Theory NSA (Isabelle2005: October 2005)

# Theory NSA

Up to index of Isabelle/HOL/HOL-Complex

theory NSA
imports HyperArith RComplete
begin

    (*  Title       : NSA.thy
        Author      : Jacques D. Fleuriot
        Copyright   : 1998  University of Cambridge

    Converted to Isar and polished by lcp
    *)

    header{*Infinite Numbers, Infinitesimals, Infinitely Close Relation*}

    theory NSA
    imports HyperArith "../Real/RComplete"
    begin

    constdefs

      Infinitesimal  :: "hypreal set"
       "Infinitesimal == {x. ∀r ∈ Reals. 0 < r --> abs x < r}"

      HFinite :: "hypreal set"
       "HFinite == {x. ∃r ∈ Reals. abs x < r}"

      HInfinite :: "hypreal set"
       "HInfinite == {x. ∀r ∈ Reals. r < abs x}"

      approx :: "[hypreal, hypreal] => bool"    (infixl "@=" 50)
        --{*the `infinitely close' relation*}
       "x @= y       == (x + -y) ∈ Infinitesimal"

      st        :: "hypreal => hypreal"
        --{*the standard part of a hyperreal*}
       "st           == (%x. @r. x ∈ HFinite & r ∈ Reals & r @= x)"

      monad     :: "hypreal => hypreal set"
       "monad x      == {y. x @= y}"

      galaxy    :: "hypreal => hypreal set"
       "galaxy x     == {y. (x + -y) ∈ HFinite}"

    defs (overloaded)

       SReal_def:      "Reals == {x. ∃r. x = hypreal_of_real r}"
         --{*the standard real numbers as a subset of the hyperreals*}

    syntax (xsymbols)
        approx :: "[hypreal, hypreal] => bool"    (infixl "≈" 50)

    syntax (HTML output)
        approx :: "[hypreal, hypreal] => bool"    (infixl "≈" 50)

    subsection{*Closure Laws for the Standard Reals*}

    lemma SReal_add [simp]:
         "[| (x::hypreal) ∈ Reals; y ∈ Reals |] ==> x + y ∈ Reals"
    apply (auto simp add: SReal_def)
    apply (rule_tac x = "r + ra" in exI, simp)
    done

    lemma SReal_mult: "[| (x::hypreal) ∈ Reals; y ∈ Reals |] ==> x * y ∈ Reals"
    apply (simp add: SReal_def, safe)
    apply (rule_tac x = "r * ra" in exI)
    apply (simp (no_asm))
    done

    lemma SReal_inverse: "(x::hypreal) ∈ Reals ==> inverse x ∈ Reals"
    apply (simp add: SReal_def)
    apply (blast intro: star_of_inverse [symmetric])
    done

    lemma SReal_divide: "[| (x::hypreal) ∈ Reals;  y ∈ Reals |] ==> x/y ∈ Reals"
    by (simp (no_asm_simp) add: SReal_mult SReal_inverse divide_inverse)

    lemma SReal_minus: "(x::hypreal) ∈ Reals ==> -x ∈ Reals"
    apply (simp add: SReal_def)
    apply (blast intro: star_of_minus [symmetric])
    done

    lemma SReal_minus_iff [simp]: "(-x ∈ Reals) = ((x::hypreal) ∈ Reals)"
    apply auto
    apply (erule_tac [2] SReal_minus)
    apply (drule SReal_minus, auto)
    done

    lemma SReal_add_cancel:
         "[| (x::hypreal) + y ∈ Reals; y ∈ Reals |] ==> x ∈ Reals"
    apply (drule_tac x = y in SReal_minus)
    apply (drule SReal_add, assumption, auto)
    done

    lemma SReal_hrabs: "(x::hypreal) ∈ Reals ==> abs x ∈ Reals"
    apply (auto simp add: SReal_def)
    apply (rule_tac x="abs r" in exI)
    apply simp
    done

    lemma SReal_hypreal_of_real [simp]: "hypreal_of_real x ∈ Reals"
    by (simp add: SReal_def)

    lemma SReal_number_of [simp]: "(number_of w ::hypreal) ∈ Reals"
    apply (simp only: star_of_number_of [symmetric])
    apply (rule SReal_hypreal_of_real)
    done

    (** As always with numerals, 0 and 1 are special cases **)

    lemma Reals_0 [simp]: "(0::hypreal) ∈ Reals"
    apply (subst numeral_0_eq_0 [symmetric])
    apply (rule SReal_number_of)
    done

    lemma Reals_1 [simp]: "(1::hypreal) ∈ Reals"
    apply (subst numeral_1_eq_1 [symmetric])
    apply (rule SReal_number_of)
    done

    lemma SReal_divide_number_of: "r ∈ Reals ==> r/(number_of w::hypreal) ∈ Reals"
    apply (simp only: divide_inverse)
    apply (blast intro!: SReal_number_of SReal_mult SReal_inverse)
    done

    text{*epsilon is not in Reals because it is an infinitesimal*}
    lemma SReal_epsilon_not_mem: "epsilon ∉ Reals"
    apply (simp add: SReal_def)
    apply (auto simp add: hypreal_of_real_not_eq_epsilon [THEN not_sym])
    done

    lemma SReal_omega_not_mem: "omega ∉ Reals"
    apply (simp add: SReal_def)
    apply (auto simp add: hypreal_of_real_not_eq_omega [THEN not_sym])
    done

    lemma SReal_UNIV_real: "{x. hypreal_of_real x ∈ Reals} = (UNIV::real set)"
    by (simp add: SReal_def)

    lemma SReal_iff: "(x ∈ Reals) = (∃y. x = hypreal_of_real y)"
    by (simp add: SReal_def)

    lemma hypreal_of_real_image: "hypreal_of_real `(UNIV::real set) = Reals"
    by (auto simp add: SReal_def)

    lemma inv_hypreal_of_real_image: "inv hypreal_of_real ` Reals = UNIV"
    apply (auto simp add: SReal_def)
    apply (rule inj_hypreal_of_real [THEN inv_f_f, THEN subst], blast)
    done

    lemma SReal_hypreal_of_real_image:
          "[| ∃x. x: P; P ⊆ Reals |] ==> ∃Q. P = hypreal_of_real ` Q"
    apply (simp add: SReal_def, blast)
    done

    lemma SReal_dense:
         "[| (x::hypreal) ∈ Reals; y ∈ Reals;  x<y |] ==> ∃r ∈ Reals. x<r & r<y"
    apply (auto simp add: SReal_iff)
    apply (drule dense, safe)
    apply (rule_tac x = "hypreal_of_real r" in bexI, auto)
    done

    text{*Completeness of Reals, but both lemmas are unused.*}

    lemma SReal_sup_lemma:
         "P ⊆ Reals ==> ((∃x ∈ P. y < x) =
          (∃X. hypreal_of_real X ∈ P & y < hypreal_of_real X))"
    by (blast dest!: SReal_iff [THEN iffD1])

    lemma SReal_sup_lemma2:
         "[| P ⊆ Reals; ∃x. x ∈ P; ∃y ∈ Reals. ∀x ∈ P. x < y |]
          ==> (∃X. X ∈ {w. hypreal_of_real w ∈ P}) &
              (∃Y. ∀X ∈ {w. hypreal_of_real w ∈ P}. X < Y)"
    apply (rule conjI)
    apply (fast dest!: SReal_iff [THEN iffD1])
    apply (auto, frule subsetD, assumption)
    apply (drule SReal_iff [THEN iffD1])
    apply (auto, rule_tac x = ya in exI, auto)
    done

    subsection{*Lifting of the Ub and Lub Properties*}

    lemma hypreal_of_real_isUb_iff:
          "(isUb (Reals) (hypreal_of_real ` Q) (hypreal_of_real Y)) =
           (isUb (UNIV :: real set) Q Y)"
    by (simp add: isUb_def setle_def)

    lemma hypreal_of_real_isLub1:
         "isLub Reals (hypreal_of_real ` Q) (hypreal_of_real Y)
          ==> isLub (UNIV :: real set) Q Y"
    apply (simp add: isLub_def leastP_def)
    apply (auto intro: hypreal_of_real_isUb_iff [THEN iffD2]
                simp add: hypreal_of_real_isUb_iff setge_def)
    done

    lemma hypreal_of_real_isLub2:
          "isLub (UNIV :: real set) Q Y
           ==> isLub Reals (hypreal_of_real ` Q) (hypreal_of_real Y)"
    apply (simp add: isLub_def leastP_def)
    apply (auto simp add: hypreal_of_real_isUb_iff setge_def)
    apply (frule_tac x2 = x in isUbD2a [THEN SReal_iff [THEN iffD1], THEN exE])
     prefer 2 apply assumption
    apply (drule_tac x = xa in spec)
    apply (auto simp add: hypreal_of_real_isUb_iff)
    done

    lemma hypreal_of_real_isLub_iff:
         "(isLub Reals (hypreal_of_real ` Q) (hypreal_of_real Y)) =
          (isLub (UNIV :: real set) Q Y)"
    by (blast intro: hypreal_of_real_isLub1 hypreal_of_real_isLub2)

    lemma lemma_isUb_hypreal_of_real:
         "isUb Reals P Y ==> ∃Yo. isUb Reals P (hypreal_of_real Yo)"
    by (auto simp add: SReal_iff isUb_def)

    lemma lemma_isLub_hypreal_of_real:
         "isLub Reals P Y ==> ∃Yo. isLub Reals P (hypreal_of_real Yo)"
    by (auto simp add: isLub_def leastP_def isUb_def SReal_iff)

    lemma lemma_isLub_hypreal_of_real2:
         "∃Yo. isLub Reals P (hypreal_of_real Yo) ==> ∃Y. isLub Reals P Y"
    by (auto simp add: isLub_def leastP_def isUb_def)

    lemma SReal_complete:
         "[| P ⊆ Reals;  ∃x. x ∈ P;  ∃Y. isUb Reals P Y |]
          ==> ∃t::hypreal. isLub Reals P t"
    apply (frule SReal_hypreal_of_real_image)
    apply (auto, drule lemma_isUb_hypreal_of_real)
    apply (auto intro!: reals_complete lemma_isLub_hypreal_of_real2
                simp add: hypreal_of_real_isLub_iff hypreal_of_real_isUb_iff)
    done

    subsection{* Set of Finite Elements is a Subring of the Extended Reals*}

    lemma HFinite_add: "[|x ∈ HFinite; y ∈ HFinite|] ==> (x+y) ∈ HFinite"
    apply (simp add: HFinite_def)
    apply (blast intro!: SReal_add hrabs_add_less)
    done

    lemma HFinite_mult: "[|x ∈ HFinite; y ∈ HFinite|] ==> x*y ∈ HFinite"
    apply (simp add: HFinite_def abs_mult)
    apply (blast intro!: SReal_mult abs_mult_less)
    done

    lemma HFinite_minus_iff: "(-x ∈ HFinite) = (x ∈ HFinite)"
    by (simp add: HFinite_def)

    lemma SReal_subset_HFinite: "Reals ⊆ HFinite"
    apply (auto simp add: SReal_def HFinite_def)
    apply (rule_tac x = "1 + abs (hypreal_of_real r) " in exI)
    apply (rule conjI, rule_tac x = "1 + abs r" in exI)
    apply simp_all
    done

    lemma HFinite_hypreal_of_real [simp]: "hypreal_of_real x ∈ HFinite"
    by (auto intro: SReal_subset_HFinite [THEN subsetD])

    lemma HFiniteD: "x ∈ HFinite ==> ∃t ∈ Reals. abs x < t"
    by (simp add: HFinite_def)

    lemma HFinite_hrabs_iff [iff]: "(abs x ∈ HFinite) = (x ∈ HFinite)"
    by (simp add: HFinite_def)

    lemma HFinite_number_of [simp]: "number_of w ∈ HFinite"
    by (rule SReal_number_of [THEN SReal_subset_HFinite [THEN subsetD]])

    (** As always with numerals, 0 and 1 are special cases **)

    lemma HFinite_0 [simp]: "0 ∈ HFinite"
    apply (subst numeral_0_eq_0 [symmetric])
    apply (rule HFinite_number_of)
    done

    lemma HFinite_1 [simp]: "1 ∈ HFinite"
    apply (subst numeral_1_eq_1 [symmetric])
    apply (rule HFinite_number_of)
    done

    lemma HFinite_bounded: "[|x ∈ HFinite; y ≤ x; 0 ≤ y |] ==> y ∈ HFinite"
    apply (case_tac "x ≤ 0")
    apply (drule_tac y = x in order_trans)
    apply (drule_tac [2] order_antisym)
    apply (auto simp add: linorder_not_le)
    apply (auto intro: order_le_less_trans simp add: abs_if HFinite_def)
    done

    subsection{* Set of Infinitesimals is a Subring of the Hyperreals*}

    lemma InfinitesimalD:
          "x ∈ Infinitesimal ==> ∀r ∈ Reals. 0 < r --> abs x < r"
    by (simp add: Infinitesimal_def)

    lemma Infinitesimal_zero [iff]: "0 ∈ Infinitesimal"
    by (simp add: Infinitesimal_def)

    lemma hypreal_sum_of_halves: "x/(2::hypreal) + x/(2::hypreal) = x"
    by auto

    lemma Infinitesimal_add:
         "[| x ∈ Infinitesimal; y ∈ Infinitesimal |] ==> (x+y) ∈ Infinitesimal"
    apply (auto simp add: Infinitesimal_def)
    apply (rule hypreal_sum_of_halves [THEN subst])
    apply (drule half_gt_zero)
    apply (blast intro: hrabs_add_less SReal_divide_number_of)
    done

    lemma Infinitesimal_minus_iff [simp]: "(-x:Infinitesimal) = (x:Infinitesimal)"
    by (simp add: Infinitesimal_def)

    lemma Infinitesimal_diff:
         "[| x ∈ Infinitesimal;  y ∈ Infinitesimal |] ==> x-y ∈ Infinitesimal"
    by (simp add: diff_def Infinitesimal_add)

    lemma Infinitesimal_mult:
         "[| x ∈ Infinitesimal; y ∈ Infinitesimal |] ==> (x * y) ∈ Infinitesimal"
    apply (auto simp add: Infinitesimal_def abs_mult)
    apply (case_tac "y=0", simp)
    apply (cut_tac a = "abs x" and b = 1 and c = "abs y" and d = r
           in mult_strict_mono, auto)
    done

    lemma Infinitesimal_HFinite_mult:
         "[| x ∈ Infinitesimal; y ∈ HFinite |] ==> (x * y) ∈ Infinitesimal"
    apply (auto dest!: HFiniteD simp add: Infinitesimal_def abs_mult)
    apply (frule hrabs_less_gt_zero)
    apply (drule_tac x = "r/t" in bspec)
    apply (blast intro: SReal_divide)
    apply (cut_tac a = "abs x" and b = "r/t" and c = "abs y" in mult_strict_mono)
    apply (auto simp add: zero_less_divide_iff)
    done

    lemma Infinitesimal_HFinite_mult2:
         "[| x ∈ Infinitesimal; y ∈ HFinite |] ==> (y * x) ∈ Infinitesimal"
    by (auto dest: Infinitesimal_HFinite_mult simp add: mult_commute)

    (*** rather long proof ***)
    lemma HInfinite_inverse_Infinitesimal:
         "x ∈ HInfinite ==> inverse x: Infinitesimal"
    apply (auto simp add: HInfinite_def Infinitesimal_def)
    apply (erule_tac x = "inverse r" in ballE)
    apply (frule_tac a1 = r and z = "abs x" in positive_imp_inverse_positive [THEN order_less_trans], assumption)
    apply (drule inverse_inverse_eq [symmetric, THEN subst])
    apply (rule inverse_less_iff_less [THEN iffD1])
    apply (auto simp add: SReal_inverse)
    done

    lemma HInfinite_mult: "[|x ∈ HInfinite;y ∈ HInfinite|] ==> (x*y) ∈ HInfinite"
    apply (auto simp add: HInfinite_def  abs_mult)
    apply (erule_tac x = 1 in ballE)
    apply (erule_tac x = r in ballE)
    apply (case_tac "y=0", simp)
    apply (cut_tac c = 1 and d = "abs x" and a = r and b = "abs y" in mult_strict_mono)
    apply (auto simp add: mult_ac)
    done

    lemma hypreal_add_zero_less_le_mono: "[|r < x; (0::hypreal) ≤ y|] ==> r < x+y"
    by (auto dest: add_less_le_mono)

    lemma HInfinite_add_ge_zero:
          "[|x ∈ HInfinite; 0 ≤ y; 0 ≤ x|] ==> (x + y): HInfinite"
    by (auto intro!: hypreal_add_zero_less_le_mono
           simp add: abs_if add_commute add_nonneg_nonneg HInfinite_def)

    lemma HInfinite_add_ge_zero2:
         "[|x ∈ HInfinite; 0 ≤ y; 0 ≤ x|] ==> (y + x): HInfinite"
    by (auto intro!: HInfinite_add_ge_zero simp add: add_commute)

    lemma HInfinite_add_gt_zero:
         "[|x ∈ HInfinite; 0 < y; 0 < x|] ==> (x + y): HInfinite"
    by (blast intro: HInfinite_add_ge_zero order_less_imp_le)

    lemma HInfinite_minus_iff: "(-x ∈ HInfinite) = (x ∈ HInfinite)"
    by (simp add: HInfinite_def)

    lemma HInfinite_add_le_zero:
         "[|x ∈ HInfinite; y ≤ 0; x ≤ 0|] ==> (x + y): HInfinite"
    apply (drule HInfinite_minus_iff [THEN iffD2])
    apply (rule HInfinite_minus_iff [THEN iffD1])
    apply (auto intro: HInfinite_add_ge_zero)
    done

    lemma HInfinite_add_lt_zero:
         "[|x ∈ HInfinite; y < 0; x < 0|] ==> (x + y): HInfinite"
    by (blast intro: HInfinite_add_le_zero order_less_imp_le)

    lemma HFinite_sum_squares:
         "[|a: HFinite; b: HFinite; c: HFinite|]
          ==> a*a + b*b + c*c ∈ HFinite"
    by (auto intro: HFinite_mult HFinite_add)

    lemma not_Infinitesimal_not_zero: "x ∉ Infinitesimal ==> x ≠ 0"
    by auto

    lemma not_Infinitesimal_not_zero2: "x ∈ HFinite - Infinitesimal ==> x ≠ 0"
    by auto

    lemma Infinitesimal_hrabs_iff [iff]:
         "(abs x ∈ Infinitesimal) = (x ∈ Infinitesimal)"
    by (auto simp add: abs_if)

    lemma HFinite_diff_Infinitesimal_hrabs:
         "x ∈ HFinite - Infinitesimal ==> abs x ∈ HFinite - Infinitesimal"
    by blast

    lemma hrabs_less_Infinitesimal:
          "[| e ∈ Infinitesimal; abs x < e |] ==> x ∈ Infinitesimal"
    by (auto simp add: Infinitesimal_def abs_less_iff)

    lemma hrabs_le_Infinitesimal:
         "[| e ∈ Infinitesimal; abs x ≤ e |] ==> x ∈ Infinitesimal"
    by (blast dest: order_le_imp_less_or_eq intro: hrabs_less_Infinitesimal)

    lemma Infinitesimal_interval:
          "[| e ∈ Infinitesimal; e' ∈ Infinitesimal; e' < x ; x < e |]
           ==> x ∈ Infinitesimal"
    by (auto simp add: Infinitesimal_def abs_less_iff)

    lemma Infinitesimal_interval2:
         "[| e ∈ Infinitesimal; e' ∈ Infinitesimal;
             e' ≤ x ; x ≤ e |] ==> x ∈ Infinitesimal"
    by (auto intro: Infinitesimal_interval simp add: order_le_less)

    lemma not_Infinitesimal_mult:
         "[| x ∉ Infinitesimal;  y ∉ Infinitesimal|] ==> (x*y) ∉Infinitesimal"
    apply (unfold Infinitesimal_def, clarify)
    apply (simp add: linorder_not_less abs_mult)
    apply (erule_tac x = "r*ra" in ballE)
    prefer 2 apply (fast intro: SReal_mult)
    apply (auto simp add: zero_less_mult_iff)
    apply (cut_tac c = ra and d = "abs y" and a = r and b = "abs x" in mult_mono, auto)
    done

    lemma Infinitesimal_mult_disj:
         "x*y ∈ Infinitesimal ==> x ∈ Infinitesimal | y ∈ Infinitesimal"
    apply (rule ccontr)
    apply (drule de_Morgan_disj [THEN iffD1])
    apply (fast dest: not_Infinitesimal_mult)
    done

    lemma HFinite_Infinitesimal_not_zero: "x ∈ HFinite-Infinitesimal ==> x ≠ 0"
    by blast

    lemma HFinite_Infinitesimal_diff_mult:
         "[| x ∈ HFinite - Infinitesimal;
                       y ∈ HFinite - Infinitesimal
                    |] ==> (x*y) ∈ HFinite - Infinitesimal"
    apply clarify
    apply (blast dest: HFinite_mult not_Infinitesimal_mult)
    done

    lemma Infinitesimal_subset_HFinite:
          "Infinitesimal ⊆ HFinite"
    apply (simp add: Infinitesimal_def HFinite_def, auto)
    apply (rule_tac x = 1 in bexI, auto)
    done

    lemma Infinitesimal_hypreal_of_real_mult:
         "x ∈ Infinitesimal ==> x * hypreal_of_real r ∈ Infinitesimal"
    by (erule HFinite_hypreal_of_real [THEN [2] Infinitesimal_HFinite_mult])

    lemma Infinitesimal_hypreal_of_real_mult2:
         "x ∈ Infinitesimal ==> hypreal_of_real r * x ∈ Infinitesimal"
    by (erule HFinite_hypreal_of_real [THEN [2] Infinitesimal_HFinite_mult2])

    subsection{*The Infinitely Close Relation*}

    lemma mem_infmal_iff: "(x ∈ Infinitesimal) = (x @= 0)"
    by (simp add: Infinitesimal_def approx_def)

    lemma approx_minus_iff: " (x @= y) = (x + -y @= 0)"
    by (simp add: approx_def)

    lemma approx_minus_iff2: " (x @= y) = (-y + x @= 0)"
    by (simp add: approx_def add_commute)

    lemma approx_refl [iff]: "x @= x"
    by (simp add: approx_def Infinitesimal_def)

    lemma hypreal_minus_distrib1: "-(y + -(x::hypreal)) = x + -y"
    by (simp add: add_commute)

    lemma approx_sym: "x @= y ==> y @= x"
    apply (simp add: approx_def)
    apply (rule hypreal_minus_distrib1 [THEN subst])
    apply (erule Infinitesimal_minus_iff [THEN iffD2])
    done

    lemma approx_trans: "[| x @= y; y @= z |] ==> x @= z"
    apply (simp add: approx_def)
    apply (drule Infinitesimal_add, assumption, auto)
    done

    lemma approx_trans2: "[| r @= x; s @= x |] ==> r @= s"
    by (blast intro: approx_sym approx_trans)

    lemma approx_trans3: "[| x @= r; x @= s|] ==> r @= s"
    by (blast intro: approx_sym approx_trans)

    lemma number_of_approx_reorient: "(number_of w @= x) = (x @= number_of w)"
    by (blast intro: approx_sym)

    lemma zero_approx_reorient: "(0 @= x) = (x @= 0)"
    by (blast intro: approx_sym)

    lemma one_approx_reorient: "(1 @= x) = (x @= 1)"
    by (blast intro: approx_sym)

