session-qualified theory imports: isabelle imports -U -i -d '~~/src/Benchmarks' -a;
(* 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 \<Rightarrow> real star"
where [transfer_unfold]: "hnorm = *f* norm"
definition Infinitesimal :: "('a::real_normed_vector) star set"
where "Infinitesimal = {x. \<forall>r \<in> Reals. 0 < r \<longrightarrow> hnorm x < r}"
definition HFinite :: "('a::real_normed_vector) star set"
where "HFinite = {x. \<exists>r \<in> Reals. hnorm x < r}"
definition HInfinite :: "('a::real_normed_vector) star set"
where "HInfinite = {x. \<forall>r \<in> Reals. r < hnorm x}"
definition approx :: "'a::real_normed_vector star \<Rightarrow> 'a star \<Rightarrow> bool" (infixl "\<approx>" 50)
where "x \<approx> y \<longleftrightarrow> x - y \<in> Infinitesimal"
\<comment> \<open>the ``infinitely close'' relation\<close>
definition st :: "hypreal \<Rightarrow> hypreal"
where "st = (\<lambda>x. SOME r. x \<in> HFinite \<and> r \<in> \<real> \<and> r \<approx> x)"
\<comment> \<open>the standard part of a hyperreal\<close>
definition monad :: "'a::real_normed_vector star \<Rightarrow> 'a star set"
where "monad x = {y. x \<approx> y}"
definition galaxy :: "'a::real_normed_vector star \<Rightarrow> 'a star set"
where "galaxy x = {y. (x + -y) \<in> HFinite}"
lemma SReal_def: "\<real> \<equiv> {x. \<exists>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 \<Rightarrow> 'a star \<Rightarrow> 'a::real_normed_vector star"
where [transfer_unfold]: "scaleHR = starfun2 scaleR"
lemma Standard_hnorm [simp]: "x \<in> Standard \<Longrightarrow> hnorm x \<in> 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]: "\<And>x::'a::real_normed_vector star. 0 \<le> hnorm x"
by transfer (rule norm_ge_zero)
lemma hnorm_eq_zero [simp]: "\<And>x::'a::real_normed_vector star. hnorm x = 0 \<longleftrightarrow> x = 0"
by transfer (rule norm_eq_zero)
lemma hnorm_triangle_ineq: "\<And>x y::'a::real_normed_vector star. hnorm (x + y) \<le> hnorm x + hnorm y"
by transfer (rule norm_triangle_ineq)
lemma hnorm_triangle_ineq3: "\<And>x y::'a::real_normed_vector star. \<bar>hnorm x - hnorm y\<bar> \<le> hnorm (x - y)"
by transfer (rule norm_triangle_ineq3)
lemma hnorm_scaleR: "\<And>x::'a::real_normed_vector star. hnorm (a *\<^sub>R x) = \<bar>star_of a\<bar> * hnorm x"
by transfer (rule norm_scaleR)
lemma hnorm_scaleHR: "\<And>a (x::'a::real_normed_vector star). hnorm (scaleHR a x) = \<bar>a\<bar> * hnorm x"
by transfer (rule norm_scaleR)
lemma hnorm_mult_ineq: "\<And>x y::'a::real_normed_algebra star. hnorm (x * y) \<le> hnorm x * hnorm y"
by transfer (rule norm_mult_ineq)
lemma hnorm_mult: "\<And>x y::'a::real_normed_div_algebra star. hnorm (x * y) = hnorm x * hnorm y"
by transfer (rule norm_mult)
lemma hnorm_hyperpow: "\<And>(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]: "\<And>x::'a::real_normed_vector star. 0 < hnorm x \<longleftrightarrow> x \<noteq> 0"
by transfer (rule zero_less_norm_iff)
lemma hnorm_minus_cancel [simp]: "\<And>x::'a::real_normed_vector star. hnorm (- x) = hnorm x"
by transfer (rule norm_minus_cancel)
lemma hnorm_minus_commute: "\<And>a b::'a::real_normed_vector star. hnorm (a - b) = hnorm (b - a)"
by transfer (rule norm_minus_commute)
lemma hnorm_triangle_ineq2: "\<And>a b::'a::real_normed_vector star. hnorm a - hnorm b \<le> hnorm (a - b)"
by transfer (rule norm_triangle_ineq2)
lemma hnorm_triangle_ineq4: "\<And>a b::'a::real_normed_vector star. hnorm (a - b) \<le> hnorm a + hnorm b"
by transfer (rule norm_triangle_ineq4)
lemma abs_hnorm_cancel [simp]: "\<And>a::'a::real_normed_vector star. \<bar>hnorm a\<bar> = hnorm a"
by transfer (rule abs_norm_cancel)
lemma hnorm_of_hypreal [simp]: "\<And>r. hnorm (of_hypreal r::'a::real_normed_algebra_1 star) = \<bar>r\<bar>"
by transfer (rule norm_of_real)
lemma nonzero_hnorm_inverse:
"\<And>a::'a::real_normed_div_algebra star. a \<noteq> 0 \<Longrightarrow> hnorm (inverse a) = inverse (hnorm a)"
by transfer (rule nonzero_norm_inverse)
lemma hnorm_inverse:
"\<And>a::'a::{real_normed_div_algebra, division_ring} star. hnorm (inverse a) = inverse (hnorm a)"
by transfer (rule norm_inverse)
lemma hnorm_divide: "\<And>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]: "\<And>r::hypreal. hnorm r = \<bar>r\<bar>"
by transfer (rule real_norm_def)
lemma hnorm_add_less:
"\<And>(x::'a::real_normed_vector star) y r s. hnorm x < r \<Longrightarrow> hnorm y < s \<Longrightarrow> hnorm (x + y) < r + s"
by transfer (rule norm_add_less)
lemma hnorm_mult_less:
"\<And>(x::'a::real_normed_algebra star) y r s. hnorm x < r \<Longrightarrow> hnorm y < s \<Longrightarrow> hnorm (x * y) < r * s"
by transfer (rule norm_mult_less)
lemma hnorm_scaleHR_less: "\<bar>x\<bar> < r \<Longrightarrow> hnorm y < s \<Longrightarrow> 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_minus_iff [simp]: "- x \<in> \<real> \<longleftrightarrow> x \<in> \<real>"
apply auto
apply (drule Reals_minus)
apply auto
done
lemma Reals_add_cancel: "x + y \<in> \<real> \<Longrightarrow> y \<in> \<real> \<Longrightarrow> x \<in> \<real>"
by (drule (1) Reals_diff) simp
lemma SReal_hrabs: "x \<in> \<real> \<Longrightarrow> \<bar>x\<bar> \<in> \<real>"
for x :: hypreal
by (simp add: Reals_eq_Standard)
lemma SReal_hypreal_of_real [simp]: "hypreal_of_real x \<in> \<real>"
by (simp add: Reals_eq_Standard)
lemma SReal_divide_numeral: "r \<in> \<real> \<Longrightarrow> r / (numeral w::hypreal) \<in> \<real>"
by simp
text \<open>\<open>\<epsilon>\<close> is not in Reals because it is an infinitesimal\<close>
lemma SReal_epsilon_not_mem: "\<epsilon> \<notin> \<real>"
by (auto simp: SReal_def hypreal_of_real_not_eq_epsilon [symmetric])
lemma SReal_omega_not_mem: "\<omega> \<notin> \<real>"
by (auto simp: SReal_def hypreal_of_real_not_eq_omega [symmetric])
lemma SReal_UNIV_real: "{x. hypreal_of_real x \<in> \<real>} = (UNIV::real set)"
by simp
lemma SReal_iff: "x \<in> \<real> \<longleftrightarrow> (\<exists>y. x = hypreal_of_real y)"
by (simp add: SReal_def)
lemma hypreal_of_real_image: "hypreal_of_real `(UNIV::real set) = \<real>"
by (simp add: Reals_eq_Standard Standard_def)
lemma inv_hypreal_of_real_image: "inv hypreal_of_real ` \<real> = UNIV"
apply (auto simp add: SReal_def)
apply (rule inj_star_of [THEN inv_f_f, THEN subst], blast)
done
lemma SReal_hypreal_of_real_image: "\<exists>x. x \<in> P \<Longrightarrow> P \<subseteq> \<real> \<Longrightarrow> \<exists>Q. P = hypreal_of_real ` Q"
unfolding SReal_def image_def by blast
lemma SReal_dense: "x \<in> \<real> \<Longrightarrow> y \<in> \<real> \<Longrightarrow> x < y \<Longrightarrow> \<exists>r \<in> Reals. x < r \<and> r < y"
for x y :: hypreal
apply (auto simp: SReal_def)
apply (drule dense)
apply auto
done
text \<open>Completeness of Reals, but both lemmas are unused.\<close>
lemma SReal_sup_lemma:
"P \<subseteq> \<real> \<Longrightarrow> (\<exists>x \<in> P. y < x) = (\<exists>X. hypreal_of_real X \<in> P \<and> y < hypreal_of_real X)"
by (blast dest!: SReal_iff [THEN iffD1])
lemma SReal_sup_lemma2:
"P \<subseteq> \<real> \<Longrightarrow> \<exists>x. x \<in> P \<Longrightarrow> \<exists>y \<in> Reals. \<forall>x \<in> P. x < y \<Longrightarrow>
(\<exists>X. X \<in> {w. hypreal_of_real w \<in> P}) \<and>
(\<exists>Y. \<forall>X \<in> {w. hypreal_of_real w \<in> P}. X < Y)"
apply (rule conjI)
apply (fast dest!: SReal_iff [THEN iffD1])
apply (auto, frule subsetD, assumption)
apply (drule SReal_iff [THEN iffD1])
apply (auto, rule_tac x = ya in exI, auto)
done
subsection \<open>Set of Finite Elements is a Subring of the Extended Reals\<close>
lemma HFinite_add: "x \<in> HFinite \<Longrightarrow> y \<in> HFinite \<Longrightarrow> x + y \<in> HFinite"
unfolding HFinite_def by (blast intro!: Reals_add hnorm_add_less)
lemma HFinite_mult: "x \<in> HFinite \<Longrightarrow> y \<in> HFinite \<Longrightarrow> x * y \<in> HFinite"
for x y :: "'a::real_normed_algebra star"
unfolding HFinite_def by (blast intro!: Reals_mult hnorm_mult_less)
lemma HFinite_scaleHR: "x \<in> HFinite \<Longrightarrow> y \<in> HFinite \<Longrightarrow> scaleHR x y \<in> HFinite"
by (auto simp: HFinite_def intro!: Reals_mult hnorm_scaleHR_less)
lemma HFinite_minus_iff: "- x \<in> HFinite \<longleftrightarrow> x \<in> HFinite"
by (simp add: HFinite_def)
lemma HFinite_star_of [simp]: "star_of x \<in> HFinite"
apply (simp add: HFinite_def)
apply (rule_tac x="star_of (norm x) + 1" in bexI)
apply (transfer, simp)
apply (blast intro: Reals_add SReal_hypreal_of_real Reals_1)
done
lemma SReal_subset_HFinite: "(\<real>::hypreal set) \<subseteq> HFinite"
by (auto simp add: SReal_def)
lemma HFiniteD: "x \<in> HFinite \<Longrightarrow> \<exists>t \<in> Reals. hnorm x < t"
by (simp add: HFinite_def)
lemma HFinite_hrabs_iff [iff]: "\<bar>x\<bar> \<in> HFinite \<longleftrightarrow> x \<in> HFinite"
for x :: hypreal
by (simp add: HFinite_def)
lemma HFinite_hnorm_iff [iff]: "hnorm x \<in> HFinite \<longleftrightarrow> x \<in> HFinite"
for x :: hypreal
by (simp add: HFinite_def)
lemma HFinite_numeral [simp]: "numeral w \<in> 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 \<in> HFinite"
unfolding star_zero_def by (rule HFinite_star_of)
lemma HFinite_1 [simp]: "1 \<in> HFinite"
unfolding star_one_def by (rule HFinite_star_of)
lemma hrealpow_HFinite: "x \<in> HFinite \<Longrightarrow> x ^ n \<in> HFinite"
for x :: "'a::{real_normed_algebra,monoid_mult} star"
by (induct n) (auto simp add: power_Suc intro: HFinite_mult)
lemma HFinite_bounded: "x \<in> HFinite \<Longrightarrow> y \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<in> HFinite"
for x y :: hypreal
apply (cases "x \<le> 0")
apply (drule_tac y = x in order_trans)
apply (drule_tac [2] order_antisym)
apply (auto simp add: linorder_not_le)
apply (auto intro: order_le_less_trans simp add: abs_if HFinite_def)
done
subsection \<open>Set of Infinitesimals is a Subring of the Hyperreals\<close>
lemma InfinitesimalI: "(\<And>r. r \<in> \<real> \<Longrightarrow> 0 < r \<Longrightarrow> hnorm x < r) \<Longrightarrow> x \<in> Infinitesimal"
by (simp add: Infinitesimal_def)
lemma InfinitesimalD: "x \<in> Infinitesimal \<Longrightarrow> \<forall>r \<in> Reals. 0 < r \<longrightarrow> hnorm x < r"
by (simp add: Infinitesimal_def)
