# Theory NSA

theory NSA
imports HyperDef Lub_Glb
```(*  Title:      HOL/Nonstandard_Analysis/NSA.thy
Author:     Jacques D. Fleuriot, University of Cambridge
Author:     Lawrence C Paulson, University of Cambridge
*)

section ‹Infinite Numbers, Infinitesimals, Infinitely Close Relation›

theory NSA
imports HyperDef "HOL-Library.Lub_Glb"
begin

definition hnorm :: "'a::real_normed_vector star ⇒ real star"
where [transfer_unfold]: "hnorm = *f* norm"

definition Infinitesimal  :: "('a::real_normed_vector) star set"
where "Infinitesimal = {x. ∀r ∈ Reals. 0 < r ⟶ hnorm x < r}"

definition HFinite :: "('a::real_normed_vector) star set"
where "HFinite = {x. ∃r ∈ Reals. hnorm x < r}"

definition HInfinite :: "('a::real_normed_vector) star set"
where "HInfinite = {x. ∀r ∈ Reals. r < hnorm x}"

definition approx :: "'a::real_normed_vector star ⇒ 'a star ⇒ bool"  (infixl "≈" 50)
where "x ≈ y ⟷ x - y ∈ Infinitesimal"
― ‹the ``infinitely close'' relation›

definition st :: "hypreal ⇒ hypreal"
where "st = (λx. SOME r. x ∈ HFinite ∧ r ∈ ℝ ∧ r ≈ x)"
― ‹the standard part of a hyperreal›

definition monad :: "'a::real_normed_vector star ⇒ 'a star set"
where "monad x = {y. x ≈ y}"

definition galaxy :: "'a::real_normed_vector star ⇒ 'a star set"
where "galaxy x = {y. (x + -y) ∈ HFinite}"

lemma SReal_def: "ℝ ≡ {x. ∃r. x = hypreal_of_real r}"
by (simp add: Reals_def image_def)

subsection ‹Nonstandard Extension of the Norm Function›

definition scaleHR :: "real star ⇒ 'a star ⇒ 'a::real_normed_vector star"
where [transfer_unfold]: "scaleHR = starfun2 scaleR"

lemma Standard_hnorm [simp]: "x ∈ Standard ⟹ hnorm x ∈ Standard"
by (simp add: hnorm_def)

lemma star_of_norm [simp]: "star_of (norm x) = hnorm (star_of x)"
by transfer (rule refl)

lemma hnorm_ge_zero [simp]: "⋀x::'a::real_normed_vector star. 0 ≤ hnorm x"
by transfer (rule norm_ge_zero)

lemma hnorm_eq_zero [simp]: "⋀x::'a::real_normed_vector star. hnorm x = 0 ⟷ x = 0"
by transfer (rule norm_eq_zero)

lemma hnorm_triangle_ineq: "⋀x y::'a::real_normed_vector star. hnorm (x + y) ≤ hnorm x + hnorm y"
by transfer (rule norm_triangle_ineq)

lemma hnorm_triangle_ineq3: "⋀x y::'a::real_normed_vector star. ¦hnorm x - hnorm y¦ ≤ hnorm (x - y)"
by transfer (rule norm_triangle_ineq3)

lemma hnorm_scaleR: "⋀x::'a::real_normed_vector star. hnorm (a *⇩R x) = ¦star_of a¦ * hnorm x"
by transfer (rule norm_scaleR)

lemma hnorm_scaleHR: "⋀a (x::'a::real_normed_vector star). hnorm (scaleHR a x) = ¦a¦ * hnorm x"
by transfer (rule norm_scaleR)

lemma hnorm_mult_ineq: "⋀x y::'a::real_normed_algebra star. hnorm (x * y) ≤ hnorm x * hnorm y"
by transfer (rule norm_mult_ineq)

lemma hnorm_mult: "⋀x y::'a::real_normed_div_algebra star. hnorm (x * y) = hnorm x * hnorm y"
by transfer (rule norm_mult)

lemma hnorm_hyperpow: "⋀(x::'a::{real_normed_div_algebra} star) n. hnorm (x pow n) = hnorm x pow n"
by transfer (rule norm_power)

lemma hnorm_one [simp]: "hnorm (1::'a::real_normed_div_algebra star) = 1"
by transfer (rule norm_one)

lemma hnorm_zero [simp]: "hnorm (0::'a::real_normed_vector star) = 0"
by transfer (rule norm_zero)

lemma zero_less_hnorm_iff [simp]: "⋀x::'a::real_normed_vector star. 0 < hnorm x ⟷ x ≠ 0"
by transfer (rule zero_less_norm_iff)

lemma hnorm_minus_cancel [simp]: "⋀x::'a::real_normed_vector star. hnorm (- x) = hnorm x"
by transfer (rule norm_minus_cancel)

lemma hnorm_minus_commute: "⋀a b::'a::real_normed_vector star. hnorm (a - b) = hnorm (b - a)"
by transfer (rule norm_minus_commute)

lemma hnorm_triangle_ineq2: "⋀a b::'a::real_normed_vector star. hnorm a - hnorm b ≤ hnorm (a - b)"
by transfer (rule norm_triangle_ineq2)

lemma hnorm_triangle_ineq4: "⋀a b::'a::real_normed_vector star. hnorm (a - b) ≤ hnorm a + hnorm b"
by transfer (rule norm_triangle_ineq4)

lemma abs_hnorm_cancel [simp]: "⋀a::'a::real_normed_vector star. ¦hnorm a¦ = hnorm a"
by transfer (rule abs_norm_cancel)

lemma hnorm_of_hypreal [simp]: "⋀r. hnorm (of_hypreal r::'a::real_normed_algebra_1 star) = ¦r¦"
by transfer (rule norm_of_real)

lemma nonzero_hnorm_inverse:
"⋀a::'a::real_normed_div_algebra star. a ≠ 0 ⟹ hnorm (inverse a) = inverse (hnorm a)"
by transfer (rule nonzero_norm_inverse)

lemma hnorm_inverse:
"⋀a::'a::{real_normed_div_algebra, division_ring} star. hnorm (inverse a) = inverse (hnorm a)"
by transfer (rule norm_inverse)

lemma hnorm_divide: "⋀a b::'a::{real_normed_field, field} star. hnorm (a / b) = hnorm a / hnorm b"
by transfer (rule norm_divide)

lemma hypreal_hnorm_def [simp]: "⋀r::hypreal. hnorm r = ¦r¦"
by transfer (rule real_norm_def)

"⋀(x::'a::real_normed_vector star) y r s. hnorm x < r ⟹ hnorm y < s ⟹ hnorm (x + y) < r + s"
by transfer (rule norm_add_less)

lemma hnorm_mult_less:
"⋀(x::'a::real_normed_algebra star) y r s. hnorm x < r ⟹ hnorm y < s ⟹ hnorm (x * y) < r * s"
by transfer (rule norm_mult_less)

lemma hnorm_scaleHR_less: "¦x¦ < r ⟹ hnorm y < s ⟹ hnorm (scaleHR x y) < r * s"
by (simp only: hnorm_scaleHR) (simp add: mult_strict_mono')

subsection ‹Closure Laws for the Standard Reals›

lemma Reals_minus_iff [simp]: "- x ∈ ℝ ⟷ x ∈ ℝ"
apply auto
apply (drule Reals_minus)
apply auto
done

lemma Reals_add_cancel: "x + y ∈ ℝ ⟹ y ∈ ℝ ⟹ x ∈ ℝ"
by (drule (1) Reals_diff) simp

lemma SReal_hrabs: "x ∈ ℝ ⟹ ¦x¦ ∈ ℝ"
for x :: hypreal
by (simp add: Reals_eq_Standard)

lemma SReal_hypreal_of_real [simp]: "hypreal_of_real x ∈ ℝ"
by (simp add: Reals_eq_Standard)

lemma SReal_divide_numeral: "r ∈ ℝ ⟹ r / (numeral w::hypreal) ∈ ℝ"
by simp

text ‹‹ε› is not in Reals because it is an infinitesimal›
lemma SReal_epsilon_not_mem: "ε ∉ ℝ"
by (auto simp: SReal_def hypreal_of_real_not_eq_epsilon [symmetric])

lemma SReal_omega_not_mem: "ω ∉ ℝ"
by (auto simp: SReal_def hypreal_of_real_not_eq_omega [symmetric])

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

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

lemma hypreal_of_real_image: "hypreal_of_real `(UNIV::real set) = ℝ"
by (simp add: Reals_eq_Standard Standard_def)

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

lemma SReal_hypreal_of_real_image: "∃x. x ∈ P ⟹ P ⊆ ℝ ⟹ ∃Q. P = hypreal_of_real ` Q"
unfolding SReal_def image_def by blast

lemma SReal_dense: "x ∈ ℝ ⟹ y ∈ ℝ ⟹ x < y ⟹ ∃r ∈ Reals. x < r ∧ r < y"
for x y :: hypreal
apply (auto simp: SReal_def)
apply (drule dense)
apply auto
done

text ‹Completeness of Reals, but both lemmas are unused.›