    ML
    {*
    val SReal_add = thm "SReal_add";
    val SReal_mult = thm "SReal_mult";
    val SReal_inverse = thm "SReal_inverse";
    val SReal_divide = thm "SReal_divide";
    val SReal_minus = thm "SReal_minus";
    val SReal_minus_iff = thm "SReal_minus_iff";
    val SReal_add_cancel = thm "SReal_add_cancel";
    val SReal_hrabs = thm "SReal_hrabs";
    val SReal_hypreal_of_real = thm "SReal_hypreal_of_real";
    val SReal_number_of = thm "SReal_number_of";
    val Reals_0 = thm "Reals_0";
    val Reals_1 = thm "Reals_1";
    val SReal_divide_number_of = thm "SReal_divide_number_of";
    val SReal_epsilon_not_mem = thm "SReal_epsilon_not_mem";
    val SReal_omega_not_mem = thm "SReal_omega_not_mem";
    val SReal_UNIV_real = thm "SReal_UNIV_real";
    val SReal_iff = thm "SReal_iff";
    val hypreal_of_real_image = thm "hypreal_of_real_image";
    val inv_hypreal_of_real_image = thm "inv_hypreal_of_real_image";
    val SReal_hypreal_of_real_image = thm "SReal_hypreal_of_real_image";
    val SReal_dense = thm "SReal_dense";
    val hypreal_of_real_isUb_iff = thm "hypreal_of_real_isUb_iff";
    val hypreal_of_real_isLub1 = thm "hypreal_of_real_isLub1";
    val hypreal_of_real_isLub2 = thm "hypreal_of_real_isLub2";
    val hypreal_of_real_isLub_iff = thm "hypreal_of_real_isLub_iff";
    val lemma_isUb_hypreal_of_real = thm "lemma_isUb_hypreal_of_real";
    val lemma_isLub_hypreal_of_real = thm "lemma_isLub_hypreal_of_real";
    val lemma_isLub_hypreal_of_real2 = thm "lemma_isLub_hypreal_of_real2";
    val SReal_complete = thm "SReal_complete";
    val HFinite_add = thm "HFinite_add";
    val HFinite_mult = thm "HFinite_mult";
    val HFinite_minus_iff = thm "HFinite_minus_iff";
    val SReal_subset_HFinite = thm "SReal_subset_HFinite";
    val HFinite_hypreal_of_real = thm "HFinite_hypreal_of_real";
    val HFiniteD = thm "HFiniteD";
    val HFinite_hrabs_iff = thm "HFinite_hrabs_iff";
    val HFinite_number_of = thm "HFinite_number_of";
    val HFinite_0 = thm "HFinite_0";
    val HFinite_1 = thm "HFinite_1";
    val HFinite_bounded = thm "HFinite_bounded";
    val InfinitesimalD = thm "InfinitesimalD";
    val Infinitesimal_zero = thm "Infinitesimal_zero";
    val hypreal_sum_of_halves = thm "hypreal_sum_of_halves";
    val Infinitesimal_add = thm "Infinitesimal_add";
    val Infinitesimal_minus_iff = thm "Infinitesimal_minus_iff";
    val Infinitesimal_diff = thm "Infinitesimal_diff";
    val Infinitesimal_mult = thm "Infinitesimal_mult";
    val Infinitesimal_HFinite_mult = thm "Infinitesimal_HFinite_mult";
    val Infinitesimal_HFinite_mult2 = thm "Infinitesimal_HFinite_mult2";
    val HInfinite_inverse_Infinitesimal = thm "HInfinite_inverse_Infinitesimal";
    val HInfinite_mult = thm "HInfinite_mult";
    val HInfinite_add_ge_zero = thm "HInfinite_add_ge_zero";
    val HInfinite_add_ge_zero2 = thm "HInfinite_add_ge_zero2";
    val HInfinite_add_gt_zero = thm "HInfinite_add_gt_zero";
    val HInfinite_minus_iff = thm "HInfinite_minus_iff";
    val HInfinite_add_le_zero = thm "HInfinite_add_le_zero";
    val HInfinite_add_lt_zero = thm "HInfinite_add_lt_zero";
    val HFinite_sum_squares = thm "HFinite_sum_squares";
    val not_Infinitesimal_not_zero = thm "not_Infinitesimal_not_zero";
    val not_Infinitesimal_not_zero2 = thm "not_Infinitesimal_not_zero2";
    val Infinitesimal_hrabs_iff = thm "Infinitesimal_hrabs_iff";
    val HFinite_diff_Infinitesimal_hrabs = thm "HFinite_diff_Infinitesimal_hrabs";
    val hrabs_less_Infinitesimal = thm "hrabs_less_Infinitesimal";
    val hrabs_le_Infinitesimal = thm "hrabs_le_Infinitesimal";
    val Infinitesimal_interval = thm "Infinitesimal_interval";
    val Infinitesimal_interval2 = thm "Infinitesimal_interval2";
    val not_Infinitesimal_mult = thm "not_Infinitesimal_mult";
    val Infinitesimal_mult_disj = thm "Infinitesimal_mult_disj";
    val HFinite_Infinitesimal_not_zero = thm "HFinite_Infinitesimal_not_zero";
    val HFinite_Infinitesimal_diff_mult = thm "HFinite_Infinitesimal_diff_mult";
    val Infinitesimal_subset_HFinite = thm "Infinitesimal_subset_HFinite";
    val Infinitesimal_hypreal_of_real_mult = thm "Infinitesimal_hypreal_of_real_mult";
    val Infinitesimal_hypreal_of_real_mult2 = thm "Infinitesimal_hypreal_of_real_mult2";
    val mem_infmal_iff = thm "mem_infmal_iff";
    val approx_minus_iff = thm "approx_minus_iff";
    val approx_minus_iff2 = thm "approx_minus_iff2";
    val approx_refl = thm "approx_refl";
    val approx_sym = thm "approx_sym";
    val approx_trans = thm "approx_trans";
    val approx_trans2 = thm "approx_trans2";
    val approx_trans3 = thm "approx_trans3";
    val number_of_approx_reorient = thm "number_of_approx_reorient";
    val zero_approx_reorient = thm "zero_approx_reorient";
    val one_approx_reorient = thm "one_approx_reorient";

    (*** re-orientation, following HOL/Integ/Bin.ML
         We re-orient x @=y where x is 0, 1 or a numeral, unless y is as well!
     ***)

    (*reorientation simprules using ==, for the following simproc*)
    val meta_zero_approx_reorient = zero_approx_reorient RS eq_reflection;
    val meta_one_approx_reorient = one_approx_reorient RS eq_reflection;
    val meta_number_of_approx_reorient = number_of_approx_reorient RS eq_reflection

    (*reorientation simplification procedure: reorients (polymorphic)
      0 = x, 1 = x, nnn = x provided x isn't 0, 1 or a numeral.*)
    fun reorient_proc sg _ (_ $ t $ u) =
      case u of
          Const("0", _) => NONE
        | Const("1", _) => NONE
        | Const("Numeral.number_of", _) $ _ => NONE
        | _ => SOME (case t of
                    Const("0", _) => meta_zero_approx_reorient
                  | Const("1", _) => meta_one_approx_reorient
                  | Const("Numeral.number_of", _) $ _ =>
                                     meta_number_of_approx_reorient);

    val approx_reorient_simproc =
      Bin_Simprocs.prep_simproc
        ("reorient_simproc", ["0@=x", "1@=x", "number_of w @= x"], reorient_proc);

    Addsimprocs [approx_reorient_simproc];
    *}

    lemma Infinitesimal_approx_minus: "(x-y ∈ Infinitesimal) = (x @= y)"
    by (auto simp add: diff_def approx_minus_iff [symmetric] mem_infmal_iff)

    lemma approx_monad_iff: "(x @= y) = (monad(x)=monad(y))"
    apply (simp add: monad_def)
    apply (auto dest: approx_sym elim!: approx_trans equalityCE)
    done

    lemma Infinitesimal_approx:
         "[| x ∈ Infinitesimal; y ∈ Infinitesimal |] ==> x @= y"
    apply (simp add: mem_infmal_iff)
    apply (blast intro: approx_trans approx_sym)
    done

    lemma approx_add: "[| a @= b; c @= d |] ==> a+c @= b+d"
    proof (unfold approx_def)
      assume inf: "a + - b ∈ Infinitesimal" "c + - d ∈ Infinitesimal"
      have "a + c + - (b + d) = (a + - b) + (c + - d)" by arith
      also have "... ∈ Infinitesimal" using inf by (rule Infinitesimal_add)
      finally show "a + c + - (b + d) ∈ Infinitesimal" .
    qed

    lemma approx_minus: "a @= b ==> -a @= -b"
    apply (rule approx_minus_iff [THEN iffD2, THEN approx_sym])
    apply (drule approx_minus_iff [THEN iffD1])
    apply (simp (no_asm) add: add_commute)
    done

    lemma approx_minus2: "-a @= -b ==> a @= b"
    by (auto dest: approx_minus)

    lemma approx_minus_cancel [simp]: "(-a @= -b) = (a @= b)"
    by (blast intro: approx_minus approx_minus2)

    lemma approx_add_minus: "[| a @= b; c @= d |] ==> a + -c @= b + -d"
    by (blast intro!: approx_add approx_minus)

    lemma approx_mult1: "[| a @= b; c: HFinite|] ==> a*c @= b*c"
    by (simp add: approx_def Infinitesimal_HFinite_mult minus_mult_left
                  left_distrib [symmetric]
             del: minus_mult_left [symmetric])

    lemma approx_mult2: "[|a @= b; c: HFinite|] ==> c*a @= c*b"
    by (simp add: approx_mult1 mult_commute)

    lemma approx_mult_subst: "[|u @= v*x; x @= y; v ∈ HFinite|] ==> u @= v*y"
    by (blast intro: approx_mult2 approx_trans)

    lemma approx_mult_subst2: "[| u @= x*v; x @= y; v ∈ HFinite |] ==> u @= y*v"
    by (blast intro: approx_mult1 approx_trans)

    lemma approx_mult_subst_SReal:
         "[| u @= x*hypreal_of_real v; x @= y |] ==> u @= y*hypreal_of_real v"
    by (auto intro: approx_mult_subst2)

    lemma approx_eq_imp: "a = b ==> a @= b"
    by (simp add: approx_def)

    lemma Infinitesimal_minus_approx: "x ∈ Infinitesimal ==> -x @= x"
    by (blast intro: Infinitesimal_minus_iff [THEN iffD2]
                        mem_infmal_iff [THEN iffD1] approx_trans2)

    lemma bex_Infinitesimal_iff: "(∃y ∈ Infinitesimal. x + -z = y) = (x @= z)"
    by (simp add: approx_def)

    lemma bex_Infinitesimal_iff2: "(∃y ∈ Infinitesimal. x = z + y) = (x @= z)"
    by (force simp add: bex_Infinitesimal_iff [symmetric])

    lemma Infinitesimal_add_approx: "[| y ∈ Infinitesimal; x + y = z |] ==> x @= z"
    apply (rule bex_Infinitesimal_iff [THEN iffD1])
    apply (drule Infinitesimal_minus_iff [THEN iffD2])
    apply (auto simp add: add_assoc [symmetric])
    done

    lemma Infinitesimal_add_approx_self: "y ∈ Infinitesimal ==> x @= x + y"
    apply (rule bex_Infinitesimal_iff [THEN iffD1])
    apply (drule Infinitesimal_minus_iff [THEN iffD2])
    apply (auto simp add: add_assoc [symmetric])
    done

    lemma Infinitesimal_add_approx_self2: "y ∈ Infinitesimal ==> x @= y + x"
    by (auto dest: Infinitesimal_add_approx_self simp add: add_commute)

    lemma Infinitesimal_add_minus_approx_self: "y ∈ Infinitesimal ==> x @= x + -y"
    by (blast intro!: Infinitesimal_add_approx_self Infinitesimal_minus_iff [THEN iffD2])

    lemma Infinitesimal_add_cancel: "[| y ∈ Infinitesimal; x+y @= z|] ==> x @= z"
    apply (drule_tac x = x in Infinitesimal_add_approx_self [THEN approx_sym])
    apply (erule approx_trans3 [THEN approx_sym], assumption)
    done

    lemma Infinitesimal_add_right_cancel:
         "[| y ∈ Infinitesimal; x @= z + y|] ==> x @= z"
    apply (drule_tac x = z in Infinitesimal_add_approx_self2 [THEN approx_sym])
    apply (erule approx_trans3 [THEN approx_sym])
    apply (simp add: add_commute)
    apply (erule approx_sym)
    done

    lemma approx_add_left_cancel: "d + b  @= d + c ==> b @= c"
    apply (drule approx_minus_iff [THEN iffD1])
    apply (simp add: approx_minus_iff [symmetric] add_ac)
    done

    lemma approx_add_right_cancel: "b + d @= c + d ==> b @= c"
    apply (rule approx_add_left_cancel)
    apply (simp add: add_commute)
    done

    lemma approx_add_mono1: "b @= c ==> d + b @= d + c"
    apply (rule approx_minus_iff [THEN iffD2])
    apply (simp add: approx_minus_iff [symmetric] add_ac)
    done

    lemma approx_add_mono2: "b @= c ==> b + a @= c + a"
    by (simp add: add_commute approx_add_mono1)

    lemma approx_add_left_iff [simp]: "(a + b @= a + c) = (b @= c)"
    by (fast elim: approx_add_left_cancel approx_add_mono1)

    lemma approx_add_right_iff [simp]: "(b + a @= c + a) = (b @= c)"
    by (simp add: add_commute)

    lemma approx_HFinite: "[| x ∈ HFinite; x @= y |] ==> y ∈ HFinite"
    apply (drule bex_Infinitesimal_iff2 [THEN iffD2], safe)
    apply (drule Infinitesimal_subset_HFinite [THEN subsetD, THEN HFinite_minus_iff [THEN iffD2]])
    apply (drule HFinite_add)
    apply (auto simp add: add_assoc)
    done

    lemma approx_hypreal_of_real_HFinite: "x @= hypreal_of_real D ==> x ∈ HFinite"
    by (rule approx_sym [THEN [2] approx_HFinite], auto)

    lemma approx_mult_HFinite:
         "[|a @= b; c @= d; b: HFinite; d: HFinite|] ==> a*c @= b*d"
    apply (rule approx_trans)
    apply (rule_tac [2] approx_mult2)
    apply (rule approx_mult1)
    prefer 2 apply (blast intro: approx_HFinite approx_sym, auto)
    done

    lemma approx_mult_hypreal_of_real:
         "[|a @= hypreal_of_real b; c @= hypreal_of_real d |]
          ==> a*c @= hypreal_of_real b*hypreal_of_real d"
    by (blast intro!: approx_mult_HFinite approx_hypreal_of_real_HFinite
                      HFinite_hypreal_of_real)

    lemma approx_SReal_mult_cancel_zero:
         "[| a ∈ Reals; a ≠ 0; a*x @= 0 |] ==> x @= 0"
    apply (drule SReal_inverse [THEN SReal_subset_HFinite [THEN subsetD]])
    apply (auto dest: approx_mult2 simp add: mult_assoc [symmetric])
    done

    lemma approx_mult_SReal1: "[| a ∈ Reals; x @= 0 |] ==> x*a @= 0"
    by (auto dest: SReal_subset_HFinite [THEN subsetD] approx_mult1)

    lemma approx_mult_SReal2: "[| a ∈ Reals; x @= 0 |] ==> a*x @= 0"
    by (auto dest: SReal_subset_HFinite [THEN subsetD] approx_mult2)

    lemma approx_mult_SReal_zero_cancel_iff [simp]:
         "[|a ∈ Reals; a ≠ 0 |] ==> (a*x @= 0) = (x @= 0)"
    by (blast intro: approx_SReal_mult_cancel_zero approx_mult_SReal2)

    lemma approx_SReal_mult_cancel:
         "[| a ∈ Reals; a ≠ 0; a* w @= a*z |] ==> w @= z"
    apply (drule SReal_inverse [THEN SReal_subset_HFinite [THEN subsetD]])
    apply (auto dest: approx_mult2 simp add: mult_assoc [symmetric])
    done

    lemma approx_SReal_mult_cancel_iff1 [simp]:
         "[| a ∈ Reals; a ≠ 0|] ==> (a* w @= a*z) = (w @= z)"
    by (auto intro!: approx_mult2 SReal_subset_HFinite [THEN subsetD]
             intro: approx_SReal_mult_cancel)

    lemma approx_le_bound: "[| z ≤ f; f @= g; g ≤ z |] ==> f @= z"
    apply (simp add: bex_Infinitesimal_iff2 [symmetric], auto)
    apply (rule_tac x = "g+y-z" in bexI)
    apply (simp (no_asm))
    apply (rule Infinitesimal_interval2)
    apply (rule_tac [2] Infinitesimal_zero, auto)
    done

    subsection{* Zero is the Only Infinitesimal that is also a Real*}

    lemma Infinitesimal_less_SReal:
         "[| x ∈ Reals; y ∈ Infinitesimal; 0 < x |] ==> y < x"
    apply (simp add: Infinitesimal_def)
    apply (rule abs_ge_self [THEN order_le_less_trans], auto)
    done

    lemma Infinitesimal_less_SReal2:
         "y ∈ Infinitesimal ==> ∀r ∈ Reals. 0 < r --> y < r"
    by (blast intro: Infinitesimal_less_SReal)

    lemma SReal_not_Infinitesimal:
         "[| 0 < y;  y ∈ Reals|] ==> y ∉ Infinitesimal"
    apply (simp add: Infinitesimal_def)
    apply (auto simp add: abs_if)
    done

    lemma SReal_minus_not_Infinitesimal:
         "[| y < 0;  y ∈ Reals |] ==> y ∉ Infinitesimal"
    apply (subst Infinitesimal_minus_iff [symmetric])
    apply (rule SReal_not_Infinitesimal, auto)
    done

    lemma SReal_Int_Infinitesimal_zero: "Reals Int Infinitesimal = {0}"
    apply auto
    apply (cut_tac x = x and y = 0 in linorder_less_linear)
    apply (blast dest: SReal_not_Infinitesimal SReal_minus_not_Infinitesimal)
    done

    lemma SReal_Infinitesimal_zero: "[| x ∈ Reals; x ∈ Infinitesimal|] ==> x = 0"
    by (cut_tac SReal_Int_Infinitesimal_zero, blast)

    lemma SReal_HFinite_diff_Infinitesimal:
         "[| x ∈ Reals; x ≠ 0 |] ==> x ∈ HFinite - Infinitesimal"
    by (auto dest: SReal_Infinitesimal_zero SReal_subset_HFinite [THEN subsetD])

    lemma hypreal_of_real_HFinite_diff_Infinitesimal:
         "hypreal_of_real x ≠ 0 ==> hypreal_of_real x ∈ HFinite - Infinitesimal"
    by (rule SReal_HFinite_diff_Infinitesimal, auto)

    lemma hypreal_of_real_Infinitesimal_iff_0 [iff]:
         "(hypreal_of_real x ∈ Infinitesimal) = (x=0)"
    apply auto
    apply (rule ccontr)
    apply (rule hypreal_of_real_HFinite_diff_Infinitesimal [THEN DiffD2], auto)
    done

    lemma number_of_not_Infinitesimal [simp]:
         "number_of w ≠ (0::hypreal) ==> number_of w ∉ Infinitesimal"
    by (fast dest: SReal_number_of [THEN SReal_Infinitesimal_zero])