lemma InfinitesimalI2: "(\<And>r. 0 < r \<Longrightarrow> hnorm x < star_of r) \<Longrightarrow> x \<in> Infinitesimal"
by (auto simp add: Infinitesimal_def SReal_def)
lemma InfinitesimalD2: "x \<in> Infinitesimal \<Longrightarrow> 0 < r \<Longrightarrow> hnorm x < star_of r"
by (auto simp add: Infinitesimal_def SReal_def)
lemma Infinitesimal_zero [iff]: "0 \<in> Infinitesimal"
by (simp add: Infinitesimal_def)
lemma hypreal_sum_of_halves: "x / 2 + x / 2 = x"
for x :: hypreal
by auto
lemma Infinitesimal_add: "x \<in> Infinitesimal \<Longrightarrow> y \<in> Infinitesimal \<Longrightarrow> x + y \<in> Infinitesimal"
apply (rule InfinitesimalI)
apply (rule hypreal_sum_of_halves [THEN subst])
apply (drule half_gt_zero)
apply (blast intro: hnorm_add_less SReal_divide_numeral dest: InfinitesimalD)
done
lemma Infinitesimal_minus_iff [simp]: "- x \<in> Infinitesimal \<longleftrightarrow> x \<in> Infinitesimal"
by (simp add: Infinitesimal_def)
lemma Infinitesimal_hnorm_iff: "hnorm x \<in> Infinitesimal \<longleftrightarrow> x \<in> Infinitesimal"
by (simp add: Infinitesimal_def)
lemma Infinitesimal_hrabs_iff [iff]: "\<bar>x\<bar> \<in> Infinitesimal \<longleftrightarrow> x \<in> Infinitesimal"
for x :: hypreal
by (simp add: abs_if)
lemma Infinitesimal_of_hypreal_iff [simp]:
"(of_hypreal x::'a::real_normed_algebra_1 star) \<in> Infinitesimal \<longleftrightarrow> x \<in> Infinitesimal"
by (subst Infinitesimal_hnorm_iff [symmetric]) simp
lemma Infinitesimal_diff: "x \<in> Infinitesimal \<Longrightarrow> y \<in> Infinitesimal \<Longrightarrow> x - y \<in> Infinitesimal"
using Infinitesimal_add [of x "- y"] by simp
lemma Infinitesimal_mult: "x \<in> Infinitesimal \<Longrightarrow> y \<in> Infinitesimal \<Longrightarrow> x * y \<in> Infinitesimal"
for x y :: "'a::real_normed_algebra star"
apply (rule InfinitesimalI)
apply (subgoal_tac "hnorm (x * y) < 1 * r")
apply (simp only: mult_1)
apply (rule hnorm_mult_less)
apply (simp_all add: InfinitesimalD)
done
lemma Infinitesimal_HFinite_mult: "x \<in> Infinitesimal \<Longrightarrow> y \<in> HFinite \<Longrightarrow> x * y \<in> Infinitesimal"
for x y :: "'a::real_normed_algebra star"
apply (rule InfinitesimalI)
apply (drule HFiniteD, clarify)
apply (subgoal_tac "0 < t")
apply (subgoal_tac "hnorm (x * y) < (r / t) * t", simp)
apply (subgoal_tac "0 < r / t")
apply (rule hnorm_mult_less)
apply (simp add: InfinitesimalD)
apply assumption
apply simp
apply (erule order_le_less_trans [OF hnorm_ge_zero])
done
lemma Infinitesimal_HFinite_scaleHR:
"x \<in> Infinitesimal \<Longrightarrow> y \<in> HFinite \<Longrightarrow> scaleHR x y \<in> Infinitesimal"
apply (rule InfinitesimalI)
apply (drule HFiniteD, clarify)
apply (drule InfinitesimalD)
apply (simp add: hnorm_scaleHR)
apply (subgoal_tac "0 < t")
apply (subgoal_tac "\<bar>x\<bar> * hnorm y < (r / t) * t", simp)
apply (subgoal_tac "0 < r / t")
apply (rule mult_strict_mono', simp_all)
apply (erule order_le_less_trans [OF hnorm_ge_zero])
done
lemma Infinitesimal_HFinite_mult2:
"x \<in> Infinitesimal \<Longrightarrow> y \<in> HFinite \<Longrightarrow> y * x \<in> Infinitesimal"
for x y :: "'a::real_normed_algebra star"
apply (rule InfinitesimalI)
apply (drule HFiniteD, clarify)
apply (subgoal_tac "0 < t")
apply (subgoal_tac "hnorm (y * x) < t * (r / t)", simp)
apply (subgoal_tac "0 < r / t")
apply (rule hnorm_mult_less)
apply assumption
apply (simp add: InfinitesimalD)
apply simp
apply (erule order_le_less_trans [OF hnorm_ge_zero])
done
lemma Infinitesimal_scaleR2: "x \<in> Infinitesimal \<Longrightarrow> a *\<^sub>R x \<in> Infinitesimal"
apply (case_tac "a = 0", simp)
apply (rule InfinitesimalI)
apply (drule InfinitesimalD)
apply (drule_tac x="r / \<bar>star_of a\<bar>" in bspec)
apply (simp add: Reals_eq_Standard)
apply simp
apply (simp add: hnorm_scaleR pos_less_divide_eq mult.commute)
done
lemma Compl_HFinite: "- HFinite = HInfinite"
apply (auto simp add: HInfinite_def HFinite_def linorder_not_less)
apply (rule_tac y="r + 1" in order_less_le_trans, simp)
apply simp
done
lemma HInfinite_inverse_Infinitesimal: "x \<in> HInfinite \<Longrightarrow> inverse x \<in> Infinitesimal"
for x :: "'a::real_normed_div_algebra star"
apply (rule InfinitesimalI)
apply (subgoal_tac "x \<noteq> 0")
apply (rule inverse_less_imp_less)
apply (simp add: nonzero_hnorm_inverse)
apply (simp add: HInfinite_def Reals_inverse)
apply assumption
apply (clarify, simp add: Compl_HFinite [symmetric])
done
lemma HInfiniteI: "(\<And>r. r \<in> \<real> \<Longrightarrow> r < hnorm x) \<Longrightarrow> x \<in> HInfinite"
by (simp add: HInfinite_def)
lemma HInfiniteD: "x \<in> HInfinite \<Longrightarrow> r \<in> \<real> \<Longrightarrow> r < hnorm x"
by (simp add: HInfinite_def)
lemma HInfinite_mult: "x \<in> HInfinite \<Longrightarrow> y \<in> HInfinite \<Longrightarrow> x * y \<in> HInfinite"
for x y :: "'a::real_normed_div_algebra star"
apply (rule HInfiniteI, simp only: hnorm_mult)
apply (subgoal_tac "r * 1 < hnorm x * hnorm y", simp only: mult_1)
apply (case_tac "x = 0", simp add: HInfinite_def)
apply (rule mult_strict_mono)
apply (simp_all add: HInfiniteD)
done
lemma hypreal_add_zero_less_le_mono: "r < x \<Longrightarrow> 0 \<le> y \<Longrightarrow> r < x + y"
for r x y :: hypreal
by (auto dest: add_less_le_mono)
lemma HInfinite_add_ge_zero: "x \<in> HInfinite \<Longrightarrow> 0 \<le> y \<Longrightarrow> 0 \<le> x \<Longrightarrow> x + y \<in> HInfinite"
for x y :: hypreal
by (auto simp: abs_if add.commute HInfinite_def)
lemma HInfinite_add_ge_zero2: "x \<in> HInfinite \<Longrightarrow> 0 \<le> y \<Longrightarrow> 0 \<le> x \<Longrightarrow> y + x \<in> HInfinite"
for x y :: hypreal
by (auto intro!: HInfinite_add_ge_zero simp add: add.commute)
lemma HInfinite_add_gt_zero: "x \<in> HInfinite \<Longrightarrow> 0 < y \<Longrightarrow> 0 < x \<Longrightarrow> x + y \<in> HInfinite"
for x y :: hypreal
by (blast intro: HInfinite_add_ge_zero order_less_imp_le)
lemma HInfinite_minus_iff: "- x \<in> HInfinite \<longleftrightarrow> x \<in> HInfinite"
by (simp add: HInfinite_def)
lemma HInfinite_add_le_zero: "x \<in> HInfinite \<Longrightarrow> y \<le> 0 \<Longrightarrow> x \<le> 0 \<Longrightarrow> x + y \<in> HInfinite"
for x y :: hypreal
apply (drule HInfinite_minus_iff [THEN iffD2])
apply (rule HInfinite_minus_iff [THEN iffD1])
apply (simp only: minus_add add.commute)
apply (rule HInfinite_add_ge_zero)
apply simp_all
done
lemma HInfinite_add_lt_zero: "x \<in> HInfinite \<Longrightarrow> y < 0 \<Longrightarrow> x < 0 \<Longrightarrow> x + y \<in> HInfinite"
for x y :: hypreal
by (blast intro: HInfinite_add_le_zero order_less_imp_le)
lemma HFinite_sum_squares:
"a \<in> HFinite \<Longrightarrow> b \<in> HFinite \<Longrightarrow> c \<in> HFinite \<Longrightarrow> a * a + b * b + c * c \<in> HFinite"
for a b c :: "'a::real_normed_algebra star"
by (auto intro: HFinite_mult HFinite_add)
lemma not_Infinitesimal_not_zero: "x \<notin> Infinitesimal \<Longrightarrow> x \<noteq> 0"
by auto
lemma not_Infinitesimal_not_zero2: "x \<in> HFinite - Infinitesimal \<Longrightarrow> x \<noteq> 0"
by auto
lemma HFinite_diff_Infinitesimal_hrabs:
"x \<in> HFinite - Infinitesimal \<Longrightarrow> \<bar>x\<bar> \<in> HFinite - Infinitesimal"
for x :: hypreal
by blast
lemma hnorm_le_Infinitesimal: "e \<in> Infinitesimal \<Longrightarrow> hnorm x \<le> e \<Longrightarrow> x \<in> Infinitesimal"
by (auto simp: Infinitesimal_def abs_less_iff)
lemma hnorm_less_Infinitesimal: "e \<in> Infinitesimal \<Longrightarrow> hnorm x < e \<Longrightarrow> x \<in> Infinitesimal"
by (erule hnorm_le_Infinitesimal, erule order_less_imp_le)
lemma hrabs_le_Infinitesimal: "e \<in> Infinitesimal \<Longrightarrow> \<bar>x\<bar> \<le> e \<Longrightarrow> x \<in> Infinitesimal"
for x :: hypreal
by (erule hnorm_le_Infinitesimal) simp
lemma hrabs_less_Infinitesimal: "e \<in> Infinitesimal \<Longrightarrow> \<bar>x\<bar> < e \<Longrightarrow> x \<in> Infinitesimal"
for x :: hypreal
by (erule hnorm_less_Infinitesimal) simp
lemma Infinitesimal_interval:
"e \<in> Infinitesimal \<Longrightarrow> e' \<in> Infinitesimal \<Longrightarrow> e' < x \<Longrightarrow> x < e \<Longrightarrow> x \<in> Infinitesimal"
for x :: hypreal
by (auto simp add: Infinitesimal_def abs_less_iff)
lemma Infinitesimal_interval2:
"e \<in> Infinitesimal \<Longrightarrow> e' \<in> Infinitesimal \<Longrightarrow> e' \<le> x \<Longrightarrow> x \<le> e \<Longrightarrow> x \<in> Infinitesimal"
for x :: hypreal
by (auto intro: Infinitesimal_interval simp add: order_le_less)
lemma lemma_Infinitesimal_hyperpow: "x \<in> Infinitesimal \<Longrightarrow> 0 < N \<Longrightarrow> \<bar>x pow N\<bar> \<le> \<bar>x\<bar>"
for x :: hypreal
apply (unfold Infinitesimal_def)
apply (auto intro!: hyperpow_Suc_le_self2
simp: hyperpow_hrabs [symmetric] hypnat_gt_zero_iff2 abs_ge_zero)
done
lemma Infinitesimal_hyperpow: "x \<in> Infinitesimal \<Longrightarrow> 0 < N \<Longrightarrow> x pow N \<in> Infinitesimal"
for x :: hypreal
apply (rule hrabs_le_Infinitesimal)
apply (rule_tac [2] lemma_Infinitesimal_hyperpow)
apply auto
done
lemma hrealpow_hyperpow_Infinitesimal_iff:
"(x ^ n \<in> Infinitesimal) \<longleftrightarrow> x pow (hypnat_of_nat n) \<in> Infinitesimal"
by (simp only: hyperpow_hypnat_of_nat)
lemma Infinitesimal_hrealpow: "x \<in> Infinitesimal \<Longrightarrow> 0 < n \<Longrightarrow> x ^ n \<in> Infinitesimal"
for x :: hypreal
by (simp add: hrealpow_hyperpow_Infinitesimal_iff Infinitesimal_hyperpow)
lemma not_Infinitesimal_mult:
"x \<notin> Infinitesimal \<Longrightarrow> y \<notin> Infinitesimal \<Longrightarrow> x * y \<notin> Infinitesimal"
for x y :: "'a::real_normed_div_algebra star"
apply (unfold Infinitesimal_def, clarify, rename_tac r s)
apply (simp only: linorder_not_less hnorm_mult)
apply (drule_tac x = "r * s" in bspec)
apply (fast intro: Reals_mult)
apply simp
apply (drule_tac c = s and d = "hnorm y" and a = r and b = "hnorm x" in mult_mono)
apply simp_all
done
lemma Infinitesimal_mult_disj: "x * y \<in> Infinitesimal \<Longrightarrow> x \<in> Infinitesimal \<or> y \<in> Infinitesimal"
for x y :: "'a::real_normed_div_algebra star"
apply (rule ccontr)
apply (drule de_Morgan_disj [THEN iffD1])
apply (fast dest: not_Infinitesimal_mult)
done
lemma HFinite_Infinitesimal_not_zero: "x \<in> HFinite-Infinitesimal \<Longrightarrow> x \<noteq> 0"
