--- a/src/HOL/Real/Hyperreal/Star.ML Sat Dec 30 22:03:46 2000 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,490 +0,0 @@
-(* Title : STAR.ML
- Author : Jacques D. Fleuriot
- Copyright : 1998 University of Cambridge
- Description : *-transforms
-*)
-
-(*--------------------------------------------------------
- Preamble - Pulling "EX" over "ALL"
- ---------------------------------------------------------*)
-
-(* This proof does not need AC and was suggested by the
- referee for the JCM Paper: let f(x) be least y such
- that Q(x,y)
-*)
-Goal "ALL x. EX y. Q x y ==> EX (f :: nat => nat). ALL x. Q x (f x)";
-by (res_inst_tac [("x","%x. LEAST y. Q x y")] exI 1);
-by (blast_tac (claset() addIs [LeastI]) 1);
-qed "no_choice";
-
-(*------------------------------------------------------------
- Properties of the *-transform applied to sets of reals
- ------------------------------------------------------------*)
-
-Goalw [starset_def] "*s*(UNIV::real set) = (UNIV::hypreal set)";
-by (Auto_tac);
-qed "STAR_real_set";
-Addsimps [STAR_real_set];
-
-Goalw [starset_def] "*s* {} = {}";
-by (Step_tac 1);
-by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
-by (dres_inst_tac [("x","%n. xa n")] bspec 1);
-by (Auto_tac);
-qed "STAR_empty_set";
-Addsimps [STAR_empty_set];
-
-Goalw [starset_def] "*s* (A Un B) = *s* A Un *s* B";
-by (Auto_tac);
-by (REPEAT(blast_tac (claset() addIs [FreeUltrafilterNat_subset]) 2));
-by (dtac FreeUltrafilterNat_Compl_mem 1);
-by (dtac bspec 1 THEN assume_tac 1);
-by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
-by (Auto_tac);
-by (Fuf_tac 1);
-qed "STAR_Un";
-
-Goalw [starset_def] "*s* (A Int B) = *s* A Int *s* B";
-by (Auto_tac);
-by (blast_tac (claset() addIs [FreeUltrafilterNat_Int,
- FreeUltrafilterNat_subset]) 3);
-by (REPEAT(blast_tac (claset() addIs [FreeUltrafilterNat_subset]) 1));
-qed "STAR_Int";
-
-Goalw [starset_def] "*s* -A = -(*s* A)";
-by (Auto_tac);
-by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
-by (res_inst_tac [("z","x")] eq_Abs_hypreal 2);
-by (REPEAT(Step_tac 1) THEN Auto_tac);
-by (Fuf_empty_tac 1);
-by (dtac FreeUltrafilterNat_Compl_mem 1);
-by (Fuf_tac 1);
-qed "STAR_Compl";
-
-goal Set.thy "(A - B) = (A Int (- B))";
-by (Step_tac 1);
-qed "set_diff_iff2";
-
-Goal "x ~: *s* F ==> x : *s* (- F)";
-by (auto_tac (claset(),simpset() addsimps [STAR_Compl]));
-qed "STAR_mem_Compl";
-
-Goal "*s* (A - B) = *s* A - *s* B";
-by (auto_tac (claset(),simpset() addsimps
- [set_diff_iff2,STAR_Int,STAR_Compl]));
-qed "STAR_diff";
-
-Goalw [starset_def] "A <= B ==> *s* A <= *s* B";
-by (REPEAT(blast_tac (claset() addIs [FreeUltrafilterNat_subset]) 1));
-qed "STAR_subset";
-
-Goalw [starset_def,hypreal_of_real_def]
- "a : A ==> hypreal_of_real a : *s* A";
-by (auto_tac (claset() addIs [FreeUltrafilterNat_subset],simpset()));
-qed "STAR_mem";
-
-Goalw [starset_def] "hypreal_of_real `` A <= *s* A";
-by (auto_tac (claset(),simpset() addsimps [hypreal_of_real_def]));
-by (blast_tac (claset() addIs [FreeUltrafilterNat_subset]) 1);
-qed "STAR_hypreal_of_real_image_subset";
-
-Goalw [starset_def] "*s* X Int SReal = hypreal_of_real `` X";
-by (auto_tac (claset(),simpset() addsimps [hypreal_of_real_def,SReal_def]));
-by (fold_tac [hypreal_of_real_def]);
-by (rtac imageI 1 THEN rtac ccontr 1);
-by (dtac bspec 1);
-by (rtac lemma_hyprel_refl 1);
-by (blast_tac (claset() addIs [FreeUltrafilterNat_subset]) 2);
-by (Auto_tac);
-qed "STAR_hypreal_of_real_Int";
-
-Goal "x ~: hypreal_of_real `` A ==> ALL y: A. x ~= hypreal_of_real y";
-by (Auto_tac);
-qed "lemma_not_hyprealA";
-
-Goal "- {n. X n = xa} = {n. X n ~= xa}";
-by (Auto_tac);
-qed "lemma_Compl_eq";
-
-Goalw [starset_def]
- "ALL n. (X n) ~: M \
-\ ==> Abs_hypreal(hyprel^^{X}) ~: *s* M";
