# Theory NSComplex

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

section‹Nonstandard Complex Numbers›

theory NSComplex
imports 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 ⇒ hcomplex" where
"hcomplex_of_hypreal ≡ of_hypreal"

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

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

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 hExp_def HComplex_def

lemma Standard_hRe [simp]: "x ∈ Standard ⟹ hRe x ∈ Standard"

lemma Standard_hIm [simp]: "x ∈ Standard ⟹ hIm x ∈ Standard"

lemma Standard_iii [simp]: "iii ∈ Standard"

lemma Standard_hcnj [simp]: "x ∈ Standard ⟹ hcnj x ∈ Standard"

lemma Standard_hsgn [simp]: "x ∈ Standard ⟹ hsgn x ∈ Standard"

lemma Standard_harg [simp]: "x ∈ Standard ⟹ harg x ∈ Standard"

lemma Standard_hcis [simp]: "r ∈ Standard ⟹ hcis r ∈ Standard"

lemma Standard_hExp [simp]: "x ∈ Standard ⟹ hExp x ∈ Standard"

lemma Standard_hrcis [simp]:
"⟦r ∈ Standard; s ∈ Standard⟧ ⟹ hrcis r s ∈ Standard"

lemma Standard_HComplex [simp]:
"⟦r ∈ Standard; s ∈ Standard⟧ ⟹ HComplex r s ∈ Standard"

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 simp

lemma hcomplex_hIm_zero [simp]: "hIm 0 = 0"
by transfer simp

lemma hcomplex_hRe_one [simp]: "hRe 1 = 1"
by transfer simp

lemma hcomplex_hIm_one [simp]: "hIm 1 = 0"
by transfer simp

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

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

subsection‹More Minus Laws›

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

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

"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_squared)

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 ≠ 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) ≠ (0::hcomplex) ==> (c*a=c*b) = (a=b)"
by simp

lemma hcomplex_mult_right_cancel:
"(c::hcomplex) ≠ (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 ε ≠ 0"

subsection‹HComplex theorems›

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

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

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*)]:
"(⋀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 ¦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)

"!!x1 y1 x2 y2. HComplex x1 y1 + HComplex x2 y2 = HComplex (x1+x2) (y1+y2)"

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)

"!!x y r. HComplex x y + hcomplex_of_hypreal r = HComplex (x+r) y"

"!!r x y. hcomplex_of_hypreal r + HComplex x y = HComplex (r+x) y"

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)"

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 ≤ hcmod a"
by transfer (rule complex_mod_triangle_ineq2)

lemma hcmod_diff_ineq [simp]: "!!a b. hcmod(a) - hcmod(b) ≤ 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 simp

lemma hcpow_mult:
"((r::hcomplex) * s) pow n = (r pow n) * (s pow n)"
by (fact hyperpow_mult)

lemma hcpow_zero2 [simp]:
"⋀n. 0 pow (hSuc n) = (0::'a::semiring_1 star)"
by transfer (rule power_0_Suc)

lemma hcpow_not_zero [simp,intro]:
"!!r n. r ≠ 0 ==> r pow n ≠ (0::hcomplex)"
by (fact 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)"

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

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"

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:
"!!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) = ¦y¦"

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

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

lemma cos_harg_i_mult_zero [simp]:
"!!y. y ≠ 0 ==> ( *f* cos) (harg(HComplex 0 y)) = 0"
by transfer simp

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:
"∀n. ∃r a. (z n) =  complex_of_real r * (Complex (cos a) (sin a))"
by (auto intro: complex_split_polar)

lemma hcomplex_split_polar:
"!!z. ∃r a. z = hcomplex_of_hypreal r * (HComplex(( *f* cos) a)(( *f* sin) a))"

lemma hcis_eq:
"!!a. hcis a =
(hcomplex_of_hypreal(( *f* cos) a) +
iii * hcomplex_of_hypreal(( *f* sin) a))"

lemma hrcis_Ex: "!!z. ∃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"

lemma hcmod_complex_polar [simp]:
"!!r a. hcmod (hcomplex_of_hypreal r * HComplex (( *f* cos) a) (( *f* sin) a)) = ¦r¦"

lemma hcmod_hrcis [simp]: "!!r a. hcmod(hrcis r a) = ¦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
done

lemma NSDeMoivre: "!!a. (hcis a) ^ n = hcis (hypreal_of_nat n * a)"
apply transfer
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 (simp add: distrib_right cis_mult)

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

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

lemma DeMoivre2_ext:
"!! a r n. (hrcis r a) pow n = hrcis (r pow n) (hypreal_of_hypnat n * a)"
by transfer (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 simp

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

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

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

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

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

lemma hExp_add: "!!a b. hExp(a + b) = hExp(a) * hExp(b)"

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

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_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_numeral:
"hcomplex_of_complex (numeral w) = hcomplex_of_hypreal(numeral w)"
by transfer (rule of_real_numeral [symmetric])

lemma hcomplex_hypreal_neg_numeral:
"hcomplex_of_complex (- numeral w) = hcomplex_of_hypreal(- numeral w)"
by transfer (rule of_real_neg_numeral [symmetric])

lemma hcomplex_numeral_hcnj [simp]:
"hcnj (numeral v :: hcomplex) = numeral v"
by transfer (rule complex_cnj_numeral)

lemma hcomplex_numeral_hcmod [simp]:
"hcmod(numeral v :: hcomplex) = (numeral v :: hypreal)"
by transfer (rule norm_numeral)

lemma hcomplex_neg_numeral_hcmod [simp]:
"hcmod(- numeral v :: hcomplex) = (numeral v :: hypreal)"
by transfer (rule norm_neg_numeral)

lemma hcomplex_numeral_hRe [simp]:
"hRe(numeral v :: hcomplex) = numeral v"
by transfer (rule complex_Re_numeral)

lemma hcomplex_numeral_hIm [simp]:
"hIm(numeral v :: hcomplex) = 0"
by transfer (rule complex_Im_numeral)

end
```