by blast
lemma HFinite_Infinitesimal_diff_mult:
"x \<in> HFinite - Infinitesimal \<Longrightarrow> y \<in> HFinite - Infinitesimal \<Longrightarrow> x * y \<in> HFinite - Infinitesimal"
for x y :: "'a::real_normed_div_algebra star"
apply clarify
apply (blast dest: HFinite_mult not_Infinitesimal_mult)
done
lemma Infinitesimal_subset_HFinite: "Infinitesimal \<subseteq> HFinite"
apply (simp add: Infinitesimal_def HFinite_def)
apply auto
apply (rule_tac x = 1 in bexI)
apply auto
done
lemma Infinitesimal_star_of_mult: "x \<in> Infinitesimal \<Longrightarrow> x * star_of r \<in> Infinitesimal"
for x :: "'a::real_normed_algebra star"
by (erule HFinite_star_of [THEN [2] Infinitesimal_HFinite_mult])
lemma Infinitesimal_star_of_mult2: "x \<in> Infinitesimal \<Longrightarrow> star_of r * x \<in> 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 \<in> Infinitesimal \<longleftrightarrow> x \<approx> 0"
by (simp add: Infinitesimal_def approx_def)
lemma approx_minus_iff: "x \<approx> y \<longleftrightarrow> x - y \<approx> 0"
by (simp add: approx_def)
lemma approx_minus_iff2: "x \<approx> y \<longleftrightarrow> - y + x \<approx> 0"
by (simp add: approx_def add.commute)
lemma approx_refl [iff]: "x \<approx> x"
by (simp add: approx_def Infinitesimal_def)
lemma hypreal_minus_distrib1: "- (y + - x) = x + -y"
for x y :: "'a::ab_group_add"
by (simp add: add.commute)
lemma approx_sym: "x \<approx> y \<Longrightarrow> y \<approx> x"
apply (simp add: approx_def)
apply (drule Infinitesimal_minus_iff [THEN iffD2])
apply simp
done
lemma approx_trans: "x \<approx> y \<Longrightarrow> y \<approx> z \<Longrightarrow> x \<approx> z"
apply (simp add: approx_def)
apply (drule (1) Infinitesimal_add)
apply simp
done
lemma approx_trans2: "r \<approx> x \<Longrightarrow> s \<approx> x \<Longrightarrow> r \<approx> s"
by (blast intro: approx_sym approx_trans)
lemma approx_trans3: "x \<approx> r \<Longrightarrow> x \<approx> s \<Longrightarrow> r \<approx> s"
by (blast intro: approx_sym approx_trans)
lemma approx_reorient: "x \<approx> y \<longleftrightarrow> y \<approx> 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 \<approx> x" | "1 \<approx> y" | "numeral w \<approx> z" | "- 1 \<approx> y" | "- numeral w \<approx> r") =
\<open>
let val rule = @{thm approx_reorient} RS eq_reflection
fun proc phi ss ct =
case Thm.term_of ct of
_ $ t $ u => if can HOLogic.dest_number u then NONE
else if can HOLogic.dest_number t then SOME rule else NONE
| _ => NONE
in proc end
\<close>
lemma Infinitesimal_approx_minus: "x - y \<in> Infinitesimal \<longleftrightarrow> x \<approx> y"
by (simp add: approx_minus_iff [symmetric] mem_infmal_iff)
lemma approx_monad_iff: "x \<approx> y \<longleftrightarrow> monad x = monad y"
by (auto simp add: monad_def dest: approx_sym elim!: approx_trans equalityCE)
lemma Infinitesimal_approx: "x \<in> Infinitesimal \<Longrightarrow> y \<in> Infinitesimal \<Longrightarrow> x \<approx> y"
apply (simp add: mem_infmal_iff)
apply (blast intro: approx_trans approx_sym)
done
lemma approx_add: "a \<approx> b \<Longrightarrow> c \<approx> d \<Longrightarrow> a + c \<approx> b + d"
proof (unfold approx_def)
assume inf: "a - b \<in> Infinitesimal" "c - d \<in> Infinitesimal"
have "a + c - (b + d) = (a - b) + (c - d)" by simp
also have "... \<in> Infinitesimal"
using inf by (rule Infinitesimal_add)
finally show "a + c - (b + d) \<in> Infinitesimal" .
qed
lemma approx_minus: "a \<approx> b \<Longrightarrow> - a \<approx> - b"
apply (rule approx_minus_iff [THEN iffD2, THEN approx_sym])
apply (drule approx_minus_iff [THEN iffD1])
apply (simp add: add.commute)
done
lemma approx_minus2: "- a \<approx> - b \<Longrightarrow> a \<approx> b"
by (auto dest: approx_minus)
lemma approx_minus_cancel [simp]: "- a \<approx> - b \<longleftrightarrow> a \<approx> b"
by (blast intro: approx_minus approx_minus2)
lemma approx_add_minus: "a \<approx> b \<Longrightarrow> c \<approx> d \<Longrightarrow> a + - c \<approx> b + - d"
by (blast intro!: approx_add approx_minus)
lemma approx_diff: "a \<approx> b \<Longrightarrow> c \<approx> d \<Longrightarrow> a - c \<approx> b - d"
using approx_add [of a b "- c" "- d"] by simp
lemma approx_mult1: "a \<approx> b \<Longrightarrow> c \<in> HFinite \<Longrightarrow> a * c \<approx> 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 \<approx> b \<Longrightarrow> c \<in> HFinite \<Longrightarrow> c * a \<approx> 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 \<approx> v * x \<Longrightarrow> x \<approx> y \<Longrightarrow> v \<in> HFinite \<Longrightarrow> u \<approx> v * y"
for u v x y :: "'a::real_normed_algebra star"
by (blast intro: approx_mult2 approx_trans)
lemma approx_mult_subst2: "u \<approx> x * v \<Longrightarrow> x \<approx> y \<Longrightarrow> v \<in> HFinite \<Longrightarrow> u \<approx> 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 \<approx> x * star_of v \<Longrightarrow> x \<approx> y \<Longrightarrow> u \<approx> 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 \<Longrightarrow> a \<approx> b"
by (simp add: approx_def)
lemma Infinitesimal_minus_approx: "x \<in> Infinitesimal \<Longrightarrow> - x \<approx> x"
by (blast intro: Infinitesimal_minus_iff [THEN iffD2] mem_infmal_iff [THEN iffD1] approx_trans2)
lemma bex_Infinitesimal_iff: "(\<exists>y \<in> Infinitesimal. x - z = y) \<longleftrightarrow> x \<approx> z"
by (simp add: approx_def)
lemma bex_Infinitesimal_iff2: "(\<exists>y \<in> Infinitesimal. x = z + y) \<longleftrightarrow> x \<approx> z"
by (force simp add: bex_Infinitesimal_iff [symmetric])
lemma Infinitesimal_add_approx: "y \<in> Infinitesimal \<Longrightarrow> x + y = z \<Longrightarrow> x \<approx> z"
apply (rule bex_Infinitesimal_iff [THEN iffD1])
apply (drule Infinitesimal_minus_iff [THEN iffD2])
apply (auto simp add: add.assoc [symmetric])
done
lemma Infinitesimal_add_approx_self: "y \<in> Infinitesimal \<Longrightarrow> x \<approx> x + y"
apply (rule bex_Infinitesimal_iff [THEN iffD1])
apply (drule Infinitesimal_minus_iff [THEN iffD2])
apply (auto simp add: add.assoc [symmetric])
done
lemma Infinitesimal_add_approx_self2: "y \<in> Infinitesimal \<Longrightarrow> x \<approx> y + x"
by (auto dest: Infinitesimal_add_approx_self simp add: add.commute)
lemma Infinitesimal_add_minus_approx_self: "y \<in> Infinitesimal \<Longrightarrow> x \<approx> x + - y"
by (blast intro!: Infinitesimal_add_approx_self Infinitesimal_minus_iff [THEN iffD2])
lemma Infinitesimal_add_cancel: "y \<in> Infinitesimal \<Longrightarrow> x + y \<approx> z \<Longrightarrow> x \<approx> z"
apply (drule_tac x = x in Infinitesimal_add_approx_self [THEN approx_sym])
apply (erule approx_trans3 [THEN approx_sym], assumption)
done
lemma Infinitesimal_add_right_cancel: "y \<in> Infinitesimal \<Longrightarrow> x \<approx> z + y \<Longrightarrow> x \<approx> z"
apply (drule_tac x = z in Infinitesimal_add_approx_self2 [THEN approx_sym])
apply (erule approx_trans3 [THEN approx_sym])
apply (simp add: add.commute)
apply (erule approx_sym)
done
lemma approx_add_left_cancel: "d + b \<approx> d + c \<Longrightarrow> b \<approx> c"
apply (drule approx_minus_iff [THEN iffD1])
apply (simp add: approx_minus_iff [symmetric] ac_simps)
done
lemma approx_add_right_cancel: "b + d \<approx> c + d \<Longrightarrow> b \<approx> c"
apply (rule approx_add_left_cancel)
apply (simp add: add.commute)
done
lemma approx_add_mono1: "b \<approx> c \<Longrightarrow> d + b \<approx> d + c"
apply (rule approx_minus_iff [THEN iffD2])
apply (simp add: approx_minus_iff [symmetric] ac_simps)
done
lemma approx_add_mono2: "b \<approx> c \<Longrightarrow> b + a \<approx> c + a"
by (simp add: add.commute approx_add_mono1)
lemma approx_add_left_iff [simp]: "a + b \<approx> a + c \<longleftrightarrow> b \<approx> c"
by (fast elim: approx_add_left_cancel approx_add_mono1)
lemma approx_add_right_iff [simp]: "b + a \<approx> c + a \<longleftrightarrow> b \<approx> c"
by (simp add: add.commute)
lemma approx_HFinite: "x \<in> HFinite \<Longrightarrow> x \<approx> y \<Longrightarrow> y \<in> HFinite"
apply (drule bex_Infinitesimal_iff2 [THEN iffD2], safe)
apply (drule Infinitesimal_subset_HFinite [THEN subsetD, THEN HFinite_minus_iff [THEN iffD2]])
apply (drule HFinite_add)
apply (auto simp add: add.assoc)
done
lemma approx_star_of_HFinite: "x \<approx> star_of D \<Longrightarrow> x \<in> HFinite"
by (rule approx_sym [THEN [2] approx_HFinite], auto)
lemma approx_mult_HFinite: "a \<approx> b \<Longrightarrow> c \<approx> d \<Longrightarrow> b \<in> HFinite \<Longrightarrow> d \<in> HFinite \<Longrightarrow> a * c \<approx> b * d"
for a b c d :: "'a::real_normed_algebra star"
apply (rule approx_trans)
apply (rule_tac [2] approx_mult2)
apply (rule approx_mult1)
prefer 2 apply (blast intro: approx_HFinite approx_sym, auto)
done
lemma scaleHR_left_diff_distrib: "\<And>a b x. scaleHR (a - b) x = scaleHR a x - scaleHR b x"
by transfer (rule scaleR_left_diff_distrib)
lemma approx_scaleR1: "a \<approx> star_of b \<Longrightarrow> c \<in> HFinite \<Longrightarrow> scaleHR a c \<approx> b *\<^sub>R c"
apply (unfold approx_def)
apply (drule (1) Infinitesimal_HFinite_scaleHR)
apply (simp only: scaleHR_left_diff_distrib)
apply (simp add: scaleHR_def star_scaleR_def [symmetric])
done
lemma approx_scaleR2: "a \<approx> b \<Longrightarrow> c *\<^sub>R a \<approx> c *\<^sub>R b"
by (simp add: approx_def Infinitesimal_scaleR2 scaleR_right_diff_distrib [symmetric])
lemma approx_scaleR_HFinite: "a \<approx> star_of b \<Longrightarrow> c \<approx> d \<Longrightarrow> d \<in> HFinite \<Longrightarrow> scaleHR a c \<approx> b *\<^sub>R d"
apply (rule approx_trans)
apply (rule_tac [2] approx_scaleR2)
apply (rule approx_scaleR1)
prefer 2 apply (blast intro: approx_HFinite approx_sym, auto)
done
lemma approx_mult_star_of: "a \<approx> star_of b \<Longrightarrow> c \<approx> star_of d \<Longrightarrow> a * c \<approx> star_of b * star_of d"
for a c :: "'a::real_normed_algebra star"
by (blast intro!: approx_mult_HFinite approx_star_of_HFinite HFinite_star_of)
lemma approx_SReal_mult_cancel_zero: "a \<in> \<real> \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> a * x \<approx> 0 \<Longrightarrow> x \<approx> 0"
for a x :: hypreal