    (*again: 1 is a special case, but not 0 this time*)
    lemma one_not_Infinitesimal [simp]: "1 ∉ Infinitesimal"
    apply (subst numeral_1_eq_1 [symmetric])
    apply (rule number_of_not_Infinitesimal)
    apply (simp (no_asm))
    done

    lemma approx_SReal_not_zero: "[| y ∈ Reals; x @= y; y≠ 0 |] ==> x ≠ 0"
    apply (cut_tac x = 0 and y = y in linorder_less_linear, simp)
    apply (blast dest: approx_sym [THEN mem_infmal_iff [THEN iffD2]] SReal_not_Infinitesimal SReal_minus_not_Infinitesimal)
    done

    lemma HFinite_diff_Infinitesimal_approx:
         "[| x @= y; y ∈ HFinite - Infinitesimal |]
          ==> x ∈ HFinite - Infinitesimal"
    apply (auto intro: approx_sym [THEN [2] approx_HFinite]
                simp add: mem_infmal_iff)
    apply (drule approx_trans3, assumption)
    apply (blast dest: approx_sym)
    done

    (*The premise y≠0 is essential; otherwise x/y =0 and we lose the
      HFinite premise.*)
    lemma Infinitesimal_ratio:
         "[| y ≠ 0;  y ∈ Infinitesimal;  x/y ∈ HFinite |] ==> x ∈ Infinitesimal"
    apply (drule Infinitesimal_HFinite_mult2, assumption)
    apply (simp add: divide_inverse mult_assoc)
    done

    lemma Infinitesimal_SReal_divide:
      "[| x ∈ Infinitesimal; y ∈ Reals |] ==> x/y ∈ Infinitesimal"
    apply (simp add: divide_inverse)
    apply (auto intro!: Infinitesimal_HFinite_mult
                dest!: SReal_inverse [THEN SReal_subset_HFinite [THEN subsetD]])
    done

    (*------------------------------------------------------------------
           Standard Part Theorem: Every finite x: R* is infinitely
           close to a unique real number (i.e a member of Reals)
     ------------------------------------------------------------------*)

    subsection{* Uniqueness: Two Infinitely Close Reals are Equal*}

    lemma SReal_approx_iff: "[|x ∈ Reals; y ∈ Reals|] ==> (x @= y) = (x = y)"
    apply auto
    apply (simp add: approx_def)
    apply (drule_tac x = y in SReal_minus)
    apply (drule SReal_add, assumption)
    apply (drule SReal_Infinitesimal_zero, assumption)
    apply (drule sym)
    apply (simp add: hypreal_eq_minus_iff [symmetric])
    done

    lemma number_of_approx_iff [simp]:
         "(number_of v @= number_of w) = (number_of v = (number_of w :: hypreal))"
    by (auto simp add: SReal_approx_iff)

    (*And also for 0 @= #nn and 1 @= #nn, #nn @= 0 and #nn @= 1.*)
    lemma [simp]: "(0 @= number_of w) = ((number_of w :: hypreal) = 0)"
                  "(number_of w @= 0) = ((number_of w :: hypreal) = 0)"
                  "(1 @= number_of w) = ((number_of w :: hypreal) = 1)"
                  "(number_of w @= 1) = ((number_of w :: hypreal) = 1)"
                  "~ (0 @= 1)" "~ (1 @= 0)"
    by (auto simp only: SReal_number_of SReal_approx_iff Reals_0 Reals_1)

    lemma hypreal_of_real_approx_iff [simp]:
         "(hypreal_of_real k @= hypreal_of_real m) = (k = m)"
    apply auto
    apply (rule inj_hypreal_of_real [THEN injD])
    apply (rule SReal_approx_iff [THEN iffD1], auto)
    done

    lemma hypreal_of_real_approx_number_of_iff [simp]:
         "(hypreal_of_real k @= number_of w) = (k = number_of w)"
    by (subst hypreal_of_real_approx_iff [symmetric], auto)

    (*And also for 0 and 1.*)
    (*And also for 0 @= #nn and 1 @= #nn, #nn @= 0 and #nn @= 1.*)
    lemma [simp]: "(hypreal_of_real k @= 0) = (k = 0)"
                  "(hypreal_of_real k @= 1) = (k = 1)"
      by (simp_all add:  hypreal_of_real_approx_iff [symmetric])

    lemma approx_unique_real:
         "[| r ∈ Reals; s ∈ Reals; r @= x; s @= x|] ==> r = s"
    by (blast intro: SReal_approx_iff [THEN iffD1] approx_trans2)

    subsection{* Existence of Unique Real Infinitely Close*}

    (* lemma about lubs *)
    lemma hypreal_isLub_unique:
         "[| isLub R S x; isLub R S y |] ==> x = (y::hypreal)"
    apply (frule isLub_isUb)
    apply (frule_tac x = y in isLub_isUb)
    apply (blast intro!: order_antisym dest!: isLub_le_isUb)
    done

    lemma lemma_st_part_ub:
         "x ∈ HFinite ==> ∃u. isUb Reals {s. s ∈ Reals & s < x} u"
    apply (drule HFiniteD, safe)
    apply (rule exI, rule isUbI)
    apply (auto intro: setleI isUbI simp add: abs_less_iff)
    done

    lemma lemma_st_part_nonempty: "x ∈ HFinite ==> ∃y. y ∈ {s. s ∈ Reals & s < x}"
    apply (drule HFiniteD, safe)
    apply (drule SReal_minus)
    apply (rule_tac x = "-t" in exI)
    apply (auto simp add: abs_less_iff)
    done

    lemma lemma_st_part_subset: "{s. s ∈ Reals & s < x} ⊆ Reals"
    by auto

    lemma lemma_st_part_lub:
         "x ∈ HFinite ==> ∃t. isLub Reals {s. s ∈ Reals & s < x} t"
    by (blast intro!: SReal_complete lemma_st_part_ub lemma_st_part_nonempty lemma_st_part_subset)

    lemma lemma_hypreal_le_left_cancel: "((t::hypreal) + r ≤ t) = (r ≤ 0)"
    apply safe
    apply (drule_tac c = "-t" in add_left_mono)
    apply (drule_tac [2] c = t in add_left_mono)
    apply (auto simp add: add_assoc [symmetric])
    done

    lemma lemma_st_part_le1:
         "[| x ∈ HFinite;  isLub Reals {s. s ∈ Reals & s < x} t;
             r ∈ Reals;  0 < r |] ==> x ≤ t + r"
    apply (frule isLubD1a)
    apply (rule ccontr, drule linorder_not_le [THEN iffD2])
    apply (drule_tac x = t in SReal_add, assumption)
    apply (drule_tac y = "t + r" in isLubD1 [THEN setleD], auto)
    done

    lemma hypreal_setle_less_trans:
         "!!x::hypreal. [| S *<= x; x < y |] ==> S *<= y"
    apply (simp add: setle_def)
    apply (auto dest!: bspec order_le_less_trans intro: order_less_imp_le)
    done

    lemma hypreal_gt_isUb:
         "!!x::hypreal. [| isUb R S x; x < y; y ∈ R |] ==> isUb R S y"
    apply (simp add: isUb_def)
    apply (blast intro: hypreal_setle_less_trans)
    done

    lemma lemma_st_part_gt_ub:
         "[| x ∈ HFinite; x < y; y ∈ Reals |]
          ==> isUb Reals {s. s ∈ Reals & s < x} y"
    by (auto dest: order_less_trans intro: order_less_imp_le intro!: isUbI setleI)

    lemma lemma_minus_le_zero: "t ≤ t + -r ==> r ≤ (0::hypreal)"
    apply (drule_tac c = "-t" in add_left_mono)
    apply (auto simp add: add_assoc [symmetric])
    done

    lemma lemma_st_part_le2:
         "[| x ∈ HFinite;
             isLub Reals {s. s ∈ Reals & s < x} t;
             r ∈ Reals; 0 < r |]
          ==> t + -r ≤ x"
    apply (frule isLubD1a)
    apply (rule ccontr, drule linorder_not_le [THEN iffD1])
    apply (drule SReal_minus, drule_tac x = t in SReal_add, assumption)
    apply (drule lemma_st_part_gt_ub, assumption+)
    apply (drule isLub_le_isUb, assumption)
    apply (drule lemma_minus_le_zero)
    apply (auto dest: order_less_le_trans)
    done

    lemma lemma_st_part1a:
         "[| x ∈ HFinite;
             isLub Reals {s. s ∈ Reals & s < x} t;
             r ∈ Reals; 0 < r |]
          ==> x + -t ≤ r"
    apply (subgoal_tac "x ≤ t+r")
    apply (auto intro: lemma_st_part_le1)
    done

    lemma lemma_st_part2a:
         "[| x ∈ HFinite;
             isLub Reals {s. s ∈ Reals & s < x} t;
             r ∈ Reals;  0 < r |]
          ==> -(x + -t) ≤ r"
    apply (subgoal_tac "(t + -r ≤ x)")
    apply (auto intro: lemma_st_part_le2)
    done

    lemma lemma_SReal_ub:
         "(x::hypreal) ∈ Reals ==> isUb Reals {s. s ∈ Reals & s < x} x"
    by (auto intro: isUbI setleI order_less_imp_le)

    lemma lemma_SReal_lub:
         "(x::hypreal) ∈ Reals ==> isLub Reals {s. s ∈ Reals & s < x} x"
    apply (auto intro!: isLubI2 lemma_SReal_ub setgeI)
    apply (frule isUbD2a)
    apply (rule_tac x = x and y = y in linorder_cases)
    apply (auto intro!: order_less_imp_le)
    apply (drule SReal_dense, assumption, assumption, safe)
    apply (drule_tac y = r in isUbD)
    apply (auto dest: order_less_le_trans)
    done

    lemma lemma_st_part_not_eq1:
         "[| x ∈ HFinite;
             isLub Reals {s. s ∈ Reals & s < x} t;
             r ∈ Reals; 0 < r |]
          ==> x + -t ≠ r"
    apply auto
    apply (frule isLubD1a [THEN SReal_minus])
    apply (drule SReal_add_cancel, assumption)
    apply (drule_tac x = x in lemma_SReal_lub)
    apply (drule hypreal_isLub_unique, assumption, auto)
    done

    lemma lemma_st_part_not_eq2:
         "[| x ∈ HFinite;
             isLub Reals {s. s ∈ Reals & s < x} t;
             r ∈ Reals; 0 < r |]
          ==> -(x + -t) ≠ r"
    apply (auto)
    apply (frule isLubD1a)
    apply (drule SReal_add_cancel, assumption)
    apply (drule_tac x = "-x" in SReal_minus, simp)
    apply (drule_tac x = x in lemma_SReal_lub)
    apply (drule hypreal_isLub_unique, assumption, auto)
    done

    lemma lemma_st_part_major:
         "[| x ∈ HFinite;
             isLub Reals {s. s ∈ Reals & s < x} t;
             r ∈ Reals; 0 < r |]
          ==> abs (x + -t) < r"
    apply (frule lemma_st_part1a)
    apply (frule_tac [4] lemma_st_part2a, auto)
    apply (drule order_le_imp_less_or_eq)+
    apply (auto dest: lemma_st_part_not_eq1 lemma_st_part_not_eq2 simp add: abs_less_iff)
    done

    lemma lemma_st_part_major2:
         "[| x ∈ HFinite; isLub Reals {s. s ∈ Reals & s < x} t |]
          ==> ∀r ∈ Reals. 0 < r --> abs (x + -t) < r"
    by (blast dest!: lemma_st_part_major)

    text{*Existence of real and Standard Part Theorem*}
    lemma lemma_st_part_Ex:
         "x ∈ HFinite ==> ∃t ∈ Reals. ∀r ∈ Reals. 0 < r --> abs (x + -t) < r"
    apply (frule lemma_st_part_lub, safe)
    apply (frule isLubD1a)
    apply (blast dest: lemma_st_part_major2)
    done

    lemma st_part_Ex:
         "x ∈ HFinite ==> ∃t ∈ Reals. x @= t"
    apply (simp add: approx_def Infinitesimal_def)
    apply (drule lemma_st_part_Ex, auto)
    done

    text{*There is a unique real infinitely close*}
    lemma st_part_Ex1: "x ∈ HFinite ==> EX! t. t ∈ Reals & x @= t"
    apply (drule st_part_Ex, safe)
    apply (drule_tac [2] approx_sym, drule_tac [2] approx_sym, drule_tac [2] approx_sym)
    apply (auto intro!: approx_unique_real)
    done

    subsection{* Finite, Infinite and Infinitesimal*}

    lemma HFinite_Int_HInfinite_empty [simp]: "HFinite Int HInfinite = {}"
    apply (simp add: HFinite_def HInfinite_def)
    apply (auto dest: order_less_trans)
    done

    lemma HFinite_not_HInfinite:
      assumes x: "x ∈ HFinite" shows "x ∉ HInfinite"
    proof
      assume x': "x ∈ HInfinite"
      with x have "x ∈ HFinite ∩ HInfinite" by blast
      thus False by auto
    qed

    lemma not_HFinite_HInfinite: "x∉ HFinite ==> x ∈ HInfinite"
    apply (simp add: HInfinite_def HFinite_def, auto)
    apply (drule_tac x = "r + 1" in bspec)
    apply (auto)
    done

    lemma HInfinite_HFinite_disj: "x ∈ HInfinite | x ∈ HFinite"
    by (blast intro: not_HFinite_HInfinite)

    lemma HInfinite_HFinite_iff: "(x ∈ HInfinite) = (x ∉ HFinite)"
    by (blast dest: HFinite_not_HInfinite not_HFinite_HInfinite)

    lemma HFinite_HInfinite_iff: "(x ∈ HFinite) = (x ∉ HInfinite)"
    by (simp add: HInfinite_HFinite_iff)

    lemma HInfinite_diff_HFinite_Infinitesimal_disj:
         "x ∉ Infinitesimal ==> x ∈ HInfinite | x ∈ HFinite - Infinitesimal"
    by (fast intro: not_HFinite_HInfinite)

    lemma HFinite_inverse:
         "[| x ∈ HFinite; x ∉ Infinitesimal |] ==> inverse x ∈ HFinite"
    apply (cut_tac x = "inverse x" in HInfinite_HFinite_disj)
    apply (auto dest!: HInfinite_inverse_Infinitesimal)
    done

    lemma HFinite_inverse2: "x ∈ HFinite - Infinitesimal ==> inverse x ∈ HFinite"
    by (blast intro: HFinite_inverse)

    (* stronger statement possible in fact *)
    lemma Infinitesimal_inverse_HFinite:
         "x ∉ Infinitesimal ==> inverse(x) ∈ HFinite"
    apply (drule HInfinite_diff_HFinite_Infinitesimal_disj)
    apply (blast intro: HFinite_inverse HInfinite_inverse_Infinitesimal Infinitesimal_subset_HFinite [THEN subsetD])
    done

    lemma HFinite_not_Infinitesimal_inverse:
         "x ∈ HFinite - Infinitesimal ==> inverse x ∈ HFinite - Infinitesimal"
    apply (auto intro: Infinitesimal_inverse_HFinite)
    apply (drule Infinitesimal_HFinite_mult2, assumption)
    apply (simp add: not_Infinitesimal_not_zero right_inverse)
    done

    lemma approx_inverse:
         "[| x @= y; y ∈  HFinite - Infinitesimal |]
          ==> inverse x @= inverse y"
    apply (frule HFinite_diff_Infinitesimal_approx, assumption)
    apply (frule not_Infinitesimal_not_zero2)
    apply (frule_tac x = x in not_Infinitesimal_not_zero2)
    apply (drule HFinite_inverse2)+
    apply (drule approx_mult2, assumption, auto)
    apply (drule_tac c = "inverse x" in approx_mult1, assumption)
    apply (auto intro: approx_sym simp add: mult_assoc)
    done

    (*Used for NSLIM_inverse, NSLIMSEQ_inverse*)
    lemmas hypreal_of_real_approx_inverse =  hypreal_of_real_HFinite_diff_Infinitesimal [THEN [2] approx_inverse]

    lemma inverse_add_Infinitesimal_approx:
         "[| x ∈ HFinite - Infinitesimal;
             h ∈ Infinitesimal |] ==> inverse(x + h) @= inverse x"
    apply (auto intro: approx_inverse approx_sym Infinitesimal_add_approx_self)
    done

    lemma inverse_add_Infinitesimal_approx2:
         "[| x ∈ HFinite - Infinitesimal;
             h ∈ Infinitesimal |] ==> inverse(h + x) @= inverse x"
    apply (rule add_commute [THEN subst])
    apply (blast intro: inverse_add_Infinitesimal_approx)
    done

    lemma inverse_add_Infinitesimal_approx_Infinitesimal:
         "[| x ∈ HFinite - Infinitesimal;
             h ∈ Infinitesimal |] ==> inverse(x + h) + -inverse x @= h"
    apply (rule approx_trans2)
    apply (auto intro: inverse_add_Infinitesimal_approx
                simp add: mem_infmal_iff approx_minus_iff [symmetric])
    done

    lemma Infinitesimal_square_iff: "(x ∈ Infinitesimal) = (x*x ∈ Infinitesimal)"
    apply (auto intro: Infinitesimal_mult)
    apply (rule ccontr, frule Infinitesimal_inverse_HFinite)
    apply (frule not_Infinitesimal_not_zero)
    apply (auto dest: Infinitesimal_HFinite_mult simp add: mult_assoc)
    done
    declare Infinitesimal_square_iff [symmetric, simp]

    lemma HFinite_square_iff [simp]: "(x*x ∈ HFinite) = (x ∈ HFinite)"
    apply (auto intro: HFinite_mult)
    apply (auto dest: HInfinite_mult simp add: HFinite_HInfinite_iff)
    done

    lemma HInfinite_square_iff [simp]: "(x*x ∈ HInfinite) = (x ∈ HInfinite)"
    by (auto simp add: HInfinite_HFinite_iff)

    lemma approx_HFinite_mult_cancel:
         "[| a: HFinite-Infinitesimal; a* w @= a*z |] ==> w @= z"
    apply safe
    apply (frule HFinite_inverse, assumption)
    apply (drule not_Infinitesimal_not_zero)
    apply (auto dest: approx_mult2 simp add: mult_assoc [symmetric])
    done

    lemma approx_HFinite_mult_cancel_iff1:
         "a: HFinite-Infinitesimal ==> (a * w @= a * z) = (w @= z)"
    by (auto intro: approx_mult2 approx_HFinite_mult_cancel)

    lemma HInfinite_HFinite_add_cancel:
         "[| x + y ∈ HInfinite; y ∈ HFinite |] ==> x ∈ HInfinite"
    apply (rule ccontr)
    apply (drule HFinite_HInfinite_iff [THEN iffD2])
    apply (auto dest: HFinite_add simp add: HInfinite_HFinite_iff)
    done

    lemma HInfinite_HFinite_add:
         "[| x ∈ HInfinite; y ∈ HFinite |] ==> x + y ∈ HInfinite"
    apply (rule_tac y = "-y" in HInfinite_HFinite_add_cancel)
    apply (auto simp add: add_assoc HFinite_minus_iff)
    done

