src/HOL/NSA/Star.thy

-(*  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\<open>Star-Transforms in Non-Standard Analysis\<close>
-
-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. \<exists>As. X = *sn* As}"
-
-definition
-  (* nonstandard extension of function *)
-  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)) = (eventually (\<lambda>n. 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. \<exists>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: "\<forall>x. \<exists>y. Q x y ==> \<exists>(f :: 'a => nat). \<forall>x. Q x (f x)"
-apply (rule_tac x = "%x. LEAST y. Q x y" in exI)
-apply (blast intro: LeastI)
-done
-
-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 Int \<real> = 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 \<notin> hypreal_of_real ` A ==> \<forall>y \<in> A. x \<noteq> hypreal_of_real y"
-by auto
-
-lemma lemma_not_starA: "x \<notin> star_of ` A ==> \<forall>y \<in> A. x \<noteq> star_of y"
-by auto
-
-lemma lemma_Compl_eq: "- {n. X n = xa} = {n. X n \<noteq> xa}"
-by auto
-
-lemma STAR_real_seq_to_hypreal:
-    "\<forall>n. (X n) \<notin> M ==> star_n X \<notin> *s* M"
-apply (unfold starset_def star_of_def)
-apply (simp add: Iset_star_n FreeUltrafilterNat.proper)
-done
-
-lemma STAR_singleton: "*s* {x} = {star_of x}"
-by simp
-
-lemma STAR_not_mem: "x \<notin> F ==> star_of x \<notin> *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: "\<forall>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: "\<forall>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)
-done
-
-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 \<noteq> star_of x
-       ==> ( *f* (\<lambda>z. 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\<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)
-
-(* this is trivial, given starfun_Id *)
-lemma starfun_Idfun_approx:
-  "x \<approx> star_of a ==> ( *f* (%x. x)) x \<approx> star_of a"
-by (simp only: starfun_Id)
-
-text\<open>The Star-function is a (nonstandard) extension of the function\<close>
-
-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)
-done
-
-text\<open>Any nonstandard extension is in fact the Star-function\<close>
-
-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])
-done
-
-lemma is_starext_starfun_iff: "(is_starext F f) = (F = *f* f)"
-by (blast intro: is_starfun_starext is_starext_starfun)
-
-text\<open>extented 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) \<approx> 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 \<approx> l; ( *f* g) x \<approx> m;
-                  l: HFinite; m: HFinite
-               |] ==>  ( *f* (%x. f x * g x)) x \<approx> l * m"
-apply (drule (3) approx_mult_HFinite)
-apply (auto intro: approx_HFinite [OF _ approx_sym])
-done
-
-lemma starfun_add_approx: "[| ( *f* f) x \<approx> l; ( *f* g) x \<approx> m
-               |] ==>  ( *f* (%x. f x + g x)) x \<approx> l + m"
-by (auto intro: approx_add)
-
-text\<open>Examples: hrabs is nonstandard extension of rabs
-              inverse is nonstandard extension of inverse\<close>
-
-(* 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\<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  \<in> A}"
-by (transfer, simp)
-
-text\<open>Alternative definition for hrabs with rabs function
-   applied entrywise to equivalence class representative.
-   This is easily proved using starfun and ns extension thm\<close>
-lemma hypreal_hrabs: "\<bar>star_n X\<bar> = star_n (%n. \<bar>X n\<bar>)"
-by (simp only: starfun_rabs_hrabs [symmetric] starfun)
-
-text\<open>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.\<close>
-lemma STAR_rabs_add_minus:
-   "*s* {x. \<bar>x + - y\<bar> < r} = {x. \<bar>x + -star_of y\<bar> < star_of r}"
-by (transfer, rule refl)
-
-lemma STAR_starfun_rabs_add_minus:
-  "*s* {x. \<bar>f x + - y\<bar> < r} =
-       {x. \<bar>( *f* f) x + -star_of y\<bar> < 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 \<in> Infinitesimal) = (\<forall>m. eventually (\<lambda>n. norm(X n) < inverse(real(Suc m))) \<U>)"
-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]:
-     "n \<in> HNatInfinite ==> inverse (hypreal_of_hypnat n) \<in> 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 \<approx> star_n Y =
-      (\<forall>r>0. eventually (\<lambda>n. norm (X n - Y n) < r) \<U>)"
-apply (subst approx_minus_iff)
-apply (rule mem_infmal_iff [THEN subst])
-apply (simp add: star_n_diff)
-apply (simp add: Infinitesimal_FreeUltrafilterNat_iff)
-done
-
-lemma approx_FreeUltrafilterNat_iff2: "star_n X \<approx> star_n Y =
-      (\<forall>m. eventually (\<lambda>n. norm (X n - Y n) < inverse(real(Suc m))) \<U>)"
-apply (subst approx_minus_iff)
-apply (rule mem_infmal_iff [THEN subst])
-apply (simp add: star_n_diff)
-apply (simp add: Infinitesimal_FreeUltrafilterNat_iff2)
-done
-
-lemma 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 FreeUltrafilterNat.proper)
-done
-
-end