src/HOL/Hyperreal/Star.thy
author nipkow
Wed Aug 18 11:09:40 2004 +0200 (2004-08-18)
changeset 15140 322485b816ac
parent 15131 c69542757a4d
child 15169 2b5da07a0b89
permissions -rw-r--r--
import -> imports
     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 constdefs
    14     (* nonstandard extension of sets *)
    15     starset :: "real set => hypreal set"          ("*s* _" [80] 80)
    16     "*s* A  == {x. \<forall>X \<in> Rep_hypreal(x). {n::nat. X n  \<in> A}: FreeUltrafilterNat}"
    17 
    18     (* internal sets *)
    19     starset_n :: "(nat => real set) => hypreal set"        ("*sn* _" [80] 80)
    20     "*sn* As  == {x. \<forall>X \<in> Rep_hypreal(x). {n::nat. X n : (As n)}: FreeUltrafilterNat}"
    21 
    22     InternalSets :: "hypreal set set"
    23     "InternalSets == {X. \<exists>As. X = *sn* As}"
    24 
    25     (* nonstandard extension of function *)
    26     is_starext  :: "[hypreal => hypreal, real => real] => bool"
    27     "is_starext F f == (\<forall>x y. \<exists>X \<in> Rep_hypreal(x). \<exists>Y \<in> Rep_hypreal(y).
    28                         ((y = (F x)) = ({n. Y n = f(X n)} : FreeUltrafilterNat)))"
    29 
    30     starfun :: "(real => real) => hypreal => hypreal"       ("*f* _" [80] 80)
    31     "*f* f  == (%x. Abs_hypreal(\<Union>X \<in> Rep_hypreal(x). hyprel``{%n. f(X n)}))"
    32 
    33     (* internal functions *)
    34     starfun_n :: "(nat => (real => real)) => hypreal => hypreal"
    35                  ("*fn* _" [80] 80)
    36     "*fn* F  == (%x. Abs_hypreal(\<Union>X \<in> Rep_hypreal(x). hyprel``{%n. (F n)(X n)}))"
    37 
    38     InternalFuns :: "(hypreal => hypreal) set"
    39     "InternalFuns == {X. \<exists>F. X = *fn* F}"
    40 
    41 
    42 
    43 (*--------------------------------------------------------
    44    Preamble - Pulling "EX" over "ALL"
    45  ---------------------------------------------------------*)
    46 
    47 (* This proof does not need AC and was suggested by the
    48    referee for the JCM Paper: let f(x) be least y such
    49    that  Q(x,y)
    50 *)
    51 lemma no_choice: "\<forall>x. \<exists>y. Q x y ==> \<exists>(f :: nat => nat). \<forall>x. Q x (f x)"
    52 apply (rule_tac x = "%x. LEAST y. Q x y" in exI)
    53 apply (blast intro: LeastI)
    54 done
    55 
    56 (*------------------------------------------------------------
    57     Properties of the *-transform applied to sets of reals
    58  ------------------------------------------------------------*)
    59 
    60 lemma STAR_real_set: "*s*(UNIV::real set) = (UNIV::hypreal set)"
    61 by (simp add: starset_def)
    62 declare STAR_real_set [simp]
    63 
    64 lemma STAR_empty_set: "*s* {} = {}"
    65 by (simp add: starset_def)
    66 declare STAR_empty_set [simp]
    67 
    68 lemma STAR_Un: "*s* (A Un B) = *s* A Un *s* B"
    69 apply (auto simp add: starset_def)
    70   prefer 3 apply (blast intro: FreeUltrafilterNat_subset)
    71  prefer 2 apply (blast intro: FreeUltrafilterNat_subset)
    72 apply (drule FreeUltrafilterNat_Compl_mem)
    73 apply (drule bspec, assumption)
    74 apply (rule_tac z = x in eq_Abs_hypreal, auto, ultra)
    75 done
    76 
    77 lemma STAR_Int: "*s* (A Int B) = *s* A Int *s* B"
    78 apply (simp add: starset_def, auto)
    79 prefer 3 apply (blast intro: FreeUltrafilterNat_Int FreeUltrafilterNat_subset)
    80 apply (blast intro: FreeUltrafilterNat_subset)+
    81 done
    82 
    83 lemma STAR_Compl: "*s* -A = -( *s* A)"
    84 apply (auto simp add: starset_def)
    85 apply (rule_tac [!] z = x in eq_Abs_hypreal)
    86 apply (auto dest!: bspec, ultra)
    87 apply (drule FreeUltrafilterNat_Compl_mem, ultra)
    88 done
    89 
