# Theory CStar

theory CStar
imports NSCA
```(*  Title       : CStar.thy
Author      : Jacques D. Fleuriot
Copyright   : 2001 University of Edinburgh
*)

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 Int SComplex = hcomplex_of_complex ` X"
by (auto simp add: 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* (%z. 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* (%i x. f i x - g i x)) z"
apply (cases z)
apply (simp add: starfun_n star_n_diff)
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* (%x. Re(f x))) x = hRe (z)) &
(( *f* (%x. 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* (%x. Re(f x))) x ≈ hRe (z)) &
(( *f* (%x. Im(f x))) x ≈ hIm (z)))"
by (simp add: hcomplex_approx_iff starfun_Re starfun_Im)

end
```