-by (Auto_tac THEN rtac bexI 1 THEN rtac lemma_hyprel_refl 2);
-by (Auto_tac);
-qed "STAR_real_seq_to_hypreal";
-
-Goalw [starset_def] "*s* {x} = {hypreal_of_real x}";
-by (auto_tac (claset(),simpset() addsimps [hypreal_of_real_def]));
-by (res_inst_tac [("z","xa")] eq_Abs_hypreal 1);
-by (auto_tac (claset() addIs [FreeUltrafilterNat_subset],simpset()));
-qed "STAR_singleton";
-Addsimps [STAR_singleton];
-
-Goal "x ~: F ==> hypreal_of_real x ~: *s* F";
-by (auto_tac (claset(),simpset() addsimps
- [starset_def,hypreal_of_real_def]));
-by (rtac bexI 1 THEN rtac lemma_hyprel_refl 2);
-by (Auto_tac);
-qed "STAR_not_mem";
-
-Goal "[| x : *s* A; A <= B |] ==> x : *s* B";
-by (blast_tac (claset() addDs [STAR_subset]) 1);
-qed "STAR_subset_closed";
-
-(*------------------------------------------------------------------
- Nonstandard extension of a set (defined using a constant
- sequence) as a special case of an internal set
- -----------------------------------------------------------------*)
-
-Goalw [starset_n_def,starset_def]
- "ALL n. (As n = A) ==> *sn* As = *s* A";
-by (Auto_tac);
-qed "starset_n_starset";
-
-
-(*----------------------------------------------------------------*)
-(* Theorems about nonstandard extensions of functions *)
-(*----------------------------------------------------------------*)
-
-(*----------------------------------------------------------------*)
-(* Nonstandard extension of a function (defined using a *)
-(* constant sequence) as a special case of an internal function *)
-(*----------------------------------------------------------------*)
-
-Goalw [starfun_n_def,starfun_def]
- "ALL n. (F n = f) ==> *fn* F = *f* f";
-by (Auto_tac);
-qed "starfun_n_starfun";
-
-
-(*
- Prove that hrabs is a nonstandard extension of rabs without
- use of congruence property (proved after this for general
- nonstandard extensions of real valued functions). This makes
- proof much longer- see comments at end of HREALABS.thy where
- we proved a congruence theorem for hrabs.
-
- NEW!!! No need to prove all the lemmas anymore. Use the ultrafilter
- tactic!
-*)
-
-Goalw [is_starext_def] "is_starext abs abs";
-by (Step_tac 1);
-by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
-by (res_inst_tac [("z","y")] eq_Abs_hypreal 1);
-by Auto_tac;
-by (rtac bexI 1 THEN rtac lemma_hyprel_refl 2);
-by (rtac bexI 1 THEN rtac lemma_hyprel_refl 2);
-by (auto_tac (claset() addSDs [spec],simpset() addsimps [hypreal_minus,hrabs_def,
- hypreal_zero_def,hypreal_le_def,hypreal_less_def]));
-by (TRYALL(Ultra_tac));
-by (TRYALL(arith_tac));
-qed "hrabs_is_starext_rabs";
-
-Goal "[| X: Rep_hypreal z; Y: Rep_hypreal z |] \
-\ ==> {n. X n = Y n} : FreeUltrafilterNat";
-by (res_inst_tac [("z","z")] eq_Abs_hypreal 1);
-by (Auto_tac THEN Fuf_tac 1);
-qed "Rep_hypreal_FreeUltrafilterNat";
-
-(*-----------------------------------------------------------------------
- Nonstandard extension of functions- congruence
- -----------------------------------------------------------------------*)
-
-Goalw [congruent_def] "congruent hyprel (%X. hyprel^^{%n. f (X n)})";
-by (safe_tac (claset()));
-by (ALLGOALS(Fuf_tac));
-qed "starfun_congruent";
-
-Goalw [starfun_def]
- "(*f* f) (Abs_hypreal(hyprel^^{%n. X n})) = \
-\ Abs_hypreal(hyprel ^^ {%n. f (X n)})";
-by (res_inst_tac [("f","Abs_hypreal")] arg_cong 1);
-by (simp_tac (simpset() addsimps
- [hyprel_in_hypreal RS Abs_hypreal_inverse,[equiv_hyprel,
- starfun_congruent] MRS UN_equiv_class]) 1);
-qed "starfun";
-
-(*-------------------------------------------
- multiplication: ( *f ) x ( *g ) = *(f x g)
- ------------------------------------------*)
-Goal "(*f* f) xa * (*f* g) xa = (*f* (%x. f x * g x)) xa";
-by (res_inst_tac [("z","xa")] eq_Abs_hypreal 1);
-by (auto_tac (claset(),simpset() addsimps [starfun,hypreal_mult]));