    90 lemma STAR_mem_Compl: "x \<notin> *s* F ==> x : *s* (- F)"
    91 by (auto simp add: STAR_Compl)
    92 
    93 lemma STAR_diff: "*s* (A - B) = *s* A - *s* B"
    94 by (auto simp add: Diff_eq STAR_Int STAR_Compl)
    95 
    96 lemma STAR_subset: "A <= B ==> *s* A <= *s* B"
    97 apply (simp add: starset_def)
    98 apply (blast intro: FreeUltrafilterNat_subset)+
    99 done
   100 
   101 lemma STAR_mem: "a  \<in> A ==> hypreal_of_real a : *s* A"
   102 apply (simp add: starset_def hypreal_of_real_def)
   103 apply (auto intro: FreeUltrafilterNat_subset)
   104 done
   105 
   106 lemma STAR_hypreal_of_real_image_subset: "hypreal_of_real ` A <= *s* A"
   107 apply (simp add: starset_def)
   108 apply (auto simp add: hypreal_of_real_def)
   109 apply (blast intro: FreeUltrafilterNat_subset)
   110 done
   111 
   112 lemma STAR_hypreal_of_real_Int: "*s* X Int Reals = hypreal_of_real ` X"
   113 apply (simp add: starset_def)
   114 apply (auto simp add: hypreal_of_real_def SReal_def)
   115 apply (simp add: hypreal_of_real_def [symmetric])
   116 apply (rule imageI, rule ccontr)
   117 apply (drule bspec)
   118 apply (rule lemma_hyprel_refl)
   119 prefer 2 apply (blast intro: FreeUltrafilterNat_subset, auto)
   120 done
   121 
   122 lemma lemma_not_hyprealA: "x \<notin> hypreal_of_real ` A ==> \<forall>y \<in> A. x \<noteq> hypreal_of_real y"
   123 by auto
   124 
   125 lemma lemma_Compl_eq: "- {n. X n = xa} = {n. X n \<noteq> xa}"
   126 by auto
   127 
   128 lemma STAR_real_seq_to_hypreal:
   129     "\<forall>n. (X n) \<notin> M
   130           ==> Abs_hypreal(hyprel``{X}) \<notin> *s* M"
   131 apply (simp add: starset_def)
   132 apply (rule bexI, rule_tac [2] lemma_hyprel_refl, auto)
   133 done
   134 
   135 lemma STAR_singleton: "*s* {x} = {hypreal_of_real x}"
   136 apply (simp add: starset_def)
   137 apply (auto simp add: hypreal_of_real_def)
   138 apply (rule_tac z = xa in eq_Abs_hypreal)
   139 apply (auto intro: FreeUltrafilterNat_subset)
   140 done
   141 declare STAR_singleton [simp]
   142 
   143 lemma STAR_not_mem: "x \<notin> F ==> hypreal_of_real x \<notin> *s* F"
   144 apply (auto simp add: starset_def hypreal_of_real_def)
   145 apply (rule bexI, rule_tac [2] lemma_hyprel_refl, auto)
   146 done
   147 
   148 lemma STAR_subset_closed: "[| x : *s* A; A <= B |] ==> x : *s* B"
   149 by (blast dest: STAR_subset)
   150 
   151 (*------------------------------------------------------------------
   152    Nonstandard extension of a set (defined using a constant
   153    sequence) as a special case of an internal set
   154  -----------------------------------------------------------------*)
   155 
   156 lemma starset_n_starset: "\<forall>n. (As n = A) ==> *sn* As = *s* A"
   157 by (simp add: starset_n_def starset_def)
   158 
   159 
   160 (*----------------------------------------------------------------*)
   161 (* Theorems about nonstandard extensions of functions             *)
   162 (*----------------------------------------------------------------*)
   163 
   164 (*----------------------------------------------------------------*)
   165 (* Nonstandard extension of a function (defined using a           *)
   166 (* constant sequence) as a special case of an internal function   *)
   167 (*----------------------------------------------------------------*)
   168 
   169 lemma starfun_n_starfun: "\<forall>n. (F n = f) ==> *fn* F = *f* f"
   170 by (simp add: starfun_n_def starfun_def)
   171 
   172 
   173 (*
   174    Prove that abs for hypreal is a nonstandard extension of abs for real w/o
   175    use of congruence property (proved after this for general
   176    nonstandard extensions of real valued functions). 
