# Theory Star

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theory Star
imports NSA
`(*  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 Starimports NSAbegindefinition    (* internal sets *)  starset_n :: "(nat => 'a set) => 'a star set" ("*sn* _" [80] 80) where  "*sn* As = Iset (star_n As)"definition  InternalSets :: "'a star set set" where  "InternalSets = {X. ∃As. X = *sn* As}"definition  (* nonstandard extension of function *)  is_starext  :: "['a star => 'a star, 'a => 'a] => bool" where  "is_starext F f = (∀x y. ∃X ∈ Rep_star(x). ∃Y ∈ Rep_star(y).                        ((y = (F x)) = ({n. Y n = f(X n)} : FreeUltrafilterNat)))"definition  (* internal functions *)  starfun_n :: "(nat => ('a => 'b)) => 'a star => 'b star"   ("*fn* _" [80] 80) where  "*fn* F = Ifun (star_n F)"definition  InternalFuns :: "('a star => 'b star) set" where  "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 :: 'a => nat). ∀x. Q x (f x)"apply (rule_tac x = "%x. LEAST y. Q x y" in exI)apply (blast intro: LeastI)donesubsection{*Properties of the Star-transform Applied to Sets of Reals*}lemma STAR_star_of_image_subset: "star_of ` A <= *s* A"by autolemma STAR_hypreal_of_real_Int: "*s* X Int Reals = hypreal_of_real ` X"by (auto simp add: SReal_def)lemma STAR_star_of_Int: "*s* X Int Standard = star_of ` X"by (auto simp add: Standard_def)lemma lemma_not_hyprealA: "x ∉ hypreal_of_real ` A ==> ∀y ∈ A. x ≠ hypreal_of_real y"by autolemma lemma_not_starA: "x ∉ star_of ` A ==> ∀y ∈ A. x ≠ star_of y"by autolemma lemma_Compl_eq: "- {n. X n = xa} = {n. X n ≠ xa}"by autolemma STAR_real_seq_to_hypreal:    "∀n. (X n) ∉ M ==> star_n X ∉ *s* M"apply (unfold starset_def star_of_def)apply (simp add: Iset_star_n)donelemma STAR_singleton: "*s* {x} = {star_of x}"by simplemma STAR_not_mem: "x ∉ F ==> star_of x ∉ *s* F"by transferlemma STAR_subset_closed: "[| x : *s* A; A <= B |] ==> x : *s* B"by (erule rev_subsetD, simp)text{*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"apply (drule fun_eq_iff [THEN iffD2])apply (simp add: starset_n_def starset_def star_of_def)done(*----------------------------------------------------------------*)(* 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"apply (drule fun_eq_iff [THEN iffD2])apply (simp add: starfun_n_def starfun_def star_of_def)done(*   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 x=x in star_cases)apply (rule_tac x=y in star_cases)apply (unfold star_n_def, auto)apply (rule bexI, rule_tac [2] lemma_starrel_refl)apply (rule bexI, rule_tac [2] lemma_starrel_refl)apply (fold star_n_def)apply (unfold star_abs_def starfun_def star_of_def)apply (simp add: Ifun_star_n star_n_eq_iff)donetext{*Nonstandard extension of functions*}lemma starfun:      "( *f* f) (star_n X) = star_n (%n. f (X n))"by (rule starfun_star_n)lemma starfun_if_eq:     "!!w. w ≠ star_of x       ==> ( *f* (λz. if z = x then a else g z)) w = ( *f* g) w"by (transfer, simp)(*-------------------------------------------  multiplication: ( *f) x ( *g) = *(f x g) ------------------------------------------*)lemma starfun_mult: "!!x. ( *f* f) x * ( *f* g) x = ( *f* (%x. f x * g x)) x"by (transfer, rule refl)declare starfun_mult [symmetric, simp](*---------------------------------------  addition: ( *f) + ( *g) = *(f + g) ---------------------------------------*)lemma starfun_add: "!!x. ( *f* f) x + ( *f* g) x = ( *f* (%x. f x + g x)) x"by (transfer, rule refl)declare starfun_add [symmetric, simp](*--------------------------------------------  subtraction: ( *f) + -( *g) = *(f + -g) -------------------------------------------*)lemma starfun_minus: "!!x. - ( *f* f) x = ( *f* (%x. - f x)) x"by (transfer, rule refl)declare starfun_minus [symmetric, simp](*FIXME: delete*)lemma starfun_add_minus: "!!x. ( *f* f) x + -( *f* g) x = ( *f* (%x. f x + -g x)) x"by (transfer, rule refl)declare starfun_add_minus [symmetric, simp]lemma starfun_diff: "!!x. ( *f* f) x  - ( *f* g) x = ( *f* (%x. f x - g x)) x"by (transfer, rule refl)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))"by (transfer, rule refl)lemma starfun_o: "( *f* f) o ( *f* g) = ( *f* (f o g))"by (transfer o_def, rule refl)text{*NS extension of constant function*}lemma starfun_const_fun [simp]: "!!x. ( *f* (%x. k)) x = star_of k"by (transfer, rule refl)text{*the NS extension of the identity function*}lemma starfun_Id [simp]: "!!x. ( *f* (%x. x)) x = x"by (transfer, rule refl)(* this is trivial, given starfun_Id *)lemma starfun_Idfun_approx:  "x @= star_of a ==> ( *f* (%x. x)) x @= star_of a"by (simp only: starfun_Id)text{*The