lemma SReal_sup_lemma:
"P ⊆ ℝ ⟹ (∃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 ⊆ ℝ ⟹ ∃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 ‹Set of Finite Elements is a Subring of the Extended Reals›

lemma HFinite_add: "x ∈ HFinite ⟹ y ∈ HFinite ⟹ x + y ∈ HFinite"
unfolding HFinite_def by (blast intro!: Reals_add hnorm_add_less)

lemma HFinite_mult: "x ∈ HFinite ⟹ y ∈ HFinite ⟹ x * y ∈ HFinite"
for x y :: "'a::real_normed_algebra star"
unfolding HFinite_def by (blast intro!: Reals_mult hnorm_mult_less)

lemma HFinite_scaleHR: "x ∈ HFinite ⟹ y ∈ HFinite ⟹ scaleHR x y ∈ HFinite"
by (auto simp: HFinite_def intro!: Reals_mult hnorm_scaleHR_less)

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

lemma HFinite_star_of [simp]: "star_of x ∈ HFinite"
apply (simp add: HFinite_def)
apply (rule_tac x="star_of (norm x) + 1" in bexI)
apply (transfer, simp)
apply (blast intro: Reals_add SReal_hypreal_of_real Reals_1)
done

lemma SReal_subset_HFinite: "(ℝ::hypreal set) ⊆ HFinite"
by (auto simp add: SReal_def)

lemma HFiniteD: "x ∈ HFinite ⟹ ∃t ∈ Reals. hnorm x < t"
by (simp add: HFinite_def)

lemma HFinite_hrabs_iff [iff]: "¦x¦ ∈ HFinite ⟷ x ∈ HFinite"
for x :: hypreal
by (simp add: HFinite_def)

lemma HFinite_hnorm_iff [iff]: "hnorm x ∈ HFinite ⟷ x ∈ HFinite"
for x :: hypreal
by (simp add: HFinite_def)

lemma HFinite_numeral [simp]: "numeral w ∈ HFinite"
unfolding star_numeral_def by (rule HFinite_star_of)

text ‹As always with numerals, ‹0› and ‹1› are special cases.›

lemma HFinite_0 [simp]: "0 ∈ HFinite"
unfolding star_zero_def by (rule HFinite_star_of)

lemma HFinite_1 [simp]: "1 ∈ HFinite"
unfolding star_one_def by (rule HFinite_star_of)

lemma hrealpow_HFinite: "x ∈ HFinite ⟹ x ^ n ∈ HFinite"
for x :: "'a::{real_normed_algebra,monoid_mult} star"
by (induct n) (auto simp add: power_Suc intro: HFinite_mult)

lemma HFinite_bounded: "x ∈ HFinite ⟹ y ≤ x ⟹ 0 ≤ y ⟹ y ∈ HFinite"
for x y :: hypreal
apply (cases "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 InfinitesimalI: "(⋀r. r ∈ ℝ ⟹ 0 < r ⟹ hnorm x < r) ⟹ x ∈ Infinitesimal"
by (simp add: Infinitesimal_def)

lemma InfinitesimalD: "x ∈ Infinitesimal ⟹ ∀r ∈ Reals. 0 < r ⟶ hnorm x < r"
by (simp add: Infinitesimal_def)

lemma InfinitesimalI2: "(⋀r. 0 < r ⟹ hnorm x < star_of r) ⟹ x ∈ Infinitesimal"
by (auto simp add: Infinitesimal_def SReal_def)

lemma InfinitesimalD2: "x ∈ Infinitesimal ⟹ 0 < r ⟹ hnorm x < star_of r"
by (auto simp add: Infinitesimal_def SReal_def)

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

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

lemma Infinitesimal_add: "x ∈ Infinitesimal ⟹ y ∈ Infinitesimal ⟹ x + y ∈ Infinitesimal"
apply (rule InfinitesimalI)
apply (rule hypreal_sum_of_halves [THEN subst])
apply (drule half_gt_zero)
apply (blast intro: hnorm_add_less SReal_divide_numeral dest: InfinitesimalD)
done

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

lemma Infinitesimal_hnorm_iff: "hnorm x ∈ Infinitesimal ⟷ x ∈ Infinitesimal"
by (simp add: Infinitesimal_def)

lemma Infinitesimal_hrabs_iff [iff]: "¦x¦ ∈ Infinitesimal ⟷ x ∈ Infinitesimal"
for x :: hypreal
by (simp add: abs_if)

lemma Infinitesimal_of_hypreal_iff [simp]:
"(of_hypreal x::'a::real_normed_algebra_1 star) ∈ Infinitesimal ⟷ x ∈ Infinitesimal"
by (subst Infinitesimal_hnorm_iff [symmetric]) simp

lemma Infinitesimal_diff: "x ∈ Infinitesimal ⟹ y ∈ Infinitesimal ⟹ x - y ∈ Infinitesimal"
using Infinitesimal_add [of x "- y"] by simp

lemma Infinitesimal_mult: "x ∈ Infinitesimal ⟹ y ∈ Infinitesimal ⟹ x * y ∈ Infinitesimal"
for x y :: "'a::real_normed_algebra star"
apply (rule InfinitesimalI)
apply (subgoal_tac "hnorm (x * y) < 1 * r")
apply (simp only: mult_1)
apply (rule hnorm_mult_less)
apply (simp_all add: InfinitesimalD)
done

lemma Infinitesimal_HFinite_mult: "x ∈ Infinitesimal ⟹ y ∈ HFinite ⟹ x * y ∈ Infinitesimal"
for x y :: "'a::real_normed_algebra star"
apply (rule InfinitesimalI)
apply (drule HFiniteD, clarify)
apply (subgoal_tac "0 < t")
apply (subgoal_tac "hnorm (x * y) < (r / t) * t", simp)
apply (subgoal_tac "0 < r / t")
apply (rule hnorm_mult_less)
apply (simp add: InfinitesimalD)
apply assumption
apply simp
apply (erule order_le_less_trans [OF hnorm_ge_zero])
done

lemma Infinitesimal_HFinite_scaleHR:
"x ∈ Infinitesimal ⟹ y ∈ HFinite ⟹ scaleHR x y ∈ Infinitesimal"
apply (rule InfinitesimalI)
apply (drule HFiniteD, clarify)
apply (drule InfinitesimalD)
apply (simp add: hnorm_scaleHR)
apply (subgoal_tac "0 < t")
apply (subgoal_tac "¦x¦ * hnorm y < (r / t) * t", simp)
apply (subgoal_tac "0 < r / t")
apply (rule mult_strict_mono', simp_all)
apply (erule order_le_less_trans [OF hnorm_ge_zero])
done

lemma Infinitesimal_HFinite_mult2:
"x ∈ Infinitesimal ⟹ y ∈ HFinite ⟹ y * x ∈ Infinitesimal"
for x y :: "'a::real_normed_algebra star"
apply (rule InfinitesimalI)
apply (drule HFiniteD, clarify)
apply (subgoal_tac "0 < t")
apply (subgoal_tac "hnorm (y * x) < t * (r / t)", simp)
apply (subgoal_tac "0 < r / t")
apply (rule hnorm_mult_less)
apply assumption
apply (simp add: InfinitesimalD)
apply simp
apply (erule order_le_less_trans [OF hnorm_ge_zero])
done

lemma Infinitesimal_scaleR2: "x ∈ Infinitesimal ⟹ a *⇩R x ∈ Infinitesimal"
apply (case_tac "a = 0", simp)
apply (rule InfinitesimalI)
apply (drule InfinitesimalD)
apply (drule_tac x="r / ¦star_of a¦" in bspec)
apply (simp add: Reals_eq_Standard)
apply simp
apply (simp add: hnorm_scaleR pos_less_divide_eq mult.commute)
done

lemma Compl_HFinite: "- HFinite = HInfinite"
apply (auto simp add: HInfinite_def HFinite_def linorder_not_less)
apply (rule_tac y="r + 1" in order_less_le_trans, simp)
apply simp
done

lemma HInfinite_inverse_Infinitesimal: "x ∈ HInfinite ⟹ inverse x ∈ Infinitesimal"
for x :: "'a::real_normed_div_algebra star"
apply (rule InfinitesimalI)
apply (subgoal_tac "x ≠ 0")
apply (rule inverse_less_imp_less)
apply (simp add: nonzero_hnorm_inverse)
apply (simp add: HInfinite_def Reals_inverse)
apply assumption
apply (clarify, simp add: Compl_HFinite [symmetric])
done

lemma HInfiniteI: "(⋀r. r ∈ ℝ ⟹ r < hnorm x) ⟹ x ∈ HInfinite"
by (simp add: HInfinite_def)

lemma HInfiniteD: "x ∈ HInfinite ⟹ r ∈ ℝ ⟹ r < hnorm x"
by (simp add: HInfinite_def)

lemma HInfinite_mult: "x ∈ HInfinite ⟹ y ∈ HInfinite ⟹ x * y ∈ HInfinite"
for x y :: "'a::real_normed_div_algebra star"
