# Theory CStar

```(*  Title:      HOL/Nonstandard_Analysis/CStar.thy
Author:     Jacques D. Fleuriot
*)

section ‹Star-transforms in NSA, Extending Sets of Complex Numbers and Complex Functions›

theory CStar
imports NSCA
begin

subsection ‹Properties of the ‹*›-Transform Applied to Sets of Reals›

lemma STARC_hcomplex_of_complex_Int: "*s* X ∩ SComplex = hcomplex_of_complex ` X"
by (auto simp: Standard_def)

lemma lemma_not_hcomplexA: "x ∉ hcomplex_of_complex ` A ⟹ ∀y ∈ A. x ≠ hcomplex_of_complex y"
by auto

subsection ‹Theorems about Nonstandard Extensions of Functions›

lemma starfunC_hcpow: "⋀Z. ( *f* (λz. z ^ n)) Z = Z pow hypnat_of_nat n"
by transfer (rule refl)

lemma starfunCR_cmod: "*f* cmod = hcmod"
by transfer (rule refl)

subsection ‹Internal Functions - Some Redundancy With ‹*f*› Now›

(** subtraction: ( *fn) - ( *gn) = *(fn - gn) **)
(*
lemma starfun_n_diff:
"( *fn* f) z - ( *fn* g) z = ( *fn* (λi x. f i x - g i x)) z"
apply (cases z)
done
*)
(** composition: ( *fn) o ( *gn) = *(fn o gn) **)

lemma starfun_Re: "( *f* (λx. Re (f x))) = (λx. hRe (( *f* f) x))"
by transfer (rule refl)

lemma starfun_Im: "( *f* (λx. Im (f x))) = (λx. hIm (( *f* f) x))"
by transfer (rule refl)

lemma starfunC_eq_Re_Im_iff:
"( *f* f) x = z ⟷ ( *f* (λx. Re (f x))) x = hRe z ∧ ( *f* (λx. Im (f x))) x = hIm z"
by (simp add: hcomplex_hRe_hIm_cancel_iff starfun_Re starfun_Im)

lemma starfunC_approx_Re_Im_iff:
"( *f* f) x ≈ z ⟷ ( *f* (λx. Re (f x))) x ≈ hRe z ∧ ( *f* (λx. Im (f x))) x ≈ hIm z"
by (simp add: hcomplex_approx_iff starfun_Re starfun_Im)

end
```