author | bulwahn |
Fri, 03 Dec 2010 08:40:47 +0100 | |
changeset 40905 | 647142607448 |
parent 39302 | d7728f65b353 |
child 58878 | f962e42e324d |
permissions | -rw-r--r-- |
27468 | 1 |
(* Title : Star.thy |
2 |
Author : Jacques D. Fleuriot |
|
3 |
Copyright : 1998 University of Cambridge |
|
4 |
Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4 |
|
5 |
*) |
|
6 |
||
7 |
header{*Star-Transforms in Non-Standard Analysis*} |
|
8 |
||
9 |
theory Star |
|
10 |
imports NSA |
|
11 |
begin |
|
12 |
||
13 |
definition |
|
14 |
(* internal sets *) |
|
15 |
starset_n :: "(nat => 'a set) => 'a star set" ("*sn* _" [80] 80) where |
|
16 |
"*sn* As = Iset (star_n As)" |
|
17 |
||
18 |
definition |
|
19 |
InternalSets :: "'a star set set" where |
|
37765 | 20 |
"InternalSets = {X. \<exists>As. X = *sn* As}" |
27468 | 21 |
|
22 |
definition |
|
23 |
(* nonstandard extension of function *) |
|
24 |
is_starext :: "['a star => 'a star, 'a => 'a] => bool" where |
|
37765 | 25 |
"is_starext F f = (\<forall>x y. \<exists>X \<in> Rep_star(x). \<exists>Y \<in> Rep_star(y). |
27468 | 26 |
((y = (F x)) = ({n. Y n = f(X n)} : FreeUltrafilterNat)))" |
27 |
||
28 |
definition |
|
29 |
(* internal functions *) |
|
30 |
starfun_n :: "(nat => ('a => 'b)) => 'a star => 'b star" ("*fn* _" [80] 80) where |
|
31 |
"*fn* F = Ifun (star_n F)" |
|
32 |
||
33 |
definition |
|
34 |
InternalFuns :: "('a star => 'b star) set" where |
|
37765 | 35 |
"InternalFuns = {X. \<exists>F. X = *fn* F}" |
27468 | 36 |
|
37 |
||
38 |
(*-------------------------------------------------------- |
|
39 |
Preamble - Pulling "EX" over "ALL" |
|
40 |
---------------------------------------------------------*) |
|
41 |
||
42 |
(* This proof does not need AC and was suggested by the |
|
43 |
referee for the JCM Paper: let f(x) be least y such |
|
44 |
that Q(x,y) |
|
45 |
*) |
|
46 |
lemma no_choice: "\<forall>x. \<exists>y. Q x y ==> \<exists>(f :: 'a => nat). \<forall>x. Q x (f x)" |
|
47 |
apply (rule_tac x = "%x. LEAST y. Q x y" in exI) |
|
48 |
apply (blast intro: LeastI) |
|
49 |
done |
|
50 |
||
51 |
subsection{*Properties of the Star-transform Applied to Sets of Reals*} |
|
52 |
||
53 |
lemma STAR_star_of_image_subset: "star_of ` A <= *s* A" |
|
54 |
by auto |
|
55 |
||
56 |
lemma STAR_hypreal_of_real_Int: "*s* X Int Reals = hypreal_of_real ` X" |
|
57 |
by (auto simp add: SReal_def) |
|
58 |
||
59 |
lemma STAR_star_of_Int: "*s* X Int Standard = star_of ` X" |
|
60 |
by (auto simp add: Standard_def) |
|
61 |
||
62 |
lemma lemma_not_hyprealA: "x \<notin> hypreal_of_real ` A ==> \<forall>y \<in> A. x \<noteq> hypreal_of_real y" |
|
63 |
by auto |
|
64 |
||
65 |
lemma lemma_not_starA: "x \<notin> star_of ` A ==> \<forall>y \<in> A. x \<noteq> star_of y" |
|
66 |
by auto |
|
67 |
||
68 |
lemma lemma_Compl_eq: "- {n. X n = xa} = {n. X n \<noteq> xa}" |
|
69 |
by auto |
|
70 |
||
71 |
lemma STAR_real_seq_to_hypreal: |
|
72 |
"\<forall>n. (X n) \<notin> M ==> star_n X \<notin> *s* M" |
|
73 |
apply (unfold starset_def star_of_def) |
