src/HOL/NSA/NSComplex.thy

refined Graph implementation: more abstract/scalable Graph.Keys instead of plain lists -- order of adjacency is now standardized wrt. Key.ord;

(*  Title:      HOL/NSA/NSComplex.thy
    Author:     Jacques D. Fleuriot, University of Edinburgh
    Author:     Lawrence C Paulson
*)

header{*Nonstandard Complex Numbers*}

theory NSComplex
imports Complex NSA
begin

type_synonym 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 -----*)

abbreviation
  hcomplex_of_hypreal :: "hypreal \<Rightarrow> hcomplex" where
  "hcomplex_of_hypreal \<equiv> of_hypreal"

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"

lemmas hcomplex_defs [transfer_unfold] =
  hRe_def hIm_def iii_def hcnj_def hsgn_def harg_def hcis_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 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 minus_unique)
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)"
(* TODO: delete *)
by (rule diff_eq_eq)


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

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

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


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 power2_i)

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_power)

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::{power,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_norm)

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)

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)"
(* TODO: delete *)
by (rule inj_star_of)

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 of_real_number_of_eq [symmetric])

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

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

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

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

end