Quelle  NSA.thy

(*  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)

lemma hnorm_add_less:
  "∧(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_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: "ψ ∉ 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"
  by (simp add: Reals_eq_Standard Standard_def inj_star_of)

lemma SReal_dense: "x ∈ ℝ ==> y ∈ ℝ ==> x < y ==> ∃r ∈ Reals. x < r ∧ r < y"
  for x y :: hypreal
  using dense by (fastforce simp add: SReal_def)


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"
  by (simp add: HFinite_def) (metis SReal_hypreal_of_real gt_ex star_of_less star_of_norm)

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 intro: HFinite_mult)

lemma HFinite_bounded: 
  fixes x y :: hypreal
  assumes "x ∈ HFinite" and y: "y ≤ x" "0 ≤ y" shows "y ∈ HFinite"
proof (cases "x ≤ 0")
  case True
  then have "y = 0"
    using y by auto
  then show ?thesis
    by simp
next
  case False
  then show ?thesis
    using assms le_less_trans by (auto simp: HFinite_def)
qed


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 Infinitesimal_add:
  assumes "x ∈ Infinitesimal" "y ∈ Infinitesimal"
  shows "x + y ∈ Infinitesimal"
proof (rule InfinitesimalI)
  show "hnorm (x + y) < r"
    if "r ∈ ℝ" and "0 < r" for r :: "real star"
  proof -
    have "hnorm x < r/2" "hnorm y < r/2"
      using InfinitesimalD SReal_divide_numeral assms half_gt_zero that by blast+
    then show ?thesis
      using hnorm_add_less by fastforce
  qed
qed

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: 
  fixes x y :: "'a::real_normed_algebra star"
  assumes "x ∈ Infinitesimal" "y ∈ Infinitesimal"
  shows "x * y ∈ Infinitesimal"
  proof (rule InfinitesimalI)
  show "hnorm (x * y) < r"
    if "r ∈ ℝ" and "0 < r" for r :: "real star"
  proof -
    have "hnorm x < 1" "hnorm y < r"
      using assms that  by (auto simp add: InfinitesimalD)
    then show ?thesis
      using hnorm_mult_less by fastforce
  qed
qed

lemma Infinitesimal_HFinite_mult:
  fixes x y :: "'a::real_normed_algebra star"
  assumes "x ∈ Infinitesimal" "y ∈ HFinite"
  shows "x * y ∈ Infinitesimal"
proof (rule InfinitesimalI)
  obtain t where "hnorm y < t" "t ∈ Reals"
    using HFiniteD ‹y ∈ HFinite› by blast
  then have "t > 0"
    using hnorm_ge_zero le_less_trans by blast
  show "hnorm (x * y) < r"
    if "r ∈ ℝ" and "0 < r" for r :: "real star"
  proof -
    have "hnorm x < r/t"
      by (meson InfinitesimalD Reals_divide ‹hnorm y 🚫› ‹t ∈ ℝ› assms(1) divide_pos_pos hnorm_ge_zero le_less_trans that) 
    then have "hnorm (x * y) < (r / t) * t"
      using ‹hnorm y 🚫› hnorm_mult_less by blast
    then show ?thesis
      using ‹0 🚫› by auto
  qed
qed

lemma Infinitesimal_HFinite_scaleHR:
  assumes "x ∈ Infinitesimal" "y ∈ HFinite"
  shows "scaleHR x y ∈ Infinitesimal"
proof (rule InfinitesimalI)
  obtain t where "hnorm y < t" "t ∈ Reals"
    using HFiniteD ‹y ∈ HFinite› by blast
  then have "t > 0"
    using hnorm_ge_zero le_less_trans by blast
  show "hnorm (scaleHR x y) < r"
    if "r ∈ ℝ" and "0 < r" for r :: "real star"
  proof -
    have "∣x∣ * hnorm y < (r / t) * t"
      by (metis InfinitesimalD Reals_divide ‹0 🚫› ‹hnorm y 🚫› ‹t ∈ ℝ› assms(1) divide_pos_pos hnorm_ge_zero hypreal_hnorm_def mult_strict_mono' that)
    then show ?thesis
      by (simp add: ‹0 🚫› hnorm_scaleHR less_imp_not_eq2)
  qed
qed

lemma Infinitesimal_HFinite_mult2:
  fixes x y :: "'a::real_normed_algebra star"
  assumes "x ∈ Infinitesimal" "y ∈ HFinite"
  shows  "y * x ∈ Infinitesimal"
proof (rule InfinitesimalI)
  obtain t where "hnorm y < t" "t ∈ Reals"
    using HFiniteD ‹y ∈ HFinite› by blast
  then have "t > 0"
    using hnorm_ge_zero le_less_trans by blast
  show "hnorm (y * x) < r"
    if "r ∈ ℝ" and "0 < r" for r :: "real star"
  proof -
    have "hnorm x < r/t"
      by (meson InfinitesimalD Reals_divide ‹hnorm y 🚫› ‹t ∈ ℝ› assms(1) divide_pos_pos hnorm_ge_zero le_less_trans that) 
    then have "hnorm (y * x) < t * (r / t)"
      using ‹hnorm y 🚫› hnorm_mult_less by blast
    then show ?thesis
      using ‹0 🚫› by auto
  qed
qed

