(* Title : Star.thy Author : Jacques D. Fleuriot Copyright : 1998 University of Cambridge Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4 *) header{*Star-Transforms in Non-Standard Analysis*} theory Star import NSA begin constdefs (* nonstandard extension of sets *) starset :: "real set => hypreal set" ("*s* _" [80] 80) "*s* A == {x. \X \ Rep_hypreal(x). {n::nat. X n \ A}: FreeUltrafilterNat}" (* internal sets *) starset_n :: "(nat => real set) => hypreal set" ("*sn* _" [80] 80) "*sn* As == {x. \X \ Rep_hypreal(x). {n::nat. X n : (As n)}: FreeUltrafilterNat}" InternalSets :: "hypreal set set" "InternalSets == {X. \As. X = *sn* As}" (* nonstandard extension of function *) is_starext :: "[hypreal => hypreal, real => real] => bool" "is_starext F f == (\x y. \X \ Rep_hypreal(x). \Y \ Rep_hypreal(y). ((y = (F x)) = ({n. Y n = f(X n)} : FreeUltrafilterNat)))" starfun :: "(real => real) => hypreal => hypreal" ("*f* _" [80] 80) "*f* f == (%x. Abs_hypreal(\X \ Rep_hypreal(x). hyprel``{%n. f(X n)}))" (* internal functions *) starfun_n :: "(nat => (real => real)) => hypreal => hypreal" ("*fn* _" [80] 80) "*fn* F == (%x. Abs_hypreal(\X \ Rep_hypreal(x). hyprel``{%n. (F n)(X n)}))" InternalFuns :: "(hypreal => hypreal) set" "InternalFuns == {X. \F. X = *fn* F}" (*-------------------------------------------------------- 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) *) lemma no_choice: "\x. \y. Q x y ==> \(f :: nat => nat). \x. Q x (f x)" apply (rule_tac x = "%x. LEAST y. Q x y" in exI) apply (blast intro: LeastI) done (*------------------------------------------------------------ Properties of the *-transform applied to sets of reals ------------------------------------------------------------*) lemma STAR_real_set: "*s*(UNIV::real set) = (UNIV::hypreal set)" by (simp add: starset_def) declare STAR_real_set [simp] lemma STAR_empty_set: "*s* {} = {}" by (simp add: starset_def) declare STAR_empty_set [simp] lemma STAR_Un: "*s* (A Un B) = *s* A Un *s* B" apply (auto simp add: starset_def) prefer 3 apply (blast intro: FreeUltrafilterNat_subset) prefer 2 apply (blast intro: FreeUltrafilterNat_subset) apply (drule FreeUltrafilterNat_Compl_mem) apply (drule bspec, assumption) apply (rule_tac z = x in eq_Abs_hypreal, auto, ultra) done lemma STAR_Int: "*s* (A Int B) = *s* A Int *s* B" apply (simp add: starset_def, auto) prefer 3 apply (blast intro: FreeUltrafilterNat_Int FreeUltrafilterNat_subset) apply (blast intro: FreeUltrafilterNat_subset)+ done lemma STAR_Compl: "*s* -A = -( *s* A)" apply (auto simp add: starset_def) apply (rule_tac [!] z = x in eq_Abs_hypreal) apply (auto dest!: bspec, ultra) apply (drule FreeUltrafilterNat_Compl_mem, ultra) done lemma STAR_mem_Compl: "x \ *s* F ==> x : *s* (- F)" by (auto simp add: STAR_Compl) lemma STAR_diff: "*s* (A - B) = *s* A - *s* B" by (auto simp add: Diff_eq STAR_Int STAR_Compl) lemma STAR_subset: "A <= B ==> *s* A <= *s* B" apply (simp add: starset_def) apply (blast intro: FreeUltrafilterNat_subset)+ done lemma STAR_mem: "a \ A ==> hypreal_of_real a : *s* A" apply (simp add: starset_def hypreal_of_real_def) apply (auto intro: FreeUltrafilterNat_subset) done lemma STAR_hypreal_of_real_image_subset: "hypreal_of_real ` A <= *s* A" apply (simp add: starset_def) apply (auto simp add: hypreal_of_real_def) apply (blast intro: FreeUltrafilterNat_subset) done lemma STAR_hypreal_of_real_Int: "*s* X Int Reals = hypreal_of_real ` X" apply (simp add: starset_def) apply (auto simp add: hypreal_of_real_def SReal_def) apply (simp add: hypreal_of_real_def [symmetric]) apply (rule imageI, rule ccontr) apply (drule bspec) apply (rule lemma_hyprel_refl) prefer 2 apply (blast intro: FreeUltrafilterNat_subset, auto) done