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products/sources/formale Sprachen/Isabelle/HOL/Nonstandard_Analysis/ (Beweissystem Isabelle Version 2025-1^©) Datei vom 16.11.2025 mit Größe 75 kB

Quelle NSA.thy Sprache: Isabelle

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

section \<open>Infinite Numbers, Infinitesimals, Infinitely Close Relation\<close>

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"
    \<comment> \<open>the ``infinitely close'' relation\<close>

definition st :: "hypreal \ hypreal"
  where "st = (\x. SOME r. x \ HFinite \ r \ \ \ r \ x)"
    \<comment> \<open>the standard part of a hyperreal\<close>

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: "\ \ {x. \r. x = hypreal_of_real r}"
  by (simp add: Reals_def image_def)

subsection \<open>Nonstandard Extension of the Norm Function\<close>

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 *\<^sub>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 \<open>Closure Laws for the Standard Reals\<close>

lemma Reals_add_cancel: "x + y \ \ \ y \ \ \ x \ \"
  by (drule (1) Reals_diff) simp

lemma SReal_hrabs: "x \ \ \ \x\ \ \"
  for x :: hypreal
  by (simp add: Reals_eq_Standard)

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

lemma SReal_divide_numeral: "r \ \ \ r / (numeral w::hypreal) \ \"
  by simp

text \<open>\<open>\<epsilon>\<close> is not in Reals because it is an infinitesimal\<close>
lemma SReal_epsilon_not_mem: "\ \ \"
  by (auto simp: SReal_def hypreal_of_real_not_eq_epsilon [symmetric])

lemma SReal_omega_not_mem: "\ \ \"
  by (auto simp: SReal_def hypreal_of_real_not_eq_omega [symmetric])

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

lemma SReal_iff: "x \ \ \ (\y. x = hypreal_of_real y)"
  by (simp add: SReal_def)

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

lemma inv_hypreal_of_real_image: "inv hypreal_of_real ` \ = UNIV"
  by (simp add: Reals_eq_Standard Standard_def inj_star_of)

lemma SReal_dense: "x \ \ \ y \ \ \ x < y \ \r \ Reals. x < r \ r < y"
  for x y :: hypreal
  using dense by (fastforce simp add: SReal_def)

subsection \<open>Set of Finite Elements is a Subring of the Extended Reals\<close>

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: "(\::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 \<open>As always with numerals, \<open>0\<close> and \<open>1\<close> are special cases.\<close>

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 \<open>Set of Infinitesimals is a Subring of the Hyperreals\<close>

lemma InfinitesimalI: "(\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 \ \" 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 \ \" 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 \<open>y \<in> HFinite\<close> by blast
  then have "t > 0"
    using hnorm_ge_zero le_less_trans by blast
  show "hnorm (x * y) < r"
    if "r \ \" and "0 < r" for r :: "real star"
  proof -
    have "hnorm x < r/t"
      by (meson InfinitesimalD Reals_divide \<open>hnorm y < t\<close> \<open>t \<in> \<real>\<close> assms(1) divide_pos_pos hnorm_ge_zero le_less_trans that)
    then have "hnorm (x * y) < (r / t) * t"
      using \<open>hnorm y < t\<close> hnorm_mult_less by blast
    then show ?thesis
      using \<open>0 < t\<close> 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 \<open>y \<in> HFinite\<close> by blast
  then have "t > 0"
    using hnorm_ge_zero le_less_trans by blast
  show "hnorm (scaleHR x y) < r"
    if "r \ \" and "0 < r" for r :: "real star"
  proof -
    have "\x\ * hnorm y < (r / t) * t"
      by (metis InfinitesimalD Reals_divide \<open>0 < t\<close> \<open>hnorm y < t\<close> \<open>t \<in> \<real>\<close> assms(1) divide_pos_pos hnorm_ge_zero hypreal_hnorm_def mult_strict_mono' that)
    then show ?thesis
      by (simp add: \<open>0 < t\<close> 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 \<open>y \<in> HFinite\<close> by blast
  then have "t > 0"
    using hnorm_ge_zero le_less_trans by blast
  show "hnorm (y * x) < r"
    if "r \ \" and "0 < r" for r :: "real star"
  proof -
    have "hnorm x < r/t"
      by (meson InfinitesimalD Reals_divide \<open>hnorm y < t\<close> \<open>t \<in> \<real>\<close> assms(1) divide_pos_pos hnorm_ge_zero le_less_trans that)
    then have "hnorm (y * x) < t * (r / t)"
      using \<open>hnorm y < t\<close> hnorm_mult_less by blast
    then show ?thesis
      using \<open>0 < t\<close> by auto
  qed
qed