lemma Infinitesimal_scaleR2: 
  assumes "x ∈ Infinitesimal" shows "a *🪙R x ∈ Infinitesimal"
    by (metis HFinite_star_of Infinitesimal_HFinite_mult2 Infinitesimal_hnorm_iff assms hnorm_scaleR hypreal_hnorm_def star_of_norm)

lemma Compl_HFinite: "- HFinite = HInfinite"
  proof -
  have "r < hnorm x" if *: "∧s. s ∈ ℝ ==> s ≤ hnorm x" and "r ∈ ℝ"
    for x :: "'a star" and r :: hypreal
    using * [of "r+1"] ‹r ∈ ℝ› by auto
  then show ?thesis
    by (auto simp add: HInfinite_def HFinite_def linorder_not_less)
qed

lemma HInfinite_inverse_Infinitesimal:
  "x ∈ HInfinite ==> inverse x ∈ Infinitesimal"
  for x :: "'a::real_normed_div_algebra star"
  by (simp add: HInfinite_def InfinitesimalI hnorm_inverse inverse_less_imp_less)

lemma inverse_Infinitesimal_iff_HInfinite:
  "x ≠ 0 ==> inverse x ∈ Infinitesimal ⟷ x ∈ HInfinite"
  for x :: "'a::real_normed_div_algebra star"
  by (metis Compl_HFinite Compl_iff HInfinite_inverse_Infinitesimal InfinitesimalD Infinitesimal_HFinite_mult Reals_1 hnorm_one left_inverse less_irrefl zero_less_one)

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: 
  fixes x y :: "'a::real_normed_div_algebra star"
  assumes "x ∈ HInfinite" "y ∈ HInfinite" shows "x * y ∈ HInfinite"
proof (rule HInfiniteI, simp only: hnorm_mult)
  have "x ≠ 0"
    using Compl_HFinite HFinite_0 assms by blast
  show "r < hnorm x * hnorm y"
    if "r ∈ ℝ" for r :: "real star"
  proof -
    have "r = r * 1"
      by simp
    also have "… < hnorm x * hnorm y"
      by (meson HInfiniteD Reals_1 ‹x ≠ 0› assms le_numeral_extra(1) mult_strict_mono that zero_less_hnorm_iff)
    finally show ?thesis .
  qed
qed

lemma hypreal_add_zero_less_le_mono: "r < x ==> 0 ≤ y ==> r < x + y"
  for r x y :: hypreal
  by simp

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
  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"
  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
  by (metis (no_types, lifting) HInfinite_add_ge_zero2 HInfinite_minus_iff add.inverse_distrib_swap neg_0_le_iff_le)

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 not_Infinitesimal_not_zero: "x ∉ 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 (clarsimp simp: Infinitesimal_def)
  by (metis Reals_1 abs_ge_zero hyperpow_Suc_le_self2 hyperpow_hrabs hypnat_gt_zero_iff2 zero_less_one)

lemma Infinitesimal_hyperpow: "x ∈ Infinitesimal ==> 0 < N ==> x pow N ∈ Infinitesimal"
  for x :: hypreal
  using hrabs_le_Infinitesimal lemma_Infinitesimal_hyperpow by blast

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"
  by (metis (no_types, lifting) inverse_Infinitesimal_iff_HInfinite ComplI Compl_HFinite Infinitesimal_HFinite_mult divide_inverse eq_divide_imp inverse_inverse_eq mult_zero_right)

lemma Infinitesimal_mult_disj: "x * y ∈ Infinitesimal ==> x ∈ Infinitesimal ∨ y ∈ Infinitesimal"
  for x y :: "'a::real_normed_div_algebra star"
  using not_Infinitesimal_mult by blast

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"
  by (simp add: HFinite_mult not_Infinitesimal_mult)

lemma Infinitesimal_subset_HFinite: "Infinitesimal ⊆ HFinite"
  using HFinite_def InfinitesimalD Reals_1 zero_less_one by blast

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"
  by (simp add: approx_def add.commute)

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

lemma approx_sym: "x ≈ y ==> y ≈ x"
  by (metis Infinitesimal_minus_iff approx_def minus_diff_eq)

lemma approx_trans:
  assumes "x ≈ y" "y ≈ z" shows "x ≈ z"
proof -
  have "x - y ∈ Infinitesimal" "z - y ∈ Infinitesimal"
    using assms approx_def approx_sym by auto
  then have "x - z ∈ Infinitesimal"
    using Infinitesimal_diff by force
  then show ?thesis
    by (simp add: approx_def)
qed

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 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 K (K 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"
  apply (simp add: monad_def set_eq_iff)
  using approx_reorient approx_trans by blast

lemma Infinitesimal_approx: "x ∈ Infinitesimal ==> y ∈ Infinitesimal ==> x ≈ y"
  by (simp add: Infinitesimal_diff approx_def)

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"
  by (metis approx_def approx_sym minus_diff_eq minus_diff_minus)

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"
  using approx_sym bex_Infinitesimal_iff2 by blast

lemma Infinitesimal_add_approx_self: "y ∈ Infinitesimal ==> x ≈ x + y"
  by (simp add: Infinitesimal_add_approx)