apply (drule Reals_inverse [THEN SReal_subset_HFinite [THEN subsetD]])
apply (auto dest: approx_mult2 simp add: mult.assoc [symmetric])
done
lemma approx_mult_SReal1: "a \<in> \<real> \<Longrightarrow> x \<approx> 0 \<Longrightarrow> x * a \<approx> 0"
for a x :: hypreal
by (auto dest: SReal_subset_HFinite [THEN subsetD] approx_mult1)
lemma approx_mult_SReal2: "a \<in> \<real> \<Longrightarrow> x \<approx> 0 \<Longrightarrow> a * x \<approx> 0"
for a x :: hypreal
by (auto dest: SReal_subset_HFinite [THEN subsetD] approx_mult2)
lemma approx_mult_SReal_zero_cancel_iff [simp]: "a \<in> \<real> \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> a * x \<approx> 0 \<longleftrightarrow> x \<approx> 0"
for a x :: hypreal
by (blast intro: approx_SReal_mult_cancel_zero approx_mult_SReal2)
lemma approx_SReal_mult_cancel: "a \<in> \<real> \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> a * w \<approx> a * z \<Longrightarrow> w \<approx> z"
for a w z :: hypreal
apply (drule Reals_inverse [THEN SReal_subset_HFinite [THEN subsetD]])
apply (auto dest: approx_mult2 simp add: mult.assoc [symmetric])
done
lemma approx_SReal_mult_cancel_iff1 [simp]: "a \<in> \<real> \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> a * w \<approx> a * z \<longleftrightarrow> w \<approx> z"
for a w z :: hypreal
by (auto intro!: approx_mult2 SReal_subset_HFinite [THEN subsetD]
intro: approx_SReal_mult_cancel)
lemma approx_le_bound: "z \<le> f \<Longrightarrow> f \<approx> g \<Longrightarrow> g \<le> z ==> f \<approx> z"
for z :: hypreal
apply (simp add: bex_Infinitesimal_iff2 [symmetric], auto)
apply (rule_tac x = "g + y - z" in bexI)
apply simp
apply (rule Infinitesimal_interval2)
apply (rule_tac [2] Infinitesimal_zero, auto)
done
lemma approx_hnorm: "x \<approx> y \<Longrightarrow> hnorm x \<approx> hnorm y"
for x y :: "'a::real_normed_vector star"
proof (unfold approx_def)
assume "x - y \<in> Infinitesimal"
then have "hnorm (x - y) \<in> Infinitesimal"
by (simp only: Infinitesimal_hnorm_iff)
moreover have "(0::real star) \<in> Infinitesimal"
by (rule Infinitesimal_zero)
moreover have "0 \<le> \<bar>hnorm x - hnorm y\<bar>"
by (rule abs_ge_zero)
moreover have "\<bar>hnorm x - hnorm y\<bar> \<le> hnorm (x - y)"
by (rule hnorm_triangle_ineq3)
ultimately have "\<bar>hnorm x - hnorm y\<bar> \<in> Infinitesimal"
by (rule Infinitesimal_interval2)
then show "hnorm x - hnorm y \<in> 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 \<in> \<real> \<Longrightarrow> y \<in> Infinitesimal \<Longrightarrow> 0 < x \<Longrightarrow> y < x"
for x y :: hypreal
apply (simp add: Infinitesimal_def)
apply (rule abs_ge_self [THEN order_le_less_trans], auto)
done
lemma Infinitesimal_less_SReal2: "y \<in> Infinitesimal \<Longrightarrow> \<forall>r \<in> Reals. 0 < r \<longrightarrow> y < r"
for y :: hypreal
by (blast intro: Infinitesimal_less_SReal)
lemma SReal_not_Infinitesimal: "0 < y \<Longrightarrow> y \<in> \<real> ==> y \<notin> Infinitesimal"
for y :: hypreal
apply (simp add: Infinitesimal_def)
apply (auto simp add: abs_if)
done
lemma SReal_minus_not_Infinitesimal: "y < 0 \<Longrightarrow> y \<in> \<real> \<Longrightarrow> y \<notin> Infinitesimal"
for y :: hypreal
apply (subst Infinitesimal_minus_iff [symmetric])
apply (rule SReal_not_Infinitesimal, auto)
done
lemma SReal_Int_Infinitesimal_zero: "\<real> Int Infinitesimal = {0::hypreal}"
apply auto
apply (cut_tac x = x and y = 0 in linorder_less_linear)
apply (blast dest: SReal_not_Infinitesimal SReal_minus_not_Infinitesimal)
done
lemma SReal_Infinitesimal_zero: "x \<in> \<real> \<Longrightarrow> x \<in> Infinitesimal \<Longrightarrow> x = 0"
for x :: hypreal
using SReal_Int_Infinitesimal_zero by blast
lemma SReal_HFinite_diff_Infinitesimal: "x \<in> \<real> \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> x \<in> 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 \<noteq> 0 \<Longrightarrow> hypreal_of_real x \<in> HFinite - Infinitesimal"
by (rule SReal_HFinite_diff_Infinitesimal) auto
lemma star_of_Infinitesimal_iff_0 [iff]: "star_of x \<in> Infinitesimal \<longleftrightarrow> x = 0"
apply (auto simp add: Infinitesimal_def)
apply (drule_tac x="hnorm (star_of x)" in bspec)
apply (simp add: SReal_def)
apply (rule_tac x="norm x" in exI, simp)
apply simp
done
lemma star_of_HFinite_diff_Infinitesimal: "x \<noteq> 0 \<Longrightarrow> star_of x \<in> HFinite - Infinitesimal"
by simp
lemma numeral_not_Infinitesimal [simp]:
"numeral w \<noteq> (0::hypreal) \<Longrightarrow> (numeral w :: hypreal) \<notin> 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) \<notin> Infinitesimal"
apply (simp only: star_one_def star_of_Infinitesimal_iff_0)
apply simp
done
lemma approx_SReal_not_zero: "y \<in> \<real> \<Longrightarrow> x \<approx> y \<Longrightarrow> y \<noteq> 0 \<Longrightarrow> x \<noteq> 0"
for x y :: hypreal
apply (cut_tac x = 0 and y = y in linorder_less_linear, simp)
apply (blast dest: approx_sym [THEN mem_infmal_iff [THEN iffD2]]
SReal_not_Infinitesimal SReal_minus_not_Infinitesimal)
done
lemma HFinite_diff_Infinitesimal_approx:
"x \<approx> y \<Longrightarrow> y \<in> HFinite - Infinitesimal \<Longrightarrow> x \<in> HFinite - Infinitesimal"
apply (auto intro: approx_sym [THEN [2] approx_HFinite] simp: mem_infmal_iff)
apply (drule approx_trans3, assumption)
apply (blast dest: approx_sym)
done
text \<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 \<noteq> 0 \<Longrightarrow> y \<in> Infinitesimal \<Longrightarrow> x / y \<in> HFinite \<Longrightarrow> x \<in> Infinitesimal"
for x y :: "'a::{real_normed_div_algebra,field} star"
apply (drule Infinitesimal_HFinite_mult2, assumption)
apply (simp add: divide_inverse mult.assoc)
done
lemma Infinitesimal_SReal_divide: "x \<in> Infinitesimal \<Longrightarrow> y \<in> \<real> \<Longrightarrow> x / y \<in> Infinitesimal"
for x y :: hypreal
apply (simp add: divide_inverse)
apply (auto intro!: Infinitesimal_HFinite_mult
dest!: Reals_inverse [THEN SReal_subset_HFinite [THEN subsetD]])
done
section \<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 \<approx> star_of y \<longleftrightarrow> x = y"
apply safe
apply (simp add: approx_def)
apply (simp only: star_of_diff [symmetric])
apply (simp only: star_of_Infinitesimal_iff_0)
apply simp
done
lemma SReal_approx_iff: "x \<in> \<real> \<Longrightarrow> y \<in> \<real> \<Longrightarrow> x \<approx> y \<longleftrightarrow> x = y"
for x y :: hypreal
apply auto
apply (simp add: approx_def)
apply (drule (1) Reals_diff)
apply (drule (1) SReal_Infinitesimal_zero)
apply simp
done
lemma numeral_approx_iff [simp]:
"(numeral v \<approx> (numeral w :: 'a::{numeral,real_normed_vector} star)) =
(numeral v = (numeral w :: 'a))"
apply (unfold star_numeral_def)
apply (rule star_of_approx_iff)
done
text \<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 \<approx> (0::'a::{numeral,real_normed_vector} star)) = (numeral w = (0::'a))"
"((0::'a::{numeral,real_normed_vector} star) \<approx> numeral w) = (numeral w = (0::'a))"
"(numeral w \<approx> (1::'b::{numeral,one,real_normed_vector} star)) = (numeral w = (1::'b))"
"((1::'b::{numeral,one,real_normed_vector} star) \<approx> numeral w) = (numeral w = (1::'b))"
"\<not> (0 \<approx> (1::'c::{zero_neq_one,real_normed_vector} star))"
"\<not> (1 \<approx> (0::'c::{zero_neq_one,real_normed_vector} star))"
apply (unfold star_numeral_def star_zero_def star_one_def)
apply (unfold star_of_approx_iff)
apply (auto intro: sym)
done
lemma star_of_approx_numeral_iff [simp]: "star_of k \<approx> numeral w \<longleftrightarrow> k = numeral w"
by (subst star_of_approx_iff [symmetric]) auto
lemma star_of_approx_zero_iff [simp]: "star_of k \<approx> 0 \<longleftrightarrow> k = 0"
by (simp_all add: star_of_approx_iff [symmetric])
lemma star_of_approx_one_iff [simp]: "star_of k \<approx> 1 \<longleftrightarrow> k = 1"
by (simp_all add: star_of_approx_iff [symmetric])
lemma approx_unique_real: "r \<in> \<real> \<Longrightarrow> s \<in> \<real> \<Longrightarrow> r \<approx> x \<Longrightarrow> s \<approx> x \<Longrightarrow> 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 \<real> (hypreal_of_real ` Q) (hypreal_of_real Y) = isUb UNIV Q Y"
for Q :: "real set" and Y :: real
by (simp add: isUb_def setle_def)
lemma hypreal_of_real_isLub1: "isLub \<real> (hypreal_of_real ` Q) (hypreal_of_real Y) \<Longrightarrow> isLub UNIV Q Y"
for Q :: "real set" and Y :: real
apply (simp add: isLub_def leastP_def)
apply (auto intro: hypreal_of_real_isUb_iff [THEN iffD2]
simp add: hypreal_of_real_isUb_iff setge_def)
done
lemma hypreal_of_real_isLub2: "isLub UNIV Q Y \<Longrightarrow> isLub \<real> (hypreal_of_real ` Q) (hypreal_of_real Y)"
for Q :: "real set" and Y :: real
apply (auto simp add: isLub_def leastP_def hypreal_of_real_isUb_iff setge_def)
apply (metis SReal_iff hypreal_of_real_isUb_iff isUbD2a star_of_le)
done
lemma hypreal_of_real_isLub_iff:
"isLub \<real> (hypreal_of_real ` Q) (hypreal_of_real Y) = isLub (UNIV :: real set) Q Y"
for Q :: "real set" and Y :: real
by (blast intro: hypreal_of_real_isLub1 hypreal_of_real_isLub2)
lemma lemma_isUb_hypreal_of_real: "isUb \<real> P Y \<Longrightarrow> \<exists>Yo. isUb \<real> P (hypreal_of_real Yo)"
by (auto simp add: SReal_iff isUb_def)
lemma lemma_isLub_hypreal_of_real: "isLub \<real> P Y \<Longrightarrow> \<exists>Yo. isLub \<real> P (hypreal_of_real Yo)"
by (auto simp add: isLub_def leastP_def isUb_def SReal_iff)
lemma lemma_isLub_hypreal_of_real2: "\<exists>Yo. isLub \<real> P (hypreal_of_real Yo) \<Longrightarrow> \<exists>Y. isLub \<real> P Y"
by (auto simp add: isLub_def leastP_def isUb_def)
lemma SReal_complete: "P \<subseteq> \<real> \<Longrightarrow> \<exists>x. x \<in> P \<Longrightarrow> \<exists>Y. isUb \<real> P Y \<Longrightarrow> \<exists>t::hypreal. isLub \<real> P t"
apply (frule SReal_hypreal_of_real_image)
apply (auto, drule lemma_isUb_hypreal_of_real)
apply (auto intro!: reals_complete lemma_isLub_hypreal_of_real2
simp add: hypreal_of_real_isLub_iff hypreal_of_real_isUb_iff)
done
text \<open>Lemmas about lubs.\<close>
lemma lemma_st_part_ub: "x \<in> HFinite \<Longrightarrow> \<exists>u. isUb \<real> {s. s \<in> \<real> \<and> s < x} u"
for x :: hypreal
apply (drule HFiniteD, safe)
apply (rule exI, rule isUbI)