   177 
   178    Proof now Uses the ultrafilter tactic!
   179 *)
   180 
   181 lemma hrabs_is_starext_rabs: "is_starext abs abs"
   182 apply (simp add: is_starext_def, safe)
   183 apply (rule_tac z = x in eq_Abs_hypreal)
   184 apply (rule_tac z = y in eq_Abs_hypreal, auto)
   185 apply (rule bexI, rule_tac [2] lemma_hyprel_refl)
   186 apply (rule bexI, rule_tac [2] lemma_hyprel_refl)
   187 apply (auto dest!: spec 
   188             simp add: hypreal_minus abs_if hypreal_zero_def
   189                   hypreal_le hypreal_less)
   190 apply (arith | ultra)+
   191 done
   192 
   193 lemma Rep_hypreal_FreeUltrafilterNat:
   194      "[| X \<in> Rep_hypreal z; Y \<in> Rep_hypreal z |]
   195       ==> {n. X n = Y n} : FreeUltrafilterNat"
   196 apply (cases z)
   197 apply (auto, ultra)
   198 done
   199 
   200 (*-----------------------------------------------------------------------
   201     Nonstandard extension of functions- congruence
   202  -----------------------------------------------------------------------*)
   203 
   204 lemma starfun_congruent: "congruent hyprel (%X. hyprel``{%n. f (X n)})"
   205 by (simp add: congruent_def, auto, ultra)
   206 
   207 lemma starfun:
   208       "( *f* f) (Abs_hypreal(hyprel``{%n. X n})) =
   209        Abs_hypreal(hyprel `` {%n. f (X n)})"
   210 apply (simp add: starfun_def)
   211 apply (rule_tac f = Abs_hypreal in arg_cong)
   212 apply (simp add: hyprel_in_hypreal [THEN Abs_hypreal_inverse] 
   213                  UN_equiv_class [OF equiv_hyprel starfun_congruent])
   214 done
   215 
   216 lemma starfun_if_eq:
   217      "w \<noteq> hypreal_of_real x
   218        ==> ( *f* (\<lambda>z. if z = x then a else g z)) w = ( *f* g) w" 
   219 apply (cases w) 
   220 apply (simp add: hypreal_of_real_def starfun, ultra)
   221 done
   222 
   223 (*-------------------------------------------
   224   multiplication: ( *f) x ( *g) = *(f x g)
   225  ------------------------------------------*)
   226 lemma starfun_mult: "( *f* f) xa * ( *f* g) xa = ( *f* (%x. f x * g x)) xa"
   227 by (cases xa, simp add: starfun hypreal_mult)
   228 
   229 declare starfun_mult [symmetric, simp]
   230 
   231 (*---------------------------------------
   232   addition: ( *f) + ( *g) = *(f + g)
   233  ---------------------------------------*)
   234 lemma starfun_add: "( *f* f) xa + ( *f* g) xa = ( *f* (%x. f x + g x)) xa"
   235 by (cases xa, simp add: starfun hypreal_add)
   236 declare starfun_add [symmetric, simp]
   237 
   238 (*--------------------------------------------
   239   subtraction: ( *f) + -( *g) = *(f + -g)
   240  -------------------------------------------*)
   241 
   242 lemma starfun_minus: "- ( *f* f) x = ( *f* (%x. - f x)) x"
   243 apply (cases x)
   244 apply (auto simp add: starfun hypreal_minus)
   245 done
   246 declare starfun_minus [symmetric, simp]
   247 
   248 (*FIXME: delete*)
   249 lemma starfun_add_minus: "( *f* f) xa + -( *f* g) xa = ( *f* (%x. f x + -g x)) xa"
   250 apply (simp (no_asm))
   251 done
   252 declare starfun_add_minus [symmetric, simp]
   253 
   254 lemma starfun_diff:
   255   "( *f* f) xa  - ( *f* g) xa = ( *f* (%x. f x - g x)) xa"
   256 apply (simp add: diff_minus)
   257 done
   258 declare starfun_diff [symmetric, simp]