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"
  using Infinitesimal_add_approx approx_trans by blast

lemma Infinitesimal_add_right_cancel: "y ∈ Infinitesimal ==> x ≈ z + y ==> x ≈ z"
  by (metis Infinitesimal_add_approx_self approx_monad_iff)

lemma approx_add_left_cancel: "d + b ≈ d + c ==> b ≈ c"
  by (metis add_diff_cancel_left bex_Infinitesimal_iff)

lemma approx_add_right_cancel: "b + d ≈ c + d ==> b ≈ c"
  by (simp add: approx_def)

lemma approx_add_mono1: "b ≈ c ==> d + b ≈ d + c"
  by (simp add: approx_add)

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"
  by (metis HFinite_add Infinitesimal_subset_HFinite approx_sym subsetD bex_Infinitesimal_iff2)

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"
  by (meson approx_HFinite approx_mult2 approx_mult_subst2 approx_sym)

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"
  unfolding approx_def
  by (metis Infinitesimal_HFinite_scaleHR scaleHR_def scaleHR_left_diff_distrib star_scaleR_def starfun2_star_of)

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"
  by (meson approx_HFinite approx_scaleR1 approx_scaleR2 approx_sym approx_trans)

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:
  fixes a x :: hypreal
  assumes "a ∈ ℝ" "a ≠ 0" "a * x ≈ 0" shows "x ≈ 0"
proof -
  have "inverse a ∈ HFinite"
    using Reals_inverse SReal_subset_HFinite assms(1) by blast
  then show ?thesis
    using assms by (auto dest: approx_mult2 simp add: mult.assoc [symmetric])
qed

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:
  fixes a w z :: hypreal
  assumes "a ∈ ℝ" "a ≠ 0" "a * w ≈ a * z" shows "w ≈ z"
proof -
  have "inverse a ∈ HFinite"
    using Reals_inverse SReal_subset_HFinite assms(1) by blast
  then show ?thesis
    using assms by (auto dest: approx_mult2 simp add: mult.assoc [symmetric])
qed

lemma approx_SReal_mult_cancel_iff1 [simp]: "a ∈ ℝ ==> a ≠ 0 ==> a * w ≈ a * z ⟷ w ≈ z"
  for a w z :: hypreal
  by (meson SReal_subset_HFinite approx_SReal_mult_cancel approx_mult2 subsetD)

lemma approx_le_bound:
  fixes z :: hypreal
  assumes "z ≤ f" " f ≈ g" "g ≤ z" shows "f ≈ z"
proof -
  obtain y where "z ≤ g + y" and "y ∈ Infinitesimal" "f = g + y"
    using assms bex_Infinitesimal_iff2 by auto
  then have "z - g ∈ Infinitesimal"
    using assms(3) hrabs_le_Infinitesimal by auto 
  then show ?thesis
    by (metis approx_def approx_trans2 assms(2))
qed

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
  using InfinitesimalD by fastforce

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
  by (auto simp add: Infinitesimal_def abs_if)

lemma SReal_minus_not_Infinitesimal: "y < 0 ==> y ∈ ℝ ==> y ∉ Infinitesimal"
  for y :: hypreal
  using Infinitesimal_minus_iff Reals_minus SReal_not_Infinitesimal neg_0_less_iff_less by blast

lemma SReal_Int_Infinitesimal_zero: "ℝ Int Infinitesimal = {0::hypreal}"
  proof -
  have "x = 0" if "x ∈ ℝ" "x ∈ Infinitesimal" for x :: "real star"
    using that SReal_minus_not_Infinitesimal SReal_not_Infinitesimal not_less_iff_gr_or_eq by blast
  then show ?thesis
    by auto
qed

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"
proof
  show "x = 0" if "star_of x ∈ Infinitesimal"
  proof -
    have "hnorm (star_n (λn. x)) ∈ Standard"
      by (metis Reals_eq_Standard SReal_iff star_of_def star_of_norm)
    then show ?thesis
      by (metis InfinitesimalD2 less_irrefl star_of_norm that zero_less_norm_iff)
  qed
qed auto

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"
  by (metis star_of_Infinitesimal_iff_0 star_one_def zero_neq_one)

lemma approx_SReal_not_zero: "y ∈ ℝ ==> x ≈ y ==> y ≠ 0 ==> x ≠ 0"
  for x y :: hypreal
  using SReal_Infinitesimal_zero approx_sym mem_infmal_iff by auto

lemma HFinite_diff_Infinitesimal_approx:
  "x ≈ y ==> y ∈ HFinite - Infinitesimal ==> x ∈ HFinite - Infinitesimal"
  by (meson Diff_iff approx_HFinite approx_sym approx_trans3 mem_infmal_iff)

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"
  using Infinitesimal_HFinite_mult by fastforce

lemma Infinitesimal_SReal_divide: "x ∈ Infinitesimal ==> y ∈ ℝ ==> x / y ∈ Infinitesimal"
  for x y :: hypreal
  by (metis HFinite_star_of Infinitesimal_HFinite_mult Reals_inverse SReal_iff divide_inverse)