-qed "starfun_mult";
-Addsimps [starfun_mult RS sym];
-
-(*---------------------------------------
- addition: ( *f ) + ( *g ) = *(f + g)
- ---------------------------------------*)
-Goal "(*f* f) xa + (*f* g) xa = (*f* (%x. f x + g x)) xa";
-by (res_inst_tac [("z","xa")] eq_Abs_hypreal 1);
-by (auto_tac (claset(),simpset() addsimps [starfun,hypreal_add]));
-qed "starfun_add";
-Addsimps [starfun_add RS sym];
-
-(*--------------------------------------------
- subtraction: ( *f ) + -( *g ) = *(f + -g)
- -------------------------------------------*)
-
-Goal "- (*f* f) x = (*f* (%x. - f x)) x";
-by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
-by (auto_tac (claset(),simpset() addsimps [starfun, hypreal_minus]));
-qed "starfun_minus";
-Addsimps [starfun_minus RS sym];
-
-(*FIXME: delete*)
-Goal "(*f* f) xa + -(*f* g) xa = (*f* (%x. f x + -g x)) xa";
-by (Simp_tac 1);
-qed "starfun_add_minus";
-Addsimps [starfun_add_minus RS sym];
-
-Goalw [hypreal_diff_def,real_diff_def]
- "(*f* f) xa - (*f* g) xa = (*f* (%x. f x - g x)) xa";
-by (rtac starfun_add_minus 1);
-qed "starfun_diff";
-Addsimps [starfun_diff RS sym];
-
-(*--------------------------------------
- composition: ( *f ) o ( *g ) = *(f o g)
- ---------------------------------------*)
-
-Goal "(%x. (*f* f) ((*f* g) x)) = *f* (%x. f (g x))";
-by (rtac ext 1);
-by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
-by (auto_tac (claset(),simpset() addsimps [starfun]));
-qed "starfun_o2";
-
-Goalw [o_def] "(*f* f) o (*f* g) = (*f* (f o g))";
-by (simp_tac (simpset() addsimps [starfun_o2]) 1);
-qed "starfun_o";
-
-(*--------------------------------------
- NS extension of constant function
- --------------------------------------*)
-Goal "(*f* (%x. k)) xa = hypreal_of_real k";
-by (res_inst_tac [("z","xa")] eq_Abs_hypreal 1);
-by (auto_tac (claset(),simpset() addsimps [starfun,
- hypreal_of_real_def]));
-qed "starfun_const_fun";
-
-Addsimps [starfun_const_fun];
-
-(*----------------------------------------------------
- the NS extension of the identity function
- ----------------------------------------------------*)
-
-Goal "x @= hypreal_of_real a ==> (*f* (%x. x)) x @= hypreal_of_real a";
-by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
-by (auto_tac (claset(),simpset() addsimps [starfun]));
-qed "starfun_Idfun_inf_close";
-
-Goal "(*f* (%x. x)) x = x";
-by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
-by (auto_tac (claset(),simpset() addsimps [starfun]));
-qed "starfun_Id";
-Addsimps [starfun_Id];
-
-(*----------------------------------------------------------------------
- the *-function is a (nonstandard) extension of the function
- ----------------------------------------------------------------------*)
-
-Goalw [is_starext_def] "is_starext (*f* f) f";
-by (Auto_tac);
-by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
-by (res_inst_tac [("z","y")] eq_Abs_hypreal 1);
-by (auto_tac (claset() addSIs [bexI] ,simpset() addsimps [starfun]));
-qed "is_starext_starfun";
-
-(*----------------------------------------------------------------------
- Any nonstandard extension is in fact the *-function
- ----------------------------------------------------------------------*)
-
-Goalw [is_starext_def] "is_starext F f ==> F = *f* f";
-by (rtac ext 1);
-by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
-by (dres_inst_tac [("x","x")] spec 1);
-by (dres_inst_tac [("x","(*f* f) x")] spec 1);
-by (auto_tac (claset() addSDs [FreeUltrafilterNat_Compl_mem],
- simpset() addsimps [starfun]));
-by (Fuf_empty_tac 1);
-qed "is_starfun_starext";
-
-Goal "(is_starext F f) = (F = *f* f)";
-by (blast_tac (claset() addIs [is_starfun_starext,is_starext_starfun]) 1);
-qed "is_starext_starfun_iff";
-