   259 
   260 (*--------------------------------------
   261   composition: ( *f) o ( *g) = *(f o g)
   262  ---------------------------------------*)
   263 
   264 lemma starfun_o2: "(%x. ( *f* f) (( *f* g) x)) = *f* (%x. f (g x))"
   265 apply (rule ext)
   266 apply (rule_tac z = x in eq_Abs_hypreal)
   267 apply (auto simp add: starfun)
   268 done
   269 
   270 lemma starfun_o: "( *f* f) o ( *f* g) = ( *f* (f o g))"
   271 apply (simp add: o_def)
   272 apply (simp (no_asm) add: starfun_o2)
   273 done
   274 
   275 (*--------------------------------------
   276   NS extension of constant function
   277  --------------------------------------*)
   278 lemma starfun_const_fun: "( *f* (%x. k)) xa = hypreal_of_real  k"
   279 apply (cases xa)
   280 apply (auto simp add: starfun hypreal_of_real_def)
   281 done
   282 
   283 declare starfun_const_fun [simp]
   284 
   285 (*----------------------------------------------------
   286    the NS extension of the identity function
   287  ----------------------------------------------------*)
   288 
   289 lemma starfun_Idfun_approx: "x @= hypreal_of_real a ==> ( *f* (%x. x)) x @= hypreal_of_real  a"
   290 apply (cases x)
   291 apply (auto simp add: starfun)
   292 done
   293 
   294 lemma starfun_Id: "( *f* (%x. x)) x = x"
   295 apply (cases x)
   296 apply (auto simp add: starfun)
   297 done
   298 declare starfun_Id [simp]
   299 
   300 (*----------------------------------------------------------------------
   301       the *-function is a (nonstandard) extension of the function
   302  ----------------------------------------------------------------------*)
   303 
   304 lemma is_starext_starfun: "is_starext ( *f* f) f"
   305 apply (simp add: is_starext_def, auto)
   306 apply (rule_tac z = x in eq_Abs_hypreal)
   307 apply (rule_tac z = y in eq_Abs_hypreal)
   308 apply (auto intro!: bexI simp add: starfun)
   309 done
   310 
   311 (*----------------------------------------------------------------------
   312      Any nonstandard extension is in fact the *-function
   313  ----------------------------------------------------------------------*)
   314 
   315 lemma is_starfun_starext: "is_starext F f ==> F = *f* f"
   316 apply (simp add: is_starext_def)
   317 apply (rule ext)
   318 apply (rule_tac z = x in eq_Abs_hypreal)
   319 apply (drule_tac x = x in spec)
   320 apply (drule_tac x = "( *f* f) x" in spec)
   321 apply (auto dest!: FreeUltrafilterNat_Compl_mem simp add: starfun, ultra)
   322 done
   323 
   324 lemma is_starext_starfun_iff: "(is_starext F f) = (F = *f* f)"
   325 by (blast intro: is_starfun_starext is_starext_starfun)
   326 
   327 (*--------------------------------------------------------
   328    extented function has same solution as its standard
   329    version for real arguments. i.e they are the same
   330    for all real arguments
   331  -------------------------------------------------------*)
   332 lemma starfun_eq: "( *f* f) (hypreal_of_real a) = hypreal_of_real (f a)"
   333 by (auto simp add: starfun hypreal_of_real_def)
   334 
   335 declare starfun_eq [simp]
   336 
   337 lemma starfun_approx: "( *f* f) (hypreal_of_real a) @= hypreal_of_real (f a)"
   338 by auto
   339 