|
74 |
apply (simp add: Iset_star_n) |
|
75 |
done |
|
76 |
||
77 |
lemma STAR_singleton: "*s* {x} = {star_of x}" |
|
78 |
by simp |
|
79 |
||
80 |
lemma STAR_not_mem: "x \<notin> F ==> star_of x \<notin> *s* F" |
|
81 |
by transfer |
|
82 |
||
83 |
lemma STAR_subset_closed: "[| x : *s* A; A <= B |] ==> x : *s* B" |
|
84 |
by (erule rev_subsetD, simp) |
|
85 |
||
86 |
text{*Nonstandard extension of a set (defined using a constant |
|
87 |
sequence) as a special case of an internal set*} |
|
88 |
||
89 |
lemma starset_n_starset: "\<forall>n. (As n = A) ==> *sn* As = *s* A" |
|
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39198
diff
changeset
|
90 |
apply (drule fun_eq_iff [THEN iffD2]) |
27468 | 91 |
apply (simp add: starset_n_def starset_def star_of_def) |
92 |
done |
|
93 |
||
94 |
||
95 |
(*----------------------------------------------------------------*) |
|
96 |
(* Theorems about nonstandard extensions of functions *) |
|
97 |
(*----------------------------------------------------------------*) |
|
98 |
||
99 |
(*----------------------------------------------------------------*) |
|
100 |
(* Nonstandard extension of a function (defined using a *) |
|
101 |
(* constant sequence) as a special case of an internal function *) |
|
102 |
(*----------------------------------------------------------------*) |
|
103 |
||
104 |
lemma starfun_n_starfun: "\<forall>n. (F n = f) ==> *fn* F = *f* f" |
|
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39198
diff
changeset
|
105 |
apply (drule fun_eq_iff [THEN iffD2]) |
27468 | 106 |
apply (simp add: starfun_n_def starfun_def star_of_def) |
107 |
done |
|
108 |
||
109 |
||
110 |
(* |
|
111 |
Prove that abs for hypreal is a nonstandard extension of abs for real w/o |
|
112 |
use of congruence property (proved after this for general |
|
113 |
nonstandard extensions of real valued functions). |
|
114 |
||
115 |
Proof now Uses the ultrafilter tactic! |
|
116 |
*) |
|
117 |
||
118 |
lemma hrabs_is_starext_rabs: "is_starext abs abs" |
|
119 |
apply (simp add: is_starext_def, safe) |
|
120 |
apply (rule_tac x=x in star_cases) |
|
121 |
apply (rule_tac x=y in star_cases) |
|
122 |
apply (unfold star_n_def, auto) |
|
123 |
apply (rule bexI, rule_tac [2] lemma_starrel_refl) |
|
124 |
apply (rule bexI, rule_tac [2] lemma_starrel_refl) |
|
125 |
apply (fold star_n_def) |
|
126 |
apply (unfold star_abs_def starfun_def star_of_def) |
|
127 |
apply (simp add: Ifun_star_n star_n_eq_iff) |
|
128 |
done |
|
129 |
||
130 |
text{*Nonstandard extension of functions*} |
|
131 |
||
132 |
lemma starfun: |
|
133 |
"( *f* f) (star_n X) = star_n (%n. f (X n))" |
|
134 |
by (rule starfun_star_n) |
|
135 |
||
136 |
lemma starfun_if_eq: |
|
137 |
"!!w. w \<noteq> star_of x |
|
138 |
==> ( *f* (\<lambda>z. if z = x then a else g z)) w = ( *f* g) w" |
|
139 |
by (transfer, simp) |
|
140 |
||
141 |
(*------------------------------------------- |
|
142 |
multiplication: ( *f) x ( *g) = *(f x g) |
|
143 |