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"
  by (metis approx_def right_minus_eq star_of_Infinitesimal_iff_0 star_of_simps(2))

lemma SReal_approx_iff: "x ∈ ℝ ==> y ∈ ℝ ==> x ≈ y ⟷ x = y"
  for x y :: hypreal
  by (meson Reals_diff SReal_Infinitesimal_zero approx_def approx_refl right_minus_eq)

lemma numeral_approx_iff [simp]:
  "(numeral v ≈ (numeral w :: 'a::{numeral,real_normed_vector} star)) = (numeral v = (numeral w :: 'a))"
  by (metis star_of_approx_iff star_of_numeral)

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))"
  unfolding star_numeral_def star_zero_def star_one_def star_of_approx_iff
  by (auto intro: sym)

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_isLub_iff:
  "isLub ℝ (hypreal_of_real ` Q) (hypreal_of_real Y) = isLub (UNIV :: real set) Q Y"  (is "?lhs = ?rhs")
  for Q :: "real set" and Y :: real
proof 
  assume ?lhs
  then show ?rhs
    by (simp add: isLub_def leastP_def) (metis hypreal_of_real_isUb_iff mem_Collect_eq setge_def star_of_le)
next
  assume ?rhs
  then show ?lhs
  apply (simp add: isLub_def leastP_def hypreal_of_real_isUb_iff setge_def)
    by (metis SReal_iff hypreal_of_real_isUb_iff isUb_def star_of_le)
qed

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 SReal_complete:
  fixes P :: "hypreal set"
  assumes "isUb ℝ P Y" "P ⊆ ℝ" "P ≠ {}"
    shows "∃t. isLub ℝ P t"
proof -
  obtain Q where "P = hypreal_of_real ` Q"
    by (metis ‹P ⊆ ℝ› hypreal_of_real_image subset_imageE)
  then show ?thesis
    by (metis assms(1) ‹P ≠ {}› equals0I hypreal_of_real_isLub_iff hypreal_of_real_isUb_iff image_empty lemma_isUb_hypreal_of_real reals_complete)
qed


text ‹Lemmas about lubs.›

lemma lemma_st_part_lub:
  fixes x :: hypreal
  assumes "x ∈ HFinite"
  shows "∃t. isLub ℝ {s. s ∈ ℝ ∧ s < x} t"
proof -
  obtain t where t: "t ∈ ℝ" "hnorm x < t"
    using HFiniteD assms by blast
  then have "isUb ℝ {s. s ∈ ℝ ∧ s < x} t"
    by (simp add: abs_less_iff isUbI le_less_linear less_imp_not_less setleI)
  moreover have "∃y. y ∈ ℝ ∧ y < x"
    using t by (rule_tac x = "-t" in exI) (auto simp add: abs_less_iff)
  ultimately show ?thesis
    using SReal_complete by fastforce
qed

lemma hypreal_setle_less_trans: "S *<= x ==> x < y ==> S *<= y"
  for x y :: hypreal
  by (meson le_less_trans less_imp_le setle_def)

lemma hypreal_gt_isUb: "isUb R S x ==> x < y ==> y ∈ R ==> isUb R S y"
  for x y :: hypreal
  using hypreal_setle_less_trans isUb_def by blast

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: 
  fixes x :: hypreal
  assumes "x ∈ ℝ" shows "isLub ℝ {s. s ∈ ℝ ∧ s < x} x"
proof -
  have "x ≤ y" if "isUb ℝ {s ∈ ℝ. s < x} y" for y
  proof -
    have "y ∈ ℝ"
      using isUbD2a that by blast
    show ?thesis
    proof (cases x y rule: linorder_cases)
      case greater
      then obtain r where "y < r" "r < x"
        using dense by blast
      then show ?thesis
        using isUbD [OF that]
        by simp (meson SReal_dense ‹y ∈ ℝ› assms greater not_le)
    qed auto
  qed
  with assms show ?thesis
    by (simp add: isLubI2 isUbI setgeI setleI)
qed

lemma lemma_st_part_major:
  fixes x r t :: hypreal
  assumes x: "x ∈ HFinite" and r: "r ∈ ℝ" "0 < r" and t: "isLub ℝ {s. s ∈ ℝ ∧ s < x} t"
  shows "∣x - t∣ < r"
proof -
  have "t ∈ ℝ"
    using isLubD1a t by blast
  have lemma_st_part_gt_ub: "x < r ==> r ∈ ℝ ==> isUb ℝ {s. s ∈ ℝ ∧ s < x} r"
    for  r :: hypreal
    by (auto dest: order_less_trans intro: order_less_imp_le intro!: isUbI setleI)

  have "isUb ℝ {s ∈ ℝ. s < x} t"
    by (simp add: t isLub_isUb)
  then have "¬ r + t < x"
    by (metis (mono_tags, lifting) Reals_add ‹t ∈ ℝ› add_le_same_cancel2 isUbD leD mem_Collect_eq r)
  then have "x - t ≤ r"
    by simp
  moreover have "¬ x < t - r"
    using lemma_st_part_gt_ub isLub_le_isUb ‹t ∈ ℝ› r t x by fastforce
  then have "- (x - t) ≤ r"
    by linarith
  moreover have False if "x - t = r ∨ - (x - t) = r"
  proof -
    have "x ∈ ℝ"
      by (metis ‹t ∈ ℝ› ‹r ∈ ℝ› that Reals_add_cancel Reals_minus_iff add_uminus_conv_diff)
    then have "isLub ℝ {s ∈ ℝ. s < x} x"
      by (rule lemma_SReal_lub)
    then show False
      using r t that x isLub_unique by force
  qed
  ultimately show ?thesis
    using abs_less_iff dual_order.order_iff_strict by blast
qed