   340 (* useful for NS definition of derivatives *)
   341 lemma starfun_lambda_cancel: "( *f* (%h. f (x + h))) xa  = ( *f* f) (hypreal_of_real  x + xa)"
   342 apply (cases xa)
   343 apply (auto simp add: starfun hypreal_of_real_def hypreal_add)
   344 done
   345 
   346 lemma starfun_lambda_cancel2: "( *f* (%h. f(g(x + h)))) xa = ( *f* (f o g)) (hypreal_of_real x + xa)"
   347 apply (cases xa)
   348 apply (auto simp add: starfun hypreal_of_real_def hypreal_add)
   349 done
   350 
   351 lemma starfun_mult_HFinite_approx: "[| ( *f* f) xa @= l; ( *f* g) xa @= m;
   352                   l: HFinite; m: HFinite
   353                |] ==>  ( *f* (%x. f x * g x)) xa @= l * m"
   354 apply (drule approx_mult_HFinite, assumption+)
   355 apply (auto intro: approx_HFinite [OF _ approx_sym])
   356 done
   357 
   358 lemma starfun_add_approx: "[| ( *f* f) xa @= l; ( *f* g) xa @= m
   359                |] ==>  ( *f* (%x. f x + g x)) xa @= l + m"
   360 apply (auto intro: approx_add)
   361 done
   362 
   363 (*----------------------------------------------------
   364     Examples: hrabs is nonstandard extension of rabs
   365               inverse is nonstandard extension of inverse
   366  ---------------------------------------------------*)
   367 
   368 (* can be proved easily using theorem "starfun" and *)
   369 (* properties of ultrafilter as for inverse below we  *)
   370 (* use the theorem we proved above instead          *)
   371 
   372 lemma starfun_rabs_hrabs: "*f* abs = abs"
   373 by (rule hrabs_is_starext_rabs [THEN is_starext_starfun_iff [THEN iffD1], symmetric])
   374 
   375 lemma starfun_inverse_inverse: "( *f* inverse) x = inverse(x)"
   376 apply (cases x)
   377 apply (auto simp add: starfun hypreal_inverse hypreal_zero_def)
   378 done
   379 declare starfun_inverse_inverse [simp]
   380 
   381 lemma starfun_inverse: "inverse (( *f* f) x) = ( *f* (%x. inverse (f x))) x"
   382 apply (cases x)
   383 apply (auto simp add: starfun hypreal_inverse)
   384 done
   385 declare starfun_inverse [symmetric, simp]
   386 
   387 lemma starfun_divide: "( *f* f) xa  / ( *f* g) xa = ( *f* (%x. f x / g x)) xa"
   388 by (simp add: divide_inverse)
   389 declare starfun_divide [symmetric, simp]
   390 
   391 lemma starfun_inverse2: "inverse (( *f* f) x) = ( *f* (%x. inverse (f x))) x"
   392 apply (cases x)
   393 apply (auto intro: FreeUltrafilterNat_subset dest!: FreeUltrafilterNat_Compl_mem simp add: starfun hypreal_inverse hypreal_zero_def)
   394 done
   395 
   396 (*-------------------------------------------------------------
   397     General lemma/theorem needed for proofs in elementary
   398     topology of the reals
   399  ------------------------------------------------------------*)
   400 lemma starfun_mem_starset:
   401       "( *f* f) x : *s* A ==> x : *s* {x. f x  \<in> A}"
   402 apply (simp add: starset_def)
   403 apply (cases x)
   404 apply (auto simp add: starfun)
   405 apply (rename_tac "X")
   406 apply (drule_tac x = "%n. f (X n) " in bspec)
   407 apply (auto, ultra)
   408 done
   409 
   410 (*------------------------------------------------------------
   411    Alternative definition for hrabs with rabs function
   412    applied entrywise to equivalence class representative.