apply (rule HInfiniteI, simp only: hnorm_mult)
apply (subgoal_tac "r * 1 < hnorm x * hnorm y", simp only: mult_1)
apply (case_tac "x = 0", simp add: HInfinite_def)
apply (rule mult_strict_mono)
apply (simp_all add: HInfiniteD)
done

lemma hypreal_add_zero_less_le_mono: "r < x ⟹ 0 ≤ y ⟹ r < x + y"
for r x y :: hypreal
by (auto dest: add_less_le_mono)

lemma HInfinite_add_ge_zero: "x ∈ HInfinite ⟹ 0 ≤ y ⟹ 0 ≤ x ⟹ x + y ∈ HInfinite"
for x y :: hypreal
by (auto simp: abs_if add.commute HInfinite_def)

lemma HInfinite_add_ge_zero2: "x ∈ HInfinite ⟹ 0 ≤ y ⟹ 0 ≤ x ⟹ y + x ∈ HInfinite"
for x y :: hypreal

lemma HInfinite_add_gt_zero: "x ∈ HInfinite ⟹ 0 < y ⟹ 0 < x ⟹ x + y ∈ HInfinite"
for x y :: hypreal
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"
for x y :: hypreal
apply (drule HInfinite_minus_iff [THEN iffD2])
apply (rule HInfinite_minus_iff [THEN iffD1])
apply simp_all
done

lemma HInfinite_add_lt_zero: "x ∈ HInfinite ⟹ y < 0 ⟹ x < 0 ⟹ x + y ∈ HInfinite"
for x y :: hypreal
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"
for a b c :: "'a::real_normed_algebra star"
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 HFinite_diff_Infinitesimal_hrabs:
"x ∈ HFinite - Infinitesimal ⟹ ¦x¦ ∈ HFinite - Infinitesimal"
for x :: hypreal
by blast

lemma hnorm_le_Infinitesimal: "e ∈ Infinitesimal ⟹ hnorm x ≤ e ⟹ x ∈ Infinitesimal"
by (auto simp: Infinitesimal_def abs_less_iff)

lemma hnorm_less_Infinitesimal: "e ∈ Infinitesimal ⟹ hnorm x < e ⟹ x ∈ Infinitesimal"
by (erule hnorm_le_Infinitesimal, erule order_less_imp_le)

lemma hrabs_le_Infinitesimal: "e ∈ Infinitesimal ⟹ ¦x¦ ≤ e ⟹ x ∈ Infinitesimal"
for x :: hypreal
by (erule hnorm_le_Infinitesimal) simp

lemma hrabs_less_Infinitesimal: "e ∈ Infinitesimal ⟹ ¦x¦ < e ⟹ x ∈ Infinitesimal"
for x :: hypreal
by (erule hnorm_less_Infinitesimal) simp

lemma Infinitesimal_interval:
"e ∈ Infinitesimal ⟹ e' ∈ Infinitesimal ⟹ e' < x ⟹ x < e ⟹ x ∈ Infinitesimal"
for x :: hypreal
by (auto simp add: Infinitesimal_def abs_less_iff)

lemma Infinitesimal_interval2:
"e ∈ Infinitesimal ⟹ e' ∈ Infinitesimal ⟹ e' ≤ x ⟹ x ≤ e ⟹ x ∈ Infinitesimal"
for x :: hypreal
by (auto intro: Infinitesimal_interval simp add: order_le_less)

lemma lemma_Infinitesimal_hyperpow: "x ∈ Infinitesimal ⟹ 0 < N ⟹ ¦x pow N¦ ≤ ¦x¦"
for x :: hypreal
apply (unfold Infinitesimal_def)
apply (auto intro!: hyperpow_Suc_le_self2
simp: hyperpow_hrabs [symmetric] hypnat_gt_zero_iff2 abs_ge_zero)
done

lemma Infinitesimal_hyperpow: "x ∈ Infinitesimal ⟹ 0 < N ⟹ x pow N ∈ Infinitesimal"
for x :: hypreal
apply (rule hrabs_le_Infinitesimal)
apply (rule_tac [2] lemma_Infinitesimal_hyperpow)
apply auto
done

lemma hrealpow_hyperpow_Infinitesimal_iff:
"(x ^ n ∈ Infinitesimal) ⟷ x pow (hypnat_of_nat n) ∈ Infinitesimal"
by (simp only: hyperpow_hypnat_of_nat)

lemma Infinitesimal_hrealpow: "x ∈ Infinitesimal ⟹ 0 < n ⟹ x ^ n ∈ Infinitesimal"
for x :: hypreal
by (simp add: hrealpow_hyperpow_Infinitesimal_iff Infinitesimal_hyperpow)

lemma not_Infinitesimal_mult:
"x ∉ Infinitesimal ⟹ y ∉ Infinitesimal ⟹ x * y ∉ Infinitesimal"
for x y :: "'a::real_normed_div_algebra star"
apply (unfold Infinitesimal_def, clarify, rename_tac r s)
apply (simp only: linorder_not_less hnorm_mult)
apply (drule_tac x = "r * s" in bspec)
apply (fast intro: Reals_mult)
apply simp
apply (drule_tac c = s and d = "hnorm y" and a = r and b = "hnorm x" in mult_mono)
apply simp_all
done

lemma Infinitesimal_mult_disj: "x * y ∈ Infinitesimal ⟹ x ∈ Infinitesimal ∨ y ∈ Infinitesimal"
for x y :: "'a::real_normed_div_algebra star"
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"
for x y :: "'a::real_normed_div_algebra star"
apply clarify
apply (blast dest: HFinite_mult not_Infinitesimal_mult)
done

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

lemma Infinitesimal_star_of_mult: "x ∈ Infinitesimal ⟹ x * star_of r ∈ Infinitesimal"
for x :: "'a::real_normed_algebra star"
by (erule HFinite_star_of [THEN [2] Infinitesimal_HFinite_mult])

lemma Infinitesimal_star_of_mult2: "x ∈ Infinitesimal ⟹ star_of r * x ∈ Infinitesimal"
for x :: "'a::real_normed_algebra star"
by (erule HFinite_star_of [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"

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

lemma hypreal_minus_distrib1: "- (y + - x) = x + -y"
for x y :: "'a::ab_group_add"

lemma approx_sym: "x ≈ y ⟹ y ≈ x"
apply (simp add: approx_def)
apply (drule Infinitesimal_minus_iff [THEN iffD2])
apply simp
done

lemma approx_trans: "x ≈ y ⟹ y ≈ z ⟹ x ≈ z"
apply (simp add: approx_def)
apply (drule (1) Infinitesimal_add)
apply simp
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 approx_reorient: "x ≈ y ⟷ y ≈ x"
by (blast intro: approx_sym)

text ‹Reorientation simplification procedure: reorients (polymorphic)
‹0 = x›, ‹1 = x›, ‹nnn = x› provided ‹x› isn't ‹0›, ‹1› or a numeral.›
simproc_setup approx_reorient_simproc
("0 ≈ x" | "1 ≈ y" | "numeral w ≈ z" | "- 1 ≈ y" | "- numeral w ≈ r") =
‹
let val rule = @{thm approx_reorient} RS eq_reflection
fun proc phi ss ct =
case Thm.term_of ct of
_ \$ t \$ u => if can HOLogic.dest_number u then NONE
else if can HOLogic.dest_number t then SOME rule else NONE
| _ => NONE
in proc end
›

lemma Infinitesimal_approx_minus: "x - y ∈ Infinitesimal ⟷ x ≈ y"
by (simp add: approx_minus_iff [symmetric] mem_infmal_iff)

lemma approx_monad_iff: "x ≈ y ⟷ monad x = monad y"
by (auto simp add: monad_def dest: approx_sym elim!: approx_trans equalityCE)

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 simp
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])
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_diff: "a ≈ b ⟹ c ≈ d ⟹ a - c ≈ b - d"
using approx_add [of a b "- c" "- d"] by simp

lemma approx_mult1: "a ≈ b ⟹ c ∈ HFinite ⟹ a * c ≈ b * c"
for a b c :: "'a::real_normed_algebra star"
by (simp add: approx_def Infinitesimal_HFinite_mult left_diff_distrib [symmetric])

lemma approx_mult2: "a ≈ b ⟹ c ∈ HFinite ⟹ c * a ≈ c * b"
for a b c :: "'a::real_normed_algebra star"
by (simp add: approx_def Infinitesimal_HFinite_mult2 right_diff_distrib [symmetric])

lemma approx_mult_subst: "u ≈ v * x ⟹ x ≈ y ⟹ v ∈ HFinite ⟹ u ≈ v * y"
for u v x y :: "'a::real_normed_algebra star"
by (blast intro: approx_mult2 approx_trans)

lemma approx_mult_subst2: "u ≈ x * v ⟹ x ≈ y ⟹ v ∈ HFinite ⟹ u ≈ y * v"
for u v x y :: "'a::real_normed_algebra star"