Star-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 x = x in star_cases)apply (rule_tac x = y in star_cases)apply (auto intro!: bexI [OF _ Rep_star_star_n]            simp add: starfun star_n_eq_iff)donetext{*Any nonstandard extension is in fact the Star-function*}lemma is_starfun_starext: "is_starext F f ==> F = *f* f"apply (simp add: is_starext_def)apply (rule ext)apply (rule_tac x = x in star_cases)apply (drule_tac x = x in spec)apply (drule_tac x = "( *f* f) x" in spec)apply (auto simp add: starfun_star_n)apply (simp add: star_n_eq_iff [symmetric])apply (simp add: starfun_star_n [of f, symmetric])donelemma is_starext_starfun_iff: "(is_starext F f) = (F = *f* f)"by (blast intro: is_starfun_starext is_starext_starfun)text{*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) (star_of a) = star_of (f a)"by (rule starfun_star_of)lemma starfun_approx: "( *f* f) (star_of a) @= star_of (f a)"by simp(* useful for NS definition of derivatives *)lemma starfun_lambda_cancel:  "!!x'. ( *f* (%h. f (x + h))) x'  = ( *f* f) (star_of x + x')"by (transfer, rule refl)lemma starfun_lambda_cancel2:  "( *f* (%h. f(g(x + h)))) x' = ( *f* (f o g)) (star_of x + x')"by (unfold o_def, rule starfun_lambda_cancel)lemma starfun_mult_HFinite_approx:  fixes l m :: "'a::real_normed_algebra star"  shows "[| ( *f* f) x @= l; ( *f* g) x @= m;                  l: HFinite; m: HFinite               |] ==>  ( *f* (%x. f x * g x)) x @= l * m"apply (drule (3) approx_mult_HFinite)apply (auto intro: approx_HFinite [OF _ approx_sym])donelemma starfun_add_approx: "[| ( *f* f) x @= l; ( *f* g) x @= m               |] ==>  ( *f* (%x. f x + g x)) x @= l + m"by (auto intro: approx_add)text{*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 (simp only: star_abs_def)lemma starfun_inverse_inverse [simp]: "( *f* inverse) x = inverse(x)"by (simp only: star_inverse_def)lemma starfun_inverse: "!!x. inverse (( *f* f) x) = ( *f* (%x. inverse (f x))) x"by (transfer, rule refl)declare starfun_inverse [symmetric, simp]lemma starfun_divide: "!!x. ( *f* f) x / ( *f* g) x = ( *f* (%x. f x / g x)) x"by (transfer, rule refl)declare starfun_divide [symmetric, simp]lemma starfun_inverse2: "!!x. inverse (( *f* f) x) = ( *f* (%x. inverse (f x))) x"by (transfer, rule refl)text{*General lemma/theorem needed for proofs in elementary    topology of the reals*}lemma starfun_mem_starset:      "!!x. ( *f* f) x : *s* A ==> x : *s* {x. f x  ∈ A}"by (transfer, simp)text{*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 (star_n X) = star_n (%n. abs (X n))"by (simp only: starfun_rabs_hrabs [symmetric] starfun)text{*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. See second NS set extension below.*}lemma STAR_rabs_add_minus:   "*s* {x. abs (x + - y) < r} =     {x. abs(x + -star_of y) < star_of r}"by (transfer, rule refl)lemma STAR_starfun_rabs_add_minus:  "*s* {x. abs (f x + - y) < r} =       {x. abs(( *f* f) x + -star_of y) < star_of r}"by (transfer, rule refl)text{*Another characterization of Infinitesimal and one of @= relation.   In this theory since @{text hypreal_hrabs} proved here. Maybe   move both theorems??*}lemma Infinitesimal_FreeUltrafilterNat_iff2:     "(star_n X ∈ Infinitesimal) =      (∀m. {n. norm(X n) < inverse(real(Suc m))}                ∈  FreeUltrafilterNat)"by (simp add: Infinitesimal_hypreal_of_nat_iff star_of_def     hnorm_def star_of_nat_def starfun_star_n     star_n_inverse star_n_less real_of_nat_def)lemma HNatInfinite_inverse_Infinitesimal [simp]:     "n ∈ HNatInfinite ==> inverse (hypreal_of_hypnat n) ∈ Infinitesimal"apply (cases n)apply (auto simp add: of_hypnat_def starfun_star_n real_of_nat_def [symmetric] star_n_inverse real_norm_def      HNatInfinite_FreeUltrafilterNat_iff      Infinitesimal_FreeUltrafilterNat_iff2)apply (drule_tac x="Suc m" in spec)apply (erule ultra, simp)donelemma approx_FreeUltrafilterNat_iff: "star_n X @= star_n Y =      (∀r>0. {n. norm (X n - Y n) < r} : FreeUltrafilterNat)"apply (subst approx_minus_iff)apply (rule mem_infmal_iff [THEN subst])apply (simp add: star_n_diff)apply (simp add: Infinitesimal_FreeUltrafilterNat_iff)donelemma approx_FreeUltrafilterNat_iff2: "star_n X @= star_n Y =      (∀m. {n. norm (X n - Y n) <                  inverse(real(Suc m))} : FreeUltrafilterNat)"apply (subst approx_minus_iff)apply (rule mem_infmal_iff [THEN subst])apply (simp add: star_n_diff)apply (simp add: Infinitesimal_FreeUltrafilterNat_iff2)donelemma inj_starfun: "inj starfun"apply (rule inj_onI)apply (rule ext, rule ccontr)apply (drule_tac x = "star_n (%n. xa)" in fun_cong)apply (auto simp add: starfun star_n_eq_iff)doneend`