src/HOL/NSA/Star.thy
author huffman
Fri Mar 30 12:32:35 2012 +0200 (2012-03-30)
changeset 47220 52426c62b5d0
parent 39302 d7728f65b353
child 58878 f962e42e324d
permissions -rw-r--r--
replace lemmas eval_nat_numeral with a simpler reformulation
     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
    20   "InternalSets = {X. \<exists>As. X = *sn* As}"
    21 
    22 definition
    23   (* nonstandard extension of function *)
    24   is_starext  :: "['a star => 'a star, 'a => 'a] => bool" where
    25   "is_starext F f = (\<forall>x y. \<exists>X \<in> Rep_star(x). \<exists>Y \<in> Rep_star(y).
    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
    35   "InternalFuns = {X. \<exists>F. X = *fn* F}"
    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"
    90 apply (drule fun_eq_iff [THEN iffD2])
    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"
   105 apply (drule fun_eq_iff [THEN iffD2])
   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