# Theory Star

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
*)

section‹Star-Transforms in Non-Standard Analysis›

theory Star
imports NSA
begin

definition
(* 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)) = (eventually (λn. Y n = f(X n)) 𝒰)))"

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)
done

subsection‹Properties of the Star-transform Applied to Sets of Reals›

lemma STAR_star_of_image_subset: "star_of ` A <= *s* A"
by auto

lemma STAR_hypreal_of_real_Int: "*s* X Int ℝ = hypreal_of_real ` X"

lemma STAR_star_of_Int: "*s* X Int Standard = star_of ` X"

lemma lemma_not_hyprealA: "x ∉ hypreal_of_real ` A ==> ∀y ∈ A. x ≠ hypreal_of_real y"
by auto

lemma lemma_not_starA: "x ∉ star_of ` A ==> ∀y ∈ A. x ≠ star_of 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 ==> star_n X ∉ *s* M"
apply (unfold starset_def star_of_def)
done

lemma STAR_singleton: "*s* {x} = {star_of x}"
by simp

lemma STAR_not_mem: "x ∉ F ==> star_of x ∉ *s* F"
by transfer

lemma 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 (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)
done

text‹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)

(*--------------------------------------------
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)

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 (rule_tac x = x in star_cases)
apply (rule_tac x = y in star_cases)
apply (auto intro!: bexI [OF _ Rep_star_star_n]
done

text‹Any nonstandard extension is in fact the Star-function›

lemma is_starfun_starext: "is_starext F f ==> F = *f* f"
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 (simp add: starfun_star_n [of f, symmetric])
done

lemma 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])
done

lemma starfun_add_approx: "[| ( *f* f) x ≈ l; ( *f* g) x ≈ m
|] ==>  ( *f* (%x. f x + g x)) x ≈ l + m"

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: "¦star_n X¦ = star_n (%n. ¦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.›
"*s* {x. ¦x + - y¦ < r} = {x. ¦x + -star_of y¦ < star_of r}"
by (transfer, rule refl)

"*s* {x. ¦f x + - y¦ < r} =
{x. ¦( *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 ‹hypreal_hrabs› proved here. Maybe
move both theorems??›
lemma Infinitesimal_FreeUltrafilterNat_iff2:
"(star_n X ∈ Infinitesimal) = (∀m. eventually (λn. norm(X n) < inverse(real(Suc m))) 𝒰)"
hnorm_def star_of_nat_def starfun_star_n
star_n_inverse star_n_less)

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 star_n_inverse real_norm_def
HNatInfinite_FreeUltrafilterNat_iff
Infinitesimal_FreeUltrafilterNat_iff2)
apply (drule_tac x="Suc m" in spec)
apply (auto elim!: eventually_mono)
done

lemma approx_FreeUltrafilterNat_iff: "star_n X ≈ star_n Y =
(∀r>0. eventually (λn. norm (X n - Y n) < r) 𝒰)"
apply (subst approx_minus_iff)
apply (rule mem_infmal_iff [THEN subst])
done

lemma approx_FreeUltrafilterNat_iff2: "star_n X ≈ star_n Y =
(∀m. eventually (λn. norm (X n - Y n) < inverse(real(Suc m))) 𝒰)"
apply (subst approx_minus_iff)
apply (rule mem_infmal_iff [THEN subst])