(* 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 "!!Q. 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. 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. 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 : 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. 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]
"!!M. 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. 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. [| 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]
"!!A. 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]
"!!A. 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 "!!z. [| 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";
(*---------------------------------------
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";
(*--------------------------------------------
subtraction: ( *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_minus,hypreal_add]));
qed "starfun_add_minus";
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";
(*--------------------------------------
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];
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";
(*----------------------------------------------------
the NS extension of the identity function
----------------------------------------------------*)
Goal "!!x. 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";
(*----------------------------------------------------------------------
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] "!!f. 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* 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() addsimps [starfun_mult]));
qed "starfun_mult_HFinite_inf_close";
Goal "!!f. [| (*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() addsimps [starfun_add RS sym]));
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 "!!x. x ~= 0 ==> (*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]));
by (dtac FreeUltrafilterNat_Compl_mem 1);
by (auto_tac (claset() addEs [FreeUltrafilterNat_subset],simpset()));
qed "starfun_inverse_inverse";
(* more specifically *)
Goal "(*f* inverse) ehr = inverse (ehr)";
by (rtac (hypreal_epsilon_not_zero RS starfun_inverse_inverse) 1);
qed "starfun_inverse_epsilon";
Goal "ALL x. f x ~= 0 ==> \
\ 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";
Goal "(*f* f) x ~= 0 ==> \
\ 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";
Goal "a ~= hypreal_of_real b ==> \
\ (*f* (%z. inverse (z + -b))) a = inverse(a + -hypreal_of_real b)";
by (res_inst_tac [("z","a")] eq_Abs_hypreal 1);
by (auto_tac (claset() addIs [FreeUltrafilterNat_subset]
addSDs [FreeUltrafilterNat_Compl_mem],
simpset() addsimps [starfun,hypreal_of_real_def,hypreal_add,
hypreal_minus,hypreal_inverse,rename_numerals
(real_eq_minus_iff2 RS sym)]));
qed "starfun_inverse3";
Goal
"!!f. a + hypreal_of_real b ~= 0 ==> \
\ (*f* (%z. inverse (z + b))) a = inverse(a + hypreal_of_real b)";
by (res_inst_tac [("z","a")] eq_Abs_hypreal 1);
by (auto_tac (claset() addIs [FreeUltrafilterNat_subset]
addSDs [FreeUltrafilterNat_Compl_mem],
simpset() addsimps [starfun,hypreal_of_real_def,hypreal_add,
hypreal_inverse,hypreal_zero_def]));
qed "starfun_inverse4";
(*-------------------------------------------------------------
General lemma/theorem needed for proofs in elementary
topology of the reals
------------------------------------------------------------*)
Goalw [starset_def]
"!!A. (*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";