def] 
  "((number_of xa :: hcomplex) + iii * number_of ya = \
\    iii * number_of yb) = \
\  (((number_of xa :: hcomplex) = 0) & \
\   ((number_of ya :: hcomplex) = number_of yb))";
by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iff3,
    hcomplex_hypreal_number_of,hcomplex_of_hypreal_zero_iff]));
qed "hcomplex_number_of_eq_cancel_iff3";
Addsimps [hcomplex_number_of_eq_cancel_iff3];

Goalw [hcomplex_number_of_def] 
  "((number_of xa :: hcomplex) + number_of ya * iii= \
\    iii * number_of yb) = \
\  (((number_of xa :: hcomplex) = 0) & \
\   ((number_of ya :: hcomplex) = number_of yb))";
by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iff3a,
    hcomplex_hypreal_number_of,hcomplex_of_hypreal_zero_iff]));
qed "hcomplex_number_of_eq_cancel_iff3a";
Addsimps [hcomplex_number_of_eq_cancel_iff3a];
*)

lemma hcomplex_number_of_hcnj [simp]:
     "hcnj (number_of v :: hcomplex) = number_of v"
by transfer (rule complex_number_of_cnj)

lemma hcomplex_number_of_hcmod [simp]: 
      "hcmod(number_of v :: hcomplex) = abs (number_of v :: hypreal)"
by transfer (rule complex_number_of_cmod)

lemma hcomplex_number_of_hRe [simp]: 
      "hRe(number_of v :: hcomplex) = number_of v"
by transfer (rule complex_number_of_Re)

lemma hcomplex_number_of_hIm [simp]: 
      "hIm(number_of v :: hcomplex) = 0"
by transfer (rule complex_number_of_Im)

end