-(*--------------------------------------------------------
- extented function has same solution as its standard
- version for real arguments. i.e they are the same
- for all real arguments
- -------------------------------------------------------*)
-Goal "(*f* f) (hypreal_of_real a) = hypreal_of_real (f a)";
-by (auto_tac (claset(),simpset() addsimps
- [starfun,hypreal_of_real_def]));
-qed "starfun_eq";
-
-Addsimps [starfun_eq];
-
-Goal "(*f* f) (hypreal_of_real a) @= hypreal_of_real (f a)";
-by (Auto_tac);
-qed "starfun_inf_close";
-
-(* useful for NS definition of derivatives *)
-Goal "(*f* (%h. f (x + h))) xa = (*f* f) (hypreal_of_real x + xa)";
-by (res_inst_tac [("z","xa")] eq_Abs_hypreal 1);
-by (auto_tac (claset(),simpset() addsimps [starfun,
- hypreal_of_real_def,hypreal_add]));
-qed "starfun_lambda_cancel";
-
-Goal "(*f* (%h. f(g(x + h)))) xa = (*f* (f o g)) (hypreal_of_real x + xa)";
-by (res_inst_tac [("z","xa")] eq_Abs_hypreal 1);
-by (auto_tac (claset(),simpset() addsimps [starfun,
- hypreal_of_real_def,hypreal_add]));
-qed "starfun_lambda_cancel2";
-
-Goal "[| (*f* f) xa @= l; (*f* g) xa @= m; \
-\ l: HFinite; m: HFinite \
-\ |] ==> (*f* (%x. f x * g x)) xa @= l * m";
-by (dtac inf_close_mult_HFinite 1);
-by (REPEAT(assume_tac 1));
-by (auto_tac (claset() addIs [inf_close_sym RSN (2,inf_close_HFinite)],
- simpset()));
-qed "starfun_mult_HFinite_inf_close";
-
-Goal "[| (*f* f) xa @= l; (*f* g) xa @= m \
-\ |] ==> (*f* (%x. f x + g x)) xa @= l + m";
-by (auto_tac (claset() addIs [inf_close_add], simpset()));
-qed "starfun_add_inf_close";
-
-(*----------------------------------------------------
- Examples: hrabs is nonstandard extension of rabs
- inverse is nonstandard extension of inverse
- ---------------------------------------------------*)
-
-(* can be proved easily using theorem "starfun" and *)
-(* properties of ultrafilter as for inverse below we *)
-(* use the theorem we proved above instead *)
-
-Goal "*f* abs = abs";
-by (rtac (hrabs_is_starext_rabs RS
- (is_starext_starfun_iff RS iffD1) RS sym) 1);
-qed "starfun_rabs_hrabs";
-
-Goal "(*f* inverse) x = inverse(x)";
-by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
-by (auto_tac (claset(),
- simpset() addsimps [starfun, hypreal_inverse, hypreal_zero_def]));
-qed "starfun_inverse_inverse";
-Addsimps [starfun_inverse_inverse];
-
-Goal "inverse ((*f* f) x) = (*f* (%x. inverse (f x))) x";
-by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
-by (auto_tac (claset(),
- simpset() addsimps [starfun, hypreal_inverse]));
-qed "starfun_inverse";
-Addsimps [starfun_inverse RS sym];
-
-Goalw [hypreal_divide_def,real_divide_def]
- "(*f* f) xa / (*f* g) xa = (*f* (%x. f x / g x)) xa";
-by Auto_tac;
-qed "starfun_divide";
-Addsimps [starfun_divide RS sym];
-
-Goal "inverse ((*f* f) x) = (*f* (%x. inverse (f x))) x";
-by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
-by (auto_tac (claset() addIs [FreeUltrafilterNat_subset]
- addSDs [FreeUltrafilterNat_Compl_mem],
- simpset() addsimps [starfun, hypreal_inverse, hypreal_zero_def]));
-qed "starfun_inverse2";
-
-(*-------------------------------------------------------------
- General lemma/theorem needed for proofs in elementary
- topology of the reals
- ------------------------------------------------------------*)
-Goalw [starset_def]
- "(*f* f) x : *s* A ==> x : *s* {x. f x : A}";
-by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
-by (auto_tac (claset(),simpset() addsimps [starfun]));
-by (dres_inst_tac [("x","%n. f (Xa n)")] bspec 1);
-by (Auto_tac THEN Fuf_tac 1);
-qed "starfun_mem_starset";
-
-(*------------------------------------------------------------
- Alternative definition for hrabs with rabs function
- applied entrywise to equivalence class representative.