by (blast intro: approx_mult1 approx_trans)

lemma approx_mult_subst_star_of: "u ≈ x * star_of v ⟹ x ≈ y ⟹ u ≈ y * star_of v"
for u x y :: "'a::real_normed_algebra star"
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])
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])
done

lemma Infinitesimal_add_approx_self2: "y ∈ Infinitesimal ⟹ x ≈ y + x"

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 (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] ac_simps)
done

lemma approx_add_right_cancel: "b + d ≈ c + d ⟹ b ≈ c"
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] ac_simps)
done

lemma approx_add_mono2: "b ≈ c ⟹ b + a ≈ c + a"

lemma approx_add_left_iff [simp]: "a + b ≈ a + c ⟷ b ≈ c"

lemma approx_add_right_iff [simp]: "b + a ≈ c + a ⟷ b ≈ c"

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]])
done

lemma approx_star_of_HFinite: "x ≈ star_of 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"
for a b c d :: "'a::real_normed_algebra star"
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 scaleHR_left_diff_distrib: "⋀a b x. scaleHR (a - b) x = scaleHR a x - scaleHR b x"
by transfer (rule scaleR_left_diff_distrib)

lemma approx_scaleR1: "a ≈ star_of b ⟹ c ∈ HFinite ⟹ scaleHR a c ≈ b *⇩R c"
apply (unfold approx_def)
apply (drule (1) Infinitesimal_HFinite_scaleHR)
apply (simp only: scaleHR_left_diff_distrib)
apply (simp add: scaleHR_def star_scaleR_def [symmetric])
done

lemma approx_scaleR2: "a ≈ b ⟹ c *⇩R a ≈ c *⇩R b"
by (simp add: approx_def Infinitesimal_scaleR2 scaleR_right_diff_distrib [symmetric])

lemma approx_scaleR_HFinite: "a ≈ star_of b ⟹ c ≈ d ⟹ d ∈ HFinite ⟹ scaleHR a c ≈ b *⇩R d"
apply (rule approx_trans)
apply (rule_tac [2] approx_scaleR2)
apply (rule approx_scaleR1)
prefer 2 apply (blast intro: approx_HFinite approx_sym, auto)
done

lemma approx_mult_star_of: "a ≈ star_of b ⟹ c ≈ star_of d ⟹ a * c ≈ star_of b * star_of d"
for a c :: "'a::real_normed_algebra star"
by (blast intro!: approx_mult_HFinite approx_star_of_HFinite HFinite_star_of)

lemma approx_SReal_mult_cancel_zero: "a ∈ ℝ ⟹ a ≠ 0 ⟹ a * x ≈ 0 ⟹ x ≈ 0"
for a x :: hypreal
apply (drule Reals_inverse [THEN SReal_subset_HFinite [THEN subsetD]])
apply (auto dest: approx_mult2 simp add: mult.assoc [symmetric])
done

lemma approx_mult_SReal1: "a ∈ ℝ ⟹ x ≈ 0 ⟹ x * a ≈ 0"
for a x :: hypreal
by (auto dest: SReal_subset_HFinite [THEN subsetD] approx_mult1)

lemma approx_mult_SReal2: "a ∈ ℝ ⟹ x ≈ 0 ⟹ a * x ≈ 0"
for a x :: hypreal
by (auto dest: SReal_subset_HFinite [THEN subsetD] approx_mult2)

lemma approx_mult_SReal_zero_cancel_iff [simp]: "a ∈ ℝ ⟹ a ≠ 0 ⟹ a * x ≈ 0 ⟷ x ≈ 0"
for a x :: hypreal
by (blast intro: approx_SReal_mult_cancel_zero approx_mult_SReal2)

lemma approx_SReal_mult_cancel: "a ∈ ℝ ⟹ a ≠ 0 ⟹ a * w ≈ a * z ⟹ w ≈ z"
for a w z :: hypreal
apply (drule Reals_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 ∈ ℝ ⟹ a ≠ 0 ⟹ a * w ≈ a * z ⟷ w ≈ z"
for a w z :: hypreal
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"
for z :: hypreal
apply (simp add: bex_Infinitesimal_iff2 [symmetric], auto)
apply (rule_tac x = "g + y - z" in bexI)
apply simp
apply (rule Infinitesimal_interval2)
apply (rule_tac [2] Infinitesimal_zero, auto)
done

lemma approx_hnorm: "x ≈ y ⟹ hnorm x ≈ hnorm y"
for x y :: "'a::real_normed_vector star"
proof (unfold approx_def)
assume "x - y ∈ Infinitesimal"
then have "hnorm (x - y) ∈ Infinitesimal"
by (simp only: Infinitesimal_hnorm_iff)
moreover have "(0::real star) ∈ Infinitesimal"
by (rule Infinitesimal_zero)
moreover have "0 ≤ ¦hnorm x - hnorm y¦"
by (rule abs_ge_zero)
moreover have "¦hnorm x - hnorm y¦ ≤ hnorm (x - y)"
by (rule hnorm_triangle_ineq3)
ultimately have "¦hnorm x - hnorm y¦ ∈ Infinitesimal"
by (rule Infinitesimal_interval2)
then show "hnorm x - hnorm y ∈ Infinitesimal"
by (simp only: Infinitesimal_hrabs_iff)
qed

subsection ‹Zero is the Only Infinitesimal that is also a Real›

lemma Infinitesimal_less_SReal: "x ∈ ℝ ⟹ y ∈ Infinitesimal ⟹ 0 < x ⟹ y < x"
for x y :: hypreal
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"
for y :: hypreal
by (blast intro: Infinitesimal_less_SReal)

lemma SReal_not_Infinitesimal: "0 < y ⟹ y ∈ ℝ ==> y ∉ Infinitesimal"
for y :: hypreal
apply (simp add: Infinitesimal_def)
apply (auto simp add: abs_if)
done

lemma SReal_minus_not_Infinitesimal: "y < 0 ⟹ y ∈ ℝ ⟹ y ∉ Infinitesimal"
for y :: hypreal
apply (subst Infinitesimal_minus_iff [symmetric])
apply (rule SReal_not_Infinitesimal, auto)
done

lemma SReal_Int_Infinitesimal_zero: "ℝ Int Infinitesimal = {0::hypreal}"
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 ∈ ℝ ⟹ x ∈ Infinitesimal ⟹ x = 0"
for x :: hypreal
using SReal_Int_Infinitesimal_zero by blast

lemma SReal_HFinite_diff_Infinitesimal: "x ∈ ℝ ⟹ x ≠ 0 ⟹ x ∈ HFinite - Infinitesimal"
for x :: hypreal
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 star_of_Infinitesimal_iff_0 [iff]: "star_of x ∈ Infinitesimal ⟷ x = 0"
apply (auto simp add: Infinitesimal_def)
apply (drule_tac x="hnorm (star_of x)" in bspec)
apply (simp add: SReal_def)
apply (rule_tac x="norm x" in exI, simp)
apply simp
done

lemma star_of_HFinite_diff_Infinitesimal: "x ≠ 0 ⟹ star_of x ∈ HFinite - Infinitesimal"
by simp

lemma numeral_not_Infinitesimal [simp]:
"numeral w ≠ (0::hypreal) ⟹ (numeral w :: hypreal) ∉ Infinitesimal"
by (fast dest: Reals_numeral [THEN SReal_Infinitesimal_zero])

text ‹Again: ‹1› is a special case, but not ‹0› this time.›
lemma one_not_Infinitesimal [simp]:
"(1::'a::{real_normed_vector,zero_neq_one} star) ∉ Infinitesimal"
apply (simp only: star_one_def star_of_Infinitesimal_iff_0)
apply simp
done

lemma approx_SReal_not_zero: "y ∈ ℝ ⟹ x ≈ y ⟹ y ≠ 0 ⟹ x ≠ 0"
for x y :: hypreal
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: mem_infmal_iff)
apply (drule approx_trans3, assumption)
apply (blast dest: approx_sym)
done

text ‹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"
for x y :: "'a::{real_normed_div_algebra,field} star"
apply (drule Infinitesimal_HFinite_mult2, assumption)
apply (simp add: divide_inverse mult.assoc)
done

lemma Infinitesimal_SReal_divide: "x ∈ Infinitesimal ⟹ y ∈ ℝ ⟹ x / y ∈ Infinitesimal"
for x y :: hypreal
apply (simp add: divide_inverse)
apply (auto intro!: Infinitesimal_HFinite_mult
dest!: Reals_inverse [THEN SReal_subset_HFinite [THEN subsetD]])
done

section ‹Standard Part Theorem›

text ‹