lemma Infinitesimal_scaleR2:
  assumes "x \ Infinitesimal" shows "a *\<^sub>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 \ \ \ s \ hnorm x" and "r \ \"
    for x :: "'a star" and r :: hypreal
    using * [of "r+1"] \<open>r \<in> \<real>\<close> 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 < hnorm x) \ x \ HInfinite"
  by (simp add: HInfinite_def)

lemma HInfiniteD: "x \ HInfinite \ 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 \ \" for r :: "real star"
  proof -
    have "r = r * 1"
      by simp
    also have "\ < hnorm x * hnorm y"
      by (meson HInfiniteD Reals_1 \<open>x \<noteq> 0\<close> 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 \<open>The Infinitely Close Relation\<close>

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 \<open>Reorientation simplification procedure: reorients (polymorphic)
  \<open>0 = x\<close>, \<open>1 = x\<close>, \<open>nnn = x\<close> provided \<open>x\<close> isn't \<open>0\<close>, \<open>1\<close> or a numeral.\<close>
simproc_setup approx_reorient_simproc
  ("0 \ x" | "1 \ y" | "numeral w \ z" | "- 1 \ y" | "- numeral w \ r") =
\<open>
  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
\<close>

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 *\<^sub>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 *\<^sub>R a \ c *\<^sub>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 *\<^sub>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 \ \" "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 \ \ \ x \ 0 \ x * a \ 0"
  for a x :: hypreal
  by (auto dest: SReal_subset_HFinite [THEN subsetD] approx_mult1)

lemma approx_mult_SReal2: "a \ \ \ 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 \ \ \ 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 \ \" "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 \ \ \ 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 \<open>Zero is the Only Infinitesimal that is also a Real\<close>

lemma Infinitesimal_less_SReal: "x \ \ \ 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 \ \ ==> y \ Infinitesimal"
  for y :: hypreal
  by (auto simp add: Infinitesimal_def abs_if)

lemma SReal_minus_not_Infinitesimal: "y < 0 \ y \ \ \ 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: "\ Int Infinitesimal = {0::hypreal}"
  proof -
  have "x = 0" if "x \ \" "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 \ \ \ x \ Infinitesimal \ x = 0"
  for x :: hypreal
  using SReal_Int_Infinitesimal_zero by blast

lemma SReal_HFinite_diff_Infinitesimal: "x \ \ \ 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 (\n. 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 \<open>Again: \<open>1\<close> is a special case, but not \<open>0\<close> this time.\<close>
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 \ \ \ 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 \<open>The premise \<open>y \<noteq> 0\<close> is essential; otherwise \<open>x / y = 0\<close> and we lose the
  \<open>HFinite\<close> premise.\<close>
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 \ \ \ x / y \ Infinitesimal"
  for x y :: hypreal
  by (metis HFinite_star_of Infinitesimal_HFinite_mult Reals_inverse SReal_iff divide_inverse)

section \<open>Standard Part Theorem\<close>

text \<open>
  Every finite \<open>x \<in> R*\<close> is infinitely close to a unique real number
  (i.e. a member of \<open>Reals\<close>).
\<close>

subsection \<open>Uniqueness: Two Infinitely Close Reals are Equal\<close>

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 \ \ \ y \ \ \ 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 \<open>And also for \<open>0 \<approx> #nn\<close> and \<open>1 \<approx> #nn\<close>, \<open>#nn \<approx> 0\<close> and \<open>#nn \<approx> 1\<close>.\<close>
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 \ \ \ s \ \ \ r \ x \ s \ x \ r = s"
  for r s :: hypreal
  by (blast intro: SReal_approx_iff [THEN iffD1] approx_trans2)

subsection \<open>Existence of Unique Real Infinitely Close\<close>

subsubsection \<open>Lifting of the Ub and Lub Properties\<close>

lemma hypreal_of_real_isUb_iff: "isUb \ (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 \ (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 \ P Y \ \Yo. isUb \ P (hypreal_of_real Yo)"
  by (auto simp add: SReal_iff isUb_def)