apply (auto intro: setleI isUbI simp add: abs_less_iff)
done
lemma lemma_st_part_nonempty: "x \<in> HFinite \<Longrightarrow> \<exists>y. y \<in> {s. s \<in> \<real> \<and> s < x}"
for x :: hypreal
apply (drule HFiniteD, safe)
apply (drule Reals_minus)
apply (rule_tac x = "-t" in exI)
apply (auto simp add: abs_less_iff)
done
lemma lemma_st_part_lub: "x \<in> HFinite \<Longrightarrow> \<exists>t. isLub \<real> {s. s \<in> \<real> \<and> s < x} t"
for x :: hypreal
by (blast intro!: SReal_complete lemma_st_part_ub lemma_st_part_nonempty Collect_restrict)
lemma lemma_st_part_le1:
"x \<in> HFinite \<Longrightarrow> isLub \<real> {s. s \<in> \<real> \<and> s < x} t \<Longrightarrow> r \<in> \<real> \<Longrightarrow> 0 < r \<Longrightarrow> x \<le> t + r"
for x r t :: hypreal
apply (frule isLubD1a)
apply (rule ccontr, drule linorder_not_le [THEN iffD2])
apply (drule (1) Reals_add)
apply (drule_tac y = "r + t" in isLubD1 [THEN setleD], auto)
done
lemma hypreal_setle_less_trans: "S *<= x \<Longrightarrow> x < y \<Longrightarrow> S *<= y"
for x y :: hypreal
apply (simp add: setle_def)
apply (auto dest!: bspec order_le_less_trans intro: order_less_imp_le)
done
lemma hypreal_gt_isUb: "isUb R S x \<Longrightarrow> x < y \<Longrightarrow> y \<in> R \<Longrightarrow> isUb R S y"
for x y :: hypreal
apply (simp add: isUb_def)
apply (blast intro: hypreal_setle_less_trans)
done
lemma lemma_st_part_gt_ub: "x \<in> HFinite \<Longrightarrow> x < y \<Longrightarrow> y \<in> \<real> \<Longrightarrow> isUb \<real> {s. s \<in> \<real> \<and> s < x} y"
for x y :: hypreal
by (auto dest: order_less_trans intro: order_less_imp_le intro!: isUbI setleI)
lemma lemma_minus_le_zero: "t \<le> t + -r \<Longrightarrow> r \<le> 0"
for r t :: hypreal
apply (drule_tac c = "-t" in add_left_mono)
apply (auto simp add: add.assoc [symmetric])
done
lemma lemma_st_part_le2:
"x \<in> HFinite \<Longrightarrow> isLub \<real> {s. s \<in> \<real> \<and> s < x} t \<Longrightarrow> r \<in> \<real> \<Longrightarrow> 0 < r \<Longrightarrow> t + -r \<le> x"
for x r t :: hypreal
apply (frule isLubD1a)
apply (rule ccontr, drule linorder_not_le [THEN iffD1])
apply (drule Reals_minus, drule_tac a = t in Reals_add, assumption)
apply (drule lemma_st_part_gt_ub, assumption+)
apply (drule isLub_le_isUb, assumption)
apply (drule lemma_minus_le_zero)
apply (auto dest: order_less_le_trans)
done
lemma lemma_st_part1a:
"x \<in> HFinite \<Longrightarrow> isLub \<real> {s. s \<in> \<real> \<and> s < x} t \<Longrightarrow> r \<in> \<real> \<Longrightarrow> 0 < r \<Longrightarrow> x + -t \<le> r"
for x r t :: hypreal
apply (subgoal_tac "x \<le> t + r")
apply (auto intro: lemma_st_part_le1)
done
lemma lemma_st_part2a:
"x \<in> HFinite \<Longrightarrow> isLub \<real> {s. s \<in> \<real> \<and> s < x} t \<Longrightarrow> r \<in> \<real> \<Longrightarrow> 0 < r \<Longrightarrow> - (x + -t) \<le> r"
for x r t :: hypreal
apply (subgoal_tac "(t + -r \<le> x)")
apply simp
apply (rule lemma_st_part_le2)
apply auto
done
lemma lemma_SReal_ub: "x \<in> \<real> \<Longrightarrow> isUb \<real> {s. s \<in> \<real> \<and> s < x} x"
for x :: hypreal
by (auto intro: isUbI setleI order_less_imp_le)
lemma lemma_SReal_lub: "x \<in> \<real> \<Longrightarrow> isLub \<real> {s. s \<in> \<real> \<and> s < x} x"
for x :: hypreal
apply (auto intro!: isLubI2 lemma_SReal_ub setgeI)
apply (frule isUbD2a)
apply (rule_tac x = x and y = y in linorder_cases)
apply (auto intro!: order_less_imp_le)
apply (drule SReal_dense, assumption, assumption, safe)
apply (drule_tac y = r in isUbD)
apply (auto dest: order_less_le_trans)
done
lemma lemma_st_part_not_eq1:
"x \<in> HFinite \<Longrightarrow> isLub \<real> {s. s \<in> \<real> \<and> s < x} t \<Longrightarrow> r \<in> \<real> \<Longrightarrow> 0 < r \<Longrightarrow> x + - t \<noteq> r"
for x r t :: hypreal
apply auto
apply (frule isLubD1a [THEN Reals_minus])
using Reals_add_cancel [of x "- t"] apply simp
apply (drule_tac x = x in lemma_SReal_lub)
apply (drule isLub_unique, assumption, auto)
done
lemma lemma_st_part_not_eq2:
"x \<in> HFinite \<Longrightarrow> isLub \<real> {s. s \<in> \<real> \<and> s < x} t \<Longrightarrow> r \<in> \<real> \<Longrightarrow> 0 < r \<Longrightarrow> - (x + -t) \<noteq> r"
for x r t :: hypreal
apply (auto)
apply (frule isLubD1a)
using Reals_add_cancel [of "- x" t] apply simp
apply (drule_tac x = x in lemma_SReal_lub)
apply (drule isLub_unique, assumption, auto)
done
lemma lemma_st_part_major:
"x \<in> HFinite \<Longrightarrow> isLub \<real> {s. s \<in> \<real> \<and> s < x} t \<Longrightarrow> r \<in> \<real> \<Longrightarrow> 0 < r \<Longrightarrow> \<bar>x - t\<bar> < r"
for x r t :: hypreal
apply (frule lemma_st_part1a)
apply (frule_tac [4] lemma_st_part2a, auto)
apply (drule order_le_imp_less_or_eq)+
apply auto
using lemma_st_part_not_eq2 apply fastforce
using lemma_st_part_not_eq1 apply fastforce
done
lemma lemma_st_part_major2:
"x \<in> HFinite \<Longrightarrow> isLub \<real> {s. s \<in> \<real> \<and> s < x} t \<Longrightarrow> \<forall>r \<in> Reals. 0 < r \<longrightarrow> \<bar>x - t\<bar> < 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 \<in> HFinite \<Longrightarrow> \<exists>t\<in>Reals. \<forall>r \<in> Reals. 0 < r \<longrightarrow> \<bar>x - t\<bar> < r"
for x :: hypreal
apply (frule lemma_st_part_lub, safe)
apply (frule isLubD1a)
apply (blast dest: lemma_st_part_major2)
done
lemma st_part_Ex: "x \<in> HFinite \<Longrightarrow> \<exists>t\<in>Reals. x \<approx> t"
for x :: hypreal
apply (simp add: approx_def Infinitesimal_def)
apply (drule lemma_st_part_Ex, auto)
done
text \<open>There is a unique real infinitely close.\<close>
lemma st_part_Ex1: "x \<in> HFinite \<Longrightarrow> \<exists>!t::hypreal. t \<in> \<real> \<and> x \<approx> t"
apply (drule st_part_Ex, safe)
apply (drule_tac [2] approx_sym, drule_tac [2] approx_sym, drule_tac [2] approx_sym)
apply (auto intro!: approx_unique_real)
done
subsection \<open>Finite, Infinite and Infinitesimal\<close>
lemma HFinite_Int_HInfinite_empty [simp]: "HFinite Int HInfinite = {}"
apply (simp add: HFinite_def HInfinite_def)
apply (auto dest: order_less_trans)
done
lemma HFinite_not_HInfinite:
assumes x: "x \<in> HFinite"
shows "x \<notin> HInfinite"
proof
assume x': "x \<in> HInfinite"
with x have "x \<in> HFinite \<inter> HInfinite" by blast
then show False by auto
qed
lemma not_HFinite_HInfinite: "x \<notin> HFinite \<Longrightarrow> x \<in> HInfinite"
apply (simp add: HInfinite_def HFinite_def, auto)
apply (drule_tac x = "r + 1" in bspec)
apply (auto)
done
lemma HInfinite_HFinite_disj: "x \<in> HInfinite \<or> x \<in> HFinite"
by (blast intro: not_HFinite_HInfinite)
lemma HInfinite_HFinite_iff: "x \<in> HInfinite \<longleftrightarrow> x \<notin> HFinite"
by (blast dest: HFinite_not_HInfinite not_HFinite_HInfinite)
lemma HFinite_HInfinite_iff: "x \<in> HFinite \<longleftrightarrow> x \<notin> HInfinite"
by (simp add: HInfinite_HFinite_iff)
lemma HInfinite_diff_HFinite_Infinitesimal_disj:
"x \<notin> Infinitesimal \<Longrightarrow> x \<in> HInfinite \<or> x \<in> HFinite - Infinitesimal"
by (fast intro: not_HFinite_HInfinite)
lemma HFinite_inverse: "x \<in> HFinite \<Longrightarrow> x \<notin> Infinitesimal \<Longrightarrow> inverse x \<in> HFinite"
for x :: "'a::real_normed_div_algebra star"
apply (subgoal_tac "x \<noteq> 0")
apply (cut_tac x = "inverse x" in HInfinite_HFinite_disj)
apply (auto dest!: HInfinite_inverse_Infinitesimal simp: nonzero_inverse_inverse_eq)
done
lemma HFinite_inverse2: "x \<in> HFinite - Infinitesimal \<Longrightarrow> inverse x \<in> 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 \<notin> Infinitesimal \<Longrightarrow> inverse x \<in> HFinite"
for x :: "'a::real_normed_div_algebra star"
apply (drule HInfinite_diff_HFinite_Infinitesimal_disj)
apply (blast intro: HFinite_inverse HInfinite_inverse_Infinitesimal Infinitesimal_subset_HFinite [THEN subsetD])
done
lemma HFinite_not_Infinitesimal_inverse:
"x \<in> HFinite - Infinitesimal \<Longrightarrow> inverse x \<in> HFinite - Infinitesimal"
for x :: "'a::real_normed_div_algebra star"
apply (auto intro: Infinitesimal_inverse_HFinite)
apply (drule Infinitesimal_HFinite_mult2, assumption)
apply (simp add: not_Infinitesimal_not_zero)
done
lemma approx_inverse: "x \<approx> y \<Longrightarrow> y \<in> HFinite - Infinitesimal \<Longrightarrow> inverse x \<approx> inverse y"
for x y :: "'a::real_normed_div_algebra star"
apply (frule HFinite_diff_Infinitesimal_approx, assumption)
apply (frule not_Infinitesimal_not_zero2)
apply (frule_tac x = x in not_Infinitesimal_not_zero2)
apply (drule HFinite_inverse2)+
apply (drule approx_mult2, assumption, auto)
apply (drule_tac c = "inverse x" in approx_mult1, assumption)
apply (auto intro: approx_sym simp add: mult.assoc)
done
(*Used for NSLIM_inverse, NSLIMSEQ_inverse*)
lemmas star_of_approx_inverse = star_of_HFinite_diff_Infinitesimal [THEN [2] approx_inverse]
lemmas hypreal_of_real_approx_inverse = hypreal_of_real_HFinite_diff_Infinitesimal [THEN [2] approx_inverse]
lemma inverse_add_Infinitesimal_approx:
"x \<in> HFinite - Infinitesimal \<Longrightarrow> h \<in> Infinitesimal \<Longrightarrow> inverse (x + h) \<approx> 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 \<in> HFinite - Infinitesimal \<Longrightarrow> h \<in> Infinitesimal \<Longrightarrow> inverse (h + x) \<approx> inverse x"
for x h :: "'a::real_normed_div_algebra star"
apply (rule add.commute [THEN subst])
apply (blast intro: inverse_add_Infinitesimal_approx)
done
lemma inverse_add_Infinitesimal_approx_Infinitesimal:
"x \<in> HFinite - Infinitesimal \<Longrightarrow> h \<in> Infinitesimal \<Longrightarrow> inverse (x + h) - inverse x \<approx> h"
for x h :: "'a::real_normed_div_algebra star"
apply (rule approx_trans2)
apply (auto intro: inverse_add_Infinitesimal_approx
simp add: mem_infmal_iff approx_minus_iff [symmetric])
done
lemma Infinitesimal_square_iff: "x \<in> Infinitesimal \<longleftrightarrow> x * x \<in> Infinitesimal"
for x :: "'a::real_normed_div_algebra star"
apply (auto intro: Infinitesimal_mult)
apply (rule ccontr, frule Infinitesimal_inverse_HFinite)
apply (frule not_Infinitesimal_not_zero)
apply (auto dest: Infinitesimal_HFinite_mult simp add: mult.assoc)
done