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 star_of_approx_iff [simp]: "star_of x ≈ star_of y ⟷ x = y"
apply safe
apply (simp add: approx_def)
apply (simp only: star_of_diff [symmetric])
apply (simp only: star_of_Infinitesimal_iff_0)
apply simp
done

lemma SReal_approx_iff: "x ∈ ℝ ⟹ y ∈ ℝ ⟹ x ≈ y ⟷ x = y"
for x y :: hypreal
apply auto
apply (simp add: approx_def)
apply (drule (1) Reals_diff)
apply (drule (1) SReal_Infinitesimal_zero)
apply simp
done

lemma numeral_approx_iff [simp]:
"(numeral v ≈ (numeral w :: 'a::{numeral,real_normed_vector} star)) =
(numeral v = (numeral w :: 'a))"
apply (unfold star_numeral_def)
apply (rule star_of_approx_iff)
done

text ‹And also for ‹0 ≈ #nn› and ‹1 ≈ #nn›, ‹#nn ≈ 0› and ‹#nn ≈ 1›.›
lemma [simp]:
"(numeral w ≈ (0::'a::{numeral,real_normed_vector} star)) = (numeral w = (0::'a))"
"((0::'a::{numeral,real_normed_vector} star) ≈ numeral w) = (numeral w = (0::'a))"
"(numeral w ≈ (1::'b::{numeral,one,real_normed_vector} star)) = (numeral w = (1::'b))"
"((1::'b::{numeral,one,real_normed_vector} star) ≈ numeral w) = (numeral w = (1::'b))"
"¬ (0 ≈ (1::'c::{zero_neq_one,real_normed_vector} star))"
"¬ (1 ≈ (0::'c::{zero_neq_one,real_normed_vector} star))"
apply (unfold star_numeral_def star_zero_def star_one_def)
apply (unfold star_of_approx_iff)
apply (auto intro: sym)
done

lemma star_of_approx_numeral_iff [simp]: "star_of k ≈ numeral w ⟷ k = numeral w"
by (subst star_of_approx_iff [symmetric]) auto

lemma star_of_approx_zero_iff [simp]: "star_of k ≈ 0 ⟷ k = 0"
by (simp_all add: star_of_approx_iff [symmetric])

lemma star_of_approx_one_iff [simp]: "star_of k ≈ 1 ⟷ k = 1"
by (simp_all add: star_of_approx_iff [symmetric])

lemma approx_unique_real: "r ∈ ℝ ⟹ s ∈ ℝ ⟹ r ≈ x ⟹ s ≈ x ⟹ r = s"
for r s :: hypreal
by (blast intro: SReal_approx_iff [THEN iffD1] approx_trans2)

subsection ‹Existence of Unique Real Infinitely Close›

subsubsection ‹Lifting of the Ub and Lub Properties›

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

lemma hypreal_of_real_isLub1: "isLub ℝ (hypreal_of_real ` Q) (hypreal_of_real Y) ⟹ isLub UNIV Q Y"
for Q :: "real set" and Y :: real
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 Q Y ⟹ isLub ℝ (hypreal_of_real ` Q) (hypreal_of_real Y)"
for Q :: "real set" and Y :: real
apply (auto simp add: isLub_def leastP_def hypreal_of_real_isUb_iff setge_def)
apply (metis SReal_iff hypreal_of_real_isUb_iff isUbD2a star_of_le)
done

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

lemma lemma_isUb_hypreal_of_real: "isUb ℝ P Y ⟹ ∃Yo. isUb ℝ P (hypreal_of_real Yo)"
by (auto simp add: SReal_iff isUb_def)

lemma lemma_isLub_hypreal_of_real: "isLub ℝ P Y ⟹ ∃Yo. isLub ℝ 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 ℝ P (hypreal_of_real Yo) ⟹ ∃Y. isLub ℝ P Y"
by (auto simp add: isLub_def leastP_def isUb_def)

lemma SReal_complete: "P ⊆ ℝ ⟹ ∃x. x ∈ P ⟹ ∃Y. isUb ℝ P Y ⟹ ∃t::hypreal. isLub ℝ 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

text ‹Lemmas about lubs.›

lemma lemma_st_part_ub: "x ∈ HFinite ⟹ ∃u. isUb ℝ {s. s ∈ ℝ ∧ s < x} u"
for x :: hypreal
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 ∈ ℝ ∧ s < x}"
for x :: hypreal
apply (drule HFiniteD, safe)
apply (drule Reals_minus)
apply (rule_tac x = "-t" in exI)
apply (auto simp add: abs_less_iff)
done

lemma lemma_st_part_lub: "x ∈ HFinite ⟹ ∃t. isLub ℝ {s. s ∈ ℝ ∧ s < x} t"
for x :: hypreal
by (blast intro!: SReal_complete lemma_st_part_ub lemma_st_part_nonempty Collect_restrict)

lemma lemma_st_part_le1:
"x ∈ HFinite ⟹ isLub ℝ {s. s ∈ ℝ ∧ s < x} t ⟹ r ∈ ℝ ⟹ 0 < r ⟹ x ≤ t + r"
for x r t :: hypreal
apply (frule isLubD1a)
apply (rule ccontr, drule linorder_not_le [THEN iffD2])
apply (drule (1) Reals_add)
apply (drule_tac y = "r + t" in isLubD1 [THEN setleD], auto)
done

lemma hypreal_setle_less_trans: "S *<= x ⟹ x < y ⟹ S *<= y"
for x y :: hypreal
apply (simp add: setle_def)
apply (auto dest!: bspec order_le_less_trans intro: order_less_imp_le)
done

lemma hypreal_gt_isUb: "isUb R S x ⟹ x < y ⟹ y ∈ R ⟹ isUb R S y"
for x y :: hypreal
apply (simp add: isUb_def)
apply (blast intro: hypreal_setle_less_trans)
done

lemma lemma_st_part_gt_ub: "x ∈ HFinite ⟹ x < y ⟹ y ∈ ℝ ⟹ isUb ℝ {s. s ∈ ℝ ∧ s < x} y"
for x y :: hypreal
by (auto dest: order_less_trans intro: order_less_imp_le intro!: isUbI setleI)

lemma lemma_minus_le_zero: "t ≤ t + -r ⟹ r ≤ 0"
for r t :: hypreal
apply (drule_tac c = "-t" in add_left_mono)
done

lemma lemma_st_part_le2:
"x ∈ HFinite ⟹ isLub ℝ {s. s ∈ ℝ ∧ s < x} t ⟹ r ∈ ℝ ⟹ 0 < r ⟹ t + -r ≤ x"
for x r t :: hypreal
apply (frule isLubD1a)
apply (rule ccontr, drule linorder_not_le [THEN iffD1])
apply (drule Reals_minus, drule_tac a = t in Reals_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 ℝ {s. s ∈ ℝ ∧ s < x} t ⟹ r ∈ ℝ ⟹ 0 < r ⟹ x + -t ≤ r"
for x r t :: hypreal
apply (subgoal_tac "x ≤ t + r")
apply (auto intro: lemma_st_part_le1)
done

lemma lemma_st_part2a:
"x ∈ HFinite ⟹ isLub ℝ {s. s ∈ ℝ ∧ s < x} t ⟹ r ∈ ℝ ⟹ 0 < r ⟹ - (x + -t) ≤ r"
for x r t :: hypreal
apply (subgoal_tac "(t + -r ≤ x)")
apply simp
apply (rule lemma_st_part_le2)
apply auto
done

lemma lemma_SReal_ub: "x ∈ ℝ ⟹ isUb ℝ {s. s ∈ ℝ ∧ s < x} x"
for x :: hypreal
by (auto intro: isUbI setleI order_less_imp_le)

lemma lemma_SReal_lub: "x ∈ ℝ ⟹ isLub ℝ {s. s ∈ ℝ ∧ s < x} x"
for x :: hypreal
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 ℝ {s. s ∈ ℝ ∧ s < x} t ⟹ r ∈ ℝ ⟹ 0 < r ⟹ x + - t ≠ r"
for x r t :: hypreal
apply auto
apply (frule isLubD1a [THEN Reals_minus])
using Reals_add_cancel [of x "- t"] apply simp
apply (drule_tac x = x in lemma_SReal_lub)
apply (drule isLub_unique, assumption, auto)
done

lemma lemma_st_part_not_eq2:
"x ∈ HFinite ⟹ isLub ℝ {s. s ∈ ℝ ∧ s < x} t ⟹ r ∈ ℝ ⟹ 0 < r ⟹ - (x + -t) ≠ r"
for x r t :: hypreal
apply (auto)
apply (frule isLubD1a)
using Reals_add_cancel [of "- x" t] apply simp
apply (drule_tac x = x in lemma_SReal_lub)
apply (drule isLub_unique, assumption, auto)
done

lemma lemma_st_part_major:
"x ∈ HFinite ⟹ isLub ℝ {s. s ∈ ℝ ∧ s < x} t ⟹ r ∈ ℝ ⟹ 0 < r ⟹ ¦x - t¦ < r"
for x r t :: hypreal
apply (frule lemma_st_part1a)
apply (frule_tac [4] lemma_st_part2a, auto)
apply (drule order_le_imp_less_or_eq)+
apply auto
using lemma_st_part_not_eq2 apply fastforce
using lemma_st_part_not_eq1 apply fastforce
done

lemma lemma_st_part_major2:
"x ∈ HFinite ⟹ isLub ℝ {s. s ∈ ℝ ∧ s < x} t ⟹ ∀r ∈ Reals. 0 < r ⟶ ¦x - t¦ < r"