- This is easily proved using starfun and ns extension thm
- ------------------------------------------------------------*)
-Goal "abs (Abs_hypreal (hyprel ^^ {X})) = \
-\ Abs_hypreal(hyprel ^^ {%n. abs (X n)})";
-by (simp_tac (simpset() addsimps [starfun_rabs_hrabs RS sym,starfun]) 1);
-qed "hypreal_hrabs";
-
-(*----------------------------------------------------------------
- nonstandard extension of set through nonstandard extension
- of rabs function i.e hrabs. A more general result should be
- where we replace rabs by some arbitrary function f and hrabs
- by its NS extenson ( *f* f). See second NS set extension below.
- ----------------------------------------------------------------*)
-Goalw [starset_def]
- "*s* {x. abs (x + - y) < r} = \
-\ {x. abs(x + -hypreal_of_real y) < hypreal_of_real r}";
-by (Step_tac 1);
-by (ALLGOALS(res_inst_tac [("z","x")] eq_Abs_hypreal));
-by (auto_tac (claset() addSIs [exI] addSDs [bspec],
- simpset() addsimps [hypreal_minus, hypreal_of_real_def,hypreal_add,
- hypreal_hrabs,hypreal_less_def]));
-by (Fuf_tac 1);
-qed "STAR_rabs_add_minus";
-
-Goalw [starset_def]
- "*s* {x. abs (f x + - y) < r} = \
-\ {x. abs((*f* f) x + -hypreal_of_real y) < hypreal_of_real r}";
-by (Step_tac 1);
-by (ALLGOALS(res_inst_tac [("z","x")] eq_Abs_hypreal));
-by (auto_tac (claset() addSIs [exI] addSDs [bspec],
- simpset() addsimps [hypreal_minus, hypreal_of_real_def,hypreal_add,
- hypreal_hrabs,hypreal_less_def,starfun]));
-by (Fuf_tac 1);
-qed "STAR_starfun_rabs_add_minus";
-
-(*-------------------------------------------------------------------
- Another charaterization of Infinitesimal and one of @= relation.
- In this theory since hypreal_hrabs proved here. (To Check:) Maybe
- move both if possible?
- -------------------------------------------------------------------*)
-Goal "(x:Infinitesimal) = \
-\ (EX X:Rep_hypreal(x). \
-\ ALL m. {n. abs(X n) < inverse(real_of_posnat m)}:FreeUltrafilterNat)";
-by (res_inst_tac [("z","x")] eq_Abs_hypreal 1);
-by (auto_tac (claset() addSIs [bexI,lemma_hyprel_refl],
- simpset() addsimps [Infinitesimal_hypreal_of_posnat_iff,
- hypreal_of_real_of_posnat,hypreal_of_real_def,hypreal_inverse,
- hypreal_hrabs,hypreal_less]));
-by (dres_inst_tac [("x","n")] spec 1);
-by (Fuf_tac 1);
-qed "Infinitesimal_FreeUltrafilterNat_iff2";
-
-Goal "(Abs_hypreal(hyprel^^{X}) @= Abs_hypreal(hyprel^^{Y})) = \
-\ (ALL m. {n. abs (X n + - Y n) < \
-\ inverse(real_of_posnat m)} : FreeUltrafilterNat)";
-by (rtac (inf_close_minus_iff RS ssubst) 1);
-by (rtac (mem_infmal_iff RS subst) 1);
-by (auto_tac (claset(),
- simpset() addsimps [hypreal_minus, hypreal_add,
- Infinitesimal_FreeUltrafilterNat_iff2]));
-by (dres_inst_tac [("x","m")] spec 1);
-by (Fuf_tac 1);
-qed "inf_close_FreeUltrafilterNat_iff";
-
-Goal "inj starfun";
-by (rtac injI 1);
-by (rtac ext 1 THEN rtac ccontr 1);
-by (dres_inst_tac [("x","Abs_hypreal(hyprel ^^{%n. xa})")] fun_cong 1);
-by (auto_tac (claset(),simpset() addsimps [starfun]));
-qed "inj_starfun";
-
-
-
-
-