(* Title: HOL/Nonstandard_Analysis/Star.thy
Author: Jacques D. Fleuriot
Copyright: 1998 University of Cambridge
Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4
*)
section \<open>Star-Transforms in Non-Standard Analysis\<close>
theory Star
imports NSA
begin
definition \<comment> \<open>internal sets\<close>
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 \<comment> \<open>nonstandard extension of function\<close>
is_starext :: "('a star \ 'a star) \ ('a \ 'a) \ bool"
where "is_starext F f \
(\<forall>x y. \<exists>X \<in> Rep_star x. \<exists>Y \<in> Rep_star y. y = F x \<longleftrightarrow> eventually (\<lambda>n. Y n = f(X n)) \<U>)"
definition \<comment> \<open>internal functions\<close>
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}"
subsection \<open>Preamble - Pulling \<open>\<exists>\<close> over \<open>\<forall>\<close>\<close>
text \<open>This proof does not need AC and was suggested by the
referee for the JCM Paper: let \<open>f x\<close> be least \<open>y\<close> such
that \<open>Q x y\<close>.\<close>
lemma no_choice: "\x. \y. Q x y \ \f :: 'a \ nat. \x. Q x (f x)"
by (rule exI [where x = "\x. LEAST y. Q x y"]) (blast intro: LeastI)
subsection \<open>Properties of the Star-transform Applied to Sets of Reals\<close>
lemma STAR_star_of_image_subset: "star_of ` A \ *s* A"
by auto
lemma STAR_hypreal_of_real_Int: "*s* X \ \ = hypreal_of_real ` X"
by (auto simp add: SReal_def)
lemma STAR_star_of_Int: "*s* X \ 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 auto
lemma lemma_not_starA: "x \ star_of ` A \ \y \ A. x \ star_of y"
by auto
lemma STAR_real_seq_to_hypreal: "\n. (X n) \ M \ star_n X \ *s* M"
by (simp add: starset_def star_of_def Iset_star_n FreeUltrafilterNat.proper)
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 \<open>Nonstandard extension of a set (defined using a constant
sequence) as a special case of an internal set.\<close>
lemma starset_n_starset: "\n. As n = A \ *sn* As = *s* A"
by (drule fun_eq_iff [THEN iffD2]) (simp add: starset_n_def starset_def star_of_def)
subsection \<open>Theorems about nonstandard extensions of functions\<close>
text \<open>Nonstandard extension of a function (defined using a
constant sequence) as a special case of an internal function.\<close>
lemma starfun_n_starfun: "F = (\n. f) \ *fn* F = *f* f"
by (simp add: starfun_n_def starfun_def star_of_def)
text \<open>Prove that \<open>abs\<close> 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!\<close>
lemma hrabs_is_starext_rabs: "is_starext abs abs"
proof -
have "\f\Rep_star (star_n h). \g\Rep_star (star_n k). (star_n k = \star_n h\) = (\\<^sub>F n in \. (g n::'a) = \f n\)"
for x y :: "'a star" and h k
by (metis (full_types) Rep_star_star_n star_n_abs star_n_eq_iff)
then show ?thesis
unfolding is_starext_def by (metis star_cases)
qed
text \<open>Nonstandard extension of functions.\<close>
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* (\z. if z = x then a else g z)) w = ( *f* g) w"
by transfer simp
text \<open>Multiplication: \<open>( *f) x ( *g) = *(f x g)\<close>\<close>
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]
text \<open>Addition: \<open>( *f) + ( *g) = *(f + g)\<close>\<close>
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]
text \<open>Subtraction: \<open>( *f) + -( *g) = *(f + -g)\<close>\<close>
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]
text \<open>Composition: \<open>( *f) \<circ> ( *g) = *(f \<circ> g)\<close>\<close>
lemma starfun_o2: "(\x. ( *f* f) (( *f* g) x)) = *f* (\x. f (g x))"
by transfer (rule refl)
lemma starfun_o: "( *f* f) \ ( *f* g) = ( *f* (f \ g))"
by (transfer o_def) (rule refl)
text \<open>NS extension of constant function.\<close>
lemma starfun_const_fun [simp]: "\x. ( *f* (\x. k)) x = star_of k"
by transfer (rule refl)
text \<open>The NS extension of the identity function.\<close>
lemma starfun_Id [simp]: "\x. ( *f* (\x. x)) x = x"
by transfer (rule refl)
text \<open>The Star-function is a (nonstandard) extension of the function.\<close>
lemma is_starext_starfun: "is_starext ( *f* f) f"
proof -
have "\X\Rep_star x. \Y\Rep_star y. (y = (*f* f) x) = (\\<^sub>F n in \. Y n = f (X n))"
for x y
by (metis (mono_tags) Rep_star_star_n star_cases star_n_eq_iff starfun_star_n)
then show ?thesis
by (auto simp: is_starext_def)
qed
text \<open>Any nonstandard extension is in fact the Star-function.\<close>
lemma is_starfun_starext:
assumes "is_starext F f"
shows "F = *f* f"