------------------------------------------*) |
|
144 |
lemma starfun_mult: "!!x. ( *f* f) x * ( *f* g) x = ( *f* (%x. f x * g x)) x" |
|
145 |
by (transfer, rule refl) |
|
146 |
declare starfun_mult [symmetric, simp] |
|
147 |
||
148 |
(*--------------------------------------- |
|
149 |
addition: ( *f) + ( *g) = *(f + g) |
|
150 |
---------------------------------------*) |
|
151 |
lemma starfun_add: "!!x. ( *f* f) x + ( *f* g) x = ( *f* (%x. f x + g x)) x" |
|
152 |
by (transfer, rule refl) |
|
153 |
declare starfun_add [symmetric, simp] |
|
154 |
||
155 |
(*-------------------------------------------- |
|
156 |
subtraction: ( *f) + -( *g) = *(f + -g) |
|
157 |
-------------------------------------------*) |
|
158 |
lemma starfun_minus: "!!x. - ( *f* f) x = ( *f* (%x. - f x)) x" |
|
159 |
by (transfer, rule refl) |
|
160 |
declare starfun_minus [symmetric, simp] |
|
161 |
||
162 |
(*FIXME: delete*) |
|
163 |
lemma starfun_add_minus: "!!x. ( *f* f) x + -( *f* g) x = ( *f* (%x. f x + -g x)) x" |
|
164 |
by (transfer, rule refl) |
|
165 |
declare starfun_add_minus [symmetric, simp] |
|
166 |
||
167 |
lemma starfun_diff: "!!x. ( *f* f) x - ( *f* g) x = ( *f* (%x. f x - g x)) x" |
|
168 |
by (transfer, rule refl) |
|
169 |
declare starfun_diff [symmetric, simp] |
|
170 |
||
171 |
(*-------------------------------------- |
|
172 |
composition: ( *f) o ( *g) = *(f o g) |
|
173 |
---------------------------------------*) |
|
174 |
||
175 |
lemma starfun_o2: "(%x. ( *f* f) (( *f* g) x)) = *f* (%x. f (g x))" |
|
176 |
by (transfer, rule refl) |
|
177 |
||
178 |
lemma starfun_o: "( *f* f) o ( *f* g) = ( *f* (f o g))" |
|
179 |
by (transfer o_def, rule refl) |
|
180 |
||
181 |
text{*NS extension of constant function*} |
|
182 |
lemma starfun_const_fun [simp]: "!!x. ( *f* (%x. k)) x = star_of k" |
|
183 |
by (transfer, rule refl) |
|
184 |
||
185 |
text{*the NS extension of the identity function*} |
|
186 |
||
187 |
lemma starfun_Id [simp]: "!!x. ( *f* (%x. x)) x = x" |
|
188 |
by (transfer, rule refl) |
|
189 |
||
190 |
(* this is trivial, given starfun_Id *) |
|
191 |
lemma starfun_Idfun_approx: |
|
192 |
"x @= star_of a ==> ( *f* (%x. x)) x @= star_of a" |
|
193 |
by (simp only: starfun_Id) |
|
194 |
||
195 |
text{*The Star-function is a (nonstandard) extension of the function*} |
|
196 |
||
197 |
lemma is_starext_starfun: "is_starext ( *f* f) f" |
|
198 |
apply (simp add: is_starext_def, auto) |
|
199 |
apply (rule_tac x = x in star_cases) |
|
200 |
apply (rule_tac x = y in star_cases) |
|
201 |
apply (auto intro!: bexI [OF _ Rep_star_star_n] |
|
202 |
simp add: starfun star_n_eq_iff) |
|
203 |
done |
|
204 |
||
205 |
text{*Any nonstandard extension is in fact the Star-function*} |
|
206 |
||
207 |
lemma is_starfun_starext: "is_starext F f ==> F = *f* f" |
|
208 |
apply (simp add: is_starext_def) |
|
209 |
apply (rule ext) |
|
210 |
apply (rule_tac x = x in star_cases) |
|
211 |
apply (drule_tac x = x in spec) |
|
212 |
apply (drule_tac x = "( *f* f) x" in spec) |
|
213 |