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
  by (meson isLubD1a lemma_st_part_lub lemma_st_part_major2)

lemma st_part_Ex: "x ∈ HFinite ==> ∃t∈Reals. x ≈ t"
  for x :: hypreal
  by (metis InfinitesimalI approx_def hypreal_hnorm_def lemma_st_part_Ex)

text ‹There is a unique real infinitely close.›
lemma st_part_Ex1: "x ∈ HFinite ==> ∃!t::hypreal. t ∈ ℝ ∧ x ≈ t"
  by (meson SReal_approx_iff approx_trans2 st_part_Ex)


subsection ‹Finite, Infinite and Infinitesimal›

lemma HFinite_Int_HInfinite_empty [simp]: "HFinite Int HInfinite = {}"
  using Compl_HFinite by blast

lemma HFinite_not_HInfinite:
  assumes x: "x ∈ HFinite" shows "x ∉ HInfinite"
  using Compl_HFinite x by blast

lemma not_HFinite_HInfinite: "x ∉ HFinite ==> x ∈ HInfinite"
  using Compl_HFinite by blast

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"
  using HInfinite_inverse_Infinitesimal not_HFinite_HInfinite by force

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"
  using HFinite_HInfinite_iff HInfinite_inverse_Infinitesimal by fastforce

lemma HFinite_not_Infinitesimal_inverse:
  "x ∈ HFinite - Infinitesimal ==> inverse x ∈ HFinite - Infinitesimal"
  for x :: "'a::real_normed_div_algebra star"
  using HFinite_Infinitesimal_not_zero HFinite_inverse2 Infinitesimal_HFinite_mult2 by fastforce

lemma approx_inverse: 
  fixes x y :: "'a::real_normed_div_algebra star"
  assumes "x ≈ y" and y: "y ∈ HFinite - Infinitesimal" shows "inverse x ≈ inverse y"
proof -
  have x: "x ∈ HFinite - Infinitesimal"
    using HFinite_diff_Infinitesimal_approx assms(1) y by blast
  with y HFinite_inverse2 have "inverse x ∈ HFinite" "inverse y ∈ HFinite"
    by blast+
  then have "inverse y * x ≈ 1"
    by (metis Diff_iff approx_mult2 assms(1) left_inverse not_Infinitesimal_not_zero y)
  then show ?thesis
    by (metis (no_types, lifting) DiffD2 HFinite_Infinitesimal_not_zero Infinitesimal_mult_disj x approx_def approx_sym left_diff_distrib left_inverse)
qed

(*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]

lemma inverse_add_Infinitesimal_approx:
  "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)

lemma inverse_add_Infinitesimal_approx2:
  "x ∈ HFinite - Infinitesimal ==> h ∈ Infinitesimal ==> inverse (h + x) ≈ inverse x"
  for x h :: "'a::real_normed_div_algebra star"
  by (metis add.commute inverse_add_Infinitesimal_approx)

lemma inverse_add_Infinitesimal_approx_Infinitesimal:
  "x ∈ HFinite - Infinitesimal ==> h ∈ Infinitesimal ==> inverse (x + h) - inverse x ≈ h"
  for x h :: "'a::real_normed_div_algebra star"
  by (meson Infinitesimal_approx bex_Infinitesimal_iff inverse_add_Infinitesimal_approx)

lemma Infinitesimal_square_iff: "x ∈ Infinitesimal ⟷ x * x ∈ Infinitesimal"
  for x :: "'a::real_normed_div_algebra star"
  using Infinitesimal_mult Infinitesimal_mult_disj by auto
declare Infinitesimal_square_iff [symmetric, simp]

lemma HFinite_square_iff [simp]: "x * x ∈ HFinite ⟷ x ∈ HFinite"
  for x :: "'a::real_normed_div_algebra star"
  using HFinite_HInfinite_iff HFinite_mult HInfinite_mult by blast

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"
  by (metis DiffD2 Infinitesimal_mult_disj bex_Infinitesimal_iff right_diff_distrib)

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"
  using HFinite_add HInfinite_HFinite_iff by blast

lemma HInfinite_HFinite_add: "x ∈ HInfinite ==> y ∈ HFinite ==> x + y ∈ HInfinite"
  by (metis (no_types, opaque_lifting) HFinite_HInfinite_iff HFinite_add HFinite_minus_iff add.commute add_minus_cancel)

