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theory HyperDef(* Title : HOL/Hyperreal/HyperDef.thy ID : $Id: HyperDef.thy,v 1.72 2007/07/30 22:56:28 wenzelm Exp $ Author : Jacques D. Fleuriot Copyright : 1998 University of Cambridge Conversion to Isar and new proofs by Lawrence C Paulson, 2004 *) header{*Construction of Hyperreals Using Ultrafilters*} theory HyperDef imports HyperNat "../Real/Real" uses ("hypreal_arith.ML") begin types hypreal = "real star" abbreviation hypreal_of_real :: "real => real star" where "hypreal_of_real == star_of" abbreviation hypreal_of_hypnat :: "hypnat => hypreal" where "hypreal_of_hypnat ≡ of_hypnat" definition omega :: hypreal where -- {*an infinite number @{text "= [<1,2,3,...>]"} *} "omega = star_n (λn. real (Suc n))" definition epsilon :: hypreal where -- {*an infinitesimal number @{text "= [<1,1/2,1/3,...>]"} *} "epsilon = star_n (λn. inverse (real (Suc n)))" notation (xsymbols) omega ("ω") and epsilon ("ε") notation (HTML output) omega ("ω") and epsilon ("ε") subsection {* Real vector class instances *} instance star :: (scaleR) scaleR .. defs (overloaded) star_scaleR_def [transfer_unfold]: "scaleR r ≡ *f* (scaleR r)" lemma Standard_scaleR [simp]: "x ∈ Standard ==> scaleR r x ∈ Standard" by (simp add: star_scaleR_def) lemma star_of_scaleR [simp]: "star_of (scaleR r x) = scaleR r (star_of x)" by transfer (rule refl) instance star :: (real_vector) real_vector proof fix a b :: real show "!!x y::'a star. scaleR a (x + y) = scaleR a x + scaleR a y" by transfer (rule scaleR_right_distrib) show "!!x::'a star. scaleR (a + b) x = scaleR a x + scaleR b x" by transfer (rule scaleR_left_distrib) show "!!x::'a star. scaleR a (scaleR b x) = scaleR (a * b) x" by transfer (rule scaleR_scaleR) show "!!x::'a star. scaleR 1 x = x" by transfer (rule scaleR_one) qed instance star :: (real_algebra) real_algebra proof fix a :: real show "!!x y::'a star. scaleR a x * y = scaleR a (x * y)" by transfer (rule mult_scaleR_left) show "!!x y::'a star. x * scaleR a y = scaleR a (x * y)" by transfer (rule mult_scaleR_right) qed instance star :: (real_algebra_1) real_algebra_1 .. instance star :: (real_div_algebra) real_div_algebra .. instance star :: (real_field) real_field .. lemma star_of_real_def [transfer_unfold]: "of_real r = star_of (of_real r)" by (unfold of_real_def, transfer, rule refl) lemma Standard_of_real [simp]: "of_real r ∈ Standard" by (simp add: star_of_real_def) lemma star_of_of_real [simp]: "star_of (of_real r) = of_real r" by transfer (rule refl) lemma of_real_eq_star_of [simp]: "of_real = star_of" proof fix r :: real show "of_real r = star_of r" by transfer simp qed lemma Reals_eq_Standard: "(Reals :: hypreal set) = Standard" by (simp add: Reals_def Standard_def) subsection {* Injection from @{typ hypreal} *} definition of_hypreal :: "hypreal => 'a::real_algebra_1 star" where "of_hypreal = *f* of_real" declare of_hypreal_def [transfer_unfold] lemma Standard_of_hypreal [simp]: "r ∈ Standard ==> of_hypreal r ∈ Standard" by (simp add: of_hypreal_def) lemma of_hypreal_0 [simp]: "of_hypreal 0 = 0" by transfer (rule of_real_0) lemma of_hypreal_1 [simp]: "of_hypreal 1 = 1" by transfer (rule of_real_1) lemma of_hypreal_add [simp]: "!!x y. of_hypreal (x + y) = of_hypreal x + of_hypreal y" by transfer (rule of_real_add) lemma of_hypreal_minus [simp]: "!!x. of_hypreal (- x) = - of_hypreal x" by transfer (rule of_real_minus) lemma of_hypreal_diff [simp]: "!!x y. of_hypreal (x - y) = of_hypreal x - of_hypreal y" by transfer (rule of_real_diff) lemma of_hypreal_mult [simp]: "!!x y. of_hypreal (x * y) = of_hypreal x * of_hypreal y" by transfer (rule of_real_mult) lemma of_hypreal_inverse [simp]: "!!x. of_hypreal (inverse x) = inverse (of_hypreal x :: 'a::{real_div_algebra,division_by_zero} star)" by transfer (rule of_real_inverse) lemma of_hypreal_divide [simp]: "!!x y. of_hypreal (x / y) = (of_hypreal x / of_hypreal y :: 'a::{real_field,division_by_zero} star)" by transfer (rule of_real_divide) lemma of_hypreal_eq_iff [simp]: "!!x y. (of_hypreal x = of_hypreal y) = (x = y)" by transfer (rule of_real_eq_iff) lemma of_hypreal_eq_0_iff [simp]: "!!x. (of_hypreal x = 0) = (x = 0)" by transfer (rule of_real_eq_0_iff) subsection{*Properties