header{*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*}
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