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"
  by (metis Infinitesimal_HFinite_mult not_HFinite_HInfinite one_not_Infinitesimal right_inverse)

lemma HInfinite_HFinite_not_Infinitesimal_mult:
  "x ∈ HInfinite ==> y ∈ HFinite - Infinitesimal ==> x * y ∈ HInfinite"
  for x y :: "'a::real_normed_div_algebra star"
  by (metis (no_types, opaque_lifting) HFinite_HInfinite_iff HFinite_Infinitesimal_not_zero HFinite_inverse2 HFinite_mult mult.assoc mult.right_neutral right_inverse)

lemma HInfinite_HFinite_not_Infinitesimal_mult2:
  "x ∈ HInfinite ==> y ∈ HFinite - Infinitesimal ==> y * x ∈ HInfinite"
  for x y :: "'a::real_normed_div_algebra star"
  by (metis Diff_iff HInfinite_HFinite_iff HInfinite_inverse_Infinitesimal Infinitesimal_HFinite_mult2 divide_inverse mult_zero_right nonzero_eq_divide_eq)

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
  by (simp add: abs_if)


subsection ‹Theorems about Monads›

lemma monad_hrabs_Un_subset: "monad ∣x∣ ≤ monad x ∪ monad (- x)"
  for x :: hypreal
  by (simp add: abs_if)

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"
  by (simp add: Infinitesimal_monad_zero_iff [symmetric])

lemma monad_zero_hrabs_iff: "x ∈ monad 0 ⟷ ∣x∣ ∈ monad 0"
  for x :: hypreal
  using Infinitesimal_monad_zero_iff by blast

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


subsection ‹Proof that 🍋‹x ≈ y› implies 🍋‹∣x∣ ≈ ∣y∣›\<close>

lemma approx_subset_monad: "x ≈ y ==> {x, y} ≤ monad x"
  by (simp (no_asm)) (simp add: approx_monad_iff)

lemma approx_subset_monad2: "x ≈ y ==> {x, y} ≤ monad y"
  using approx_subset_monad approx_sym by auto

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"
  using approx_mem_monad approx_sym by blast

lemma approx_mem_monad_zero: "x ≈ y ==> x ∈ monad 0 ==> y ∈ monad 0"
  using approx_trans monad_def by blast

lemma Infinitesimal_approx_hrabs: "x ≈ y ==> x ∈ Infinitesimal ==> ∣x∣ ≈ ∣y∣"
  for x y :: hypreal
  using approx_hnorm by fastforce

lemma less_Infinitesimal_less: "0 < x ==> x ∉ Infinitesimal ==> e ∈ Infinitesimal ==> e < x"
  for x :: hypreal
  using Infinitesimal_interval less_linear by blast

lemma Ball_mem_monad_gt_zero: "0 < x ==> x ∉ Infinitesimal ==> u ∈ monad x ==> 0 < u"
  for u x :: hypreal
  by (metis bex_Infinitesimal_iff2 less_Infinitesimal_less less_add_same_cancel2 mem_monad_approx)

lemma Ball_mem_monad_less_zero: "x < 0 ==> x ∉ Infinitesimal ==> u ∈ monad x ==> u < 0"
  for u x :: hypreal
  by (metis Ball_mem_monad_gt_zero approx_monad_iff less_asym linorder_neqE_linordered_idom mem_infmal_iff mem_monad_approx mem_monad_self)

lemma lemma_approx_gt_zero: "0 < x ==> x ∉ Infinitesimal ==> x ≈ y ==> 0 < y"
  for x y :: hypreal
  by (blast dest: Ball_mem_monad_gt_zero approx_subset_monad)

lemma lemma_approx_less_zero: "x < 0 ==> x ∉ Infinitesimal ==> x ≈ y ==> y < 0"
  for x y :: hypreal
  by (blast dest: Ball_mem_monad_less_zero approx_subset_monad)

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 mem_infmal_iff by blast

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)

lemma hrabs_add_Infinitesimal_cancel:
  "e ∈ Infinitesimal ==> e' ∈ Infinitesimal ==> ∣x + e∣ = ∣y + e'∣ ==> ∣x∣ ≈ ∣y∣"
  for e e' x y :: hypreal
  by (metis approx_hrabs_add_Infinitesimal approx_trans2)

lemma hrabs_add_minus_Infinitesimal_cancel:
  "e ∈ Infinitesimal ==> e' ∈ Infinitesimal ==> ∣x + -e∣ = ∣y + -e'∣ ==> ∣x∣ ≈ ∣y∣"
  for e e' x y :: hypreal
  by (meson Infinitesimal_minus_iff hrabs_add_Infinitesimal_cancel)


subsection ‹More 🍋‹HFinite› and 🍋‹Infinitesimal› Theorems›

text ‹
  Interesting slightly counterintuitive theorem: necessary
  for proving that an open interval is an NS open set.
 ›
lemma Infinitesimal_add_hypreal_of_real_less:
  assumes "x < y" and u: "u ∈ Infinitesimal"
  shows "hypreal_of_real x + u < hypreal_of_real y"
proof -
  have "∣u∣ < hypreal_of_real y - hypreal_of_real x"
    using InfinitesimalD ‹x 🚫› u by fastforce
  then show ?thesis
    by (simp add: abs_less_iff)
qed

lemma Infinitesimal_add_hrabs_hypreal_of_real_less:
  "x ∈ Infinitesimal ==> ∣hypreal_of_real r∣ < hypreal_of_real y ==>
    ∣hypreal_of_real r + x∣ < hypreal_of_real y"
  by (metis Infinitesimal_add_hypreal_of_real_less approx_hrabs_add_Infinitesimal approx_sym bex_Infinitesimal_iff2 star_of_abs star_of_less)