declare Infinitesimal_square_iff [symmetric, simp]
lemma HFinite_square_iff [simp]: "x * x \<in> HFinite \<longleftrightarrow> x \<in> HFinite"
for x :: "'a::real_normed_div_algebra star"
apply (auto intro: HFinite_mult)
apply (auto dest: HInfinite_mult simp add: HFinite_HInfinite_iff)
done
lemma HInfinite_square_iff [simp]: "x * x \<in> HInfinite \<longleftrightarrow> x \<in> HInfinite"
for x :: "'a::real_normed_div_algebra star"
by (auto simp add: HInfinite_HFinite_iff)
lemma approx_HFinite_mult_cancel: "a \<in> HFinite - Infinitesimal \<Longrightarrow> a * w \<approx> a * z \<Longrightarrow> w \<approx> z"
for a w z :: "'a::real_normed_div_algebra star"
apply safe
apply (frule HFinite_inverse, assumption)
apply (drule not_Infinitesimal_not_zero)
apply (auto dest: approx_mult2 simp add: mult.assoc [symmetric])
done
lemma approx_HFinite_mult_cancel_iff1: "a \<in> HFinite - Infinitesimal \<Longrightarrow> a * w \<approx> a * z \<longleftrightarrow> w \<approx> 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 \<in> HInfinite \<Longrightarrow> y \<in> HFinite \<Longrightarrow> x \<in> HInfinite"
apply (rule ccontr)
apply (drule HFinite_HInfinite_iff [THEN iffD2])
apply (auto dest: HFinite_add simp add: HInfinite_HFinite_iff)
done
lemma HInfinite_HFinite_add: "x \<in> HInfinite \<Longrightarrow> y \<in> HFinite \<Longrightarrow> x + y \<in> HInfinite"
apply (rule_tac y = "-y" in HInfinite_HFinite_add_cancel)
apply (auto simp add: add.assoc HFinite_minus_iff)
done
lemma HInfinite_ge_HInfinite: "x \<in> HInfinite \<Longrightarrow> x \<le> y \<Longrightarrow> 0 \<le> x \<Longrightarrow> y \<in> HInfinite"
for x y :: hypreal
by (auto intro: HFinite_bounded simp add: HInfinite_HFinite_iff)
lemma Infinitesimal_inverse_HInfinite: "x \<in> Infinitesimal \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> inverse x \<in> HInfinite"
for x :: "'a::real_normed_div_algebra star"
apply (rule ccontr, drule HFinite_HInfinite_iff [THEN iffD2])
apply (auto dest: Infinitesimal_HFinite_mult2)
done
lemma HInfinite_HFinite_not_Infinitesimal_mult:
"x \<in> HInfinite \<Longrightarrow> y \<in> HFinite - Infinitesimal \<Longrightarrow> x * y \<in> HInfinite"
for x y :: "'a::real_normed_div_algebra star"
apply (rule ccontr, drule HFinite_HInfinite_iff [THEN iffD2])
apply (frule HFinite_Infinitesimal_not_zero)
apply (drule HFinite_not_Infinitesimal_inverse)
apply (safe, drule HFinite_mult)
apply (auto simp add: mult.assoc HFinite_HInfinite_iff)
done
lemma HInfinite_HFinite_not_Infinitesimal_mult2:
"x \<in> HInfinite \<Longrightarrow> y \<in> HFinite - Infinitesimal \<Longrightarrow> y * x \<in> HInfinite"
for x y :: "'a::real_normed_div_algebra star"
apply (rule ccontr, drule HFinite_HInfinite_iff [THEN iffD2])
apply (frule HFinite_Infinitesimal_not_zero)
apply (drule HFinite_not_Infinitesimal_inverse)
apply (safe, drule_tac x="inverse y" in HFinite_mult)
apply assumption
apply (auto simp add: mult.assoc [symmetric] HFinite_HInfinite_iff)
done
lemma HInfinite_gt_SReal: "x \<in> HInfinite \<Longrightarrow> 0 < x \<Longrightarrow> y \<in> \<real> \<Longrightarrow> 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 \<in> HInfinite \<Longrightarrow> 0 < x \<Longrightarrow> 1 < x"
for x :: hypreal
by (auto intro: HInfinite_gt_SReal)
lemma not_HInfinite_one [simp]: "1 \<notin> HInfinite"
by (simp add: HInfinite_HFinite_iff)
lemma approx_hrabs_disj: "\<bar>x\<bar> \<approx> x \<or> \<bar>x\<bar> \<approx> -x"
for x :: hypreal
using hrabs_disj [of x] by auto
subsection \<open>Theorems about Monads\<close>
lemma monad_hrabs_Un_subset: "monad \<bar>x\<bar> \<le> monad x \<union> monad (- x)"
for x :: hypreal
by (rule hrabs_disj [of x, THEN disjE]) auto
lemma Infinitesimal_monad_eq: "e \<in> Infinitesimal \<Longrightarrow> 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 \<in> monad x \<longleftrightarrow> - u \<in> monad (- x)"
by (simp add: monad_def)
lemma Infinitesimal_monad_zero_iff: "x \<in> Infinitesimal \<longleftrightarrow> x \<in> monad 0"
by (auto intro: approx_sym simp add: monad_def mem_infmal_iff)
lemma monad_zero_minus_iff: "x \<in> monad 0 \<longleftrightarrow> - x \<in> monad 0"
by (simp add: Infinitesimal_monad_zero_iff [symmetric])
lemma monad_zero_hrabs_iff: "x \<in> monad 0 \<longleftrightarrow> \<bar>x\<bar> \<in> monad 0"
for x :: hypreal
by (rule hrabs_disj [of x, THEN disjE]) (auto simp: monad_zero_minus_iff [symmetric])
lemma mem_monad_self [simp]: "x \<in> monad x"
by (simp add: monad_def)
subsection \<open>Proof that @{term "x \<approx> y"} implies @{term"\<bar>x\<bar> \<approx> \<bar>y\<bar>"}\<close>
lemma approx_subset_monad: "x \<approx> y \<Longrightarrow> {x, y} \<le> monad x"
by (simp (no_asm)) (simp add: approx_monad_iff)
lemma approx_subset_monad2: "x \<approx> y \<Longrightarrow> {x, y} \<le> monad y"
apply (drule approx_sym)
apply (fast dest: approx_subset_monad)
done
lemma mem_monad_approx: "u \<in> monad x \<Longrightarrow> x \<approx> u"
by (simp add: monad_def)
lemma approx_mem_monad: "x \<approx> u \<Longrightarrow> u \<in> monad x"
by (simp add: monad_def)
lemma approx_mem_monad2: "x \<approx> u \<Longrightarrow> x \<in> monad u"
apply (simp add: monad_def)
apply (blast intro!: approx_sym)
done
lemma approx_mem_monad_zero: "x \<approx> y \<Longrightarrow> x \<in> monad 0 \<Longrightarrow> y \<in> monad 0"
apply (drule mem_monad_approx)
apply (fast intro: approx_mem_monad approx_trans)
done
lemma Infinitesimal_approx_hrabs: "x \<approx> y \<Longrightarrow> x \<in> Infinitesimal \<Longrightarrow> \<bar>x\<bar> \<approx> \<bar>y\<bar>"
for x y :: hypreal
apply (drule Infinitesimal_monad_zero_iff [THEN iffD1])
apply (blast intro: approx_mem_monad_zero monad_zero_hrabs_iff [THEN iffD1]
mem_monad_approx approx_trans3)
done
lemma less_Infinitesimal_less: "0 < x \<Longrightarrow> x \<notin> Infinitesimal \<Longrightarrow> e \<in> Infinitesimal \<Longrightarrow> e < x"
for x :: hypreal
apply (rule ccontr)
apply (auto intro: Infinitesimal_zero [THEN [2] Infinitesimal_interval]
dest!: order_le_imp_less_or_eq simp add: linorder_not_less)
done
lemma Ball_mem_monad_gt_zero: "0 < x \<Longrightarrow> x \<notin> Infinitesimal \<Longrightarrow> u \<in> monad x \<Longrightarrow> 0 < u"
for u x :: hypreal
apply (drule mem_monad_approx [THEN approx_sym])
apply (erule bex_Infinitesimal_iff2 [THEN iffD2, THEN bexE])
apply (drule_tac e = "-xa" in less_Infinitesimal_less, auto)
done
lemma Ball_mem_monad_less_zero: "x < 0 \<Longrightarrow> x \<notin> Infinitesimal \<Longrightarrow> u \<in> monad x \<Longrightarrow> u < 0"
for u x :: hypreal
apply (drule mem_monad_approx [THEN approx_sym])
apply (erule bex_Infinitesimal_iff [THEN iffD2, THEN bexE])
apply (cut_tac x = "-x" and e = xa in less_Infinitesimal_less, auto)
done
lemma lemma_approx_gt_zero: "0 < x \<Longrightarrow> x \<notin> Infinitesimal \<Longrightarrow> x \<approx> y \<Longrightarrow> 0 < y"
for x y :: hypreal
by (blast dest: Ball_mem_monad_gt_zero approx_subset_monad)
lemma lemma_approx_less_zero: "x < 0 \<Longrightarrow> x \<notin> Infinitesimal \<Longrightarrow> x \<approx> y \<Longrightarrow> y < 0"
for x y :: hypreal
by (blast dest: Ball_mem_monad_less_zero approx_subset_monad)
lemma approx_hrabs: "x \<approx> y \<Longrightarrow> \<bar>x\<bar> \<approx> \<bar>y\<bar>"
for x y :: hypreal
by (drule approx_hnorm) simp
lemma approx_hrabs_zero_cancel: "\<bar>x\<bar> \<approx> 0 \<Longrightarrow> x \<approx> 0"
for x :: hypreal
using hrabs_disj [of x] by (auto dest: approx_minus)
lemma approx_hrabs_add_Infinitesimal: "e \<in> Infinitesimal \<Longrightarrow> \<bar>x\<bar> \<approx> \<bar>x + e\<bar>"
for e x :: hypreal
by (fast intro: approx_hrabs Infinitesimal_add_approx_self)
lemma approx_hrabs_add_minus_Infinitesimal: "e \<in> Infinitesimal ==> \<bar>x\<bar> \<approx> \<bar>x + -e\<bar>"
for e x :: hypreal
by (fast intro: approx_hrabs Infinitesimal_add_minus_approx_self)
lemma hrabs_add_Infinitesimal_cancel:
"e \<in> Infinitesimal \<Longrightarrow> e' \<in> Infinitesimal \<Longrightarrow> \<bar>x + e\<bar> = \<bar>y + e'\<bar> \<Longrightarrow> \<bar>x\<bar> \<approx> \<bar>y\<bar>"
for e e' x y :: hypreal
apply (drule_tac x = x in approx_hrabs_add_Infinitesimal)
apply (drule_tac x = y in approx_hrabs_add_Infinitesimal)
apply (auto intro: approx_trans2)
done
lemma hrabs_add_minus_Infinitesimal_cancel:
"e \<in> Infinitesimal \<Longrightarrow> e' \<in> Infinitesimal \<Longrightarrow> \<bar>x + -e\<bar> = \<bar>y + -e'\<bar> \<Longrightarrow> \<bar>x\<bar> \<approx> \<bar>y\<bar>"
for e e' x y :: hypreal
apply (drule_tac x = x in approx_hrabs_add_minus_Infinitesimal)
apply (drule_tac x = y in approx_hrabs_add_minus_Infinitesimal)
apply (auto intro: approx_trans2)
done
subsection \<open>More @{term HFinite} and @{term Infinitesimal} 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:
"x < y \<Longrightarrow> u \<in> Infinitesimal \<Longrightarrow> hypreal_of_real x + u < hypreal_of_real y"
apply (simp add: Infinitesimal_def)
apply (drule_tac x = "hypreal_of_real y + -hypreal_of_real x" in bspec, simp)
apply (simp add: abs_less_iff)
done
lemma Infinitesimal_add_hrabs_hypreal_of_real_less:
"x \<in> Infinitesimal \<Longrightarrow> \<bar>hypreal_of_real r\<bar> < hypreal_of_real y \<Longrightarrow>
\<bar>hypreal_of_real r + x\<bar> < hypreal_of_real y"
apply (drule_tac x = "hypreal_of_real r" in approx_hrabs_add_Infinitesimal)
apply (drule approx_sym [THEN bex_Infinitesimal_iff2 [THEN iffD2]])
apply (auto intro!: Infinitesimal_add_hypreal_of_real_less
simp del: star_of_abs simp add: star_of_abs [symmetric])
done
lemma Infinitesimal_add_hrabs_hypreal_of_real_less2:
"x \<in> Infinitesimal \<Longrightarrow> \<bar>hypreal_of_real r\<bar> < hypreal_of_real y \<Longrightarrow>
\<bar>x + hypreal_of_real r\<bar> < hypreal_of_real y"
apply (rule add.commute [THEN subst])
apply (erule Infinitesimal_add_hrabs_hypreal_of_real_less, assumption)
done
lemma hypreal_of_real_le_add_Infininitesimal_cancel:
"u \<in> Infinitesimal \<Longrightarrow> v \<in> Infinitesimal \<Longrightarrow>
hypreal_of_real x + u \<le> hypreal_of_real y + v \<Longrightarrow>
hypreal_of_real x \<le> hypreal_of_real y"
apply (simp add: linorder_not_less [symmetric], auto)
apply (drule_tac u = "v-u" in Infinitesimal_add_hypreal_of_real_less)
