1(*  Title:      HOL/Nonstandard_Analysis/CStar.thy
2    Author:     Jacques D. Fleuriot
3    Copyright:  2001 University of Edinburgh
4*)
5
6section \<open>Star-transforms in NSA, Extending Sets of Complex Numbers and Complex Functions\<close>
7
8theory CStar
9  imports NSCA
10begin
11
12subsection \<open>Properties of the \<open>*\<close>-Transform Applied to Sets of Reals\<close>
13
14lemma STARC_hcomplex_of_complex_Int: "*s* X \<inter> SComplex = hcomplex_of_complex ` X"
15  by (auto simp: Standard_def)
16
17lemma lemma_not_hcomplexA: "x \<notin> hcomplex_of_complex ` A \<Longrightarrow> \<forall>y \<in> A. x \<noteq> hcomplex_of_complex y"
18  by auto
19
20
21subsection \<open>Theorems about Nonstandard Extensions of Functions\<close>
22
23lemma starfunC_hcpow: "\<And>Z. ( *f* (\<lambda>z. z ^ n)) Z = Z pow hypnat_of_nat n"
24  by transfer (rule refl)
25
26lemma starfunCR_cmod: "*f* cmod = hcmod"
27  by transfer (rule refl)
28
29
30subsection \<open>Internal Functions - Some Redundancy With \<open>*f*\<close> Now\<close>
31
32(** subtraction: ( *fn) - ( *gn) = *(fn - gn) **)
33(*
34lemma starfun_n_diff:
35   "( *fn* f) z - ( *fn* g) z = ( *fn* (\<lambda>i x. f i x - g i x)) z"
36apply (cases z)
37apply (simp add: starfun_n star_n_diff)
38done
39*)
40(** composition: ( *fn) o ( *gn) = *(fn o gn) **)
41
42lemma starfun_Re: "( *f* (\<lambda>x. Re (f x))) = (\<lambda>x. hRe (( *f* f) x))"
43  by transfer (rule refl)
44
45lemma starfun_Im: "( *f* (\<lambda>x. Im (f x))) = (\<lambda>x. hIm (( *f* f) x))"
46  by transfer (rule refl)
47
48lemma starfunC_eq_Re_Im_iff:
49  "( *f* f) x = z \<longleftrightarrow> ( *f* (\<lambda>x. Re (f x))) x = hRe z \<and> ( *f* (\<lambda>x. Im (f x))) x = hIm z"
50  by (simp add: hcomplex_hRe_hIm_cancel_iff starfun_Re starfun_Im)
51
52lemma starfunC_approx_Re_Im_iff:
53  "( *f* f) x \<approx> z \<longleftrightarrow> ( *f* (\<lambda>x. Re (f x))) x \<approx> hRe z \<and> ( *f* (\<lambda>x. Im (f x))) x \<approx> hIm z"
54  by (simp add: hcomplex_approx_iff starfun_Re starfun_Im)
55
56end
57