lemma Infinitesimal_add_hrabs_hypreal_of_real_less2:
  "x ∈ Infinitesimal ==> ∣hypreal_of_real r∣ < hypreal_of_real y ==>
    ∣x + hypreal_of_real r∣ < hypreal_of_real y"
  using Infinitesimal_add_hrabs_hypreal_of_real_less by fastforce

lemma hypreal_of_real_le_add_Infininitesimal_cancel:
  assumes le: "hypreal_of_real x + u ≤ hypreal_of_real y + v" 
      and u: "u ∈ Infinitesimal" and v: "v ∈ Infinitesimal"
  shows "hypreal_of_real x ≤ hypreal_of_real y"
proof (rule ccontr)
  assume "¬ hypreal_of_real x ≤ hypreal_of_real y"
  then have "hypreal_of_real y + (v - u) < hypreal_of_real x"
    by (simp add: Infinitesimal_add_hypreal_of_real_less Infinitesimal_diff u v)
  then show False
    by (simp add: add_diff_eq add_le_imp_le_diff le leD)
qed

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"
  by (metis Infinitesimal_interval eq_iff le_less_linear star_of_Infinitesimal_iff_0 star_of_eq_0)

lemma Infinitesimal_add_not_zero: "h ∈ Infinitesimal ==> x ≠ 0 ==> star_of x + h ≠ 0"
  by (metis Infinitesimal_add_approx_self star_of_approx_zero_iff)

lemma monad_hrabs_less: "y ∈ monad x ==> 0 < hypreal_of_real e ==> ∣y - x∣ < hypreal_of_real e"
  by (simp add: Infinitesimal_approx_minus approx_sym less_Infinitesimal_less mem_monad_approx)

lemma mem_monad_SReal_HFinite: "x ∈ monad (hypreal_of_real a) ==> x ∈ HFinite"
  using HFinite_star_of approx_HFinite mem_monad_approx by blast


subsection ‹Theorems about Standard Part›

lemma st_approx_self: "x ∈ HFinite ==> st x ≈ x"
  by (metis (no_types, lifting) approx_refl approx_trans3 someI_ex st_def st_part_Ex st_part_Ex1)

lemma st_SReal: "x ∈ HFinite ==> st x ∈ ℝ"
  by (metis (mono_tags, lifting) approx_sym someI_ex st_def st_part_Ex)

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"
  by (meson SReal_subset_HFinite approx_HFinite approx_unique_real st_SReal st_approx_self subsetD)

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"
  by (simp add: Infinitesimal_add_approx_self st_unique)

lemma st_Infinitesimal_add_SReal2: "x ∈ ℝ ==> e ∈ Infinitesimal ==> st (e + x) = x"
  by (metis add.commute st_Infinitesimal_add_SReal)

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"
  using st_unique by auto

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)"
  by (simp add: approx_inverse st_SReal st_approx_self st_not_Infinitesimal st_unique)

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"
  by (metis Infinitesimal_add_hypreal_of_real_less SReal_iff st_SReal star_of_less)

lemma Infinitesimal_add_st_le_cancel:
  "x ∈ HFinite ==> y ∈ HFinite ==> u ∈ Infinitesimal ==>
    st x ≤ st y + u ==> st x ≤ st y"
  by (meson Infinitesimal_add_st_less leD le_less_linear)

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"
  by (metis HFinite_0 st_0 st_le)

lemma st_zero_ge: "x ≤ 0 ==> x ∈ HFinite ==> st x ≤ 0"
  by (metis HFinite_0 st_0 st_le)

lemma st_hrabs: "x ∈ HFinite ==> ∣st x∣ = st ∣x∣"
  by (simp add: order_class.order.antisym st_zero_ge linorder_not_le st_zero_le abs_if st_minus linorder_not_less)


subsection ‹Alternative Definitions using Free Ultrafilter›

subsubsection ‹🍋‹HFinite›\›

lemma HFinite_FreeUltrafilterNat:
  assumes "star_n X ∈ HFinite"
  shows "∃u. eventually (λn. norm (X n) < u) U"
proof -
  obtain r where "hnorm (star_n X) < hypreal_of_real r"
    using HFiniteD SReal_iff assms by fastforce
  then have "∀🪙F n in U. norm (X n) < r"
    by (simp add: hnorm_def star_n_less star_of_def starfun_star_n)
  then show ?thesis ..
qed

lemma FreeUltrafilterNat_HFinite:
  assumes "eventually (λn. norm (X n) < u) U"
  shows "star_n X ∈ HFinite"
proof -
  have "hnorm (star_n X) < hypreal_of_real u"
    by (simp add: assms hnorm_def star_n_less star_of_def starfun_star_n)
  then show ?thesis
    by (meson HInfiniteD SReal_hypreal_of_real less_asym not_HFinite_HInfinite)
qed