    lemma HInfinite_ge_HInfinite:
         "[| x ∈ HInfinite; x ≤ y; 0 ≤ x |] ==> y ∈ HInfinite"
    by (auto intro: HFinite_bounded simp add: HInfinite_HFinite_iff)

    lemma Infinitesimal_inverse_HInfinite:
         "[| x ∈ Infinitesimal; x ≠ 0 |] ==> inverse x ∈ HInfinite"
    apply (rule ccontr, drule HFinite_HInfinite_iff [THEN iffD2])
    apply (auto dest: Infinitesimal_HFinite_mult2)
    done

    lemma HInfinite_HFinite_not_Infinitesimal_mult:
         "[| x ∈ HInfinite; y ∈ HFinite - Infinitesimal |]
          ==> x * y ∈ HInfinite"
    apply (rule ccontr, drule HFinite_HInfinite_iff [THEN iffD2])
    apply (frule HFinite_Infinitesimal_not_zero)
    apply (drule HFinite_not_Infinitesimal_inverse)
    apply (safe, drule HFinite_mult)
    apply (auto simp add: mult_assoc HFinite_HInfinite_iff)
    done

    lemma HInfinite_HFinite_not_Infinitesimal_mult2:
         "[| x ∈ HInfinite; y ∈ HFinite - Infinitesimal |]
          ==> y * x ∈ HInfinite"
    by (auto simp add: mult_commute HInfinite_HFinite_not_Infinitesimal_mult)

    lemma HInfinite_gt_SReal: "[| x ∈ HInfinite; 0 < x; y ∈ Reals |] ==> y < x"
    by (auto dest!: bspec simp add: HInfinite_def abs_if order_less_imp_le)

    lemma HInfinite_gt_zero_gt_one: "[| x ∈ HInfinite; 0 < x |] ==> 1 < x"
    by (auto intro: HInfinite_gt_SReal)

    lemma not_HInfinite_one [simp]: "1 ∉ HInfinite"
    apply (simp (no_asm) add: HInfinite_HFinite_iff)
    done

    lemma approx_hrabs_disj: "abs x @= x | abs x @= -x"
    by (cut_tac x = x in hrabs_disj, auto)

    subsection{*Theorems about Monads*}

    lemma monad_hrabs_Un_subset: "monad (abs x) ≤ monad(x) Un monad(-x)"
    by (rule_tac x1 = x in hrabs_disj [THEN disjE], auto)

    lemma Infinitesimal_monad_eq: "e ∈ Infinitesimal ==> monad (x+e) = monad x"
    by (fast intro!: Infinitesimal_add_approx_self [THEN approx_sym] approx_monad_iff [THEN iffD1])

    lemma mem_monad_iff: "(u ∈ monad x) = (-u ∈ monad (-x))"
    by (simp add: monad_def)

    lemma Infinitesimal_monad_zero_iff: "(x ∈ Infinitesimal) = (x ∈ monad 0)"
    by (auto intro: approx_sym simp add: monad_def mem_infmal_iff)

    lemma monad_zero_minus_iff: "(x ∈ monad 0) = (-x ∈ monad 0)"
    apply (simp (no_asm) add: Infinitesimal_monad_zero_iff [symmetric])
    done

    lemma monad_zero_hrabs_iff: "(x ∈ monad 0) = (abs x ∈ monad 0)"
    apply (rule_tac x1 = x in hrabs_disj [THEN disjE])
    apply (auto simp add: monad_zero_minus_iff [symmetric])
    done

    lemma mem_monad_self [simp]: "x ∈ monad x"
    by (simp add: monad_def)

    subsection{*Proof that @{term "x @= y"} implies @{term"¦x¦ @= ¦y¦"}*}

    lemma approx_subset_monad: "x @= y ==> {x,y} ≤ monad x"
    apply (simp (no_asm))
    apply (simp add: approx_monad_iff)
    done

    lemma approx_subset_monad2: "x @= y ==> {x,y} ≤ monad y"
    apply (drule approx_sym)
    apply (fast dest: approx_subset_monad)
    done

    lemma mem_monad_approx: "u ∈ monad x ==> x @= u"
    by (simp add: monad_def)

    lemma approx_mem_monad: "x @= u ==> u ∈ monad x"
    by (simp add: monad_def)

    lemma approx_mem_monad2: "x @= u ==> x ∈ monad u"
    apply (simp add: monad_def)
    apply (blast intro!: approx_sym)
    done

    lemma approx_mem_monad_zero: "[| x @= y;x ∈ monad 0 |] ==> y ∈ monad 0"
    apply (drule mem_monad_approx)
    apply (fast intro: approx_mem_monad approx_trans)
    done

    lemma Infinitesimal_approx_hrabs:
         "[| x @= y; x ∈ Infinitesimal |] ==> abs x @= abs y"
    apply (drule Infinitesimal_monad_zero_iff [THEN iffD1])
    apply (blast intro: approx_mem_monad_zero monad_zero_hrabs_iff [THEN iffD1] mem_monad_approx approx_trans3)
    done

    lemma less_Infinitesimal_less:
         "[| 0 < x;  x ∉Infinitesimal;  e :Infinitesimal |] ==> e < x"
    apply (rule ccontr)
    apply (auto intro: Infinitesimal_zero [THEN [2] Infinitesimal_interval]
                dest!: order_le_imp_less_or_eq simp add: linorder_not_less)
    done

    lemma Ball_mem_monad_gt_zero:
         "[| 0 < x;  x ∉ Infinitesimal; u ∈ monad x |] ==> 0 < u"
    apply (drule mem_monad_approx [THEN approx_sym])
    apply (erule bex_Infinitesimal_iff2 [THEN iffD2, THEN bexE])
    apply (drule_tac e = "-xa" in less_Infinitesimal_less, auto)
    done

    lemma Ball_mem_monad_less_zero:
         "[| x < 0; x ∉ Infinitesimal; u ∈ monad x |] ==> u < 0"
    apply (drule mem_monad_approx [THEN approx_sym])
    apply (erule bex_Infinitesimal_iff [THEN iffD2, THEN bexE])
    apply (cut_tac x = "-x" and e = xa in less_Infinitesimal_less, auto)
    done

    lemma lemma_approx_gt_zero:
         "[|0 < x; x ∉ Infinitesimal; x @= y|] ==> 0 < y"
    by (blast dest: Ball_mem_monad_gt_zero approx_subset_monad)

    lemma lemma_approx_less_zero:
         "[|x < 0; x ∉ Infinitesimal; x @= y|] ==> y < 0"
    by (blast dest: Ball_mem_monad_less_zero approx_subset_monad)

    theorem approx_hrabs: "x @= y ==> abs x @= abs y"
    apply (case_tac "x ∈ Infinitesimal")
    apply (simp add: Infinitesimal_approx_hrabs)
    apply (rule linorder_cases [of 0 x])
    apply (frule lemma_approx_gt_zero [of x y])
    apply (auto simp add: lemma_approx_less_zero [of x y] abs_of_neg)
    done

    lemma approx_hrabs_zero_cancel: "abs(x) @= 0 ==> x @= 0"
    apply (cut_tac x = x in hrabs_disj)
    apply (auto dest: approx_minus)
    done

    lemma approx_hrabs_add_Infinitesimal: "e ∈ Infinitesimal ==> abs x @= abs(x+e)"
    by (fast intro: approx_hrabs Infinitesimal_add_approx_self)

    lemma approx_hrabs_add_minus_Infinitesimal:
         "e ∈ Infinitesimal ==> abs x @= abs(x + -e)"
    by (fast intro: approx_hrabs Infinitesimal_add_minus_approx_self)

    lemma hrabs_add_Infinitesimal_cancel:
         "[| e ∈ Infinitesimal; e' ∈ Infinitesimal;
             abs(x+e) = abs(y+e')|] ==> abs x @= abs y"
    apply (drule_tac x = x in approx_hrabs_add_Infinitesimal)
    apply (drule_tac x = y in approx_hrabs_add_Infinitesimal)
    apply (auto intro: approx_trans2)
    done

    lemma hrabs_add_minus_Infinitesimal_cancel:
         "[| e ∈ Infinitesimal; e' ∈ Infinitesimal;
             abs(x + -e) = abs(y + -e')|] ==> abs x @= abs y"
    apply (drule_tac x = x in approx_hrabs_add_minus_Infinitesimal)
    apply (drule_tac x = y in approx_hrabs_add_minus_Infinitesimal)
    apply (auto intro: approx_trans2)
    done

    (* interesting slightly counterintuitive theorem: necessary
       for proving that an open interval is an NS open set
    *)
    lemma Infinitesimal_add_hypreal_of_real_less:
         "[| x < y;  u ∈ Infinitesimal |]
          ==> hypreal_of_real x + u < hypreal_of_real y"
    apply (simp add: Infinitesimal_def)
    apply (drule_tac x = "hypreal_of_real y + -hypreal_of_real x" in bspec, simp)
    apply (simp add: abs_less_iff)
    done

    lemma Infinitesimal_add_hrabs_hypreal_of_real_less:
         "[| x ∈ Infinitesimal; abs(hypreal_of_real r) < hypreal_of_real y |]
          ==> abs (hypreal_of_real r + x) < hypreal_of_real y"
    apply (drule_tac x = "hypreal_of_real r" in approx_hrabs_add_Infinitesimal)
    apply (drule approx_sym [THEN bex_Infinitesimal_iff2 [THEN iffD2]])
    apply (auto intro!: Infinitesimal_add_hypreal_of_real_less
                simp del: star_of_abs
                simp add: hypreal_of_real_hrabs)
    done

    lemma Infinitesimal_add_hrabs_hypreal_of_real_less2:
         "[| x ∈ Infinitesimal;  abs(hypreal_of_real r) < hypreal_of_real y |]
          ==> abs (x + hypreal_of_real r) < hypreal_of_real y"
    apply (rule add_commute [THEN subst])
    apply (erule Infinitesimal_add_hrabs_hypreal_of_real_less, assumption)
    done

    lemma hypreal_of_real_le_add_Infininitesimal_cancel:
         "[| u ∈ Infinitesimal; v ∈ Infinitesimal;
             hypreal_of_real x + u ≤ hypreal_of_real y + v |]
          ==> hypreal_of_real x ≤ hypreal_of_real y"
    apply (simp add: linorder_not_less [symmetric], auto)
    apply (drule_tac u = "v-u" in Infinitesimal_add_hypreal_of_real_less)
    apply (auto simp add: Infinitesimal_diff)
    done

    lemma hypreal_of_real_le_add_Infininitesimal_cancel2:
         "[| u ∈ Infinitesimal; v ∈ Infinitesimal;
             hypreal_of_real x + u ≤ hypreal_of_real y + v |]
          ==> x ≤ y"
    by (blast intro: star_of_le [THEN iffD1]
              intro!: hypreal_of_real_le_add_Infininitesimal_cancel)

    lemma hypreal_of_real_less_Infinitesimal_le_zero:
        "[| hypreal_of_real x < e; e ∈ Infinitesimal |] ==> hypreal_of_real x ≤ 0"
    apply (rule linorder_not_less [THEN iffD1], safe)
    apply (drule Infinitesimal_interval)
    apply (drule_tac [4] SReal_hypreal_of_real [THEN SReal_Infinitesimal_zero], auto)
    done

    (*used once, in Lim/NSDERIV_inverse*)
    lemma Infinitesimal_add_not_zero:
         "[| h ∈ Infinitesimal; x ≠ 0 |] ==> hypreal_of_real x + h ≠ 0"
    apply auto
    apply (subgoal_tac "h = - hypreal_of_real x", auto)
    done

    lemma Infinitesimal_square_cancel [simp]:
         "x*x + y*y ∈ Infinitesimal ==> x*x ∈ Infinitesimal"
    apply (rule Infinitesimal_interval2)
    apply (rule_tac [3] zero_le_square, assumption)
    apply (auto simp add: zero_le_square)
    done

    lemma HFinite_square_cancel [simp]: "x*x + y*y ∈ HFinite ==> x*x ∈ HFinite"
    apply (rule HFinite_bounded, assumption)
    apply (auto simp add: zero_le_square)
    done

    lemma Infinitesimal_square_cancel2 [simp]:
         "x*x + y*y ∈ Infinitesimal ==> y*y ∈ Infinitesimal"
    apply (rule Infinitesimal_square_cancel)
    apply (rule add_commute [THEN subst])
    apply (simp (no_asm))
    done

    lemma HFinite_square_cancel2 [simp]: "x*x + y*y ∈ HFinite ==> y*y ∈ HFinite"
    apply (rule HFinite_square_cancel)
    apply (rule add_commute [THEN subst])
    apply (simp (no_asm))
    done

    lemma Infinitesimal_sum_square_cancel [simp]:
         "x*x + y*y + z*z ∈ Infinitesimal ==> x*x ∈ Infinitesimal"
    apply (rule Infinitesimal_interval2, assumption)
    apply (rule_tac [2] zero_le_square, simp)
    apply (insert zero_le_square [of y])
    apply (insert zero_le_square [of z], simp)
    done

    lemma HFinite_sum_square_cancel [simp]:
         "x*x + y*y + z*z ∈ HFinite ==> x*x ∈ HFinite"
    apply (rule HFinite_bounded, assumption)
    apply (rule_tac [2] zero_le_square)
    apply (insert zero_le_square [of y])
    apply (insert zero_le_square [of z], simp)
    done

    lemma Infinitesimal_sum_square_cancel2 [simp]:
         "y*y + x*x + z*z ∈ Infinitesimal ==> x*x ∈ Infinitesimal"
    apply (rule Infinitesimal_sum_square_cancel)
    apply (simp add: add_ac)
    done

    lemma HFinite_sum_square_cancel2 [simp]:
         "y*y + x*x + z*z ∈ HFinite ==> x*x ∈ HFinite"
    apply (rule HFinite_sum_square_cancel)
    apply (simp add: add_ac)
    done

    lemma Infinitesimal_sum_square_cancel3 [simp]:
         "z*z + y*y + x*x ∈ Infinitesimal ==> x*x ∈ Infinitesimal"
    apply (rule Infinitesimal_sum_square_cancel)
    apply (simp add: add_ac)
    done

    lemma HFinite_sum_square_cancel3 [simp]:
         "z*z + y*y + x*x ∈ HFinite ==> x*x ∈ HFinite"
    apply (rule HFinite_sum_square_cancel)
    apply (simp add: add_ac)
    done

    lemma monad_hrabs_less:
         "[| y ∈ monad x; 0 < hypreal_of_real e |]
          ==> abs (y + -x) < hypreal_of_real e"
    apply (drule mem_monad_approx [THEN approx_sym])
    apply (drule bex_Infinitesimal_iff [THEN iffD2])
    apply (auto dest!: InfinitesimalD)
    done

    lemma mem_monad_SReal_HFinite:
         "x ∈ monad (hypreal_of_real  a) ==> x ∈ HFinite"
    apply (drule mem_monad_approx [THEN approx_sym])
    apply (drule bex_Infinitesimal_iff2 [THEN iffD2])
    apply (safe dest!: Infinitesimal_subset_HFinite [THEN subsetD])
    apply (erule SReal_hypreal_of_real [THEN SReal_subset_HFinite [THEN subsetD], THEN HFinite_add])
    done

    subsection{* Theorems about Standard Part*}

    lemma st_approx_self: "x ∈ HFinite ==> st x @= x"
    apply (simp add: st_def)
    apply (frule st_part_Ex, safe)
    apply (rule someI2)
    apply (auto intro: approx_sym)
    done

    lemma st_SReal: "x ∈ HFinite ==> st x ∈ Reals"
    apply (simp add: st_def)
    apply (frule st_part_Ex, safe)
    apply (rule someI2)
    apply (auto intro: approx_sym)
    done

    lemma st_HFinite: "x ∈ HFinite ==> st x ∈ HFinite"
    by (erule st_SReal [THEN SReal_subset_HFinite [THEN subsetD]])

    lemma st_SReal_eq: "x ∈ Reals ==> st x = x"
    apply (simp add: st_def)
    apply (rule some_equality)
    apply (fast intro: SReal_subset_HFinite [THEN subsetD])
    apply (blast dest: SReal_approx_iff [THEN iffD1])
    done

    lemma st_hypreal_of_real [simp]: "st (hypreal_of_real x) = hypreal_of_real x"
    by (rule SReal_hypreal_of_real [THEN st_SReal_eq])

    lemma st_eq_approx: "[| x ∈ HFinite; y ∈ HFinite; st x = st y |] ==> x @= y"
    by (auto dest!: st_approx_self elim!: approx_trans3)

    lemma approx_st_eq:
      assumes "x ∈ HFinite" and "y ∈ HFinite" and "x @= y"
      shows "st x = st y"
    proof -
      have "st x @= x" "st y @= y" "st x ∈ Reals" "st y ∈ Reals"
        by (simp_all add: st_approx_self st_SReal prems)
      with prems show ?thesis
        by (fast elim: approx_trans approx_trans2 SReal_approx_iff [THEN iffD1])
    qed

    lemma st_eq_approx_iff:
         "[| x ∈ HFinite; y ∈ HFinite|]
                       ==> (x @= y) = (st x = st y)"
    by (blast intro: approx_st_eq st_eq_approx)

    lemma st_Infinitesimal_add_SReal:
         "[| x ∈ Reals; e ∈ Infinitesimal |] ==> st(x + e) = x"
    apply (frule st_SReal_eq [THEN subst])
    prefer 2 apply assumption
    apply (frule SReal_subset_HFinite [THEN subsetD])
    apply (frule Infinitesimal_subset_HFinite [THEN subsetD])
    apply (drule st_SReal_eq)
    apply (rule approx_st_eq)
    apply (auto intro: HFinite_add simp add: Infinitesimal_add_approx_self [THEN approx_sym])
    done

    lemma st_Infinitesimal_add_SReal2:
         "[| x ∈ Reals; e ∈ Infinitesimal |] ==> st(e + x) = x"
    apply (rule add_commute [THEN subst])
    apply (blast intro!: st_Infinitesimal_add_SReal)
    done

    lemma HFinite_st_Infinitesimal_add:
         "x ∈ HFinite ==> ∃e ∈ Infinitesimal. x = st(x) + e"
    by (blast dest!: st_approx_self [THEN approx_sym] bex_Infinitesimal_iff2 [THEN iffD2])

    lemma st_add:
      assumes x: "x ∈ HFinite" and y: "y ∈ HFinite"
      shows "st (x + y) = st(x) + st(y)"
    proof -
      from HFinite_st_Infinitesimal_add [OF x]
      obtain ex where ex: "ex ∈ Infinitesimal" "st x + ex = x"
        by (blast intro: sym)
      from HFinite_st_Infinitesimal_add [OF y]
      obtain ey where ey: "ey ∈ Infinitesimal" "st y + ey = y"
        by (blast intro: sym)
      have "st (x + y) = st ((st x + ex) + (st y + ey))"
        by (simp add: ex ey)
      also have "... = st ((ex + ey) + (st x + st y))" by (simp add: add_ac)
      also have "... = st x + st y"
        by (simp add: prems st_SReal Infinitesimal_add
                      st_Infinitesimal_add_SReal2)
      finally show ?thesis .
    qed

    lemma st_number_of [simp]: "st (number_of w) = number_of w"
    by (rule SReal_number_of [THEN st_SReal_eq])