for x t :: hypreal
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 ⟶ ¦x - t¦ < r"
for x :: hypreal
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"
for x :: hypreal
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 ⟹ ∃!t::hypreal. t ∈ ℝ ∧ 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
then show 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"
for x :: "'a::real_normed_div_algebra star"
apply (subgoal_tac "x ≠ 0")
apply (cut_tac x = "inverse x" in HInfinite_HFinite_disj)
apply (auto dest!: HInfinite_inverse_Infinitesimal simp: nonzero_inverse_inverse_eq)
done

lemma HFinite_inverse2: "x ∈ HFinite - Infinitesimal ⟹ inverse x ∈ HFinite"
for x :: "'a::real_normed_div_algebra star"
by (blast intro: HFinite_inverse)

text ‹Stronger statement possible in fact.›
lemma Infinitesimal_inverse_HFinite: "x ∉ Infinitesimal ⟹ inverse x ∈ HFinite"
for x :: "'a::real_normed_div_algebra star"
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"
for x :: "'a::real_normed_div_algebra star"
apply (auto intro: Infinitesimal_inverse_HFinite)
apply (drule Infinitesimal_HFinite_mult2, assumption)
apply (simp add: not_Infinitesimal_not_zero)
done

lemma approx_inverse: "x ≈ y ⟹ y ∈ HFinite - Infinitesimal ⟹ inverse x ≈ inverse y"
for x y :: "'a::real_normed_div_algebra star"
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 star_of_approx_inverse = star_of_HFinite_diff_Infinitesimal [THEN [2] approx_inverse]
lemmas hypreal_of_real_approx_inverse =  hypreal_of_real_HFinite_diff_Infinitesimal [THEN [2] approx_inverse]

"x ∈ HFinite - Infinitesimal ⟹ h ∈ Infinitesimal ⟹ inverse (x + h) ≈ inverse x"
for x h :: "'a::real_normed_div_algebra star"
by (auto intro: approx_inverse approx_sym Infinitesimal_add_approx_self)

"x ∈ HFinite - Infinitesimal ⟹ h ∈ Infinitesimal ⟹ inverse (h + x) ≈ inverse x"
for x h :: "'a::real_normed_div_algebra star"
apply (rule add.commute [THEN subst])
apply (blast intro: inverse_add_Infinitesimal_approx)
done

"x ∈ HFinite - Infinitesimal ⟹ h ∈ Infinitesimal ⟹ inverse (x + h) - inverse x ≈ h"
for x h :: "'a::real_normed_div_algebra star"
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"
for x :: "'a::real_normed_div_algebra star"
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"
for x :: "'a::real_normed_div_algebra star"
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"
for x :: "'a::real_normed_div_algebra star"
by (auto simp add: HInfinite_HFinite_iff)

lemma approx_HFinite_mult_cancel: "a ∈ HFinite - Infinitesimal ⟹ a * w ≈ a * z ⟹ w ≈ z"
for a w z :: "'a::real_normed_div_algebra star"
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"
for a w z :: "'a::real_normed_div_algebra star"
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)
done

lemma HInfinite_ge_HInfinite: "x ∈ HInfinite ⟹ x ≤ y ⟹ 0 ≤ x ⟹ y ∈ HInfinite"
for x y :: hypreal
by (auto intro: HFinite_bounded simp add: HInfinite_HFinite_iff)

lemma Infinitesimal_inverse_HInfinite: "x ∈ Infinitesimal ⟹ x ≠ 0 ⟹ inverse x ∈ HInfinite"
for x :: "'a::real_normed_div_algebra star"
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"
for x y :: "'a::real_normed_div_algebra star"
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"
for x y :: "'a::real_normed_div_algebra star"
apply (rule ccontr, drule HFinite_HInfinite_iff [THEN iffD2])
apply (frule HFinite_Infinitesimal_not_zero)
apply (drule HFinite_not_Infinitesimal_inverse)
apply (safe, drule_tac x="inverse y" in HFinite_mult)
apply assumption
apply (auto simp add: mult.assoc [symmetric] HFinite_HInfinite_iff)
done

lemma HInfinite_gt_SReal: "x ∈ HInfinite ⟹ 0 < x ⟹ y ∈ ℝ ⟹ y < x"
for x y :: hypreal
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"
for x :: hypreal
by (auto intro: HInfinite_gt_SReal)

lemma not_HInfinite_one [simp]: "1 ∉ HInfinite"
by (simp add: HInfinite_HFinite_iff)

lemma approx_hrabs_disj: "¦x¦ ≈ x ∨ ¦x¦ ≈ -x"
for x :: hypreal
using hrabs_disj [of x] by auto

for x :: hypreal
by (rule hrabs_disj [of x, 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)"

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"

lemma monad_zero_hrabs_iff: "x ∈ monad 0 ⟷ ¦x¦ ∈ monad 0"
for x :: hypreal
by (rule hrabs_disj [of x, THEN disjE]) (auto simp: monad_zero_minus_iff [symmetric])

lemma mem_monad_self [simp]: "x ∈ monad x"

subsection ‹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"
apply (drule approx_sym)
apply (fast dest: approx_subset_monad)
done

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"
apply (blast intro!: approx_sym)
done

lemma approx_mem_monad_zero: "x ≈ y ⟹ x ∈ monad 0 ⟹ y ∈ monad 0"
apply (fast intro: approx_mem_monad approx_trans)
done

lemma Infinitesimal_approx_hrabs: "x ≈ y ⟹ x ∈ Infinitesimal ⟹ ¦x¦ ≈ ¦y¦"
for x y :: hypreal
apply (drule Infinitesimal_monad_zero_iff [THEN iffD1])
apply (blast intro: approx_mem_monad_zero monad_zero_hrabs_iff [THEN iffD1]
done

lemma less_Infinitesimal_less: "0 < x ⟹ x ∉ Infinitesimal ⟹ e ∈ Infinitesimal ⟹ e < x"
for x :: hypreal
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"
for u x :: hypreal
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"
for u x :: hypreal
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"
for x y :: hypreal

lemma lemma_approx_less_zero: "x < 0 ⟹ x ∉ Infinitesimal ⟹ x ≈ y ⟹ y < 0"
for x y :: hypreal

lemma approx_hrabs: "x ≈ y ⟹ ¦x¦ ≈ ¦y¦"
for x y :: hypreal
by (drule approx_hnorm) simp

lemma approx_hrabs_zero_cancel: "¦x¦ ≈ 0 ⟹ x ≈ 0"
for x :: hypreal
using hrabs_disj [of x] by (auto dest: approx_minus)

lemma approx_hrabs_add_Infinitesimal: "e ∈ Infinitesimal ⟹ ¦x¦ ≈ ¦x + e¦"
for e x :: hypreal
by (fast intro: approx_hrabs Infinitesimal_add_approx_self)

lemma approx_hrabs_add_minus_Infinitesimal: "e ∈ Infinitesimal ==> ¦x¦ ≈ ¦x + -e¦"
for e x :: hypreal
by (fast intro: approx_hrabs Infinitesimal_add_minus_approx_self)

"e ∈ Infinitesimal ⟹ e' ∈ Infinitesimal ⟹ ¦x + e¦ = ¦y + e'¦ ⟹ ¦x¦ ≈ ¦y¦"
for e e' x y :: hypreal
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

"e ∈ Infinitesimal ⟹ e' ∈ Infinitesimal ⟹ ¦x + -e¦ = ¦y + -e'¦ ⟹ ¦x¦ ≈ ¦y¦"
for e e' x y :: hypreal
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

subsection ‹More @{term HFinite} and @{term Infinitesimal} Theorems›

text ‹
Interesting slightly counterintuitive theorem: necessary
for proving that an open interval is an NS open set.
