# Theory Star

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theory Star
imports NSA
begin

    (*  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
    imports NSA
    begin

    constdefs
        (* internal sets *)
        starset_n :: "(nat => 'a set) => 'a star set"        ("*sn* _" [80] 80)
        "*sn* As == Iset (star_n As)"

        InternalSets :: "'a star set set"
        "InternalSets == {X. ∃As. X = *sn* As}"

        (* nonstandard extension of function *)
        is_starext  :: "['a star => 'a star, 'a => 'a] => bool"
        "is_starext F f == (∀x y. ∃X ∈ Rep_star(x). ∃Y ∈ Rep_star(y).
                            ((y = (F x)) = ({n. Y n = f(X n)} : FreeUltrafilterNat)))"
        (* internal functions *)
        starfun_n :: "(nat => ('a => 'b)) => 'a star => 'b star"
                     ("*fn* _" [80] 80)
        "*fn* F == Ifun (star_n F)"

        InternalFuns :: "('a star => 'b star) 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

    subsection{*Properties of the Star-transform Applied to Sets of Reals*}

    lemma STAR_UNIV_set: "*s*(UNIV::'a set) = (UNIV::'a star set)"
    by (transfer UNIV_def, rule refl)

    lemma STAR_empty_set: "*s* {} = {}"
    by (transfer empty_def, rule refl)

    lemma STAR_Un: "*s* (A Un B) = *s* A Un *s* B"
    by (transfer Un_def, rule refl)

    lemma STAR_Int: "*s* (A Int B) = *s* A Int *s* B"
    by (transfer Int_def, rule refl)

    lemma STAR_Compl: "*s* -A = -( *s* A)"
    by (transfer Compl_def, rule refl)

    lemma STAR_mem_Compl: "!!x. x ∉ *s* F ==> x : *s* (- F)"
    by (transfer Compl_def, simp)

    lemma STAR_diff: "*s* (A - B) = *s* A - *s* B"
    by (transfer set_diff_def, rule refl)

    lemma STAR_subset: "A <= B ==> *s* A <= *s* B"
    by (transfer subset_def, simp)

    lemma STAR_mem: "a ∈ A ==> star_of a : *s* A"
    by transfer

    lemma STAR_mem_iff: "(star_of x ∈ *s* A) = (x ∈ A)"
    by (transfer, rule refl)

    lemma STAR_star_of_image_subset: "star_of ` A <= *s* A"
    by (auto simp add: STAR_mem)

    lemma STAR_hypreal_of_real_Int: "*s* X Int Reals = hypreal_of_real ` X"
    by (auto simp add: SReal_def STAR_mem_iff)

    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 ==> star_n X ∉ *s* M"
    apply (unfold starset_def star_of_def)
    apply (simp add: Iset_star_n)
    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 (blast dest: STAR_subset)

    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 expand_fun_eq [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 expand_fun_eq [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

    lemma Rep_star_FreeUltrafilterNat:
         "[| X ∈ Rep_star z; Y ∈ Rep_star z |]
          ==> {n. X n = Y n} : FreeUltrafilterNat"
    by (rule FreeUltrafilterNat_Rep_hypreal)

    text{*Nonstandard extension of functions*}

    lemma starfun:
          "( *f* f) (star_n X) = star_n (%n. f (X n))"
    by (simp add: starfun_def star_of_def Ifun_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 @= hypreal_of_real a ==> ( *f* (%x. x)) x @= hypreal_of_real 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)
    done

    text{*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 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)

    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 [simp]: "( *f* f) (star_of a) = star_of (f a)"
    by (transfer, rule refl)

    lemma starfun_approx: "( *f* f) (star_of a) @= hypreal_of_real (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: "[| ( *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"
    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 (rule hrabs_is_starext_rabs [THEN is_starext_starfun_iff [THEN iffD1], symmetric])

    lemma starfun_inverse_inverse [simp]: "( *f* inverse) x = inverse(x)"
    by (unfold star_inverse_def, rule refl)

    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 + -hypreal_of_real y) < hypreal_of_real r}"
    by (transfer, rule refl)

    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}"
    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:
         "(x ∈ Infinitesimal) =
          (∃X ∈ Rep_star(x).
            ∀m. {n. abs(X n) < inverse(real(Suc m))}
                    ∈  FreeUltrafilterNat)"
    apply (cases x)
    apply (auto intro!: bexI lemma_starrel_refl
                simp add: Infinitesimal_hypreal_of_nat_iff star_of_def
         star_n_inverse star_n_abs star_n_less hypreal_of_nat_eq)
    apply (drule_tac x = n in spec, ultra)
    done

    lemma approx_FreeUltrafilterNat_iff: "star_n X @= star_n 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: star_n_minus star_n_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 = "star_n (%n. xa)" in fun_cong)
    apply (auto simp add: starfun star_n_eq_iff)
    done

    ML
    {*
    val starset_n_def = thm"starset_n_def";
    val InternalSets_def = thm"InternalSets_def";
    val is_starext_def = thm"is_starext_def";
    val starfun_n_def = thm"starfun_n_def";
    val InternalFuns_def = thm"InternalFuns_def";

    val no_choice = thm "no_choice";
    val STAR_UNIV_set = thm "STAR_UNIV_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_star_of_image_subset = thm "STAR_star_of_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_star_FreeUltrafilterNat = thm "Rep_star_FreeUltrafilterNat";
    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

lemma no_choice:

      ∀x. ∃y. Q x y ==> ∃f. ∀x. Q x (f x)

### Properties of the Star-transform Applied to Sets of Reals

lemma STAR_UNIV_set:

      *s* UNIV = UNIV

lemma STAR_empty_set:

      *s* {} = {}

lemma STAR_Un:

      *s* (A ∪ B) = *s* A ∪ *s* B

lemma STAR_Int:

      *s* (A ∩ B) = *s* A ∩ *s* B

lemma STAR_Compl:

      *s* - A = - (*s* A)

lemma STAR_mem_Compl:

      x ∉ *s* F ==> x ∈ *s* - F

lemma STAR_diff:

      *s* (A - B) = *s* A - *s* B

lemma STAR_subset:

      A ⊆ B ==> *s* A ⊆ *s* B

lemma STAR_mem:

      a ∈ A ==> star_of a ∈ *s* A

lemma STAR_mem_iff:

      (star_of x ∈ *s* A) = (x ∈ A)

lemma STAR_star_of_image_subset:

      star_of ` A ⊆ *s* A

lemma STAR_hypreal_of_real_Int:

      *s* X ∩ Reals = star_of ` X

lemma lemma_not_hyprealA:

      x ∉ star_of ` A ==> ∀y∈A. x ≠ star_of y

lemma lemma_Compl_eq:

      - {n. X n = xa} = {n. X n ≠ xa}

lemma STAR_real_seq_to_hypreal:

      ∀n. X n ∉ M ==> star_n X ∉ *s* M

lemma STAR_singleton:

      *s* {x} = {star_of x}

lemma STAR_not_mem:

      x ∉ F ==> star_of x ∉ *s* F

lemma STAR_subset_closed:

      [| x ∈ *s* A; A ⊆ B |] ==> x ∈ *s* B

lemma starset_n_starset:

      ∀n. As n = A ==> *sn* As = *s* A

lemma starfun_n_starfun:

      ∀n. F n = f ==> *fn* F = *f* f

lemma hrabs_is_starext_rabs:

      is_starext abs abs

lemma Rep_star_FreeUltrafilterNat:

      [| X ∈ Rep_star z; Y ∈ Rep_star z |] ==> {n. X n = Y n} ∈ \<U>

lemma starfun:

      (*f* f) (star_n X) = star_n (%n. f (X n))

lemma starfun_if_eq:

      w ≠ star_of x ==> (*f* (%z. if z = x then a else g z)) w = (*f* g) w

lemma starfun_mult:

      (*f* f) x * (*f* g) x = (*f* (%x. f x * g x)) x

lemma starfun_add:

      (*f* f) x + (*f* g) x = (*f* (%x. f x + g x)) x

lemma starfun_minus:

      - (*f* f) x = (*f* (%x. - f x)) x

lemma starfun_add_minus:

      (*f* f) x + - (*f* g) x = (*f* (%x. f x + - g x)) x

lemma starfun_diff:

      (*f* f) x - (*f* g) x = (*f* (%x. f x - g x)) x

lemma starfun_o2:

      (%x. (*f* f) ((*f* g) x)) = *f* (%x. f (g x))

lemma starfun_o:

      *f* f o *f* g = *f* (f o g)

lemma starfun_const_fun:

      (*f* (%x. k)) x = star_of k

lemma starfun_Id:

      (*f* (%x. x)) x = x

lemma starfun_Idfun_approx:

      x ≈ star_of a ==> (*f* (%x. x)) x ≈ star_of a

lemma is_starext_starfun:

      is_starext (*f* f) f

lemma is_starfun_starext:

      is_starext F f ==> F = *f* f

lemma is_starext_starfun_iff:

      is_starext F f = (F = *f* f)

lemma starfun_eq:

      (*f* f) (star_of a) = star_of (f a)

lemma starfun_approx:

      (*f* f) (star_of a) ≈ star_of (f a)

lemma starfun_lambda_cancel:

      (*f* (%h. f (x + h))) x' = (*f* f) (star_of x + x')

lemma starfun_lambda_cancel2:

      (*f* (%h. f (g (x + h)))) x' = (*f* (f o g)) (star_of x + x')

lemma starfun_mult_HFinite_approx:

      [| (*f* f) x ≈ l; (*f* g) x ≈ m; l ∈ HFinite; m ∈ HFinite |]
      ==> (*f* (%x. f x * g x)) x ≈ l * m

lemma starfun_add_approx:

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

lemma starfun_rabs_hrabs:

      *f* abs = abs

lemma starfun_inverse_inverse:

      (*f* inverse) x = inverse x

lemma starfun_inverse:

      inverse ((*f* f) x) = (*f* (%x. inverse (f x))) x

lemma starfun_divide:

      (*f* f) x / (*f* g) x = (*f* (%x. f x / g x)) x

lemma starfun_inverse2:

      inverse ((*f* f) x) = (*f* (%x. inverse (f x))) x

lemma starfun_mem_starset:

      (*f* f) x ∈ *s* A ==> x ∈ *s* {x. f x ∈ A}

lemma hypreal_hrabs:

      ¦star_n X¦ = star_n (%n. ¦X n¦)

lemma STAR_rabs_add_minus:

      *s* {x. ¦x + - y¦ < r} = {x. ¦x + - star_of y¦ < star_of r}

lemma STAR_starfun_rabs_add_minus:

      *s* {x. ¦f x + - y¦ < r} = {x. ¦(*f* f) x + - star_of y¦ < star_of r}

lemma Infinitesimal_FreeUltrafilterNat_iff2:

      (x ∈ Infinitesimal) =
      (∃X∈Rep_star x. ∀m. {n. ¦X n¦ < inverse (real (Suc m))} ∈ \<U>)

lemma approx_FreeUltrafilterNat_iff:

      (star_n X ≈ star_n Y) = (∀m. {n. ¦X n + - Y n¦ < inverse (real (Suc m))} ∈ \<U>)

lemma inj_starfun:

      inj starfun