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


subsubsection ‹🍋‹HInfinite›\›

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"
  by (simp add: FreeUltrafilterNat_HFinite [where u = "u+1"] eventually_mono)

lemma HInfinite_FreeUltrafilterNat:
  assumes "star_n X ∈ HInfinite" shows "∀🪙F n in U. u < norm (X n)"
proof -
have "¬ (∀🪙F n in U. norm (X n) < u + 1)"
  using FreeUltrafilterNat_HFinite HFinite_HInfinite_iff assms by auto
  then show ?thesis
    by (auto simp flip: FreeUltrafilterNat.eventually_not_iff elim: eventually_mono)
qed

lemma FreeUltrafilterNat_HInfinite:
  assumes "∧u. eventually (λn. u < norm (X n)) U"
  shows "star_n X ∈ HInfinite"
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 have False
      by (simp add: eventually_False FreeUltrafilterNat.proper) }
  then show ?thesis
    using HFinite_FreeUltrafilterNat HInfinite_HFinite_iff assms by blast
qed

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


subsubsection ‹🍋‹Infinitesimal›\›

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


lemma Infinitesimal_FreeUltrafilterNat_iff:
  "(star_n X ∈ Infinitesimal) = (∀u>0. eventually (λn. norm (X n) < u) U)"  (is "?lhs = ?rhs")
proof -
  have "?lhs ⟷ (∀r>0. hnorm (star_n X) < hypreal_of_real r)"
    by (simp add: Infinitesimal_def ball_SReal_eq)
  also have "... ⟷ ?rhs"
    by (simp add: hnorm_def starfun_star_n star_of_def star_less_def starP2_star_n)
  finally show ?thesis .
qed


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)))"
  by (meson inverse_positive_iff_positive less_trans of_nat_0_less_iff reals_Archimedean zero_less_Suc)

lemma lemma_Infinitesimal2:
  "(∀r ∈ Reals. 0 < r ⟶ x < r) ⟷ (∀n. x < inverse(hypreal_of_nat (Suc n)))" (is "_ = ?rhs")
proof (intro iffI strip)
  assume R: ?rhs
  fix r::hypreal
  assume "r ∈ ℝ" "0 < r"
  then obtain n y where "inverse (real (Suc n)) < y" and r: "r = hypreal_of_real y"
    by (metis SReal_iff reals_Archimedean star_of_0_less)
  then have "inverse (1 + hypreal_of_nat n) < hypreal_of_real y"
    by (metis of_nat_Suc star_of_inverse star_of_less star_of_nat_def)
  then show "x < r"
    by (metis R r le_less_trans less_imp_le of_nat_Suc)
qed (meson Reals_inverse Reals_of_nat of_nat_0_less_iff positive_imp_inverse_positive zero_less_Suc)


lemma Infinitesimal_hypreal_of_nat_iff:
  "Infinitesimal = {x. ∀n. hnorm x < inverse (hypreal_of_nat (Suc n))}"
  using Infinitesimal_def lemma_Infinitesimal2 by auto


subsection ‹Proof that ‹ψ› 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}"
proof -
  obtain m where "u < real m"
    using reals_Archimedean2 by blast
  then have "{n. real n < u} ⊆ {..
    by force
  then show ?thesis
    using finite_nat_iff_bounded by force
qed

lemma finite_real_of_nat_le_real: "finite {n::nat. real n ≤ u}"
  by (metis infinite_nat_iff_unbounded leD le_nat_floor mem_Collect_eq)

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"
proof -
  have "{n::nat. ¬ u < real n} = {n. real n ≤ u}"
    by auto
  then show ?thesis
    by (auto simp add: FreeUltrafilterNat.finite' finite_real_of_nat_le_real)
qed

text ‹The complement of ‹{n. ∣real n∣ ≤ u} = {n. u 🚫∣real n∣}› is in
  ‹U› by property of (free) ultrafilters.›

text ‹🍋‹ψ› is a member of 🍋‹HInfinite›.›
theorem HInfinite_omega [simp]: "ψ ∈ HInfinite"
proof -
  have "∀🪙F n in U. u < norm (1 + real n)" for u
    using FreeUltrafilterNat_nat_gt_real [of "u-1"] eventually_mono by fastforce
  then show ?thesis
    by (simp add: omega_def FreeUltrafilterNat_HInfinite)
qed


text ‹Epsilon is a member of Infinitesimal.›

lemma Infinitesimal_epsilon [simp]: "ε ∈ Infinitesimal"
  by (auto intro!: HInfinite_inverse_Infinitesimal HInfinite_omega
      simp add: epsilon_inverse_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 ‹[🚫n)] ≈ hypreal_of_real a› given
  that ‹∀n. |X n - a| 🚫/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"
  using less_imp_inverse_less by force

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 finite_inverse_real_of_posnat_ge_real:
  assumes "0 < u"
  shows "finite {n. u ≤ inverse (real (Suc n))}"
proof -
  have "∀na. u ≤ inverse (1 + real na) ⟶ na ≤ ceiling (inverse u)"
    by (smt (verit, best) assms ceiling_less_cancel ceiling_of_nat inverse_inverse_eq inverse_le_iff_le)
  then show ?thesis
    apply (auto simp add: finite_nat_set_iff_bounded_le)
    by (meson assms inverse_positive_iff_positive le_nat_iff less_imp_le zero_less_ceiling)
qed

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)

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| 🚫/n ==> [🚫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