›
"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

"x ∈ Infinitesimal ⟹ ¦hypreal_of_real r¦ < hypreal_of_real y ⟹
¦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: star_of_abs [symmetric])
done

"x ∈ Infinitesimal ⟹ ¦hypreal_of_real r¦ < hypreal_of_real y ⟹
¦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

"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

"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 ⟹ star_of x + h ≠ 0"
apply auto
apply (subgoal_tac "h = - star_of x")
apply (auto intro: minus_unique [symmetric])
done

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

lemma HFinite_square_cancel [simp]: "x * x + y * y ∈ HFinite ⟹ x * x ∈ HFinite"
for x y :: hypreal
apply (rule HFinite_bounded, assumption)
apply auto
done

lemma Infinitesimal_square_cancel2 [simp]: "x * x + y * y ∈ Infinitesimal ⟹ y * y ∈ Infinitesimal"
for x y :: hypreal
apply (rule Infinitesimal_square_cancel)
apply (rule add.commute [THEN subst])
apply simp
done

lemma HFinite_square_cancel2 [simp]: "x * x + y * y ∈ HFinite ⟹ y * y ∈ HFinite"
for x y :: hypreal
apply (rule HFinite_square_cancel)
apply (rule add.commute [THEN subst])
apply simp
done

lemma Infinitesimal_sum_square_cancel [simp]:
"x * x + y * y + z * z ∈ Infinitesimal ⟹ x * x ∈ Infinitesimal"
for x y z :: hypreal
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 del:zero_le_square)
done

lemma HFinite_sum_square_cancel [simp]: "x * x + y * y + z * z ∈ HFinite ⟹ x * x ∈ HFinite"
for x y z :: hypreal
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 del:zero_le_square)
done

lemma Infinitesimal_sum_square_cancel2 [simp]:
"y * y + x * x + z * z ∈ Infinitesimal ⟹ x * x ∈ Infinitesimal"
for x y z :: hypreal
apply (rule Infinitesimal_sum_square_cancel)
apply (simp add: ac_simps)
done

lemma HFinite_sum_square_cancel2 [simp]: "y * y + x * x + z * z ∈ HFinite ⟹ x * x ∈ HFinite"
for x y z :: hypreal
apply (rule HFinite_sum_square_cancel)
apply (simp add: ac_simps)
done

lemma Infinitesimal_sum_square_cancel3 [simp]:
"z * z + y * y + x * x ∈ Infinitesimal ⟹ x * x ∈ Infinitesimal"
for x y z :: hypreal
apply (rule Infinitesimal_sum_square_cancel)
apply (simp add: ac_simps)
done

lemma HFinite_sum_square_cancel3 [simp]: "z * z + y * y + x * x ∈ HFinite ⟹ x * x ∈ HFinite"
for x y z :: hypreal
apply (rule HFinite_sum_square_cancel)
apply (simp add: ac_simps)
done

lemma monad_hrabs_less: "y ∈ monad x ⟹ 0 < hypreal_of_real e ⟹ ¦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 ∈ ℝ"
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_unique: "r ∈ ℝ ⟹ r ≈ x ⟹ st x = r"
apply (frule SReal_subset_HFinite [THEN subsetD])
apply (drule (1) approx_HFinite)
apply (unfold st_def)
apply (rule some_equality)
apply (auto intro: approx_unique_real)
done

lemma st_SReal_eq: "x ∈ ℝ ⟹ st x = x"
by (metis approx_refl st_unique)

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: "x ∈ HFinite" and y: "y ∈ HFinite" and xy: "x ≈ y"
shows "st x = st y"
proof -
have "st x ≈ x" "st y ≈ y" "st x ∈ ℝ" "st y ∈ ℝ"
by (simp_all add: st_approx_self st_SReal x y)
with xy 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 ∈ ℝ ⟹ e ∈ Infinitesimal ⟹ st (x + e) = x"
apply (erule st_unique)
done

lemma st_Infinitesimal_add_SReal2: "x ∈ ℝ ⟹ e ∈ Infinitesimal ⟹ st (e + x) = x"
apply (erule st_unique)
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: "x ∈ HFinite ⟹ y ∈ HFinite ⟹ st (x + y) = st x + st y"
by (simp add: st_unique st_SReal st_approx_self approx_add)

lemma st_numeral [simp]: "st (numeral w) = numeral w"
by (rule Reals_numeral [THEN st_SReal_eq])

lemma st_neg_numeral [simp]: "st (- numeral w) = - numeral w"
proof -
from Reals_numeral have "numeral w ∈ ℝ" .
then have "- numeral w ∈ ℝ" by simp
with st_SReal_eq show ?thesis .
qed

lemma st_0 [simp]: "st 0 = 0"
by (simp add: st_SReal_eq)

lemma st_1 [simp]: "st 1 = 1"
by (simp add: st_SReal_eq)

lemma st_neg_1 [simp]: "st (- 1) = - 1"
by (simp add: st_SReal_eq)

lemma st_minus: "x ∈ HFinite ⟹ st (- x) = - st x"
by (simp add: st_unique st_SReal st_approx_self approx_minus)

lemma st_diff: "⟦x ∈ HFinite; y ∈ HFinite⟧ ⟹ st (x - y) = st x - st y"
by (simp add: st_unique st_SReal st_approx_self approx_diff)

lemma st_mult: "⟦x ∈ HFinite; y ∈ HFinite⟧ ⟹ st (x * y) = st x * st y"
by (simp add: st_unique st_SReal st_approx_self approx_mult_HFinite)

lemma st_Infinitesimal: "x ∈ Infinitesimal ⟹ st x = 0"
by (simp add: st_unique mem_infmal_iff)

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 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)

"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

"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"
by (metis approx_le_bound approx_sym linear st_SReal st_approx_self st_part_Ex1)

lemma st_zero_le: "0 ≤ x ⟹ x ∈ HFinite ⟹ 0 ≤ st x"
apply (subst st_0 [symmetric])
apply (rule st_le, auto)
done

lemma st_zero_ge: "x ≤ 0 ⟹ x ∈ HFinite ⟹ st x ≤ 0"
apply (subst st_0 [symmetric])
apply (rule st_le, auto)
done

lemma st_hrabs: "x ∈ HFinite ⟹ ¦st x¦ = st ¦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 using Free Ultrafilter›

subsubsection ‹@{term HFinite}›

lemma HFinite_FreeUltrafilterNat:
"star_n X ∈ HFinite ⟹ ∃u. eventually (λn. norm (X n) < u) 𝒰"
apply (auto simp add: HFinite_def SReal_def)
apply (rule_tac x=r in exI)
apply (simp add: hnorm_def star_of_def starfun_star_n)
apply (simp add: star_less_def starP2_star_n)
done

lemma FreeUltrafilterNat_HFinite:
"∃u. eventually (λn. norm (X n) < u) 𝒰 ⟹ star_n X ∈ HFinite"
apply (auto simp add: HFinite_def mem_Rep_star_iff)
apply (rule_tac x="star_of u" in bexI)
apply (simp add: hnorm_def starfun_star_n star_of_def)
apply (simp add: star_less_def starP2_star_n)
apply (simp add: SReal_def)
done

lemma HFinite_FreeUltrafilterNat_iff:
"star_n X ∈ HFinite ⟷ (∃u. eventually (λn. norm (X n) < u) 𝒰)"
by (blast intro!: HFinite_FreeUltrafilterNat FreeUltrafilterNat_HFinite)

subsubsection ‹@{term HInfinite}›

lemma lemma_Compl_eq: "- {n. u < norm (f n)} = {n. norm (f n) ≤ u}"
by auto

lemma lemma_Compl_eq2: "- {n. norm (f n) < u} = {n. u ≤ norm (f n)}"
by auto

lemma lemma_Int_eq1: "{n. norm (f n) ≤ u} Int {n. u ≤ norm (f n)} = {n. norm(f n) = u}"
by auto

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

text ‹Exclude this type of sets from free ultrafilter for Infinite numbers!›
lemma FreeUltrafilterNat_const_Finite:
"eventually (λn. norm (X n) = u) 𝒰 ⟹ star_n X ∈ HFinite"
apply (rule FreeUltrafilterNat_HFinite)
apply (rule_tac x = "u + 1" in exI)
apply (auto elim: eventually_mono)
done

lemma HInfinite_FreeUltrafilterNat:
"star_n X ∈ HInfinite ⟹ ∀u. eventually (λn. u < norm (X n)) 𝒰"