proof -
have "F x = (*f* f) x"
if "\x y. \X\Rep_star x. \Y\Rep_star y. (y = F x) = (\\<^sub>F n in \. Y n = f (X n))" for x
by (metis that mem_Rep_star_iff star_n_eq_iff starfun_star_n)
with assms show ?thesis
by (force simp add: is_starext_def)
qed
lemma is_starext_starfun_iff: "is_starext F f \ F = *f* f"
by (blast intro: is_starfun_starext is_starext_starfun)
text \<open>Extended function has same solution as its standard version
for real arguments. i.e they are the same for all real arguments.\<close>
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
text \<open>Useful for NS definition of derivatives.\<close>
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 \ g)) (star_of x + x')"
unfolding o_def by (rule starfun_lambda_cancel)
lemma starfun_mult_HFinite_approx:
"( *f* f) x \ l \ ( *f* g) x \ m \ l \ HFinite \ m \ HFinite \
( *f* (\<lambda>x. f x * g x)) x \<approx> l * m"
for l m :: "'a::real_normed_algebra star"
using approx_mult_HFinite by auto
lemma 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 \<open>Examples: \<open>hrabs\<close> is nonstandard extension of \<open>rabs\<close>,
\<open>inverse\<close> is nonstandard extension of \<open>inverse\<close>.\<close>
text \<open>Can be proved easily using theorem \<open>starfun\<close> and
properties of ultrafilter as for inverse below we
use the theorem we proved above instead.\<close>
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 \<open>General lemma/theorem needed for proofs in elementary topology of the reals.\<close>
lemma starfun_mem_starset: "\x. ( *f* f) x \ *s* A \ x \ *s* {x. f x \ A}"
by transfer simp
text \<open>Alternative definition for \<open>hrabs\<close> with \<open>rabs\<close> function applied
entrywise to equivalence class representative.
This is easily proved using @{thm [source] starfun} and ns extension thm.\<close>
lemma hypreal_hrabs: "\star_n X\ = star_n (\n. \X n\)"
by (simp only: starfun_rabs_hrabs [symmetric] starfun)
text \<open>Nonstandard extension of set through nonstandard extension
of \<open>rabs\<close> function i.e. \<open>hrabs\<close>. A more general result should be
where we replace \<open>rabs\<close> by some arbitrary function \<open>f\<close> and \<open>hrabs\<close>
by its NS extenson. See second NS set extension below.\<close>
lemma STAR_rabs_add_minus: "*s* {x. \x + - y\ < r} = {x. \x + -star_of y\ < star_of r}"
by transfer (rule refl)
lemma STAR_starfun_rabs_add_minus:
"*s* {x. \f x + - y\ < r} = {x. \( *f* f) x + -star_of y\ < star_of r}"
by transfer (rule refl)
text \<open>Another characterization of Infinitesimal and one of \<open>\<approx>\<close> relation.
In this theory since \<open>hypreal_hrabs\<close> proved here. Maybe move both theorems??\<close>
lemma Infinitesimal_FreeUltrafilterNat_iff2:
"star_n X \ Infinitesimal \ (\m. eventually (\n. norm (X n) < inverse (real (Suc m))) \)"
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)
lemma HNatInfinite_inverse_Infinitesimal [simp]:
assumes "n \ HNatInfinite"
shows "inverse (hypreal_of_hypnat n) \ Infinitesimal"
proof (cases n)
case (star_n X)
then have *: "\k. \\<^sub>F n in \. k < X n"
using HNatInfinite_FreeUltrafilterNat assms by blast
have "\\<^sub>F n in \. inverse (real (X n)) < inverse (1 + real m)" for m
using * [of "Suc m"] by (auto elim!: eventually_mono)
then show ?thesis
using star_n by (auto simp: of_hypnat_def starfun_star_n star_n_inverse Infinitesimal_FreeUltrafilterNat_iff2)
qed
lemma approx_FreeUltrafilterNat_iff:
"star_n X \ star_n Y \ (\r>0. eventually (\n. norm (X n - Y n) < r) \)"
(is "?lhs = ?rhs")
proof -
have "?lhs = (star_n X - star_n Y \ 0)"
using approx_minus_iff by blast
also have "... = ?rhs"
by (metis (full_types) Infinitesimal_FreeUltrafilterNat_iff mem_infmal_iff star_n_diff)
finally show ?thesis .
qed
lemma approx_FreeUltrafilterNat_iff2:
"star_n X \ star_n Y \ (\m. eventually (\n. norm (X n - Y n) < inverse (real (Suc m))) \)"
(is "?lhs = ?rhs")
proof -
have "?lhs = (star_n X - star_n Y \ 0)"
using approx_minus_iff by blast
also have "... = ?rhs"
by (metis (full_types) Infinitesimal_FreeUltrafilterNat_iff2 mem_infmal_iff star_n_diff)
finally show ?thesis .
qed
lemma inj_starfun: "inj starfun"
proof (rule inj_onI)
show "\ = \" if eq: "*f* \ = *f* \" for \ \ :: "'a \ 'b"
proof (rule ext, rule ccontr)
show False
if "\ x \ \ x" for x
by (metis eq that star_of_inject starfun_eq)
qed
qed
end
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