apply (auto simp add: starfun_star_n) |
|
214 |
apply (simp add: star_n_eq_iff [symmetric]) |
|
215 |
apply (simp add: starfun_star_n [of f, symmetric]) |
|
216 |
done |
|
217 |
||
218 |
lemma is_starext_starfun_iff: "(is_starext F f) = (F = *f* f)" |
|
219 |
by (blast intro: is_starfun_starext is_starext_starfun) |
|
220 |
||
221 |
text{*extented function has same solution as its standard |
|
222 |
version for real arguments. i.e they are the same |
|
223 |
for all real arguments*} |
|
224 |
lemma starfun_eq: "( *f* f) (star_of a) = star_of (f a)" |
|
225 |
by (rule starfun_star_of) |
|
226 |
||
227 |
lemma starfun_approx: "( *f* f) (star_of a) @= star_of (f a)" |
|
228 |
by simp |
|
229 |
||
230 |
(* useful for NS definition of derivatives *) |
|
231 |
lemma starfun_lambda_cancel: |
|
232 |
"!!x'. ( *f* (%h. f (x + h))) x' = ( *f* f) (star_of x + x')" |
|
233 |
by (transfer, rule refl) |
|
234 |
||
235 |
lemma starfun_lambda_cancel2: |
|
236 |
"( *f* (%h. f(g(x + h)))) x' = ( *f* (f o g)) (star_of x + x')" |
|
237 |
by (unfold o_def, rule starfun_lambda_cancel) |
|
238 |
||
239 |
lemma starfun_mult_HFinite_approx: |
|
240 |
fixes l m :: "'a::real_normed_algebra star" |
|
241 |
shows "[| ( *f* f) x @= l; ( *f* g) x @= m; |
|
242 |
l: HFinite; m: HFinite |
|
243 |
|] ==> ( *f* (%x. f x * g x)) x @= l * m" |
|
244 |
apply (drule (3) approx_mult_HFinite) |
|
245 |
apply (auto intro: approx_HFinite [OF _ approx_sym]) |
|
246 |
done |
|
247 |
||
248 |
lemma starfun_add_approx: "[| ( *f* f) x @= l; ( *f* g) x @= m |
|
249 |
|] ==> ( *f* (%x. f x + g x)) x @= l + m" |
|
250 |
by (auto intro: approx_add) |
|
251 |
||
252 |
text{*Examples: hrabs is nonstandard extension of rabs |
|
253 |
inverse is nonstandard extension of inverse*} |
|
254 |
||
255 |
(* can be proved easily using theorem "starfun" and *) |
|
256 |
(* properties of ultrafilter as for inverse below we *) |
|
257 |
(* use the theorem we proved above instead *) |
|
258 |
||
259 |
lemma starfun_rabs_hrabs: "*f* abs = abs" |
|
260 |
by (simp only: star_abs_def) |
|
261 |
||
262 |
lemma starfun_inverse_inverse [simp]: "( *f* inverse) x = inverse(x)" |
|
263 |
by (simp only: star_inverse_def) |
|
264 |
||
265 |
lemma starfun_inverse: "!!x. inverse (( *f* f) x) = ( *f* (%x. inverse (f x))) x" |
|
266 |
by (transfer, rule refl) |
|
267 |
declare starfun_inverse [symmetric, simp] |
|
268 |
||
269 |
lemma starfun_divide: "!!x. ( *f* f) x / ( *f* g) x = ( *f* (%x. f x / g x)) x" |
|
270 |
by (transfer, rule refl) |
|
271 |
declare starfun_divide [symmetric, simp] |
|
272 |
||
273 |
lemma starfun_inverse2: "!!x. inverse (( *f* f) x) = ( *f* (%x. inverse (f x))) x" |
|
274 |
by (transfer, rule refl) |
|
275 |
||
276 |
text{*General lemma/theorem needed for proofs in elementary |
|
277 |
topology of the reals*} |
|
278 |
lemma starfun_mem_starset: |
|
279 |
"!!x. ( *f* f) x : *s* A ==> x : *s* {x. f x \<in> A}" |
|
280 |
by (transfer, simp) |
|
281 |
||
282 |
text{*Alternative definition for hrabs with rabs function |
|
283 |