    (*the theorem above for the special cases of zero and one*)
    lemma [simp]: "st 0 = 0" "st 1 = 1"
    by (simp_all add: st_SReal_eq)

    lemma st_minus: assumes "y ∈ HFinite" shows "st(-y) = -st(y)"
    proof -
      have "st (- y) + st y = 0"
       by (simp add: prems st_add [symmetric] HFinite_minus_iff)
      thus ?thesis by arith
    qed

    lemma st_diff: "[| x ∈ HFinite; y ∈ HFinite |] ==> st (x-y) = st(x) - st(y)"
    apply (simp add: diff_def)
    apply (frule_tac y1 = y in st_minus [symmetric])
    apply (drule_tac x1 = y in HFinite_minus_iff [THEN iffD2])
    apply (simp (no_asm_simp) add: st_add)
    done

    lemma lemma_st_mult:
         "[| x ∈ HFinite; y ∈ HFinite; e ∈ Infinitesimal; ea ∈ Infinitesimal |]
          ==> e*y + x*ea + e*ea ∈ Infinitesimal"
    apply (frule_tac x = e and y = y in Infinitesimal_HFinite_mult)
    apply (frule_tac [2] x = ea and y = x in Infinitesimal_HFinite_mult)
    apply (drule_tac [3] Infinitesimal_mult)
    apply (auto intro: Infinitesimal_add simp add: add_ac mult_ac)
    done

    lemma st_mult: "[| x ∈ HFinite; y ∈ HFinite |] ==> st (x * y) = st(x) * st(y)"
    apply (frule HFinite_st_Infinitesimal_add)
    apply (frule_tac x = y in HFinite_st_Infinitesimal_add, safe)
    apply (subgoal_tac "st (x * y) = st ((st x + e) * (st y + ea))")
    apply (drule_tac [2] sym, drule_tac [2] sym)
     prefer 2 apply simp
    apply (erule_tac V = "x = st x + e" in thin_rl)
    apply (erule_tac V = "y = st y + ea" in thin_rl)
    apply (simp add: left_distrib right_distrib)
    apply (drule st_SReal)+
    apply (simp (no_asm_use) add: add_assoc)
    apply (rule st_Infinitesimal_add_SReal)
    apply (blast intro!: SReal_mult)
    apply (drule SReal_subset_HFinite [THEN subsetD])+
    apply (rule add_assoc [THEN subst])
    apply (blast intro!: lemma_st_mult)
    done

    lemma st_Infinitesimal: "x ∈ Infinitesimal ==> st x = 0"
    apply (subst numeral_0_eq_0 [symmetric])
    apply (rule st_number_of [THEN subst])
    apply (rule approx_st_eq)
    apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
                simp add: mem_infmal_iff [symmetric])
    done

    lemma st_not_Infinitesimal: "st(x) ≠ 0 ==> x ∉ Infinitesimal"
    by (fast intro: st_Infinitesimal)

    lemma st_inverse:
         "[| x ∈ HFinite; st x ≠ 0 |]
          ==> st(inverse x) = inverse (st x)"
    apply (rule_tac c1 = "st x" in hypreal_mult_left_cancel [THEN iffD1])
    apply (auto simp add: st_mult [symmetric] st_not_Infinitesimal HFinite_inverse)
    apply (subst right_inverse, auto)
    done

    lemma st_divide [simp]:
         "[| x ∈ HFinite; y ∈ HFinite; st y ≠ 0 |]
          ==> st(x/y) = (st x) / (st y)"
    by (simp add: divide_inverse st_mult st_not_Infinitesimal HFinite_inverse st_inverse)

    lemma st_idempotent [simp]: "x ∈ HFinite ==> st(st(x)) = st(x)"
    by (blast intro: st_HFinite st_approx_self approx_st_eq)

    lemma Infinitesimal_add_st_less:
         "[| x ∈ HFinite; y ∈ HFinite; u ∈ Infinitesimal; st x < st y |]
          ==> st x + u < st y"
    apply (drule st_SReal)+
    apply (auto intro!: Infinitesimal_add_hypreal_of_real_less simp add: SReal_iff)
    done

    lemma Infinitesimal_add_st_le_cancel:
         "[| x ∈ HFinite; y ∈ HFinite;
             u ∈ Infinitesimal; st x ≤ st y + u
          |] ==> st x ≤ st y"
    apply (simp add: linorder_not_less [symmetric])
    apply (auto dest: Infinitesimal_add_st_less)
    done

    lemma st_le: "[| x ∈ HFinite; y ∈ HFinite; x ≤ y |] ==> st(x) ≤ st(y)"
    apply (frule HFinite_st_Infinitesimal_add)
    apply (rotate_tac 1)
    apply (frule HFinite_st_Infinitesimal_add, safe)
    apply (rule Infinitesimal_add_st_le_cancel)
    apply (rule_tac [3] x = ea and y = e in Infinitesimal_diff)
    apply (auto simp add: add_assoc [symmetric])
    done

    lemma st_zero_le: "[| 0 ≤ x;  x ∈ HFinite |] ==> 0 ≤ st x"
    apply (subst numeral_0_eq_0 [symmetric])
    apply (rule st_number_of [THEN subst])
    apply (rule st_le, auto)
    done

    lemma st_zero_ge: "[| x ≤ 0;  x ∈ HFinite |] ==> st x ≤ 0"
    apply (subst numeral_0_eq_0 [symmetric])
    apply (rule st_number_of [THEN subst])
    apply (rule st_le, auto)
    done

    lemma st_hrabs: "x ∈ HFinite ==> abs(st x) = st(abs x)"
    apply (simp add: linorder_not_le st_zero_le abs_if st_minus
       linorder_not_less)
    apply (auto dest!: st_zero_ge [OF order_less_imp_le])
    done

    subsection{*Alternative Definitions for @{term HFinite} using Free Ultrafilter*}

    lemma FreeUltrafilterNat_Rep_hypreal:
         "[| X ∈ Rep_star x; Y ∈ Rep_star x |]
          ==> {n. X n = Y n} ∈ FreeUltrafilterNat"
    by (cases x, unfold star_n_def, auto, ultra)

    lemma HFinite_FreeUltrafilterNat:
        "x ∈ HFinite
         ==> ∃X ∈ Rep_star x. ∃u. {n. abs (X n) < u} ∈ FreeUltrafilterNat"
    apply (cases x)
    apply (auto simp add: HFinite_def abs_less_iff minus_less_iff [of x]
                  star_of_def
                  star_n_less SReal_iff star_n_minus)
    apply (rule_tac x=X in bexI)
    apply (rule_tac x=y in exI, ultra)
    apply simp
    done

    lemma FreeUltrafilterNat_HFinite:
         "∃X ∈ Rep_star x.
           ∃u. {n. abs (X n) < u} ∈ FreeUltrafilterNat
           ==>  x ∈ HFinite"
    apply (cases x)
    apply (auto simp add: HFinite_def abs_less_iff minus_less_iff [of x])
    apply (rule_tac x = "hypreal_of_real u" in bexI)
    apply (auto simp add: star_n_less SReal_iff star_n_minus star_of_def)
    apply ultra+
    done

    lemma HFinite_FreeUltrafilterNat_iff:
         "(x ∈ HFinite) = (∃X ∈ Rep_star x.
               ∃u. {n. abs (X n) < u} ∈ FreeUltrafilterNat)"
    by (blast intro!: HFinite_FreeUltrafilterNat FreeUltrafilterNat_HFinite)

    subsection{*Alternative Definitions for @{term HInfinite} using Free Ultrafilter*}

    lemma lemma_Compl_eq: "- {n. (u::real) < abs (xa n)} = {n. abs (xa n) ≤ u}"
    by auto

    lemma lemma_Compl_eq2: "- {n. abs (xa n) < (u::real)} = {n. u ≤ abs (xa n)}"
    by auto

    lemma lemma_Int_eq1:
         "{n. abs (xa n) ≤ (u::real)} Int {n. u ≤ abs (xa n)}
              = {n. abs(xa n) = u}"
    by auto

    lemma lemma_FreeUltrafilterNat_one:
         "{n. abs (xa n) = u} ≤ {n. abs (xa n) < u + (1::real)}"
    by auto

    (*-------------------------------------
      Exclude this type of sets from free
      ultrafilter for Infinite numbers!
     -------------------------------------*)
    lemma FreeUltrafilterNat_const_Finite:
         "[| xa: Rep_star x;
                      {n. abs (xa n) = u} ∈ FreeUltrafilterNat
                   |] ==> x ∈ HFinite"
    apply (rule FreeUltrafilterNat_HFinite)
    apply (rule_tac x = xa in bexI)
    apply (rule_tac x = "u + 1" in exI)
    apply (ultra, assumption)
    done

    lemma HInfinite_FreeUltrafilterNat:
         "x ∈ HInfinite ==> ∃X ∈ Rep_star x.
               ∀u. {n. u < abs (X n)} ∈ FreeUltrafilterNat"
    apply (frule HInfinite_HFinite_iff [THEN iffD1])
    apply (cut_tac x = x in Rep_hypreal_nonempty)
    apply (auto simp del: Rep_hypreal_nonempty simp add: HFinite_FreeUltrafilterNat_iff Bex_def)
    apply (drule spec)+
    apply auto
    apply (drule_tac x = u in spec)
    apply (drule FreeUltrafilterNat_Compl_mem)+
    apply (drule FreeUltrafilterNat_Int, assumption)
    apply (simp add: lemma_Compl_eq lemma_Compl_eq2 lemma_Int_eq1)
    apply (auto dest: FreeUltrafilterNat_const_Finite simp
          add: HInfinite_HFinite_iff [THEN iffD1])
    done

    lemma lemma_Int_HI:
         "{n. abs (Xa n) < u} Int {n. X n = Xa n} ⊆ {n. abs (X n) < (u::real)}"
    by auto

    lemma lemma_Int_HIa: "{n. u < abs (X n)} Int {n. abs (X n) < (u::real)} = {}"
    by (auto intro: order_less_asym)

    lemma FreeUltrafilterNat_HInfinite:
         "∃X ∈ Rep_star x. ∀u.
                   {n. u < abs (X n)} ∈ FreeUltrafilterNat
                   ==>  x ∈ HInfinite"
    apply (rule HInfinite_HFinite_iff [THEN iffD2])
    apply (safe, drule HFinite_FreeUltrafilterNat, auto)
    apply (drule_tac x = u in spec)
    apply (drule FreeUltrafilterNat_Rep_hypreal, assumption)
    apply (drule_tac Y = "{n. X n = Xa n}" in FreeUltrafilterNat_Int, simp)
    apply (drule lemma_Int_HI [THEN [2] FreeUltrafilterNat_subset])
    apply (drule_tac Y = "{n. abs (X n) < u}" in FreeUltrafilterNat_Int)
    apply (auto simp add: lemma_Int_HIa)
    done

    lemma HInfinite_FreeUltrafilterNat_iff:
         "(x ∈ HInfinite) = (∃X ∈ Rep_star x.
               ∀u. {n. u < abs (X n)} ∈ FreeUltrafilterNat)"
    by (blast intro!: HInfinite_FreeUltrafilterNat FreeUltrafilterNat_HInfinite)

    subsection{*Alternative Definitions for @{term Infinitesimal} using Free Ultrafilter*}

    lemma Infinitesimal_FreeUltrafilterNat:
              "x ∈ Infinitesimal ==> ∃X ∈ Rep_star x.
               ∀u. 0 < u --> {n. abs (X n) < u} ∈ FreeUltrafilterNat"
    apply (simp add: Infinitesimal_def)
    apply (auto simp add: abs_less_iff minus_less_iff [of x])
    apply (cases x)
    apply (auto, rule bexI [OF _ Rep_star_star_n], safe)
    apply (drule star_of_less [THEN iffD2])
    apply (drule_tac x = "hypreal_of_real u" in bspec, auto)
    apply (auto simp add: star_n_less star_n_minus star_of_def, ultra)
    done

    lemma FreeUltrafilterNat_Infinitesimal:
         "∃X ∈ Rep_star x.
                ∀u. 0 < u --> {n. abs (X n) < u} ∈ FreeUltrafilterNat
          ==> x ∈ Infinitesimal"
    apply (simp add: Infinitesimal_def)
    apply (cases x)
    apply (auto simp add: abs_less_iff abs_interval_iff minus_less_iff [of x])
    apply (auto simp add: SReal_iff)
    apply (drule_tac [!] x=y in spec)
    apply (auto simp add: star_n_less star_n_minus star_of_def, ultra+)
    done

    lemma Infinitesimal_FreeUltrafilterNat_iff:
         "(x ∈ Infinitesimal) = (∃X ∈ Rep_star x.
               ∀u. 0 < u --> {n. abs (X n) < u} ∈ FreeUltrafilterNat)"
    by (blast intro!: Infinitesimal_FreeUltrafilterNat FreeUltrafilterNat_Infinitesimal)

    (*------------------------------------------------------------------------
             Infinitesimals as smaller than 1/n for all n::nat (> 0)
     ------------------------------------------------------------------------*)

    lemma lemma_Infinitesimal:
         "(∀r. 0 < r --> x < r) = (∀n. x < inverse(real (Suc n)))"
    apply (auto simp add: real_of_nat_Suc_gt_zero)
    apply (blast dest!: reals_Archimedean intro: order_less_trans)
    done

    lemma of_nat_in_Reals [simp]: "(of_nat n::hypreal) ∈ \<real>"
    apply (induct n)
    apply (simp_all)
    done

    lemma lemma_Infinitesimal2:
         "(∀r ∈ Reals. 0 < r --> x < r) =
          (∀n. x < inverse(hypreal_of_nat (Suc n)))"
    apply safe
    apply (drule_tac x = "inverse (hypreal_of_real (real (Suc n))) " in bspec)
    apply (simp (no_asm_use) add: SReal_inverse)
    apply (rule real_of_nat_Suc_gt_zero [THEN positive_imp_inverse_positive, THEN star_of_less [THEN iffD2], THEN [2] impE])
    prefer 2 apply assumption
    apply (simp add: real_of_nat_Suc_gt_zero hypreal_of_nat_eq)
    apply (auto dest!: reals_Archimedean simp add: SReal_iff)
    apply (drule star_of_less [THEN iffD2])
    apply (simp add: real_of_nat_Suc_gt_zero hypreal_of_nat_eq)
    apply (blast intro: order_less_trans)
    done

    lemma Infinitesimal_hypreal_of_nat_iff:
         "Infinitesimal = {x. ∀n. abs x < inverse (hypreal_of_nat (Suc n))}"
    apply (simp add: Infinitesimal_def)
    apply (auto simp add: lemma_Infinitesimal2)
    done

    subsection{*Proof that @{term omega} is an infinite number*}

    text{*It will follow that epsilon is an infinitesimal number.*}

    lemma Suc_Un_eq: "{n. n < Suc m} = {n. n < m} Un {n. n = m}"
    by (auto simp add: less_Suc_eq)

    (*-------------------------------------------
      Prove that any segment is finite and
      hence cannot belong to FreeUltrafilterNat
     -------------------------------------------*)
    lemma finite_nat_segment: "finite {n::nat. n < m}"
    apply (induct "m")
    apply (auto simp add: Suc_Un_eq)
    done

    lemma finite_real_of_nat_segment: "finite {n::nat. real n < real (m::nat)}"
    by (auto intro: finite_nat_segment)

    lemma finite_real_of_nat_less_real: "finite {n::nat. real n < u}"
    apply (cut_tac x = u in reals_Archimedean2, safe)
    apply (rule finite_real_of_nat_segment [THEN [2] finite_subset])
    apply (auto dest: order_less_trans)
    done

    lemma lemma_real_le_Un_eq:
         "{n. f n ≤ u} = {n. f n < u} Un {n. u = (f n :: real)}"
    by (auto dest: order_le_imp_less_or_eq simp add: order_less_imp_le)

    lemma finite_real_of_nat_le_real: "finite {n::nat. real n ≤ u}"
    by (auto simp add: lemma_real_le_Un_eq lemma_finite_omega_set finite_real_of_nat_less_real)

    lemma finite_rabs_real_of_nat_le_real: "finite {n::nat. abs(real n) ≤ u}"
    apply (simp (no_asm) add: real_of_nat_Suc_gt_zero finite_real_of_nat_le_real)
    done

    lemma rabs_real_of_nat_le_real_FreeUltrafilterNat:
         "{n. abs(real n) ≤ u} ∉ FreeUltrafilterNat"
    by (blast intro!: FreeUltrafilterNat_finite finite_rabs_real_of_nat_le_real)

    lemma FreeUltrafilterNat_nat_gt_real: "{n. u < real n} ∈ FreeUltrafilterNat"
    apply (rule ccontr, drule FreeUltrafilterNat_Compl_mem)
    apply (subgoal_tac "- {n::nat. u < real n} = {n. real n ≤ u}")
    prefer 2 apply force
    apply (simp add: finite_real_of_nat_le_real [THEN FreeUltrafilterNat_finite])
    done

    (*--------------------------------------------------------------
     The complement of {n. abs(real n) ≤ u} =
     {n. u < abs (real n)} is in FreeUltrafilterNat
     by property of (free) ultrafilters
     --------------------------------------------------------------*)

    lemma Compl_real_le_eq: "- {n::nat. real n ≤ u} = {n. u < real n}"
    by (auto dest!: order_le_less_trans simp add: linorder_not_le)

    text{*@{term omega} is a member of @{term HInfinite}*}

    lemma FreeUltrafilterNat_omega: "{n. u < real n} ∈ FreeUltrafilterNat"
    apply (cut_tac u = u in rabs_real_of_nat_le_real_FreeUltrafilterNat)
    apply (auto dest: FreeUltrafilterNat_Compl_mem simp add: Compl_real_le_eq)
    done

    theorem HInfinite_omega [simp]: "omega ∈ HInfinite"
    apply (simp add: omega_def star_n_def)
    apply (auto intro!: FreeUltrafilterNat_HInfinite)
    apply (rule bexI)
    apply (rule_tac [2] lemma_starrel_refl, auto)
    apply (simp (no_asm) add: real_of_nat_Suc diff_less_eq [symmetric] FreeUltrafilterNat_omega)
    done

    (*-----------------------------------------------
           Epsilon is a member of Infinitesimal
     -----------------------------------------------*)

    lemma Infinitesimal_epsilon [simp]: "epsilon ∈ Infinitesimal"
    by (auto intro!: HInfinite_inverse_Infinitesimal HInfinite_omega simp add: hypreal_epsilon_inverse_omega)

    lemma HFinite_epsilon [simp]: "epsilon ∈ HFinite"
    by (auto intro: Infinitesimal_subset_HFinite [THEN subsetD])

    lemma epsilon_approx_zero [simp]: "epsilon @= 0"
    apply (simp (no_asm) add: mem_infmal_iff [symmetric])
    done

    (*------------------------------------------------------------------------
      Needed for proof that we define a hyperreal [<X(n)] @= hypreal_of_real a given
      that ∀n. |X n - a| < 1/n. Used in proof of NSLIM => LIM.
     -----------------------------------------------------------------------*)

    lemma real_of_nat_less_inverse_iff:
         "0 < u  ==> (u < inverse (real(Suc n))) = (real(Suc n) < inverse u)"
    apply (simp add: inverse_eq_divide)
    apply (subst pos_less_divide_eq, assumption)
    apply (subst pos_less_divide_eq)
     apply (simp add: real_of_nat_Suc_gt_zero)
    apply (simp add: real_mult_commute)
    done

    lemma finite_inverse_real_of_posnat_gt_real:
         "0 < u ==> finite {n. u < inverse(real(Suc n))}"
    apply (simp (no_asm_simp) add: real_of_nat_less_inverse_iff)
    apply (simp (no_asm_simp) add: real_of_nat_Suc less_diff_eq [symmetric])
    apply (rule finite_real_of_nat_less_real)
    done

    lemma lemma_real_le_Un_eq2:
         "{n. u ≤ inverse(real(Suc n))} =
         {n. u < inverse(real(Suc n))} Un {n. u = inverse(real(Suc n))}"
    apply (auto dest: order_le_imp_less_or_eq simp add: order_less_imp_le)
    done

    lemma real_of_nat_inverse_le_iff:
         "(inverse (real(Suc n)) ≤ r) = (1 ≤ r * real(Suc n))"
    apply (simp (no_asm) add: linorder_not_less [symmetric])
    apply (simp (no_asm) add: inverse_eq_divide)
    apply (subst pos_less_divide_eq)
    apply (simp (no_asm) add: real_of_nat_Suc_gt_zero)
    apply (simp (no_asm) add: real_mult_commute)
    done

    lemma real_of_nat_inverse_eq_iff:
         "(u = inverse (real(Suc n))) = (real(Suc n) = inverse u)"
    by (auto simp add: real_of_nat_Suc_gt_zero real_not_refl2 [THEN not_sym])

    lemma lemma_finite_omega_set2: "finite {n::nat. u = inverse(real(Suc n))}"
    apply (simp (no_asm_simp) add: real_of_nat_inverse_eq_iff)
    apply (cut_tac x = "inverse u - 1" in lemma_finite_omega_set)
    apply (simp add: real_of_nat_Suc diff_eq_eq [symmetric] eq_commute)
    done

    lemma finite_inverse_real_of_posnat_ge_real:
         "0 < u ==> finite {n. u ≤ inverse(real(Suc n))}"
    by (auto simp add: lemma_real_le_Un_eq2 lemma_finite_omega_set2 finite_inverse_real_of_posnat_gt_real)

    lemma inverse_real_of_posnat_ge_real_FreeUltrafilterNat:
         "0 < u ==> {n. u ≤ inverse(real(Suc n))} ∉ FreeUltrafilterNat"
    by (blast intro!: FreeUltrafilterNat_finite finite_inverse_real_of_posnat_ge_real)