apply (drule HInfinite_HFinite_iff [THEN iffD1])
apply (simp add: HFinite_FreeUltrafilterNat_iff)
apply (rule allI, drule_tac x="u + 1" in spec)
apply (simp add: FreeUltrafilterNat.eventually_not_iff[symmetric])
apply (auto elim: eventually_mono)
done

lemma lemma_Int_HI: "{n. norm (Xa n) < u} ∩ {n. X n = Xa n} ⊆ {n. norm (X n) < u}"
for u :: real
by auto

lemma lemma_Int_HIa: "{n. u < norm (X n)} ∩ {n. norm (X n) < u} = {}"
by (auto intro: order_less_asym)

lemma FreeUltrafilterNat_HInfinite:
"∀u. eventually (λn. u < norm (X n)) 𝒰 ⟹ star_n X ∈ HInfinite"
apply (rule HInfinite_HFinite_iff [THEN iffD2])
apply (safe, drule HFinite_FreeUltrafilterNat, safe)
apply (drule_tac x = u in spec)
proof -
fix u
assume "∀⇩Fn in 𝒰. norm (X n) < u" "∀⇩Fn in 𝒰. u < norm (X n)"
then have "∀⇩F x in 𝒰. False"
by eventually_elim auto
then show False
by (simp add: eventually_False FreeUltrafilterNat.proper)
qed

lemma HInfinite_FreeUltrafilterNat_iff:
"star_n X ∈ HInfinite ⟷ (∀u. eventually (λn. u < norm (X n)) 𝒰)"
by (blast intro!: HInfinite_FreeUltrafilterNat FreeUltrafilterNat_HInfinite)

subsubsection ‹@{term Infinitesimal}›

lemma ball_SReal_eq: "(∀x::hypreal ∈ Reals. P x) ⟷ (∀x::real. P (star_of x))"
by (auto simp: SReal_def)

lemma Infinitesimal_FreeUltrafilterNat:
"star_n X ∈ Infinitesimal ⟹ ∀u>0. eventually (λn. norm (X n) < u) 𝒰"
apply (simp add: Infinitesimal_def ball_SReal_eq)
apply (simp add: hnorm_def starfun_star_n star_of_def)
apply (simp add: star_less_def starP2_star_n)
done

lemma FreeUltrafilterNat_Infinitesimal:
"∀u>0. eventually (λn. norm (X n) < u) 𝒰 ⟹ star_n X ∈ Infinitesimal"
apply (simp add: Infinitesimal_def ball_SReal_eq)
apply (simp add: hnorm_def starfun_star_n star_of_def)
apply (simp add: star_less_def starP2_star_n)
done

lemma Infinitesimal_FreeUltrafilterNat_iff:
"(star_n X ∈ Infinitesimal) = (∀u>0. eventually (λn. norm (X n) < u) 𝒰)"
by (blast intro!: Infinitesimal_FreeUltrafilterNat FreeUltrafilterNat_Infinitesimal)

text ‹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 del: of_nat_Suc)
apply (blast dest!: reals_Archimedean intro: order_less_trans)
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_all
using less_imp_of_nat_less apply fastforce
apply (auto dest!: reals_Archimedean simp add: SReal_iff simp del: of_nat_Suc)
apply (drule star_of_less [THEN iffD2])
apply simp
apply (blast intro: order_less_trans)
done

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

subsection ‹Proof that ‹ω› is an infinite number›

text ‹It will follow that ‹ε› 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)

text ‹Prove that any segment is finite and hence cannot belong to ‹𝒰›.›

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

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} ∪ {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. ¦real n¦ ≤ u}"
by (simp add: finite_real_of_nat_le_real)

lemma rabs_real_of_nat_le_real_FreeUltrafilterNat:
"¬ eventually (λn. ¦real n¦ ≤ u) 𝒰"
by (blast intro!: FreeUltrafilterNat.finite finite_rabs_real_of_nat_le_real)

lemma FreeUltrafilterNat_nat_gt_real: "eventually (λn. u < real n) 𝒰"
apply (rule FreeUltrafilterNat.finite')
apply (subgoal_tac "{n::nat. ¬ u < real n} = {n. real n ≤ u}")
apply (auto simp add: finite_real_of_nat_le_real)
done

text ‹The complement of ‹{n. ¦real n¦ ≤ u} = {n. u < ¦real n¦}› is in
‹𝒰› 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 ω} is a member of @{term HInfinite}.›
theorem HInfinite_omega [simp]: "ω ∈ HInfinite"
apply (simp add: omega_def)
apply (rule FreeUltrafilterNat_HInfinite)
apply clarify
apply (rule_tac u1 = "u-1" in eventually_mono [OF FreeUltrafilterNat_nat_gt_real])
apply auto
done

text ‹Epsilon is a member of Infinitesimal.›

lemma Infinitesimal_epsilon [simp]: "ε ∈ Infinitesimal"
by (auto intro!: HInfinite_inverse_Infinitesimal HInfinite_omega

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

lemma epsilon_approx_zero [simp]: "ε ≈ 0"
by (simp add: mem_infmal_iff [symmetric])

text ‹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
apply (simp add: mult.commute)
done

lemma finite_inverse_real_of_posnat_gt_real: "0 < u ⟹ finite {n. u < inverse (real (Suc n))}"
proof (simp only: real_of_nat_less_inverse_iff)
have "{n. 1 + real n < inverse u} = {n. real n < inverse u - 1}"
by fastforce
then show "finite {n. real (Suc n) < inverse u}"
using finite_real_of_nat_less_real [of "inverse u - 1"]
by auto
qed

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

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_epsilon_set finite_inverse_real_of_posnat_gt_real
simp del: of_nat_Suc)

lemma inverse_real_of_posnat_ge_real_FreeUltrafilterNat:
"0 < u ⟹ ¬ eventually (λn. u ≤ inverse(real(Suc n))) 𝒰"
by (blast intro!: FreeUltrafilterNat.finite finite_inverse_real_of_posnat_ge_real)

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

lemma FreeUltrafilterNat_inverse_real_of_posnat:
"0 < u ⟹ eventually (λn. inverse(real(Suc n)) < u) 𝒰"
by (drule inverse_real_of_posnat_ge_real_FreeUltrafilterNat)
(simp add: FreeUltrafilterNat.eventually_not_iff not_le[symmetric])

text ‹Example of an hypersequence (i.e. an extended standard sequence)
whose term with an hypernatural suffix is an infinitesimal i.e.
the whn'nth term of the hypersequence is a member of Infinitesimal›

lemma SEQ_Infinitesimal: "( *f* (λn::nat. inverse(real(Suc n)))) whn ∈ Infinitesimal"
by (simp add: hypnat_omega_def starfun_star_n star_n_inverse Infinitesimal_FreeUltrafilterNat_iff
FreeUltrafilterNat_inverse_real_of_posnat del: of_nat_Suc)

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.›

text ‹‹|X(n) - x| < 1/n ⟹ [<X n>] - hypreal_of_real x| ∈ Infinitesimal››
lemma real_seq_to_hypreal_Infinitesimal:
"∀n. norm (X n - x) < inverse (real (Suc n)) ⟹ star_n X - star_of x ∈ Infinitesimal"
unfolding star_n_diff star_of_def Infinitesimal_FreeUltrafilterNat_iff star_n_inverse
by (auto dest!: FreeUltrafilterNat_inverse_real_of_posnat
intro: order_less_trans elim!: eventually_mono)

lemma real_seq_to_hypreal_approx:
"∀n. norm (X n - x) < inverse (real (Suc n)) ⟹ star_n X ≈ star_of x"
by (metis bex_Infinitesimal_iff real_seq_to_hypreal_Infinitesimal)

lemma real_seq_to_hypreal_approx2:
"∀n. norm (x - X n) < inverse(real(Suc n)) ⟹ star_n X ≈ star_of x"
by (metis norm_minus_commute real_seq_to_hypreal_approx)

lemma real_seq_to_hypreal_Infinitesimal2:
"∀n. norm(X n - Y n) < inverse(real(Suc n)) ⟹ star_n X - star_n Y ∈ Infinitesimal"
unfolding Infinitesimal_FreeUltrafilterNat_iff star_n_diff
by (auto dest!: FreeUltrafilterNat_inverse_real_of_posnat
intro: order_less_trans elim!: eventually_mono)

end
```