lemma lemma_not_hyprealA: "x \ hypreal_of_real ` A ==> \y \ A. x \ hypreal_of_real y" by auto lemma lemma_Compl_eq: "- {n. X n = xa} = {n. X n \ xa}" by auto lemma STAR_real_seq_to_hypreal: "\n. (X n) \ M ==> Abs_hypreal(hyprel``{X}) \ *s* M" apply (simp add: starset_def) apply (rule bexI, rule_tac [2] lemma_hyprel_refl, auto) done lemma STAR_singleton: "*s* {x} = {hypreal_of_real x}" apply (simp add: starset_def) apply (auto simp add: hypreal_of_real_def) apply (rule_tac z = xa in eq_Abs_hypreal) apply (auto intro: FreeUltrafilterNat_subset) done declare STAR_singleton [simp] lemma STAR_not_mem: "x \ F ==> hypreal_of_real x \ *s* F" apply (auto simp add: starset_def hypreal_of_real_def) apply (rule bexI, rule_tac [2] lemma_hyprel_refl, auto) done lemma STAR_subset_closed: "[| x : *s* A; A <= B |] ==> x : *s* B" by (blast dest: STAR_subset) (*------------------------------------------------------------------ Nonstandard extension of a set (defined using a constant sequence) as a special case of an internal set -----------------------------------------------------------------*) lemma starset_n_starset: "\n. (As n = A) ==> *sn* As = *s* A" by (simp add: starset_n_def starset_def) (*----------------------------------------------------------------*) (* Theorems about nonstandard extensions of functions *) (*----------------------------------------------------------------*) (*----------------------------------------------------------------*) (* Nonstandard extension of a function (defined using a *) (* constant sequence) as a special case of an internal function *) (*----------------------------------------------------------------*) lemma starfun_n_starfun: "\n. (F n = f) ==> *fn* F = *f* f" by (simp add: starfun_n_def starfun_def) (* Prove that abs for hypreal is a nonstandard extension of abs for real w/o use of congruence property (proved after this for general nonstandard extensions of real valued functions). Proof now Uses the ultrafilter tactic! *) lemma hrabs_is_starext_rabs: "is_starext abs abs" apply (simp add: is_starext_def, safe) apply (rule_tac z = x in eq_Abs_hypreal) apply (rule_tac z = y in eq_Abs_hypreal, auto) apply (rule bexI, rule_tac [2] lemma_hyprel_refl) apply (rule bexI, rule_tac [2] lemma_hyprel_refl) apply (auto dest!: spec simp add: hypreal_minus abs_if hypreal_zero_def hypreal_le hypreal_less) apply (arith | ultra)+ done lemma Rep_hypreal_FreeUltrafilterNat: "[| X \ Rep_hypreal z; Y \ Rep_hypreal z |] ==> {n. X n = Y n} : FreeUltrafilterNat" apply (cases z) apply (auto, ultra) done (*----------------------------------------------------------------------- Nonstandard extension of functions- congruence -----------------------------------------------------------------------*) lemma starfun_congruent: "congruent hyprel (%X. hyprel``{%n. f (X n)})" by (simp add: congruent_def, auto, ultra) lemma starfun: "( *f* f) (Abs_hypreal(hyprel``{%n. X n})) = Abs_hypreal(hyprel `` {%n. f (X n)})" apply (simp add: starfun_def) apply (rule_tac f = Abs_hypreal in arg_cong) apply (simp add: hyprel_in_hypreal [THEN Abs_hypreal_inverse] UN_equiv_class [OF equiv_hyprel starfun_congruent]) done lemma starfun_if_eq: "w \ hypreal_of_real x ==> ( *f* (\z. if z = x then a else g z)) w = ( *f* g) w" apply (cases w) apply (simp add: hypreal_of_real_def starfun, ultra) done (*------------------------------------------- multiplication: ( *f) x ( *g) = *(f x g) ------------------------------------------*) lemma starfun_mult: "( *f* f) xa * ( *f* g) xa = ( *f* (%x. f x * g x)) xa" by (cases xa, simp add: starfun hypreal_mult) declare starfun_mult [symmetric, simp] (*--------------------------------------- addition: ( *f) + ( *g) = *(f + g) ---------------------------------------*) lemma starfun_add: "( *f* f) xa + ( *f* g) xa = ( *f* (%x. f x + g x)) xa" by (cases xa, simp add: starfun hypreal_add) declare starfun_add [symmetric, simp] (*-------------------------------------------- subtraction: ( *f) + -( *g) = *(f + -g) -------------------------------------------*) lemma starfun_minus: "- ( *f* f) x = ( *f* (%x. - f x)) x" apply (cases x) apply (auto simp add: starfun hypreal_minus) done declare starfun_minus [symmetric, simp] (*FIXME: delete*) lemma starfun_add_minus: "( *f* f) xa + -( *f* g) xa = ( *f* (%x. f x + -g x)) xa" apply (simp (no_asm)) done declare starfun_add_minus [symmetric, simp] lemma starfun_diff: "( *f* f) xa - ( *f* g) xa = ( *f* (%x. f x - g x)) xa" apply (simp add: diff_minus) done declare starfun_diff [symmetric, simp] (*-------------------------------------- composition: ( *f) o ( *g) = *(f o g) ---------------------------------------*) lemma starfun_o2: "(%x. ( *f* f) (( *f* g) x)) = *f* (%x. f (g x))" apply (rule ext) apply (rule_tac z = x in eq_Abs_hypreal) apply (auto simp add: starfun) done lemma starfun_o: "( *f* f) o ( *f* g) = ( *f* (f o g))" apply (simp add: o_def) apply (simp (no_asm) add: starfun_o2) done (*-------------------------------------- NS extension of constant function --------------------------------------*) lemma starfun_const_fun: "( *f* (%x. k)) xa = hypreal_of_real k" apply (cases xa) apply (auto simp add: starfun hypreal_of_real_def) done declare starfun_const_fun [simp] (*---------------------------------------------------- the NS extension of the identity function ----------------------------------------------------*) lemma starfun_Idfun_approx: "x @= hypreal_of_real a ==> ( *f* (%x. x)) x @= hypreal_of_real a" apply (cases x) apply (auto simp add: starfun) done lemma starfun_Id: "( *f* (%x. x)) x = x" apply (cases x) apply (auto simp add: starfun) done declare starfun_Id [simp] (*---------------------------------------------------------------------- the *-function is a (nonstandard) extension of the function ----------------------------------------------------------------------*) lemma is_starext_starfun: "is_starext ( *f* f) f" apply (simp add: is_starext_def, auto) apply (rule_tac z = x in eq_Abs_hypreal) apply (rule_tac z = y in eq_Abs_hypreal) apply (auto intro!: bexI simp add: starfun) done (*---------------------------------------------------------------------- Any nonstandard extension is in fact the *-function ----------------------------------------------------------------------*) lemma is_starfun_starext: "is_starext F f ==> F = *f* f" apply (simp add: is_starext_def) apply (rule ext) apply (rule_tac z = x in eq_Abs_hypreal) apply (drule_tac x = x in spec) apply (drule_tac x = "( *f* f) x" in spec) apply (auto dest!: FreeUltrafilterNat_Compl_mem simp add: starfun, ultra) done lemma is_starext_starfun_iff: "(is_starext F f) = (F = *f* f)" by (blast intro: is_starfun_starext is_starext_starfun) (*-------------------------------------------------------- extented function has same solution as its standard version for real arguments. i.e they are the same for all real arguments -------------------------------------------------------*) lemma starfun_eq: "( *f* f) (hypreal_of_real a) = hypreal_of_real (f a)" by (auto simp add: starfun hypreal_of_real_def) declare starfun_eq [simp] lemma starfun_approx: "( *f* f) (hypreal_of_real a) @= hypreal_of_real (f a)" by auto (* useful for NS definition of derivatives *) lemma starfun_lambda_cancel: "( *f* (%h. f (x + h))) xa = ( *f* f) (hypreal_of_real x + xa)" apply (cases xa) apply (auto simp add: starfun hypreal_of_real_def hypreal_add) done lemma starfun_lambda_cancel2: "( *f* (%h. f(g(x + h)))) xa = ( *f* (f o g)) (hypreal_of_real x + xa)" apply (cases xa) apply (auto simp add: starfun hypreal_of_real_def hypreal_add) done lemma starfun_mult_HFinite_approx: "[| ( *f* f) xa @= l; ( *f* g) xa @= m; l: HFinite; m: HFinite |] ==> ( *f* (%x. f x * g x)) xa @= l * m" apply (drule approx_mult_HFinite, assumption+) apply (auto intro: approx_HFinite [OF _ approx_sym]) done lemma starfun_add_approx: "[| ( *f* f) xa @= l; ( *f* g) xa @= m |] ==> ( *f* (%x. f x + g x)) xa @= l + m" apply (auto intro: approx_add) done (*---------------------------------------------------- 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 *) lemma starfun_rabs_hrabs: "*f* abs = abs" by (rule hrabs_is_starext_rabs [THEN is_starext_starfun_iff [THEN iffD1], symmetric]) lemma starfun_inverse_inverse: "( *f* inverse) x = inverse(x)" apply (cases x) apply (auto simp add: starfun hypreal_inverse hypreal_zero_def) done declare starfun_inverse_inverse [simp] lemma starfun_inverse: "inverse (( *f* f) x) = ( *f* (%x. inverse (f x))) x" apply (cases x) apply (auto simp add: starfun hypreal_inverse) done declare starfun_inverse [symmetric, simp] lemma starfun_divide: "( *f* f) xa / ( *f* g) xa = ( *f* (%x. f x / g x)) xa" by (simp add: divide_inverse) declare starfun_divide [symmetric, simp] lemma starfun_inverse2: "inverse (( *f* f) x) = ( *f* (%x. inverse (f x))) x" apply (cases x) apply (auto intro: FreeUltrafilterNat_subset dest!: FreeUltrafilterNat_Compl_mem simp add: starfun hypreal_inverse hypreal_zero_def) done (*------------------------------------------------------------- General lemma/theorem needed for proofs in elementary topology of the reals ------------------------------------------------------------*) lemma starfun_mem_starset: "( *f* f) x : *s* A ==> x : *s* {x. f x \ A}" apply (simp add: starset_def) apply (cases x) apply (auto simp add: starfun) apply (rename_tac "X") apply (drule_tac x = "%n. f (X n) " in bspec) apply (auto, ultra) done (*------------------------------------------------------------ Alternative definition for hrabs with rabs function applied entrywise to equivalence class representative. This is easily proved using starfun and ns extension thm ------------------------------------------------------------*) lemma hypreal_hrabs: "abs (Abs_hypreal (hyprel `` {X})) = Abs_hypreal(hyprel `` {%n. abs (X n)})" apply (simp (no_asm) add: starfun_rabs_hrabs [symmetric] starfun) done (*---------------------------------------------------------------- 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. ----------------------------------------------------------------*) lemma STAR_rabs_add_minus: "*s* {x. abs (x + - y) < r} = {x. abs(x + -hypreal_of_real y) < hypreal_of_real r}" apply (simp add: starset_def, safe) apply (rule_tac [!] z = x in eq_Abs_hypreal) apply (auto intro!: exI dest!: bspec simp add: hypreal_minus hypreal_of_real_def hypreal_add hypreal_hrabs hypreal_less, ultra) done lemma STAR_starfun_rabs_add_minus: "*s* {x. abs (f x + - y) < r} = {x. abs(( *f* f) x + -hypreal_of_real y) < hypreal_of_real r}" apply (simp add: starset_def, safe) apply (rule_tac [!] z = x in eq_Abs_hypreal) apply (auto intro!: exI dest!: bspec simp add: hypreal_minus hypreal_of_real_def hypreal_add hypreal_hrabs hypreal_less starfun, ultra) done (*------------------------------------------------------------------- Another characterization of Infinitesimal and one of @= relation. In this theory since hypreal_hrabs proved here. (To Check:) Maybe move both if possible? -------------------------------------------------------------------*) lemma Infinitesimal_FreeUltrafilterNat_iff2: "(x \ Infinitesimal) = (\X \ Rep_hypreal(x). \m. {n. abs(X