   413    This is easily proved using starfun and ns extension thm
   414  ------------------------------------------------------------*)
   415 lemma hypreal_hrabs: "abs (Abs_hypreal (hyprel `` {X})) =
   416                   Abs_hypreal(hyprel `` {%n. abs (X n)})"
   417 apply (simp (no_asm) add: starfun_rabs_hrabs [symmetric] starfun)
   418 done
   419 
   420 (*----------------------------------------------------------------
   421    nonstandard extension of set through nonstandard extension
   422    of rabs function i.e hrabs. A more general result should be
   423    where we replace rabs by some arbitrary function f and hrabs
   424    by its NS extenson ( *f* f). See second NS set extension below.
   425  ----------------------------------------------------------------*)
   426 lemma STAR_rabs_add_minus:
   427    "*s* {x. abs (x + - y) < r} =
   428      {x. abs(x + -hypreal_of_real y) < hypreal_of_real r}"
   429 apply (simp add: starset_def, safe)
   430 apply (rule_tac [!] z = x in eq_Abs_hypreal)
   431 apply (auto intro!: exI dest!: bspec simp add: hypreal_minus hypreal_of_real_def hypreal_add hypreal_hrabs hypreal_less, ultra)
   432 done
   433 
   434 lemma STAR_starfun_rabs_add_minus:
   435   "*s* {x. abs (f x + - y) < r} =
   436        {x. abs(( *f* f) x + -hypreal_of_real y) < hypreal_of_real r}"
   437 apply (simp add: starset_def, safe)
   438 apply (rule_tac [!] z = x in eq_Abs_hypreal)
   439 apply (auto intro!: exI dest!: bspec simp add: hypreal_minus hypreal_of_real_def hypreal_add hypreal_hrabs hypreal_less starfun, ultra)
   440 done
   441 
   442 (*-------------------------------------------------------------------
   443    Another characterization of Infinitesimal and one of @= relation.
   444    In this theory since hypreal_hrabs proved here. (To Check:) Maybe
   445    move both if possible?
   446  -------------------------------------------------------------------*)
   447 lemma Infinitesimal_FreeUltrafilterNat_iff2:
   448      "(x \<in> Infinitesimal) =
   449       (\<exists>X \<in> Rep_hypreal(x).
   450         \<forall>m. {n. abs(X n) < inverse(real(Suc m))}
   451                 \<in>  FreeUltrafilterNat)"
   452 apply (cases x)
   453 apply (auto intro!: bexI lemma_hyprel_refl 
   454             simp add: Infinitesimal_hypreal_of_nat_iff hypreal_of_real_def
   455      hypreal_inverse hypreal_hrabs hypreal_less hypreal_of_nat_eq)
   456 apply (drule_tac x = n in spec, ultra)
   457 done
   458 
   459 lemma approx_FreeUltrafilterNat_iff: "(Abs_hypreal(hyprel``{X}) @= Abs_hypreal(hyprel``{Y})) =
   460       (\<forall>m. {n. abs (X n + - Y n) <
   461                   inverse(real(Suc m))} : FreeUltrafilterNat)"
   462 apply (subst approx_minus_iff)
   463 apply (rule mem_infmal_iff [THEN subst])
   464 apply (auto simp add: hypreal_minus hypreal_add Infinitesimal_FreeUltrafilterNat_iff2)
   465 apply (drule_tac x = m in spec, ultra)
   466 done
   467 
   468 lemma inj_starfun: "inj starfun"
   469 apply (rule inj_onI)
   470 apply (rule ext, rule ccontr)
   471 apply (drule_tac x = "Abs_hypreal (hyprel ``{%n. xa}) " in fun_cong)
   472 apply (auto simp add: starfun)
   473 done
   474 
   475 ML
   476 {*
   477 val starset_def = thm"starset_def";