    (*--------------------------------------------------------------
        The complement of  {n. u ≤ inverse(real(Suc n))} =
        {n. inverse(real(Suc n)) < u} is in FreeUltrafilterNat
        by property of (free) ultrafilters
     --------------------------------------------------------------*)
    lemma Compl_le_inverse_eq:
         "- {n. u ≤ inverse(real(Suc n))} =
          {n. inverse(real(Suc n)) < u}"
    apply (auto dest!: order_le_less_trans simp add: linorder_not_le)
    done

    lemma FreeUltrafilterNat_inverse_real_of_posnat:
         "0 < u ==>
          {n. inverse(real(Suc n)) < u} ∈ FreeUltrafilterNat"
    apply (cut_tac u = u in inverse_real_of_posnat_ge_real_FreeUltrafilterNat)
    apply (auto dest: FreeUltrafilterNat_Compl_mem simp add: Compl_le_inverse_eq)
    done

    text{* Example where we get a hyperreal from a real sequence
          for which a particular property holds. The theorem is
          used in proofs about equivalence of nonstandard and
          standard neighbourhoods. Also used for equivalence of
          nonstandard ans standard definitions of pointwise
          limit.*}

    (*-----------------------------------------------------
        |X(n) - x| < 1/n ==> [<X n>] - hypreal_of_real x| ∈ Infinitesimal
     -----------------------------------------------------*)
    lemma real_seq_to_hypreal_Infinitesimal:
         "∀n. abs(X n + -x) < inverse(real(Suc n))
         ==> star_n X + -hypreal_of_real x ∈ Infinitesimal"
    apply (auto intro!: bexI dest: FreeUltrafilterNat_inverse_real_of_posnat FreeUltrafilterNat_all FreeUltrafilterNat_Int intro: order_less_trans FreeUltrafilterNat_subset simp add: star_n_minus star_of_def star_n_add Infinitesimal_FreeUltrafilterNat_iff star_n_inverse)
    done

    lemma real_seq_to_hypreal_approx:
         "∀n. abs(X n + -x) < inverse(real(Suc n))
          ==> star_n X @= hypreal_of_real x"
    apply (subst approx_minus_iff)
    apply (rule mem_infmal_iff [THEN subst])
    apply (erule real_seq_to_hypreal_Infinitesimal)
    done

    lemma real_seq_to_hypreal_approx2:
         "∀n. abs(x + -X n) < inverse(real(Suc n))
                   ==> star_n X @= hypreal_of_real x"
    apply (simp add: abs_minus_add_cancel real_seq_to_hypreal_approx)
    done

    lemma real_seq_to_hypreal_Infinitesimal2:
         "∀n. abs(X n + -Y n) < inverse(real(Suc n))
          ==> star_n X + -star_n Y ∈ Infinitesimal"
    by (auto intro!: bexI
             dest: FreeUltrafilterNat_inverse_real_of_posnat
                   FreeUltrafilterNat_all FreeUltrafilterNat_Int
             intro: order_less_trans FreeUltrafilterNat_subset
             simp add: Infinitesimal_FreeUltrafilterNat_iff star_n_minus
                       star_n_add star_n_inverse)

    ML
    {*
    val Infinitesimal_def = thm"Infinitesimal_def";
    val HFinite_def = thm"HFinite_def";
    val HInfinite_def = thm"HInfinite_def";
    val st_def = thm"st_def";
    val monad_def = thm"monad_def";
    val galaxy_def = thm"galaxy_def";
    val approx_def = thm"approx_def";
    val SReal_def = thm"SReal_def";

    val Infinitesimal_approx_minus = thm "Infinitesimal_approx_minus";
    val approx_monad_iff = thm "approx_monad_iff";
    val Infinitesimal_approx = thm "Infinitesimal_approx";
    val approx_add = thm "approx_add";
    val approx_minus = thm "approx_minus";
    val approx_minus2 = thm "approx_minus2";
    val approx_minus_cancel = thm "approx_minus_cancel";
    val approx_add_minus = thm "approx_add_minus";
    val approx_mult1 = thm "approx_mult1";
    val approx_mult2 = thm "approx_mult2";
    val approx_mult_subst = thm "approx_mult_subst";
    val approx_mult_subst2 = thm "approx_mult_subst2";
    val approx_mult_subst_SReal = thm "approx_mult_subst_SReal";
    val approx_eq_imp = thm "approx_eq_imp";
    val Infinitesimal_minus_approx = thm "Infinitesimal_minus_approx";
    val bex_Infinitesimal_iff = thm "bex_Infinitesimal_iff";
    val bex_Infinitesimal_iff2 = thm "bex_Infinitesimal_iff2";
    val Infinitesimal_add_approx = thm "Infinitesimal_add_approx";
    val Infinitesimal_add_approx_self = thm "Infinitesimal_add_approx_self";
    val Infinitesimal_add_approx_self2 = thm "Infinitesimal_add_approx_self2";
    val Infinitesimal_add_minus_approx_self = thm "Infinitesimal_add_minus_approx_self";
    val Infinitesimal_add_cancel = thm "Infinitesimal_add_cancel";
    val Infinitesimal_add_right_cancel = thm "Infinitesimal_add_right_cancel";
    val approx_add_left_cancel = thm "approx_add_left_cancel";
    val approx_add_right_cancel = thm "approx_add_right_cancel";
    val approx_add_mono1 = thm "approx_add_mono1";
    val approx_add_mono2 = thm "approx_add_mono2";
    val approx_add_left_iff = thm "approx_add_left_iff";
    val approx_add_right_iff = thm "approx_add_right_iff";
    val approx_HFinite = thm "approx_HFinite";
    val approx_hypreal_of_real_HFinite = thm "approx_hypreal_of_real_HFinite";
    val approx_mult_HFinite = thm "approx_mult_HFinite";
    val approx_mult_hypreal_of_real = thm "approx_mult_hypreal_of_real";
    val approx_SReal_mult_cancel_zero = thm "approx_SReal_mult_cancel_zero";
    val approx_mult_SReal1 = thm "approx_mult_SReal1";
    val approx_mult_SReal2 = thm "approx_mult_SReal2";
    val approx_mult_SReal_zero_cancel_iff = thm "approx_mult_SReal_zero_cancel_iff";
    val approx_SReal_mult_cancel = thm "approx_SReal_mult_cancel";
    val approx_SReal_mult_cancel_iff1 = thm "approx_SReal_mult_cancel_iff1";
    val approx_le_bound = thm "approx_le_bound";
    val Infinitesimal_less_SReal = thm "Infinitesimal_less_SReal";
    val Infinitesimal_less_SReal2 = thm "Infinitesimal_less_SReal2";
    val SReal_not_Infinitesimal = thm "SReal_not_Infinitesimal";
    val SReal_minus_not_Infinitesimal = thm "SReal_minus_not_Infinitesimal";
    val SReal_Int_Infinitesimal_zero = thm "SReal_Int_Infinitesimal_zero";
    val SReal_Infinitesimal_zero = thm "SReal_Infinitesimal_zero";
    val SReal_HFinite_diff_Infinitesimal = thm "SReal_HFinite_diff_Infinitesimal";
    val hypreal_of_real_HFinite_diff_Infinitesimal = thm "hypreal_of_real_HFinite_diff_Infinitesimal";
    val hypreal_of_real_Infinitesimal_iff_0 = thm "hypreal_of_real_Infinitesimal_iff_0";
    val number_of_not_Infinitesimal = thm "number_of_not_Infinitesimal";
    val one_not_Infinitesimal = thm "one_not_Infinitesimal";
    val approx_SReal_not_zero = thm "approx_SReal_not_zero";
    val HFinite_diff_Infinitesimal_approx = thm "HFinite_diff_Infinitesimal_approx";
    val Infinitesimal_ratio = thm "Infinitesimal_ratio";
    val SReal_approx_iff = thm "SReal_approx_iff";
    val number_of_approx_iff = thm "number_of_approx_iff";
    val hypreal_of_real_approx_iff = thm "hypreal_of_real_approx_iff";
    val hypreal_of_real_approx_number_of_iff = thm "hypreal_of_real_approx_number_of_iff";
    val approx_unique_real = thm "approx_unique_real";
    val hypreal_isLub_unique = thm "hypreal_isLub_unique";
    val hypreal_setle_less_trans = thm "hypreal_setle_less_trans";
    val hypreal_gt_isUb = thm "hypreal_gt_isUb";
    val st_part_Ex = thm "st_part_Ex";
    val st_part_Ex1 = thm "st_part_Ex1";
    val HFinite_Int_HInfinite_empty = thm "HFinite_Int_HInfinite_empty";
    val HFinite_not_HInfinite = thm "HFinite_not_HInfinite";
    val not_HFinite_HInfinite = thm "not_HFinite_HInfinite";
    val HInfinite_HFinite_disj = thm "HInfinite_HFinite_disj";
    val HInfinite_HFinite_iff = thm "HInfinite_HFinite_iff";
    val HFinite_HInfinite_iff = thm "HFinite_HInfinite_iff";
    val HInfinite_diff_HFinite_Infinitesimal_disj = thm "HInfinite_diff_HFinite_Infinitesimal_disj";
    val HFinite_inverse = thm "HFinite_inverse";
    val HFinite_inverse2 = thm "HFinite_inverse2";
    val Infinitesimal_inverse_HFinite = thm "Infinitesimal_inverse_HFinite";
    val HFinite_not_Infinitesimal_inverse = thm "HFinite_not_Infinitesimal_inverse";
    val approx_inverse = thm "approx_inverse";
    val hypreal_of_real_approx_inverse = thm "hypreal_of_real_approx_inverse";
    val inverse_add_Infinitesimal_approx = thm "inverse_add_Infinitesimal_approx";
    val inverse_add_Infinitesimal_approx2 = thm "inverse_add_Infinitesimal_approx2";
    val inverse_add_Infinitesimal_approx_Infinitesimal = thm "inverse_add_Infinitesimal_approx_Infinitesimal";
    val Infinitesimal_square_iff = thm "Infinitesimal_square_iff";
    val HFinite_square_iff = thm "HFinite_square_iff";
    val HInfinite_square_iff = thm "HInfinite_square_iff";
    val approx_HFinite_mult_cancel = thm "approx_HFinite_mult_cancel";
    val approx_HFinite_mult_cancel_iff1 = thm "approx_HFinite_mult_cancel_iff1";
    val approx_hrabs_disj = thm "approx_hrabs_disj";
    val monad_hrabs_Un_subset = thm "monad_hrabs_Un_subset";
    val Infinitesimal_monad_eq = thm "Infinitesimal_monad_eq";
    val mem_monad_iff = thm "mem_monad_iff";
    val Infinitesimal_monad_zero_iff = thm "Infinitesimal_monad_zero_iff";
    val monad_zero_minus_iff = thm "monad_zero_minus_iff";
    val monad_zero_hrabs_iff = thm "monad_zero_hrabs_iff";
    val mem_monad_self = thm "mem_monad_self";
    val approx_subset_monad = thm "approx_subset_monad";
    val approx_subset_monad2 = thm "approx_subset_monad2";
    val mem_monad_approx = thm "mem_monad_approx";
    val approx_mem_monad = thm "approx_mem_monad";
    val approx_mem_monad2 = thm "approx_mem_monad2";
    val approx_mem_monad_zero = thm "approx_mem_monad_zero";
    val Infinitesimal_approx_hrabs = thm "Infinitesimal_approx_hrabs";
    val less_Infinitesimal_less = thm "less_Infinitesimal_less";
    val Ball_mem_monad_gt_zero = thm "Ball_mem_monad_gt_zero";
    val Ball_mem_monad_less_zero = thm "Ball_mem_monad_less_zero";
    val approx_hrabs = thm "approx_hrabs";
    val approx_hrabs_zero_cancel = thm "approx_hrabs_zero_cancel";
    val approx_hrabs_add_Infinitesimal = thm "approx_hrabs_add_Infinitesimal";
    val approx_hrabs_add_minus_Infinitesimal = thm "approx_hrabs_add_minus_Infinitesimal";
    val hrabs_add_Infinitesimal_cancel = thm "hrabs_add_Infinitesimal_cancel";
    val hrabs_add_minus_Infinitesimal_cancel = thm "hrabs_add_minus_Infinitesimal_cancel";
    val Infinitesimal_add_hypreal_of_real_less = thm "Infinitesimal_add_hypreal_of_real_less";
    val Infinitesimal_add_hrabs_hypreal_of_real_less = thm "Infinitesimal_add_hrabs_hypreal_of_real_less";
    val Infinitesimal_add_hrabs_hypreal_of_real_less2 = thm "Infinitesimal_add_hrabs_hypreal_of_real_less2";
    val hypreal_of_real_le_add_Infininitesimal_cancel2 = thm"hypreal_of_real_le_add_Infininitesimal_cancel2";
    val hypreal_of_real_less_Infinitesimal_le_zero = thm "hypreal_of_real_less_Infinitesimal_le_zero";
    val Infinitesimal_add_not_zero = thm "Infinitesimal_add_not_zero";
    val Infinitesimal_square_cancel = thm "Infinitesimal_square_cancel";
    val HFinite_square_cancel = thm "HFinite_square_cancel";
    val Infinitesimal_square_cancel2 = thm "Infinitesimal_square_cancel2";
    val HFinite_square_cancel2 = thm "HFinite_square_cancel2";
    val Infinitesimal_sum_square_cancel = thm "Infinitesimal_sum_square_cancel";
    val HFinite_sum_square_cancel = thm "HFinite_sum_square_cancel";
    val Infinitesimal_sum_square_cancel2 = thm "Infinitesimal_sum_square_cancel2";
    val HFinite_sum_square_cancel2 = thm "HFinite_sum_square_cancel2";
    val Infinitesimal_sum_square_cancel3 = thm "Infinitesimal_sum_square_cancel3";
    val HFinite_sum_square_cancel3 = thm "HFinite_sum_square_cancel3";
    val monad_hrabs_less = thm "monad_hrabs_less";
    val mem_monad_SReal_HFinite = thm "mem_monad_SReal_HFinite";
    val st_approx_self = thm "st_approx_self";
    val st_SReal = thm "st_SReal";
    val st_HFinite = thm "st_HFinite";
    val st_SReal_eq = thm "st_SReal_eq";
    val st_hypreal_of_real = thm "st_hypreal_of_real";
    val st_eq_approx = thm "st_eq_approx";
    val approx_st_eq = thm "approx_st_eq";
    val st_eq_approx_iff = thm "st_eq_approx_iff";
    val st_Infinitesimal_add_SReal = thm "st_Infinitesimal_add_SReal";
    val st_Infinitesimal_add_SReal2 = thm "st_Infinitesimal_add_SReal2";
    val HFinite_st_Infinitesimal_add = thm "HFinite_st_Infinitesimal_add";
    val st_add = thm "st_add";
    val st_number_of = thm "st_number_of";
    val st_minus = thm "st_minus";
    val st_diff = thm "st_diff";
    val st_mult = thm "st_mult";
    val st_Infinitesimal = thm "st_Infinitesimal";
    val st_not_Infinitesimal = thm "st_not_Infinitesimal";
    val st_inverse = thm "st_inverse";
    val st_divide = thm "st_divide";
    val st_idempotent = thm "st_idempotent";
    val Infinitesimal_add_st_less = thm "Infinitesimal_add_st_less";
    val Infinitesimal_add_st_le_cancel = thm "Infinitesimal_add_st_le_cancel";
    val st_le = thm "st_le";
    val st_zero_le = thm "st_zero_le";
    val st_zero_ge = thm "st_zero_ge";
    val st_hrabs = thm "st_hrabs";
    val FreeUltrafilterNat_HFinite = thm "FreeUltrafilterNat_HFinite";
    val HFinite_FreeUltrafilterNat_iff = thm "HFinite_FreeUltrafilterNat_iff";
    val FreeUltrafilterNat_const_Finite = thm "FreeUltrafilterNat_const_Finite";
    val FreeUltrafilterNat_HInfinite = thm "FreeUltrafilterNat_HInfinite";
    val HInfinite_FreeUltrafilterNat_iff = thm "HInfinite_FreeUltrafilterNat_iff";
    val Infinitesimal_FreeUltrafilterNat = thm "Infinitesimal_FreeUltrafilterNat";
    val FreeUltrafilterNat_Infinitesimal = thm "FreeUltrafilterNat_Infinitesimal";
    val Infinitesimal_FreeUltrafilterNat_iff = thm "Infinitesimal_FreeUltrafilterNat_iff";
    val Infinitesimal_hypreal_of_nat_iff = thm "Infinitesimal_hypreal_of_nat_iff";
    val Suc_Un_eq = thm "Suc_Un_eq";
    val finite_nat_segment = thm "finite_nat_segment";
    val finite_real_of_nat_segment = thm "finite_real_of_nat_segment";
    val finite_real_of_nat_less_real = thm "finite_real_of_nat_less_real";
    val finite_real_of_nat_le_real = thm "finite_real_of_nat_le_real";
    val finite_rabs_real_of_nat_le_real = thm "finite_rabs_real_of_nat_le_real";
    val rabs_real_of_nat_le_real_FreeUltrafilterNat = thm "rabs_real_of_nat_le_real_FreeUltrafilterNat";
    val FreeUltrafilterNat_nat_gt_real = thm "FreeUltrafilterNat_nat_gt_real";
    val FreeUltrafilterNat_omega = thm "FreeUltrafilterNat_omega";
    val HInfinite_omega = thm "HInfinite_omega";
    val Infinitesimal_epsilon = thm "Infinitesimal_epsilon";
    val HFinite_epsilon = thm "HFinite_epsilon";
    val epsilon_approx_zero = thm "epsilon_approx_zero";
    val real_of_nat_less_inverse_iff = thm "real_of_nat_less_inverse_iff";
    val finite_inverse_real_of_posnat_gt_real = thm "finite_inverse_real_of_posnat_gt_real";
    val real_of_nat_inverse_le_iff = thm "real_of_nat_inverse_le_iff";
    val real_of_nat_inverse_eq_iff = thm "real_of_nat_inverse_eq_iff";
    val finite_inverse_real_of_posnat_ge_real = thm "finite_inverse_real_of_posnat_ge_real";
    val inverse_real_of_posnat_ge_real_FreeUltrafilterNat = thm "inverse_real_of_posnat_ge_real_FreeUltrafilterNat";
    val FreeUltrafilterNat_inverse_real_of_posnat = thm "FreeUltrafilterNat_inverse_real_of_posnat";
    val real_seq_to_hypreal_Infinitesimal = thm "real_seq_to_hypreal_Infinitesimal";
    val real_seq_to_hypreal_approx = thm "real_seq_to_hypreal_approx";
    val real_seq_to_hypreal_approx2 = thm "real_seq_to_hypreal_approx2";
    val real_seq_to_hypreal_Infinitesimal2 = thm "real_seq_to_hypreal_Infinitesimal2";
    val HInfinite_HFinite_add = thm "HInfinite_HFinite_add";
    val HInfinite_ge_HInfinite = thm "HInfinite_ge_HInfinite";
    val Infinitesimal_inverse_HInfinite = thm "Infinitesimal_inverse_HInfinite";
    val HInfinite_HFinite_not_Infinitesimal_mult = thm "HInfinite_HFinite_not_Infinitesimal_mult";
    val HInfinite_HFinite_not_Infinitesimal_mult2 = thm "HInfinite_HFinite_not_Infinitesimal_mult2";
    val HInfinite_gt_SReal = thm "HInfinite_gt_SReal";
    val HInfinite_gt_zero_gt_one = thm "HInfinite_gt_zero_gt_one";
    val not_HInfinite_one = thm "not_HInfinite_one";
    *}

    end

### Closure Laws for the Standard Reals

lemma SReal_add:

      [| x ∈ Reals; y ∈ Reals |] ==> x + y ∈ Reals

lemma SReal_mult:

      [| x ∈ Reals; y ∈ Reals |] ==> x * y ∈ Reals

lemma SReal_inverse:

      x ∈ Reals ==> inverse x ∈ Reals

lemma SReal_divide:

      [| x ∈ Reals; y ∈ Reals |] ==> x / y ∈ Reals

lemma SReal_minus:

      x ∈ Reals ==> - x ∈ Reals

lemma SReal_minus_iff:

      (- x ∈ Reals) = (x ∈ Reals)

lemma SReal_add_cancel:

      [| x + y ∈ Reals; y ∈ Reals |] ==> x ∈ Reals

lemma SReal_hrabs:

      x ∈ Reals ==> ¦x¦ ∈ Reals

lemma SReal_hypreal_of_real:

      star_of x ∈ Reals

lemma SReal_number_of:

      number_of w ∈ Reals

lemma Reals_0:

      0 ∈ Reals

lemma Reals_1:

      1 ∈ Reals

lemma SReal_divide_number_of:

      r ∈ Reals ==> r / number_of w ∈ Reals

lemma SReal_epsilon_not_mem:

      ε ∉ Reals

lemma SReal_omega_not_mem:

      ω ∉ Reals

lemma SReal_UNIV_real:

      {x. star_of x ∈ Reals} = UNIV

lemma SReal_iff:

      (x ∈ Reals) = (∃y. x = star_of y)

lemma hypreal_of_real_image:

      range star_of = Reals

lemma inv_hypreal_of_real_image:

      inv star_of ` Reals = UNIV

lemma SReal_hypreal_of_real_image:

      [| ∃x. x ∈ P; P ⊆ Reals |] ==> ∃Q. P = star_of ` Q

lemma SReal_dense:

      [| x ∈ Reals; y ∈ Reals; x < y |] ==> ∃r∈Reals. x < r ∧ r < y

lemma SReal_sup_lemma:

      P ⊆ Reals ==> (∃x∈P. y < x) = (∃X. star_of X ∈ P ∧ y < star_of X)

lemma SReal_sup_lemma2:

      [| P ⊆ Reals; ∃x. x ∈ P; ∃y∈Reals. ∀x∈P. x < y |]
      ==> (∃X. X ∈ {w. star_of w ∈ P}) ∧ (∃Y. ∀X∈{w. star_of w ∈ P}. X < Y)

### Lifting of the Ub and Lub Properties

lemma hypreal_of_real_isUb_iff:

      isUb Reals (star_of ` Q) (star_of Y) = isUb UNIV Q Y

lemma hypreal_of_real_isLub1:

      isLub Reals (star_of ` Q) (star_of Y) ==> isLub UNIV Q Y

lemma hypreal_of_real_isLub2:

      isLub UNIV Q Y ==> isLub Reals (star_of ` Q) (star_of Y)

lemma hypreal_of_real_isLub_iff:

      isLub Reals (star_of ` Q) (star_of Y) = isLub UNIV Q Y

lemma lemma_isUb_hypreal_of_real:

      isUb Reals P Y ==> ∃Yo. isUb Reals P (star_of Yo)

lemma lemma_isLub_hypreal_of_real:

      isLub Reals P Y ==> ∃Yo. isLub Reals P (star_of Yo)

lemma lemma_isLub_hypreal_of_real2:

      ∃Yo. isLub Reals P (star_of Yo) ==> ∃Y. isLub Reals P Y

lemma SReal_complete:

      [| P ⊆ Reals; ∃x. x ∈ P; ∃Y. isUb Reals P Y |] ==> ∃t. isLub Reals P t

### Set of Finite Elements is a Subring of the Extended Reals

lemma HFinite_add:

      [| x ∈ HFinite; y ∈ HFinite |] ==> x + y ∈ HFinite

lemma HFinite_mult:

      [| x ∈ HFinite; y ∈ HFinite |] ==> x * y ∈ HFinite

lemma HFinite_minus_iff:

      (- x ∈ HFinite) = (x ∈ HFinite)

lemma SReal_subset_HFinite:

      Reals ⊆ HFinite

lemma HFinite_hypreal_of_real:

      star_of x ∈ HFinite

lemma HFiniteD:

      x ∈ HFinite ==> ∃t∈Reals. ¦x¦ < t

lemma HFinite_hrabs_iff:

      (¦x¦ ∈ HFinite) = (x ∈ HFinite)

lemma HFinite_number_of:

      number_of w ∈ HFinite

lemma HFinite_0:

      0 ∈ HFinite

lemma HFinite_1:

      1 ∈ HFinite

lemma HFinite_bounded:

      [| x ∈ HFinite; y ≤ x; 0 ≤ y |] ==> y ∈ HFinite

### Set of Infinitesimals is a Subring of the Hyperreals

lemma InfinitesimalD:

      x ∈ Infinitesimal ==> ∀r∈Reals. 0 < r --> ¦x¦ < r

lemma Infinitesimal_zero:

      0 ∈ Infinitesimal

lemma hypreal_sum_of_halves:

      x / 2 + x / 2 = x

lemma Infinitesimal_add:

      [| x ∈ Infinitesimal; y ∈ Infinitesimal |] ==> x + y ∈ Infinitesimal

lemma Infinitesimal_minus_iff:

      (- x ∈ Infinitesimal) = (x ∈ Infinitesimal)

lemma Infinitesimal_diff:

      [| x ∈ Infinitesimal; y ∈ Infinitesimal |] ==> x - y ∈ Infinitesimal

lemma Infinitesimal_mult:

      [| x ∈ Infinitesimal; y ∈ Infinitesimal |] ==> x * y ∈ Infinitesimal

lemma Infinitesimal_HFinite_mult:

      [| x ∈ Infinitesimal; y ∈ HFinite |] ==> x * y ∈ Infinitesimal

lemma Infinitesimal_HFinite_mult2:

      [| x ∈ Infinitesimal; y ∈ HFinite |] ==> y * x ∈ Infinitesimal

lemma HInfinite_inverse_Infinitesimal:

      x ∈ HInfinite ==> inverse x ∈ Infinitesimal

lemma HInfinite_mult:

      [| x ∈ HInfinite; y ∈ HInfinite |] ==> x * y ∈ HInfinite

lemma hypreal_add_zero_less_le_mono:

      [| r < x; 0 ≤ y |] ==> r < x + y

lemma HInfinite_add_ge_zero:

      [| x ∈ HInfinite; 0 ≤ y; 0 ≤ x |] ==> x + y ∈ HInfinite

lemma HInfinite_add_ge_zero2:

      [| x ∈ HInfinite; 0 ≤ y; 0 ≤ x |] ==> y + x ∈ HInfinite

lemma HInfinite_add_gt_zero:

      [| x ∈ HInfinite; 0 < y; 0 < x |] ==> x + y ∈ HInfinite

lemma HInfinite_minus_iff:

      (- x ∈ HInfinite) = (x ∈ HInfinite)

lemma HInfinite_add_le_zero:

      [| x ∈ HInfinite; y ≤ 0; x ≤ 0 |] ==> x + y ∈ HInfinite

lemma HInfinite_add_lt_zero:

      [| x ∈ HInfinite; y < 0; x < 0 |] ==> x + y ∈ HInfinite

lemma HFinite_sum_squares:

      [| a ∈ HFinite; b ∈ HFinite; c ∈ HFinite |] ==> a * a + b * b + c * c ∈ HFinite

lemma not_Infinitesimal_not_zero:

      x ∉ Infinitesimal ==> x ≠ 0

lemma not_Infinitesimal_not_zero2:

      x ∈ HFinite - Infinitesimal ==> x ≠ 0

lemma Infinitesimal_hrabs_iff:

      (¦x¦ ∈ Infinitesimal) = (x ∈ Infinitesimal)

lemma HFinite_diff_Infinitesimal_hrabs:

      x ∈ HFinite - Infinitesimal ==> ¦x¦ ∈ HFinite - Infinitesimal

lemma hrabs_less_Infinitesimal:

      [| e ∈ Infinitesimal; ¦x¦ < e |] ==> x ∈ Infinitesimal

lemma hrabs_le_Infinitesimal:

      [| e ∈ Infinitesimal; ¦x¦ ≤ e |] ==> x ∈ Infinitesimal

lemma Infinitesimal_interval:

      [| e ∈ Infinitesimal; e' ∈ Infinitesimal; e' < x; x < e |] ==> x ∈ Infinitesimal

lemma Infinitesimal_interval2:

      [| e ∈ Infinitesimal; e' ∈ Infinitesimal; e' ≤ x; x ≤ e |] ==> x ∈ Infinitesimal

lemma not_Infinitesimal_mult:

      [| x ∉ Infinitesimal; y ∉ Infinitesimal |] ==> x * y ∉ Infinitesimal

lemma Infinitesimal_mult_disj:

      x * y ∈ Infinitesimal ==> x ∈ Infinitesimal ∨ y ∈ Infinitesimal

lemma HFinite_Infinitesimal_not_zero:

      x ∈ HFinite - Infinitesimal ==> x ≠ 0

lemma HFinite_Infinitesimal_diff_mult:

      [| x ∈ HFinite - Infinitesimal; y ∈ HFinite - Infinitesimal |]
      ==> x * y ∈ HFinite - Infinitesimal

lemma Infinitesimal_subset_HFinite:

      Infinitesimal ⊆ HFinite

lemma Infinitesimal_hypreal_of_real_mult:

      x ∈ Infinitesimal ==> x * star_of r ∈ Infinitesimal

lemma Infinitesimal_hypreal_of_real_mult2:

      x ∈ Infinitesimal ==> star_of r * x ∈ Infinitesimal

### The Infinitely Close Relation

lemma mem_infmal_iff:

      (x ∈ Infinitesimal) = (x ≈ 0)

lemma approx_minus_iff:

      (x ≈ y) = (x + - y ≈ 0)

lemma approx_minus_iff2:

      (x ≈ y) = (- y + x ≈ 0)

lemma approx_refl:

      x ≈ x

lemma hypreal_minus_distrib1:

      - (y + - x) = x + - y

lemma approx_sym:

      x ≈ y ==> y ≈ x

lemma approx_trans:

      [| x ≈ y; y ≈ z |] ==> x ≈ z

lemma approx_trans2:

      [| r ≈ x; s ≈ x |] ==> r ≈ s

lemma approx_trans3:

      [| x ≈ r; x ≈ s |] ==> r ≈ s

lemma number_of_approx_reorient:

      (number_of w ≈ x) = (x ≈ number_of w)

lemma zero_approx_reorient:

      (0 ≈ x) = (x ≈ 0)

lemma one_approx_reorient:

      (1 ≈ x) = (x ≈ 1)

lemma Infinitesimal_approx_minus:

      (x - y ∈ Infinitesimal) = (x ≈ y)

lemma approx_monad_iff:

      (x ≈ y) = (monad x = monad y)

lemma Infinitesimal_approx:

      [| x ∈ Infinitesimal; y ∈ Infinitesimal |] ==> x ≈ y

lemma approx_add:

      [| a ≈ b; c ≈ d |] ==> a + c ≈ b + d

lemma approx_minus:

      a ≈ b ==> - a ≈ - b

lemma approx_minus2:

      - a ≈ - b ==> a ≈ b

lemma approx_minus_cancel:

      (- a ≈ - b) = (a ≈ b)

lemma approx_add_minus:

      [| a ≈ b; c ≈ d |] ==> a + - c ≈ b + - d

lemma approx_mult1:

      [| a ≈ b; c ∈ HFinite |] ==> a * c ≈ b * c

lemma approx_mult2:

      [| a ≈ b; c ∈ HFinite |] ==> c * a ≈ c * b

lemma approx_mult_subst:

      [| u ≈ v * x; x ≈ y; v ∈ HFinite |] ==> u ≈ v * y

lemma approx_mult_subst2:

      [| u ≈ x * v; x ≈ y; v ∈ HFinite |] ==> u ≈ y * v

lemma approx_mult_subst_SReal:

      [| u ≈ x * star_of v; x ≈ y |] ==> u ≈ y * star_of v

lemma approx_eq_imp:

      a = b ==> a ≈ b

lemma Infinitesimal_minus_approx:

      x ∈ Infinitesimal ==> - x ≈ x

lemma bex_Infinitesimal_iff:

      (∃y∈Infinitesimal. x + - z = y) = (x ≈ z)

lemma bex_Infinitesimal_iff2:

      (∃y∈Infinitesimal. x = z + y) = (x ≈ z)

lemma Infinitesimal_add_approx:

      [| y ∈ Infinitesimal; x + y = z |] ==> x ≈ z

lemma Infinitesimal_add_approx_self:

      y ∈ Infinitesimal ==> x ≈ x + y

lemma Infinitesimal_add_approx_self2:

      y ∈ Infinitesimal ==> x ≈ y + x

lemma Infinitesimal_add_minus_approx_self:

      y ∈ Infinitesimal ==> x ≈ x + - y

lemma Infinitesimal_add_cancel:

      [| y ∈ Infinitesimal; x + y ≈ z |] ==> x ≈ z

lemma Infinitesimal_add_right_cancel:

      [| y ∈ Infinitesimal; x ≈ z + y |] ==> x ≈ z

lemma approx_add_left_cancel:

      d + b ≈ d + c ==> b ≈ c

lemma approx_add_right_cancel:

      b + d ≈ c + d ==> b ≈ c

lemma approx_add_mono1:

      b ≈ c ==> d + b ≈ d + c

lemma approx_add_mono2:

      b ≈ c ==> b + a ≈ c + a

lemma approx_add_left_iff:

      (a + b ≈ a + c) = (b ≈ c)

lemma approx_add_right_iff:

      (b + a ≈ c + a) = (b ≈ c)

lemma approx_HFinite:

      [| x ∈ HFinite; x ≈ y |] ==> y ∈ HFinite

lemma approx_hypreal_of_real_HFinite:

      x ≈ star_of D ==> x ∈ HFinite

lemma approx_mult_HFinite:

      [| a ≈ b; c ≈ d; b ∈ HFinite; d ∈ HFinite |] ==> a * c ≈ b * d

lemma approx_mult_hypreal_of_real:

      [| a ≈ star_of b; c ≈ star_of d |] ==> a * c ≈ star_of b * star_of d

lemma approx_SReal_mult_cancel_zero:

      [| a ∈ Reals; a ≠ 0; a * x ≈ 0 |] ==> x ≈ 0

lemma approx_mult_SReal1:

      [| a ∈ Reals; x ≈ 0 |] ==> x * a ≈ 0

lemma approx_mult_SReal2:

      [| a ∈ Reals; x ≈ 0 |] ==> a * x ≈ 0

lemma approx_mult_SReal_zero_cancel_iff:

      [| a ∈ Reals; a ≠ 0 |] ==> (a * x ≈ 0) = (x ≈ 0)

lemma approx_SReal_mult_cancel:

      [| a ∈ Reals; a ≠ 0; a * w ≈ a * z |] ==> w ≈ z

lemma approx_SReal_mult_cancel_iff1:

      [| a ∈ Reals; a ≠ 0 |] ==> (a * w ≈ a * z) = (w ≈ z)

lemma approx_le_bound:

      [| z ≤ f; f ≈ g; g ≤ z |] ==> f ≈ z

### Zero is the Only Infinitesimal that is also a Real

lemma Infinitesimal_less_SReal:

      [| x ∈ Reals; y ∈ Infinitesimal; 0 < x |] ==> y < x

lemma Infinitesimal_less_SReal2:

      y ∈ Infinitesimal ==> ∀r∈Reals. 0 < r --> y < r

lemma SReal_not_Infinitesimal:

      [| 0 < y; y ∈ Reals |] ==> y ∉ Infinitesimal

lemma SReal_minus_not_Infinitesimal:

      [| y < 0; y ∈ Reals |] ==> y ∉ Infinitesimal

lemma SReal_Int_Infinitesimal_zero:

      Reals ∩ Infinitesimal = {0}

lemma SReal_Infinitesimal_zero:

      [| x ∈ Reals; x ∈ Infinitesimal |] ==> x = 0

lemma SReal_HFinite_diff_Infinitesimal:

      [| x ∈ Reals; x ≠ 0 |] ==> x ∈ HFinite - Infinitesimal

lemma hypreal_of_real_HFinite_diff_Infinitesimal:

      star_of x ≠ 0 ==> star_of x ∈ HFinite - Infinitesimal

lemma hypreal_of_real_Infinitesimal_iff_0:

      (star_of x ∈ Infinitesimal) = (x = 0)

lemma number_of_not_Infinitesimal:

      number_of w ≠ 0 ==> number_of w ∉ Infinitesimal

lemma one_not_Infinitesimal:

      1 ∉ Infinitesimal

lemma approx_SReal_not_zero:

      [| y ∈ Reals; x ≈ y; y ≠ 0 |] ==> x ≠ 0

lemma HFinite_diff_Infinitesimal_approx:

      [| x ≈ y; y ∈ HFinite - Infinitesimal |] ==> x ∈ HFinite - Infinitesimal

lemma Infinitesimal_ratio:

      [| y ≠ 0; y ∈ Infinitesimal; x / y ∈ HFinite |] ==> x ∈ Infinitesimal

lemma Infinitesimal_SReal_divide:

      [| x ∈ Infinitesimal; y ∈ Reals |] ==> x / y ∈ Infinitesimal

### Uniqueness: Two Infinitely Close Reals are Equal

lemma SReal_approx_iff:

      [| x ∈ Reals; y ∈ Reals |] ==> (x ≈ y) = (x = y)

lemma number_of_approx_iff:

      (number_of v ≈ number_of w) = (number_of v = number_of w)

lemma

      (0 ≈ number_of w) = (number_of w = 0)

      (number_of w ≈ 0) = (number_of w = 0)

      (1 ≈ number_of w) = (number_of w = 1)

      (number_of w ≈ 1) = (number_of w = 1)

      ¬ 0 ≈ 1

      ¬ 1 ≈ 0

lemma hypreal_of_real_approx_iff:

      (star_of k ≈ star_of m) = (k = m)

lemma hypreal_of_real_approx_number_of_iff:

      (star_of k ≈ number_of w) = (k = number_of w)

lemma

      (star_of k ≈ 0) = (k = 0)

      (star_of k ≈ 1) = (k = 1)

lemma approx_unique_real:

      [| r ∈ Reals; s ∈ Reals; r ≈ x; s ≈ x |] ==> r = s

### Existence of Unique Real Infinitely Close

lemma hypreal_isLub_unique:

      [| isLub R S x; isLub R S y |] ==> x = y

lemma lemma_st_part_ub:

      x ∈ HFinite ==> ∃u. isUb Reals {s : Reals. s < x} u

lemma lemma_st_part_nonempty:

      x ∈ HFinite ==> ∃y. y ∈ {s : Reals. s < x}

lemma lemma_st_part_subset:

      {s : Reals. s < x} ⊆ Reals

lemma lemma_st_part_lub:

      x ∈ HFinite ==> ∃t. isLub Reals {s : Reals. s < x} t

lemma lemma_hypreal_le_left_cancel:

      (t + r ≤ t) = (r ≤ 0)

lemma lemma_st_part_le1:

      [| x ∈ HFinite; isLub Reals {s : Reals. s < x} t; r ∈ Reals; 0 < r |]
      ==> x ≤ t + r

lemma hypreal_setle_less_trans:

      [| S *<= x; x < y |] ==> S *<= y

lemma hypreal_gt_isUb:

      [| isUb R S x; x < y; y ∈ R |] ==> isUb R S y

lemma lemma_st_part_gt_ub:

      [| x ∈ HFinite; x < y; y ∈ Reals |] ==> isUb Reals {s : Reals. s < x} y

lemma lemma_minus_le_zero:

      t ≤ t + - r ==> r ≤ 0

lemma lemma_st_part_le2:

      [| x ∈ HFinite; isLub Reals {s : Reals. s < x} t; r ∈ Reals; 0 < r |]
      ==> t + - r ≤ x

lemma lemma_st_part1a:

      [| x ∈ HFinite; isLub Reals {s : Reals. s < x} t; r ∈ Reals; 0 < r |]
      ==> x + - t ≤ r

lemma lemma_st_part2a:

      [| x ∈ HFinite; isLub Reals {s : Reals. s < x} t; r ∈ Reals; 0 < r |]
      ==> - (x + - t) ≤ r

lemma lemma_SReal_ub:

      x ∈ Reals ==> isUb Reals {s : Reals. s < x} x

lemma lemma_SReal_lub:

      x ∈ Reals ==> isLub Reals {s : Reals. s < x} x

lemma lemma_st_part_not_eq1:

      [| x ∈ HFinite; isLub Reals {s : Reals. s < x} t; r ∈ Reals; 0 < r |]
      ==> x + - t ≠ r

lemma lemma_st_part_not_eq2:

      [| x ∈ HFinite; isLub Reals {s : Reals. s < x} t; r ∈ Reals; 0 < r |]
      ==> - (x + - t) ≠ r

lemma lemma_st_part_major:

      [| x ∈ HFinite; isLub Reals {s : Reals. s < x} t; r ∈ Reals; 0 < r |]
      ==> ¦x + - t¦ < r

lemma lemma_st_part_major2:

      [| x ∈ HFinite; isLub Reals {s : Reals. s < x} t |]
      ==> ∀r∈Reals. 0 < r --> ¦x + - t¦ < r

lemma lemma_st_part_Ex:

      x ∈ HFinite ==> ∃t∈Reals. ∀r∈Reals. 0 < r --> ¦x + - t¦ < r

lemma st_part_Ex:

      x ∈ HFinite ==> ∃t∈Reals. x ≈ t

lemma st_part_Ex1:

      x ∈ HFinite ==> ∃!t. t ∈ Reals ∧ x ≈ t

### Finite, Infinite and Infinitesimal

lemma HFinite_Int_HInfinite_empty:

      HFinite ∩ HInfinite = {}

lemma HFinite_not_HInfinite:

      x ∈ HFinite ==> x ∉ HInfinite

lemma not_HFinite_HInfinite:

      x ∉ HFinite ==> x ∈ HInfinite

lemma HInfinite_HFinite_disj:

      x ∈ HInfinite ∨ x ∈ HFinite

lemma HInfinite_HFinite_iff:

      (x ∈ HInfinite) = (x ∉ HFinite)

lemma HFinite_HInfinite_iff:

      (x ∈ HFinite) = (x ∉ HInfinite)

lemma HInfinite_diff_HFinite_Infinitesimal_disj:

      x ∉ Infinitesimal ==> x ∈ HInfinite ∨ x ∈ HFinite - Infinitesimal

lemma HFinite_inverse:

      [| x ∈ HFinite; x ∉ Infinitesimal |] ==> inverse x ∈ HFinite

lemma HFinite_inverse2:

      x ∈ HFinite - Infinitesimal ==> inverse x ∈ HFinite

lemma Infinitesimal_inverse_HFinite:

      x ∉ Infinitesimal ==> inverse x ∈ HFinite

lemma HFinite_not_Infinitesimal_inverse:

      x ∈ HFinite - Infinitesimal ==> inverse x ∈ HFinite - Infinitesimal

lemma approx_inverse:

      [| x ≈ y; y ∈ HFinite - Infinitesimal |] ==> inverse x ≈ inverse y

lemmas hypreal_of_real_approx_inverse:

      [| x ≈ star_of x1; star_of x1 ≠ 0 |] ==> inverse x ≈ inverse (star_of x1)

lemmas hypreal_of_real_approx_inverse:

      [| x ≈ star_of x1; star_of x1 ≠ 0 |] ==> inverse x ≈ inverse (star_of x1)

lemma inverse_add_Infinitesimal_approx:

      [| x ∈ HFinite - Infinitesimal; h ∈ Infinitesimal |]
      ==> inverse (x + h) ≈ inverse x

lemma inverse_add_Infinitesimal_approx2:

      [| x ∈ HFinite - Infinitesimal; h ∈ Infinitesimal |]
      ==> inverse (h + x) ≈ inverse x

lemma inverse_add_Infinitesimal_approx_Infinitesimal:

      [| x ∈ HFinite - Infinitesimal; h ∈ Infinitesimal |]
      ==> inverse (x + h) + - inverse x ≈ h

lemma Infinitesimal_square_iff:

      (x ∈ Infinitesimal) = (x * x ∈ Infinitesimal)

lemma HFinite_square_iff:

      (x * x ∈ HFinite) = (x ∈ HFinite)

lemma HInfinite_square_iff:

      (x * x ∈ HInfinite) = (x ∈ HInfinite)

lemma approx_HFinite_mult_cancel:

      [| a ∈ HFinite - Infinitesimal; a * w ≈ a * z |] ==> w ≈ z

lemma approx_HFinite_mult_cancel_iff1:

      a ∈ HFinite - Infinitesimal ==> (a * w ≈ a * z) = (w ≈ z)

lemma HInfinite_HFinite_add_cancel:

      [| x + y ∈ HInfinite; y ∈ HFinite |] ==> x ∈ HInfinite

lemma HInfinite_HFinite_add:

      [| x ∈ HInfinite; y ∈ HFinite |] ==> x + y ∈ HInfinite

lemma HInfinite_ge_HInfinite:

      [| x ∈ HInfinite; x ≤ y; 0 ≤ x |] ==> y ∈ HInfinite

lemma Infinitesimal_inverse_HInfinite:

      [| x ∈ Infinitesimal; x ≠ 0 |] ==> inverse x ∈ HInfinite

lemma HInfinite_HFinite_not_Infinitesimal_mult:

      [| x ∈ HInfinite; y ∈ HFinite - Infinitesimal |] ==> x * y ∈ HInfinite

lemma HInfinite_HFinite_not_Infinitesimal_mult2:

      [| x ∈ HInfinite; y ∈ HFinite - Infinitesimal |] ==> y * x ∈ HInfinite

lemma HInfinite_gt_SReal:

      [| x ∈ HInfinite; 0 < x; y ∈ Reals |] ==> y < x

lemma HInfinite_gt_zero_gt_one:

      [| x ∈ HInfinite; 0 < x |] ==> 1 < x

lemma not_HInfinite_one:

      1 ∉ HInfinite

lemma approx_hrabs_disj:

      ¦x¦ ≈ x ∨ ¦x¦ ≈ - x

### Theorems about Monads

lemma monad_hrabs_Un_subset:

      monad ¦x¦ ⊆ monad x ∪ monad (- x)

lemma Infinitesimal_monad_eq:

      e ∈ Infinitesimal ==> monad (x + e) = monad x

lemma mem_monad_iff:

      (u ∈ monad x) = (- u ∈ monad (- x))

lemma Infinitesimal_monad_zero_iff:

      (x ∈ Infinitesimal) = (x ∈ monad 0)

lemma monad_zero_minus_iff:

      (x ∈ monad 0) = (- x ∈ monad 0)

lemma monad_zero_hrabs_iff:

      (x ∈ monad 0) = (¦x¦ ∈ monad 0)

lemma mem_monad_self:

      x ∈ monad x

### Proof that @{term "x @= y"} implies @{term"¦x¦ @= ¦y¦"}

lemma approx_subset_monad:

      x ≈ y ==> {x, y} ⊆ monad x

lemma approx_subset_monad2:

      x ≈ y ==> {x, y} ⊆ monad y

lemma mem_monad_approx:

      u ∈ monad x ==> x ≈ u

lemma approx_mem_monad:

      x ≈ u ==> u ∈ monad x

lemma approx_mem_monad2:

      x ≈ u ==> x ∈ monad u

lemma approx_mem_monad_zero:

      [| x ≈ y; x ∈ monad 0 |] ==> y ∈ monad 0

lemma Infinitesimal_approx_hrabs:

      [| x ≈ y; x ∈ Infinitesimal |] ==> ¦x¦ ≈ ¦y¦

lemma less_Infinitesimal_less:

      [| 0 < x; x ∉ Infinitesimal; e ∈ Infinitesimal |] ==> e < x

lemma Ball_mem_monad_gt_zero:

      [| 0 < x; x ∉ Infinitesimal; u ∈ monad x |] ==> 0 < u

lemma Ball_mem_monad_less_zero:

      [| x < 0; x ∉ Infinitesimal; u ∈ monad x |] ==> u < 0

lemma lemma_approx_gt_zero:

      [| 0 < x; x ∉ Infinitesimal; x ≈ y |] ==> 0 < y

lemma lemma_approx_less_zero:

      [| x < 0; x ∉ Infinitesimal; x ≈ y |] ==> y < 0

theorem approx_hrabs:

      x ≈ y ==> ¦x¦ ≈ ¦y¦

lemma approx_hrabs_zero_cancel:

      ¦x¦ ≈ 0 ==> x ≈ 0

lemma approx_hrabs_add_Infinitesimal:

      e ∈ Infinitesimal ==> ¦x¦ ≈ ¦x + e¦

lemma approx_hrabs_add_minus_Infinitesimal:

      e ∈ Infinitesimal ==> ¦x¦ ≈ ¦x + - e¦

lemma hrabs_add_Infinitesimal_cancel:

      [| e ∈ Infinitesimal; e' ∈ Infinitesimal; ¦x + e¦ = ¦y + e'¦ |] ==> ¦x¦ ≈ ¦y¦

lemma hrabs_add_minus_Infinitesimal_cancel:

      [| e ∈ Infinitesimal; e' ∈ Infinitesimal; ¦x + - e¦ = ¦y + - e'¦ |]
      ==> ¦x¦ ≈ ¦y¦

lemma Infinitesimal_add_hypreal_of_real_less:

      [| x < y; u ∈ Infinitesimal |] ==> star_of x + u < star_of y

lemma Infinitesimal_add_hrabs_hypreal_of_real_less:

      [| x ∈ Infinitesimal; ¦star_of r¦ < star_of y |] ==> ¦star_of r + x¦ < star_of y

lemma Infinitesimal_add_hrabs_hypreal_of_real_less2:

      [| x ∈ Infinitesimal; ¦star_of r¦ < star_of y |] ==> ¦x + star_of r¦ < star_of y

lemma hypreal_of_real_le_add_Infininitesimal_cancel:

      [| u ∈ Infinitesimal; v ∈ Infinitesimal; star_of x + u ≤ star_of y + v |]
      ==> star_of x ≤ star_of y

lemma hypreal_of_real_le_add_Infininitesimal_cancel2:

      [| u ∈ Infinitesimal; v ∈ Infinitesimal; star_of x + u ≤ star_of y + v |]
      ==> x ≤ y

lemma hypreal_of_real_less_Infinitesimal_le_zero:

      [| star_of x < e; e ∈ Infinitesimal |] ==> star_of x ≤ 0

lemma Infinitesimal_add_not_zero:

      [| h ∈ Infinitesimal; x ≠ 0 |] ==> star_of x + h ≠ 0

lemma Infinitesimal_square_cancel:

      x * x + y * y ∈ Infinitesimal ==> x * x ∈ Infinitesimal

lemma HFinite_square_cancel:

      x * x + y * y ∈ HFinite ==> x * x ∈ HFinite

lemma Infinitesimal_square_cancel2:

      x * x + y * y ∈ Infinitesimal ==> y * y ∈ Infinitesimal

lemma HFinite_square_cancel2:

      x * x + y * y ∈ HFinite ==> y * y ∈ HFinite

lemma Infinitesimal_sum_square_cancel:

      x * x + y * y + z * z ∈ Infinitesimal ==> x * x ∈ Infinitesimal

lemma HFinite_sum_square_cancel:

      x * x + y * y + z * z ∈ HFinite ==> x * x ∈ HFinite

lemma Infinitesimal_sum_square_cancel2:

      y * y + x * x + z * z ∈ Infinitesimal ==> x * x ∈ Infinitesimal

lemma HFinite_sum_square_cancel2:

      y * y + x * x + z * z ∈ HFinite ==> x * x ∈ HFinite

lemma Infinitesimal_sum_square_cancel3:

      z * z + y * y + x * x ∈ Infinitesimal ==> x * x ∈ Infinitesimal

lemma HFinite_sum_square_cancel3:

      z * z + y * y + x * x ∈ HFinite ==> x * x ∈ HFinite

lemma monad_hrabs_less:

      [| y ∈ monad x; 0 < star_of e |] ==> ¦y + - x¦ < star_of e

lemma mem_monad_SReal_HFinite:

      x ∈ monad (star_of a) ==> x ∈ HFinite

### Theorems about Standard Part

lemma st_approx_self:

      x ∈ HFinite ==> st x ≈ x

lemma st_SReal:

      x ∈ HFinite ==> st x ∈ Reals

lemma st_HFinite:

      x ∈ HFinite ==> st x ∈ HFinite

lemma st_SReal_eq:

      x ∈ Reals ==> st x = x

lemma st_hypreal_of_real:

      st (star_of x) = star_of x

lemma st_eq_approx:

      [| x ∈ HFinite; y ∈ HFinite; st x = st y |] ==> x ≈ y

lemma approx_st_eq:

      [| x ∈ HFinite; y ∈ HFinite; x ≈ y |] ==> st x = st y

lemma st_eq_approx_iff:

      [| x ∈ HFinite; y ∈ HFinite |] ==> (x ≈ y) = (st x = st y)

lemma st_Infinitesimal_add_SReal:

      [| x ∈ Reals; e ∈ Infinitesimal |] ==> st (x + e) = x

lemma st_Infinitesimal_add_SReal2:

      [| x ∈ Reals; e ∈ Infinitesimal |] ==> st (e + x) = x

lemma HFinite_st_Infinitesimal_add:

      x ∈ HFinite ==> ∃e∈Infinitesimal. x = st x + e

lemma st_add:

      [| x ∈ HFinite; y ∈ HFinite |] ==> st (x + y) = st x + st y

lemma st_number_of:

      st (number_of w) = number_of w

lemma

      st 0 = 0

      st 1 = 1

lemma st_minus:

      y ∈ HFinite ==> st (- y) = - st y

lemma st_diff:

      [| x ∈ HFinite; y ∈ HFinite |] ==> st (x - y) = st x - st y

lemma lemma_st_mult:

      [| x ∈ HFinite; y ∈ HFinite; e ∈ Infinitesimal; ea ∈ Infinitesimal |]
      ==> e * y + x * ea + e * ea ∈ Infinitesimal

lemma st_mult:

      [| x ∈ HFinite; y ∈ HFinite |] ==> st (x * y) = st x * st y

lemma st_Infinitesimal:

      x ∈ Infinitesimal ==> st x = 0

lemma st_not_Infinitesimal:

      st x ≠ 0 ==> x ∉ Infinitesimal

lemma st_inverse:

      [| x ∈ HFinite; st x ≠ 0 |] ==> st (inverse x) = inverse (st x)

lemma st_divide:

      [| x ∈ HFinite; y ∈ HFinite; st y ≠ 0 |] ==> st (x / y) = st x / st y

lemma st_idempotent:

      x ∈ HFinite ==> st (st x) = st x

lemma Infinitesimal_add_st_less:

      [| x ∈ HFinite; y ∈ HFinite; u ∈ Infinitesimal; st x < st y |]
      ==> st x + u < st y

lemma Infinitesimal_add_st_le_cancel:

      [| x ∈ HFinite; y ∈ HFinite; u ∈ Infinitesimal; st x ≤ st y + u |]
      ==> st x ≤ st y

lemma st_le:

      [| x ∈ HFinite; y ∈ HFinite; x ≤ y |] ==> st x ≤ st y

lemma st_zero_le:

      [| 0 ≤ x; x ∈ HFinite |] ==> 0 ≤ st x

lemma st_zero_ge:

      [| x ≤ 0; x ∈ HFinite |] ==> st x ≤ 0

lemma st_hrabs:

      x ∈ HFinite ==> ¦st x¦ = st ¦x¦

### Alternative Definitions for @{term HFinite} using Free Ultrafilter

lemma FreeUltrafilterNat_Rep_hypreal:

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

lemma HFinite_FreeUltrafilterNat:

      x ∈ HFinite ==> ∃X∈Rep_star x. ∃u. {n. ¦X n¦ < u} ∈ \<U>

lemma FreeUltrafilterNat_HFinite:

      ∃X∈Rep_star x. ∃u. {n. ¦X n¦ < u} ∈ \<U> ==> x ∈ HFinite

lemma HFinite_FreeUltrafilterNat_iff:

      (x ∈ HFinite) = (∃X∈Rep_star x. ∃u. {n. ¦X n¦ < u} ∈ \<U>)

### Alternative Definitions for @{term HInfinite} using Free Ultrafilter

lemma lemma_Compl_eq:

      - {n. u < ¦xa n¦} = {n. ¦xa n¦ ≤ u}

lemma lemma_Compl_eq2:

      - {n. ¦xa n¦ < u} = {n. u ≤ ¦xa n¦}

lemma lemma_Int_eq1:

      {n. ¦xa n¦ ≤ u} ∩ {n. u ≤ ¦xa n¦} = {n. ¦xa n¦ = u}

lemma lemma_FreeUltrafilterNat_one:

      {n. ¦xa n¦ = u} ⊆ {n. ¦xa n¦ < u + 1}

lemma FreeUltrafilterNat_const_Finite:

      [| xa ∈ Rep_star x; {n. ¦xa n¦ = u} ∈ \<U> |] ==> x ∈ HFinite

lemma HInfinite_FreeUltrafilterNat:

      x ∈ HInfinite ==> ∃X∈Rep_star x. ∀u. {n. u < ¦X n¦} ∈ \<U>

lemma lemma_Int_HI:

      {n. ¦Xa n¦ < u} ∩ {n. X n = Xa n} ⊆ {n. ¦X n¦ < u}

lemma lemma_Int_HIa:

      {n. u < ¦X n¦} ∩ {n. ¦X n¦ < u} = {}

lemma FreeUltrafilterNat_HInfinite:

      ∃X∈Rep_star x. ∀u. {n. u < ¦X n¦} ∈ \<U> ==> x ∈ HInfinite

lemma HInfinite_FreeUltrafilterNat_iff:

      (x ∈ HInfinite) = (∃X∈Rep_star x. ∀u. {n. u < ¦X n¦} ∈ \<U>)

### Alternative Definitions for @{term Infinitesimal} using Free Ultrafilter

lemma Infinitesimal_FreeUltrafilterNat:

      x ∈ Infinitesimal ==> ∃X∈Rep_star x. ∀u>0. {n. ¦X n¦ < u} ∈ \<U>

lemma FreeUltrafilterNat_Infinitesimal:

      ∃X∈Rep_star x. ∀u>0. {n. ¦X n¦ < u} ∈ \<U> ==> x ∈ Infinitesimal

lemma Infinitesimal_FreeUltrafilterNat_iff:

      (x ∈ Infinitesimal) = (∃X∈Rep_star x. ∀u>0. {n. ¦X n¦ < u} ∈ \<U>)

lemma lemma_Infinitesimal:

      (∀r>0. x < r) = (∀n. x < inverse (real (Suc n)))

lemma of_nat_in_Reals:

      of_nat n ∈ Reals

lemma lemma_Infinitesimal2:

      (∀r∈Reals. 0 < r --> x < r) = (∀n. x < inverse (hypreal_of_nat (Suc n)))

lemma Infinitesimal_hypreal_of_nat_iff:

      Infinitesimal = {x. ∀n. ¦x¦ < inverse (hypreal_of_nat (Suc n))}

### Proof that @{term omega} is an infinite number

lemma Suc_Un_eq:

      {n. n < Suc m} = {n. n < m} ∪ {n. n = m}

lemma finite_nat_segment:

      finite {n. n < m}

lemma finite_real_of_nat_segment:

      finite {n. real n < real m}

lemma finite_real_of_nat_less_real:

      finite {n. real n < u}

lemma lemma_real_le_Un_eq:

      {n. f n ≤ u} = {n. f n < u} ∪ {n. u = f n}

lemma finite_real_of_nat_le_real:

      finite {n. real n ≤ u}

lemma finite_rabs_real_of_nat_le_real:

      finite {n. ¦real n¦ ≤ u}

lemma rabs_real_of_nat_le_real_FreeUltrafilterNat:

      {n. ¦real n¦ ≤ u} ∉ \<U>

lemma FreeUltrafilterNat_nat_gt_real:

      {n. u < real n} ∈ \<U>

lemma Compl_real_le_eq:

      - {n. real n ≤ u} = {n. u < real n}

lemma FreeUltrafilterNat_omega:

      {n. u < real n} ∈ \<U>

theorem HInfinite_omega:

      ω ∈ HInfinite

lemma Infinitesimal_epsilon:

      ε ∈ Infinitesimal

lemma HFinite_epsilon:

      ε ∈ HFinite

lemma epsilon_approx_zero:

      ε ≈ 0

lemma real_of_nat_less_inverse_iff:

      0 < u ==> (u < inverse (real (Suc n))) = (real (Suc n) < inverse u)

lemma finite_inverse_real_of_posnat_gt_real:

      0 < u ==> finite {n. u < inverse (real (Suc n))}

lemma lemma_real_le_Un_eq2:

      {n. u ≤ inverse (real (Suc n))} =
      {n. u < inverse (real (Suc n))} ∪ {n. u = inverse (real (Suc n))}

lemma real_of_nat_inverse_le_iff:

      (inverse (real (Suc n)) ≤ r) = (1 ≤ r * real (Suc n))

lemma real_of_nat_inverse_eq_iff:

      (u = inverse (real (Suc n))) = (real (Suc n) = inverse u)

lemma lemma_finite_omega_set2:

      finite {n. u = inverse (real (Suc n))}

lemma finite_inverse_real_of_posnat_ge_real:

      0 < u ==> finite {n. u ≤ inverse (real (Suc n))}

lemma inverse_real_of_posnat_ge_real_FreeUltrafilterNat:

      0 < u ==> {n. u ≤ inverse (real (Suc n))} ∉ \<U>

lemma Compl_le_inverse_eq:

      - {n. u ≤ inverse (real (Suc n))} = {n. inverse (real (Suc n)) < u}

lemma FreeUltrafilterNat_inverse_real_of_posnat:

      0 < u ==> {n. inverse (real (Suc n)) < u} ∈ \<U>

lemma real_seq_to_hypreal_Infinitesimal:

      ∀n. ¦X n + - x¦ < inverse (real (Suc n))
      ==> star_n X + - star_of x ∈ Infinitesimal

lemma real_seq_to_hypreal_approx:

      ∀n. ¦X n + - x¦ < inverse (real (Suc n)) ==> star_n X ≈ star_of x

lemma real_seq_to_hypreal_approx2:

      ∀n. ¦x + - X n¦ < inverse (real (Suc n)) ==> star_n X ≈ star_of x

lemma real_seq_to_hypreal_Infinitesimal2:

      ∀n. ¦X n + - Y n¦ < inverse (real (Suc n))
      ==> star_n X + - star_n Y ∈ Infinitesimal