apply (auto simp add: Infinitesimal_diff)
done
lemma hypreal_of_real_le_add_Infininitesimal_cancel2:
"u \<in> Infinitesimal \<Longrightarrow> v \<in> Infinitesimal \<Longrightarrow>
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 \<Longrightarrow> e \<in> Infinitesimal \<Longrightarrow> hypreal_of_real x \<le> 0"
apply (rule linorder_not_less [THEN iffD1], safe)
apply (drule Infinitesimal_interval)
apply (drule_tac [4] SReal_hypreal_of_real [THEN SReal_Infinitesimal_zero], auto)
done
(*used once, in Lim/NSDERIV_inverse*)
lemma Infinitesimal_add_not_zero: "h \<in> Infinitesimal \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> star_of x + h \<noteq> 0"
apply auto
apply (subgoal_tac "h = - star_of x")
apply (auto intro: minus_unique [symmetric])
done
lemma Infinitesimal_square_cancel [simp]: "x * x + y * y \<in> Infinitesimal \<Longrightarrow> x * x \<in> Infinitesimal"
for x y :: hypreal
apply (rule Infinitesimal_interval2)
apply (rule_tac [3] zero_le_square, assumption)
apply auto
done
lemma HFinite_square_cancel [simp]: "x * x + y * y \<in> HFinite \<Longrightarrow> x * x \<in> HFinite"
for x y :: hypreal
apply (rule HFinite_bounded, assumption)
apply auto
done
lemma Infinitesimal_square_cancel2 [simp]: "x * x + y * y \<in> Infinitesimal \<Longrightarrow> y * y \<in> Infinitesimal"
for x y :: hypreal
apply (rule Infinitesimal_square_cancel)
apply (rule add.commute [THEN subst])
apply simp
done
lemma HFinite_square_cancel2 [simp]: "x * x + y * y \<in> HFinite \<Longrightarrow> y * y \<in> HFinite"
for x y :: hypreal
apply (rule HFinite_square_cancel)
apply (rule add.commute [THEN subst])
apply simp
done
lemma Infinitesimal_sum_square_cancel [simp]:
"x * x + y * y + z * z \<in> Infinitesimal \<Longrightarrow> x * x \<in> Infinitesimal"
for x y z :: hypreal
apply (rule Infinitesimal_interval2, assumption)
apply (rule_tac [2] zero_le_square, simp)
apply (insert zero_le_square [of y])
apply (insert zero_le_square [of z], simp del:zero_le_square)
done
lemma HFinite_sum_square_cancel [simp]: "x * x + y * y + z * z \<in> HFinite \<Longrightarrow> x * x \<in> HFinite"
for x y z :: hypreal
apply (rule HFinite_bounded, assumption)
apply (rule_tac [2] zero_le_square)
apply (insert zero_le_square [of y])
apply (insert zero_le_square [of z], simp del:zero_le_square)
done
lemma Infinitesimal_sum_square_cancel2 [simp]:
"y * y + x * x + z * z \<in> Infinitesimal \<Longrightarrow> x * x \<in> Infinitesimal"
for x y z :: hypreal
apply (rule Infinitesimal_sum_square_cancel)
apply (simp add: ac_simps)
done
lemma HFinite_sum_square_cancel2 [simp]: "y * y + x * x + z * z \<in> HFinite \<Longrightarrow> x * x \<in> HFinite"
for x y z :: hypreal
apply (rule HFinite_sum_square_cancel)
apply (simp add: ac_simps)
done
lemma Infinitesimal_sum_square_cancel3 [simp]:
"z * z + y * y + x * x \<in> Infinitesimal \<Longrightarrow> x * x \<in> Infinitesimal"
for x y z :: hypreal
apply (rule Infinitesimal_sum_square_cancel)
apply (simp add: ac_simps)
done
lemma HFinite_sum_square_cancel3 [simp]: "z * z + y * y + x * x \<in> HFinite \<Longrightarrow> x * x \<in> HFinite"
for x y z :: hypreal
apply (rule HFinite_sum_square_cancel)
apply (simp add: ac_simps)
done
lemma monad_hrabs_less: "y \<in> monad x \<Longrightarrow> 0 < hypreal_of_real e \<Longrightarrow> \<bar>y - x\<bar> < hypreal_of_real e"
apply (drule mem_monad_approx [THEN approx_sym])
apply (drule bex_Infinitesimal_iff [THEN iffD2])
apply (auto dest!: InfinitesimalD)
done
lemma mem_monad_SReal_HFinite: "x \<in> monad (hypreal_of_real a) \<Longrightarrow> x \<in> HFinite"
apply (drule mem_monad_approx [THEN approx_sym])
apply (drule bex_Infinitesimal_iff2 [THEN iffD2])
apply (safe dest!: Infinitesimal_subset_HFinite [THEN subsetD])
apply (erule SReal_hypreal_of_real [THEN SReal_subset_HFinite [THEN subsetD], THEN HFinite_add])
done
subsection \<open>Theorems about Standard Part\<close>
lemma st_approx_self: "x \<in> HFinite \<Longrightarrow> st x \<approx> x"
apply (simp add: st_def)
apply (frule st_part_Ex, safe)
apply (rule someI2)
apply (auto intro: approx_sym)
done
lemma st_SReal: "x \<in> HFinite \<Longrightarrow> st x \<in> \<real>"
apply (simp add: st_def)
apply (frule st_part_Ex, safe)
apply (rule someI2)
apply (auto intro: approx_sym)
done
lemma st_HFinite: "x \<in> HFinite \<Longrightarrow> st x \<in> HFinite"
by (erule st_SReal [THEN SReal_subset_HFinite [THEN subsetD]])
lemma st_unique: "r \<in> \<real> \<Longrightarrow> r \<approx> x \<Longrightarrow> st x = r"
apply (frule SReal_subset_HFinite [THEN subsetD])
apply (drule (1) approx_HFinite)
apply (unfold st_def)
apply (rule some_equality)
apply (auto intro: approx_unique_real)
done
lemma st_SReal_eq: "x \<in> \<real> \<Longrightarrow> 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 \<in> HFinite \<Longrightarrow> y \<in> HFinite \<Longrightarrow> st x = st y \<Longrightarrow> x \<approx> y"
by (auto dest!: st_approx_self elim!: approx_trans3)
lemma approx_st_eq:
assumes x: "x \<in> HFinite" and y: "y \<in> HFinite" and xy: "x \<approx> y"
shows "st x = st y"
proof -
have "st x \<approx> x" "st y \<approx> y" "st x \<in> \<real>" "st y \<in> \<real>"
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 \<in> HFinite \<Longrightarrow> y \<in> HFinite \<Longrightarrow> x \<approx> y \<longleftrightarrow> st x = st y"
by (blast intro: approx_st_eq st_eq_approx)
lemma st_Infinitesimal_add_SReal: "x \<in> \<real> \<Longrightarrow> e \<in> Infinitesimal \<Longrightarrow> st (x + e) = x"
apply (erule st_unique)
apply (erule Infinitesimal_add_approx_self)
done
lemma st_Infinitesimal_add_SReal2: "x \<in> \<real> \<Longrightarrow> e \<in> Infinitesimal \<Longrightarrow> st (e + x) = x"
apply (erule st_unique)
apply (erule Infinitesimal_add_approx_self2)
done
lemma HFinite_st_Infinitesimal_add: "x \<in> HFinite \<Longrightarrow> \<exists>e \<in> Infinitesimal. x = st(x) + e"
by (blast dest!: st_approx_self [THEN approx_sym] bex_Infinitesimal_iff2 [THEN iffD2])
lemma st_add: "x \<in> HFinite \<Longrightarrow> y \<in> HFinite \<Longrightarrow> st (x + y) = st x + st y"
by (simp add: st_unique st_SReal st_approx_self approx_add)
lemma st_numeral [simp]: "st (numeral w) = numeral w"
by (rule Reals_numeral [THEN st_SReal_eq])
lemma st_neg_numeral [simp]: "st (- numeral w) = - numeral w"
proof -
from Reals_numeral have "numeral w \<in> \<real>" .
then have "- numeral w \<in> \<real>" by simp
with st_SReal_eq show ?thesis .
qed
lemma st_0 [simp]: "st 0 = 0"
by (simp add: st_SReal_eq)
lemma st_1 [simp]: "st 1 = 1"
by (simp add: st_SReal_eq)
lemma st_neg_1 [simp]: "st (- 1) = - 1"
by (simp add: st_SReal_eq)
lemma st_minus: "x \<in> HFinite \<Longrightarrow> st (- x) = - st x"
by (simp add: st_unique st_SReal st_approx_self approx_minus)
lemma st_diff: "\<lbrakk>x \<in> HFinite; y \<in> HFinite\<rbrakk> \<Longrightarrow> st (x - y) = st x - st y"
by (simp add: st_unique st_SReal st_approx_self approx_diff)
lemma st_mult: "\<lbrakk>x \<in> HFinite; y \<in> HFinite\<rbrakk> \<Longrightarrow> st (x * y) = st x * st y"
by (simp add: st_unique st_SReal st_approx_self approx_mult_HFinite)
lemma st_Infinitesimal: "x \<in> Infinitesimal \<Longrightarrow> st x = 0"
by (simp add: st_unique mem_infmal_iff)
lemma st_not_Infinitesimal: "st(x) \<noteq> 0 \<Longrightarrow> x \<notin> Infinitesimal"
by (fast intro: st_Infinitesimal)
lemma st_inverse: "x \<in> HFinite \<Longrightarrow> st x \<noteq> 0 \<Longrightarrow> st (inverse x) = inverse (st x)"
apply (rule_tac c1 = "st x" in mult_left_cancel [THEN iffD1])
apply (auto simp add: st_mult [symmetric] st_not_Infinitesimal HFinite_inverse)
apply (subst right_inverse, auto)
done
lemma st_divide [simp]: "x \<in> HFinite \<Longrightarrow> y \<in> HFinite \<Longrightarrow> st y \<noteq> 0 \<Longrightarrow> 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 \<in> HFinite \<Longrightarrow> st (st x) = st x"
by (blast intro: st_HFinite st_approx_self approx_st_eq)
lemma Infinitesimal_add_st_less:
"x \<in> HFinite \<Longrightarrow> y \<in> HFinite \<Longrightarrow> u \<in> Infinitesimal \<Longrightarrow> st x < st y \<Longrightarrow> st x + u < st y"
apply (drule st_SReal)+
apply (auto intro!: Infinitesimal_add_hypreal_of_real_less simp add: SReal_iff)
done
lemma Infinitesimal_add_st_le_cancel:
"x \<in> HFinite \<Longrightarrow> y \<in> HFinite \<Longrightarrow> u \<in> Infinitesimal \<Longrightarrow>
st x \<le> st y + u \<Longrightarrow> st x \<le> st y"
apply (simp add: linorder_not_less [symmetric])
apply (auto dest: Infinitesimal_add_st_less)
done
lemma st_le: "x \<in> HFinite \<Longrightarrow> y \<in> HFinite \<Longrightarrow> x \<le> y \<Longrightarrow> st x \<le> st y"
by (metis approx_le_bound approx_sym linear st_SReal st_approx_self st_part_Ex1)
lemma st_zero_le: "0 \<le> x \<Longrightarrow> x \<in> HFinite \<Longrightarrow> 0 \<le> st x"
apply (subst st_0 [symmetric])
apply (rule st_le, auto)
done
lemma st_zero_ge: "x \<le> 0 \<Longrightarrow> x \<in> HFinite \<Longrightarrow> st x \<le> 0"
apply (subst st_0 [symmetric])
apply (rule st_le, auto)
done
lemma st_hrabs: "x \<in> HFinite \<Longrightarrow> \<bar>st x\<bar> = st \<bar>x\<bar>"
apply (simp add: linorder_not_le st_zero_le abs_if st_minus linorder_not_less)
apply (auto dest!: st_zero_ge [OF order_less_imp_le])
done
subsection \<open>Alternative Definitions using Free Ultrafilter\<close>
subsubsection \<open>@{term HFinite}\<close>
lemma HFinite_FreeUltrafilterNat:
"star_n X \<in> HFinite \<Longrightarrow> \<exists>u. eventually (\<lambda>n. norm (X n) < u) \<U>"
apply (auto simp add: HFinite_def SReal_def)
apply (rule_tac x=r in exI)
apply (simp add: hnorm_def star_of_def starfun_star_n)
apply (simp add: star_less_def starP2_star_n)
done
lemma FreeUltrafilterNat_HFinite:
"\<exists>u. eventually (\<lambda>n. norm (X n) < u) \<U> \<Longrightarrow> star_n X \<in> HFinite"
apply (auto simp add: HFinite_def mem_Rep_star_iff)
apply (rule_tac x="star_of u" in bexI)
apply (simp add: hnorm_def starfun_star_n star_of_def)
apply (simp add: star_less_def starP2_star_n)
apply (simp add: SReal_def)
done
lemma HFinite_FreeUltrafilterNat_iff:
"star_n X \<in> HFinite \<longleftrightarrow> (\<exists>u. eventually (\<lambda>n. norm (X n) < u) \<U>)"