lemma lemma_isLub_hypreal_of_real: "isLub \ P Y \ \Yo. isLub \ 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 \ P Y" "P \ \" "P \ {}"
    shows "\t. isLub \ P t"
proof -
  obtain Q where "P = hypreal_of_real ` Q"
    by (metis \<open>P \<subseteq> \<real>\<close> hypreal_of_real_image subset_imageE)
  then show ?thesis
    by (metis assms(1) \<open>P \<noteq> {}\<close> equals0I hypreal_of_real_isLub_iff hypreal_of_real_isUb_iff image_empty lemma_isUb_hypreal_of_real reals_complete)
qed

text \<open>Lemmas about lubs.\<close>

lemma lemma_st_part_lub:
  fixes x :: hypreal
  assumes "x \ HFinite"
  shows "\t. isLub \ {s. s \ \ \ s < x} t"
proof -
  obtain t where t: "t \ \" "hnorm x < t"
    using HFiniteD assms by blast
  then have "isUb \ {s. s \ \ \ s < x} t"
    by (simp add: abs_less_iff isUbI le_less_linear less_imp_not_less setleI)
  moreover have "\y. y \ \ \ 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 \ \ \ isUb \ {s. s \ \ \ s < x} x"
  for x :: hypreal
  by (auto intro: isUbI setleI order_less_imp_le)

lemma lemma_SReal_lub:
  fixes x :: hypreal
  assumes "x \ \" shows "isLub \ {s. s \ \ \ s < x} x"
proof -
  have "x \ y" if "isUb \ {s \ \. s < x} y" for y
  proof -
    have "y \ \"
      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 \<open>y \<in> \<real>\<close> 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 \ \" "0 < r" and t: "isLub \ {s. s \ \ \ s < x} t"
  shows "\x - t\ < r"
proof -
  have "t \ \"
    using isLubD1a t by blast
  have lemma_st_part_gt_ub: "x < r \ r \ \ \ isUb \ {s. s \ \ \ s < x} r"
    for  r :: hypreal
    by (auto dest: order_less_trans intro: order_less_imp_le intro!: isUbI setleI)

  have "isUb \ {s \ \. s < x} t"
    by (simp add: t isLub_isUb)
  then have "\ r + t < x"
    by (metis (mono_tags, lifting) Reals_add \<open>t \<in> \<real>\<close> 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 \<open>t \<in> \<real>\<close> 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 \ \"
      by (metis \<open>t \<in> \<real>\<close> \<open>r \<in> \<real>\<close> that Reals_add_cancel Reals_minus_iff add_uminus_conv_diff)
    then have "isLub \ {s \ \. 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 \ {s. s \ \ \ s < x} t \ \r \ Reals. 0 < r \ \x - t\ < r"
  for x t :: hypreal
  by (blast dest!: lemma_st_part_major)

text\<open>Existence of real and Standard Part Theorem.\<close>

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 \<open>There is a unique real infinitely close.\<close>
lemma st_part_Ex1: "x \ HFinite \ \!t::hypreal. t \ \ \ x \ t"
  by (meson SReal_approx_iff approx_trans2 st_part_Ex)

subsection \<open>Finite, Infinite and Infinitesimal\<close>

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 \<open>Stronger statement possible in fact.\<close>
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 \ \ \ 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 \<open>Theorems about Monads\<close>

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 \<open>Proof that \<^term>\<open>x \<approx> y\<close> implies \<^term>\<open>\<bar>x\<bar> \<approx> \<bar>y\<bar>\<close>\<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 \<open>More \<^term>\<open>HFinite\<close> and \<^term>\<open>Infinitesimal\<close> Theorems\<close>

text \<open>
  Interesting slightly counterintuitive theorem: necessary
  for proving that an open interval is an NS open set.
\<close>
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 \<open>x < y\<close> 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 \
    \<bar>hypreal_of_real r + x\<bar> < 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 \
    \<bar>x + hypreal_of_real r\<bar> < 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 \<le> hypreal_of_real y + v \<Longrightarrow> x \<le> 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 \<open>Theorems about Standard Part\<close>

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 \ \"
  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 \ 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 \ \ \ 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 \ \" "st y \ \"
    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 \ \ \ e \ Infinitesimal \ st (x + e) = x"
  by (simp add: Infinitesimal_add_approx_self st_unique)

lemma st_Infinitesimal_add_SReal2: "x \ \ \ 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)

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