applied entrywise to equivalence class representative. |
|
284 |
This is easily proved using starfun and ns extension thm*} |
|
285 |
lemma hypreal_hrabs: |
|
286 |
"abs (star_n X) = star_n (%n. abs (X n))" |
|
287 |
by (simp only: starfun_rabs_hrabs [symmetric] starfun) |
|
288 |
||
289 |
text{*nonstandard extension of set through nonstandard extension |
|
290 |
of rabs function i.e hrabs. A more general result should be |
|
291 |
where we replace rabs by some arbitrary function f and hrabs |
|
292 |
by its NS extenson. See second NS set extension below.*} |
|
293 |
lemma STAR_rabs_add_minus: |
|
294 |
"*s* {x. abs (x + - y) < r} = |
|
295 |
{x. abs(x + -star_of y) < star_of r}" |
|
296 |
by (transfer, rule refl) |
|
297 |
||
298 |
lemma STAR_starfun_rabs_add_minus: |
|
299 |
"*s* {x. abs (f x + - y) < r} = |
|
300 |
{x. abs(( *f* f) x + -star_of y) < star_of r}" |
|
301 |
by (transfer, rule refl) |
|
302 |
||
303 |
text{*Another characterization of Infinitesimal and one of @= relation. |
|
304 |
In this theory since @{text hypreal_hrabs} proved here. Maybe |
|
305 |
move both theorems??*} |
|
306 |
lemma Infinitesimal_FreeUltrafilterNat_iff2: |
|
307 |
"(star_n X \<in> Infinitesimal) = |
|
308 |
(\<forall>m. {n. norm(X n) < inverse(real(Suc m))} |
|
309 |
\<in> FreeUltrafilterNat)" |
|
310 |
by (simp add: Infinitesimal_hypreal_of_nat_iff star_of_def |
|
311 |
hnorm_def star_of_nat_def starfun_star_n |
|
312 |
star_n_inverse star_n_less real_of_nat_def) |
|
313 |
||
314 |
lemma HNatInfinite_inverse_Infinitesimal [simp]: |
|
315 |
"n \<in> HNatInfinite ==> inverse (hypreal_of_hypnat n) \<in> Infinitesimal" |
|
316 |
apply (cases n) |
|
317 |
apply (auto simp add: of_hypnat_def starfun_star_n real_of_nat_def [symmetric] star_n_inverse real_norm_def |
|
318 |
HNatInfinite_FreeUltrafilterNat_iff |
|
319 |
Infinitesimal_FreeUltrafilterNat_iff2) |
|
320 |
apply (drule_tac x="Suc m" in spec) |
|
321 |
apply (erule ultra, simp) |
|
322 |
done |
|
323 |
||
324 |
lemma approx_FreeUltrafilterNat_iff: "star_n X @= star_n Y = |
|
325 |
(\<forall>r>0. {n. norm (X n - Y n) < r} : FreeUltrafilterNat)" |
|
326 |
apply (subst approx_minus_iff) |
|
327 |
apply (rule mem_infmal_iff [THEN subst]) |
|
328 |
apply (simp add: star_n_diff) |
|
329 |
apply (simp add: Infinitesimal_FreeUltrafilterNat_iff) |
|
330 |
done |
|
331 |
||
332 |
lemma approx_FreeUltrafilterNat_iff2: "star_n X @= star_n Y = |
|
333 |
(\<forall>m. {n. norm (X n - Y n) < |
|
334 |
inverse(real(Suc m))} : FreeUltrafilterNat)" |
|
335 |
apply (subst approx_minus_iff) |
|
336 |
apply (rule mem_infmal_iff [THEN subst]) |
|
337 |
apply (simp add: star_n_diff) |
|
338 |
apply (simp add: Infinitesimal_FreeUltrafilterNat_iff2) |
|
339 |
done |
|
340 |
||
341 |
lemma inj_starfun: "inj starfun" |
|
342 |
apply (rule inj_onI) |
|
343 |
apply (rule ext, rule ccontr) |
|
344 |
apply (drule_tac x = "star_n (%n. xa)" in fun_cong) |
|
345 |
apply (auto simp add: starfun star_n_eq_iff) |
|
346 |
done |
|
347 |
||
348 |
end |