by (blast intro!: HFinite_FreeUltrafilterNat FreeUltrafilterNat_HFinite)
subsubsection \<open>@{term HInfinite}\<close>
lemma lemma_Compl_eq: "- {n. u < norm (f n)} = {n. norm (f n) \<le> u}"
by auto
lemma lemma_Compl_eq2: "- {n. norm (f n) < u} = {n. u \<le> norm (f n)}"
by auto
lemma lemma_Int_eq1: "{n. norm (f n) \<le> u} Int {n. u \<le> norm (f n)} = {n. norm(f n) = u}"
by auto
lemma lemma_FreeUltrafilterNat_one: "{n. norm (f n) = u} \<le> {n. norm (f n) < u + (1::real)}"
by auto
text \<open>Exclude this type of sets from free ultrafilter for Infinite numbers!\<close>
lemma FreeUltrafilterNat_const_Finite:
"eventually (\<lambda>n. norm (X n) = u) \<U> \<Longrightarrow> star_n X \<in> HFinite"
apply (rule FreeUltrafilterNat_HFinite)
apply (rule_tac x = "u + 1" in exI)
apply (auto elim: eventually_mono)
done
lemma HInfinite_FreeUltrafilterNat:
"star_n X \<in> HInfinite \<Longrightarrow> \<forall>u. eventually (\<lambda>n. u < norm (X n)) \<U>"
apply (drule HInfinite_HFinite_iff [THEN iffD1])
apply (simp add: HFinite_FreeUltrafilterNat_iff)
apply (rule allI, drule_tac x="u + 1" in spec)
apply (simp add: FreeUltrafilterNat.eventually_not_iff[symmetric])
apply (auto elim: eventually_mono)
done
lemma lemma_Int_HI: "{n. norm (Xa n) < u} \<inter> {n. X n = Xa n} \<subseteq> {n. norm (X n) < u}"
for u :: real
by auto
lemma lemma_Int_HIa: "{n. u < norm (X n)} \<inter> {n. norm (X n) < u} = {}"
by (auto intro: order_less_asym)
lemma FreeUltrafilterNat_HInfinite:
"\<forall>u. eventually (\<lambda>n. u < norm (X n)) \<U> \<Longrightarrow> star_n X \<in> HInfinite"
apply (rule HInfinite_HFinite_iff [THEN iffD2])
apply (safe, drule HFinite_FreeUltrafilterNat, safe)
apply (drule_tac x = u in spec)
proof -
fix u
assume "\<forall>\<^sub>Fn in \<U>. norm (X n) < u" "\<forall>\<^sub>Fn in \<U>. u < norm (X n)"
then have "\<forall>\<^sub>F x in \<U>. False"
by eventually_elim auto
then show False
by (simp add: eventually_False FreeUltrafilterNat.proper)
qed
lemma HInfinite_FreeUltrafilterNat_iff:
"star_n X \<in> HInfinite \<longleftrightarrow> (\<forall>u. eventually (\<lambda>n. u < norm (X n)) \<U>)"
by (blast intro!: HInfinite_FreeUltrafilterNat FreeUltrafilterNat_HInfinite)
subsubsection \<open>@{term Infinitesimal}\<close>
lemma ball_SReal_eq: "(\<forall>x::hypreal \<in> Reals. P x) \<longleftrightarrow> (\<forall>x::real. P (star_of x))"
by (auto simp: SReal_def)
lemma Infinitesimal_FreeUltrafilterNat:
"star_n X \<in> Infinitesimal \<Longrightarrow> \<forall>u>0. eventually (\<lambda>n. norm (X n) < u) \<U>"
apply (simp add: Infinitesimal_def ball_SReal_eq)
apply (simp add: hnorm_def starfun_star_n star_of_def)
apply (simp add: star_less_def starP2_star_n)
done
lemma FreeUltrafilterNat_Infinitesimal:
"\<forall>u>0. eventually (\<lambda>n. norm (X n) < u) \<U> \<Longrightarrow> star_n X \<in> Infinitesimal"
apply (simp add: Infinitesimal_def ball_SReal_eq)
apply (simp add: hnorm_def starfun_star_n star_of_def)
apply (simp add: star_less_def starP2_star_n)
done
lemma Infinitesimal_FreeUltrafilterNat_iff:
"(star_n X \<in> Infinitesimal) = (\<forall>u>0. eventually (\<lambda>n. norm (X n) < u) \<U>)"
by (blast intro!: Infinitesimal_FreeUltrafilterNat FreeUltrafilterNat_Infinitesimal)
text \<open>Infinitesimals as smaller than \<open>1/n\<close> for all \<open>n::nat (> 0)\<close>.\<close>
lemma lemma_Infinitesimal: "(\<forall>r. 0 < r \<longrightarrow> x < r) \<longleftrightarrow> (\<forall>n. x < inverse (real (Suc n)))"
apply (auto simp del: of_nat_Suc)
apply (blast dest!: reals_Archimedean intro: order_less_trans)
done
lemma lemma_Infinitesimal2:
"(\<forall>r \<in> Reals. 0 < r \<longrightarrow> x < r) \<longleftrightarrow> (\<forall>n. x < inverse(hypreal_of_nat (Suc n)))"
apply safe
apply (drule_tac x = "inverse (hypreal_of_real (real (Suc n))) " in bspec)
apply simp_all
using less_imp_of_nat_less apply fastforce
apply (auto dest!: reals_Archimedean simp add: SReal_iff simp del: of_nat_Suc)
apply (drule star_of_less [THEN iffD2])
apply simp
apply (blast intro: order_less_trans)
done
lemma Infinitesimal_hypreal_of_nat_iff:
"Infinitesimal = {x. \<forall>n. hnorm x < inverse (hypreal_of_nat (Suc n))}"
apply (simp add: Infinitesimal_def)
apply (auto simp add: lemma_Infinitesimal2)
done
subsection \<open>Proof that \<open>\<omega>\<close> is an infinite number\<close>
text \<open>It will follow that \<open>\<epsilon>\<close> is an infinitesimal number.\<close>
lemma Suc_Un_eq: "{n. n < Suc m} = {n. n < m} Un {n. n = m}"
by (auto simp add: less_Suc_eq)
text \<open>Prove that any segment is finite and hence cannot belong to \<open>\<U>\<close>.\<close>
lemma finite_real_of_nat_segment: "finite {n::nat. real n < real (m::nat)}"
by auto
lemma finite_real_of_nat_less_real: "finite {n::nat. real n < u}"
apply (cut_tac x = u in reals_Archimedean2, safe)
apply (rule finite_real_of_nat_segment [THEN [2] finite_subset])
apply (auto dest: order_less_trans)
done
lemma lemma_real_le_Un_eq: "{n. f n \<le> u} = {n. f n < u} \<union> {n. u = (f n :: real)}"
by (auto dest: order_le_imp_less_or_eq simp add: order_less_imp_le)
lemma finite_real_of_nat_le_real: "finite {n::nat. real n \<le> u}"
by (auto simp add: lemma_real_le_Un_eq lemma_finite_omega_set finite_real_of_nat_less_real)
lemma finite_rabs_real_of_nat_le_real: "finite {n::nat. \<bar>real n\<bar> \<le> u}"
by (simp add: finite_real_of_nat_le_real)
lemma rabs_real_of_nat_le_real_FreeUltrafilterNat:
"\<not> eventually (\<lambda>n. \<bar>real n\<bar> \<le> u) \<U>"
by (blast intro!: FreeUltrafilterNat.finite finite_rabs_real_of_nat_le_real)
lemma FreeUltrafilterNat_nat_gt_real: "eventually (\<lambda>n. u < real n) \<U>"
apply (rule FreeUltrafilterNat.finite')
apply (subgoal_tac "{n::nat. \<not> u < real n} = {n. real n \<le> u}")
apply (auto simp add: finite_real_of_nat_le_real)
done
text \<open>The complement of \<open>{n. \<bar>real n\<bar> \<le> u} = {n. u < \<bar>real n\<bar>}\<close> is in
\<open>\<U>\<close> by property of (free) ultrafilters.\<close>
lemma Compl_real_le_eq: "- {n::nat. real n \<le> u} = {n. u < real n}"
by (auto dest!: order_le_less_trans simp add: linorder_not_le)
text \<open>@{term \<omega>} is a member of @{term HInfinite}.\<close>
theorem HInfinite_omega [simp]: "\<omega> \<in> HInfinite"
apply (simp add: omega_def)
apply (rule FreeUltrafilterNat_HInfinite)
apply clarify
apply (rule_tac u1 = "u-1" in eventually_mono [OF FreeUltrafilterNat_nat_gt_real])
apply auto
done
text \<open>Epsilon is a member of Infinitesimal.\<close>
lemma Infinitesimal_epsilon [simp]: "\<epsilon> \<in> Infinitesimal"
by (auto intro!: HInfinite_inverse_Infinitesimal HInfinite_omega
simp add: hypreal_epsilon_inverse_omega)
lemma HFinite_epsilon [simp]: "\<epsilon> \<in> HFinite"
by (auto intro: Infinitesimal_subset_HFinite [THEN subsetD])
lemma epsilon_approx_zero [simp]: "\<epsilon> \<approx> 0"
by (simp add: mem_infmal_iff [symmetric])
text \<open>Needed for proof that we define a hyperreal \<open>[<X(n)] \<approx> hypreal_of_real a\<close> given
that \<open>\<forall>n. |X n - a| < 1/n\<close>. Used in proof of \<open>NSLIM \<Rightarrow> LIM\<close>.\<close>
lemma real_of_nat_less_inverse_iff: "0 < u \<Longrightarrow> u < inverse (real(Suc n)) \<longleftrightarrow> real(Suc n) < inverse u"
apply (simp add: inverse_eq_divide)
apply (subst pos_less_divide_eq, assumption)
apply (subst pos_less_divide_eq)
apply simp
apply (simp add: mult.commute)
done
lemma finite_inverse_real_of_posnat_gt_real: "0 < u \<Longrightarrow> finite {n. u < inverse (real (Suc n))}"
proof (simp only: real_of_nat_less_inverse_iff)
have "{n. 1 + real n < inverse u} = {n. real n < inverse u - 1}"
by fastforce
then show "finite {n. real (Suc n) < inverse u}"
using finite_real_of_nat_less_real [of "inverse u - 1"]
by auto
qed
lemma lemma_real_le_Un_eq2:
"{n. u \<le> inverse(real(Suc n))} =
{n. u < inverse(real(Suc n))} \<union> {n. u = inverse(real(Suc n))}"
by (auto dest: order_le_imp_less_or_eq simp add: order_less_imp_le)
lemma finite_inverse_real_of_posnat_ge_real: "0 < u \<Longrightarrow> finite {n. u \<le> inverse (real (Suc n))}"
by (auto simp add: lemma_real_le_Un_eq2 lemma_finite_epsilon_set finite_inverse_real_of_posnat_gt_real
simp del: of_nat_Suc)
lemma inverse_real_of_posnat_ge_real_FreeUltrafilterNat:
"0 < u \<Longrightarrow> \<not> eventually (\<lambda>n. u \<le> inverse(real(Suc n))) \<U>"
by (blast intro!: FreeUltrafilterNat.finite finite_inverse_real_of_posnat_ge_real)
text \<open>The complement of \<open>{n. u \<le> inverse(real(Suc n))} = {n. inverse (real (Suc n)) < u}\<close>
is in \<open>\<U>\<close> by property of (free) ultrafilters.\<close>
lemma Compl_le_inverse_eq: "- {n. u \<le> inverse(real(Suc n))} = {n. inverse(real(Suc n)) < u}"
by (auto dest!: order_le_less_trans simp add: linorder_not_le)
lemma FreeUltrafilterNat_inverse_real_of_posnat:
"0 < u \<Longrightarrow> eventually (\<lambda>n. 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 \<open>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\<close>
lemma SEQ_Infinitesimal: "( *f* (\<lambda>n::nat. inverse(real(Suc n)))) whn \<in> 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 \<open>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.\<close>
text \<open>\<open>|X(n) - x| < 1/n \<Longrightarrow> [<X n>] - hypreal_of_real x| \<in> Infinitesimal\<close>\<close>
lemma real_seq_to_hypreal_Infinitesimal:
"\<forall>n. norm (X n - x) < inverse (real (Suc n)) \<Longrightarrow> star_n X - star_of x \<in> 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:
"\<forall>n. norm (X n - x) < inverse (real (Suc n)) \<Longrightarrow> star_n X \<approx> star_of x"
by (metis bex_Infinitesimal_iff real_seq_to_hypreal_Infinitesimal)
lemma real_seq_to_hypreal_approx2:
"\<forall>n. norm (x - X n) < inverse(real(Suc n)) \<Longrightarrow> star_n X \<approx> star_of x"
by (metis norm_minus_commute real_seq_to_hypreal_approx)
lemma real_seq_to_hypreal_Infinitesimal2:
"\<forall>n. norm(X n - Y n) < inverse(real(Suc n)) \<Longrightarrow> star_n X - star_n Y \<in> Infinitesimal"
unfolding Infinitesimal_FreeUltrafilterNat_iff star_n_diff
by (auto dest!: FreeUltrafilterNat_inverse_real_of_posnat
intro: order_less_trans elim!: eventually_mono)
end