n) < inverse(real(Suc m))} \ FreeUltrafilterNat)" apply (cases x) apply (auto intro!: bexI lemma_hyprel_refl simp add: Infinitesimal_hypreal_of_nat_iff hypreal_of_real_def hypreal_inverse hypreal_hrabs hypreal_less hypreal_of_nat_eq) apply (drule_tac x = n in spec, ultra) done lemma approx_FreeUltrafilterNat_iff: "(Abs_hypreal(hyprel``{X}) @= Abs_hypreal(hyprel``{Y})) = (\m. {n. abs (X n + - Y n) < inverse(real(Suc m))} : FreeUltrafilterNat)" apply (subst approx_minus_iff) apply (rule mem_infmal_iff [THEN subst]) apply (auto simp add: hypreal_minus hypreal_add Infinitesimal_FreeUltrafilterNat_iff2) apply (drule_tac x = m in spec, ultra) done lemma inj_starfun: "inj starfun" apply (rule inj_onI) apply (rule ext, rule ccontr) apply (drule_tac x = "Abs_hypreal (hyprel ``{%n. xa}) " in fun_cong) apply (auto simp add: starfun) done ML {* val starset_def = thm"starset_def"; val starset_n_def = thm"starset_n_def"; val InternalSets_def = thm"InternalSets_def"; val is_starext_def = thm"is_starext_def"; val starfun_def = thm"starfun_def"; val starfun_n_def = thm"starfun_n_def"; val InternalFuns_def = thm"InternalFuns_def"; val no_choice = thm "no_choice"; val STAR_real_set = thm "STAR_real_set"; val STAR_empty_set = thm "STAR_empty_set"; val STAR_Un = thm "STAR_Un"; val STAR_Int = thm "STAR_Int"; val STAR_Compl = thm "STAR_Compl"; val STAR_mem_Compl = thm "STAR_mem_Compl"; val STAR_diff = thm "STAR_diff"; val STAR_subset = thm "STAR_subset"; val STAR_mem = thm "STAR_mem"; val STAR_hypreal_of_real_image_subset = thm "STAR_hypreal_of_real_image_subset"; val STAR_hypreal_of_real_Int = thm "STAR_hypreal_of_real_Int"; val STAR_real_seq_to_hypreal = thm "STAR_real_seq_to_hypreal"; val STAR_singleton = thm "STAR_singleton"; val STAR_not_mem = thm "STAR_not_mem"; val STAR_subset_closed = thm "STAR_subset_closed"; val starset_n_starset = thm "starset_n_starset"; val starfun_n_starfun = thm "starfun_n_starfun"; val hrabs_is_starext_rabs = thm "hrabs_is_starext_rabs"; val Rep_hypreal_FreeUltrafilterNat = thm "Rep_hypreal_FreeUltrafilterNat"; val starfun_congruent = thm "starfun_congruent"; val starfun = thm "starfun"; val starfun_mult = thm "starfun_mult"; val starfun_add = thm "starfun_add"; val starfun_minus = thm "starfun_minus"; val starfun_add_minus = thm "starfun_add_minus"; val starfun_diff = thm "starfun_diff"; val starfun_o2 = thm "starfun_o2"; val starfun_o = thm "starfun_o"; val starfun_const_fun = thm "starfun_const_fun"; val starfun_Idfun_approx = thm "starfun_Idfun_approx"; val starfun_Id = thm "starfun_Id"; val is_starext_starfun = thm "is_starext_starfun"; val is_starfun_starext = thm "is_starfun_starext"; val is_starext_starfun_iff = thm "is_starext_starfun_iff"; val starfun_eq = thm "starfun_eq"; val starfun_approx = thm "starfun_approx"; val starfun_lambda_cancel = thm "starfun_lambda_cancel"; val starfun_lambda_cancel2 = thm "starfun_lambda_cancel2"; val starfun_mult_HFinite_approx = thm "starfun_mult_HFinite_approx"; val starfun_add_approx = thm "starfun_add_approx"; val starfun_rabs_hrabs = thm "starfun_rabs_hrabs"; val starfun_inverse_inverse = thm "starfun_inverse_inverse"; val starfun_inverse = thm "starfun_inverse"; val starfun_divide = thm "starfun_divide"; val starfun_inverse2 = thm "starfun_inverse2"; val starfun_mem_starset = thm "starfun_mem_starset"; val hypreal_hrabs = thm "hypreal_hrabs"; val STAR_rabs_add_minus = thm "STAR_rabs_add_minus"; val STAR_starfun_rabs_add_minus = thm "STAR_starfun_rabs_add_minus"; val Infinitesimal_FreeUltrafilterNat_iff2 = thm "Infinitesimal_FreeUltrafilterNat_iff2"; val approx_FreeUltrafilterNat_iff = thm "approx_FreeUltrafilterNat_iff"; val inj_starfun = thm "inj_starfun"; *} end