   478 val starset_n_def = thm"starset_n_def";
   479 val InternalSets_def = thm"InternalSets_def";
   480 val is_starext_def = thm"is_starext_def";
   481 val starfun_def = thm"starfun_def";
   482 val starfun_n_def = thm"starfun_n_def";
   483 val InternalFuns_def = thm"InternalFuns_def";
   484 
   485 val no_choice = thm "no_choice";
   486 val STAR_real_set = thm "STAR_real_set";
   487 val STAR_empty_set = thm "STAR_empty_set";
   488 val STAR_Un = thm "STAR_Un";
   489 val STAR_Int = thm "STAR_Int";
   490 val STAR_Compl = thm "STAR_Compl";
   491 val STAR_mem_Compl = thm "STAR_mem_Compl";
   492 val STAR_diff = thm "STAR_diff";
   493 val STAR_subset = thm "STAR_subset";
   494 val STAR_mem = thm "STAR_mem";
   495 val STAR_hypreal_of_real_image_subset = thm "STAR_hypreal_of_real_image_subset";
   496 val STAR_hypreal_of_real_Int = thm "STAR_hypreal_of_real_Int";
   497 val STAR_real_seq_to_hypreal = thm "STAR_real_seq_to_hypreal";
   498 val STAR_singleton = thm "STAR_singleton";
   499 val STAR_not_mem = thm "STAR_not_mem";
   500 val STAR_subset_closed = thm "STAR_subset_closed";
   501 val starset_n_starset = thm "starset_n_starset";
   502 val starfun_n_starfun = thm "starfun_n_starfun";
   503 val hrabs_is_starext_rabs = thm "hrabs_is_starext_rabs";
   504 val Rep_hypreal_FreeUltrafilterNat = thm "Rep_hypreal_FreeUltrafilterNat";
   505 val starfun_congruent = thm "starfun_congruent";
   506 val starfun = thm "starfun";
   507 val starfun_mult = thm "starfun_mult";
   508 val starfun_add = thm "starfun_add";
   509 val starfun_minus = thm "starfun_minus";
   510 val starfun_add_minus = thm "starfun_add_minus";
   511 val starfun_diff = thm "starfun_diff";
   512 val starfun_o2 = thm "starfun_o2";
   513 val starfun_o = thm "starfun_o";
   514 val starfun_const_fun = thm "starfun_const_fun";
   515 val starfun_Idfun_approx = thm "starfun_Idfun_approx";
   516 val starfun_Id = thm "starfun_Id";
   517 val is_starext_starfun = thm "is_starext_starfun";
   518 val is_starfun_starext = thm "is_starfun_starext";
   519 val is_starext_starfun_iff = thm "is_starext_starfun_iff";
   520 val starfun_eq = thm "starfun_eq";
   521 val starfun_approx = thm "starfun_approx";
   522 val starfun_lambda_cancel = thm "starfun_lambda_cancel";
   523 val starfun_lambda_cancel2 = thm "starfun_lambda_cancel2";
   524 val starfun_mult_HFinite_approx = thm "starfun_mult_HFinite_approx";
   525 val starfun_add_approx = thm "starfun_add_approx";
   526 val starfun_rabs_hrabs = thm "starfun_rabs_hrabs";
   527 val starfun_inverse_inverse = thm "starfun_inverse_inverse";
   528 val starfun_inverse = thm "starfun_inverse";
   529 val starfun_divide = thm "starfun_divide";
   530 val starfun_inverse2 = thm "starfun_inverse2";
   531 val starfun_mem_starset = thm "starfun_mem_starset";
   532 val hypreal_hrabs = thm "hypreal_hrabs";
   533 val STAR_rabs_add_minus = thm "STAR_rabs_add_minus";
   534 val STAR_starfun_rabs_add_minus = thm "STAR_starfun_rabs_add_minus";
   535 val Infinitesimal_FreeUltrafilterNat_iff2 = thm "Infinitesimal_FreeUltrafilterNat_iff2";
   536 val approx_FreeUltrafilterNat_iff = thm "approx_FreeUltrafilterNat_iff";
   537 val inj_starfun = thm "inj_starfun";
   538 *}
   539 
   540 end