of @{term starrel}*} lemma lemma_starrel_refl [simp]: "x ∈ starrel `` {x}" by (simp add: starrel_def) lemma starrel_in_hypreal [simp]: "starrel``{x}:star" by (simp add: star_def starrel_def quotient_def, blast) declare Abs_star_inject [simp] Abs_star_inverse [simp] declare equiv_starrel [THEN eq_equiv_class_iff, simp] subsection{*@{term hypreal_of_real}: the Injection from @{typ real} to @{typ hypreal}*} lemma inj_star_of: "inj star_of" by (rule inj_onI, simp) lemma mem_Rep_star_iff: "(X ∈ Rep_star x) = (x = star_n X)" by (cases x, simp add: star_n_def) lemma Rep_star_star_n_iff [simp]: "(X ∈ Rep_star (star_n Y)) = ({n. Y n = X n} ∈ \<U>)" by (simp add: star_n_def) lemma Rep_star_star_n: "X ∈ Rep_star (star_n X)" by simp subsection{* Properties of @{term star_n} *} lemma star_n_add: "star_n X + star_n Y = star_n (%n. X n + Y n)" by (simp only: star_add_def starfun2_star_n) lemma star_n_minus: "- star_n X = star_n (%n. -(X n))" by (simp only: star_minus_def starfun_star_n) lemma star_n_diff: "star_n X - star_n Y = star_n (%n. X n - Y n)" by (simp only: star_diff_def starfun2_star_n) lemma star_n_mult: "star_n X * star_n Y = star_n (%n. X n * Y n)" by (simp only: star_mult_def starfun2_star_n) lemma star_n_inverse: "inverse (star_n X) = star_n (%n. inverse(X n))" by (simp only: star_inverse_def starfun_star_n) lemma star_n_le: "star_n X ≤ star_n Y = ({n. X n ≤ Y n} ∈ FreeUltrafilterNat)" by (simp only: star_le_def starP2_star_n) lemma star_n_less: "star_n X < star_n Y = ({n. X n < Y n} ∈ FreeUltrafilterNat)" by (simp only: star_less_def starP2_star_n) lemma star_n_zero_num: "0 = star_n (%n. 0)" by (simp only: star_zero_def star_of_def) lemma star_n_one_num: "1 = star_n (%n. 1)" by (simp only: star_one_def star_of_def) lemma star_n_abs: "abs (star_n X) = star_n (%n. abs (X n))" by (simp only: star_abs_def starfun_star_n) subsection{*Misc Others*} lemma hypreal_not_refl2: "!!(x::hypreal). x < y ==> x ≠ y" by (auto) lemma hypreal_eq_minus_iff: "((x::hypreal) = y) = (x + - y = 0)" by auto lemma hypreal_mult_left_cancel: "(c::hypreal) ≠ 0 ==> (c*a=c*b) = (a=b)" by auto lemma hypreal_mult_right_cancel: "(c::hypreal) ≠ 0 ==> (a*c=b*c) = (a=b)" by auto lemma hypreal_omega_gt_zero [simp]: "0 < omega" by (simp add: omega_def star_n_zero_num star_n_less) subsection{*Existence of Infinite Hyperreal Number*} text{*Existence of infinite number not corresponding to any real number. Use assumption that member @{term FreeUltrafilterNat} is not finite.*} text{*A few lemmas first*} lemma lemma_omega_empty_singleton_disj: "{n::nat. x = real n} = {} | (∃y. {n::nat. x = real n} = {y})" by force lemma lemma_finite_omega_set: "finite {n::nat. x = real n}" by (cut_tac x = x in lemma_omega_empty_singleton_disj, auto) lemma not_ex_hypreal_of_real_eq_omega: "~ (∃x. hypreal_of_real x = omega)" apply (simp add: omega_def) apply (simp add: star_of_def star_n_eq_iff) apply (auto simp add: real_of_nat_Suc diff_eq_eq [symmetric] lemma_finite_omega_set [THEN FreeUltrafilterNat.finite]) done lemma hypreal_of_real_not_eq_omega: "hypreal_of_real x ≠ omega" by (insert not_ex_hypreal_of_real_eq_omega, auto) text{*Existence of infinitesimal number also not corresponding to any real number*} lemma lemma_epsilon_empty_singleton_disj: "{n::nat. x = inverse(real(Suc n))} = {} | (∃y. {n::nat. x = inverse(real(Suc n))} = {y})" by auto lemma lemma_finite_epsilon_set: "finite {n. x = inverse(real(Suc n))}" by (cut_tac x = x in lemma_epsilon_empty_singleton_disj, auto) lemma not_ex_hypreal_of_real_eq_epsilon: "~ (∃x. hypreal_of_real x = epsilon)" by (auto simp add: epsilon_def star_of_def star_n_eq_iff lemma_finite_epsilon_set [THEN FreeUltrafilterNat.finite]) lemma hypreal_of_real_not_eq_epsilon: "hypreal_of_real x ≠ epsilon" by (insert not_ex_hypreal_of_real_eq_epsilon, auto) lemma hypreal_epsilon_not_zero: "epsilon ≠ 0" by (simp add: epsilon_def star_zero_def star_of_def star_n_eq_iff del: star_of_zero) lemma hypreal_epsilon_inverse_omega: "epsilon = inverse(omega)" by (simp add: epsilon_def omega_def star_n_inverse) lemma hypreal_epsilon_gt_zero: "0 < epsilon" by (simp add: hypreal_epsilon_inverse_omega) subsection{*Absolute Value Function for the Hyperreals*} lemma hrabs_add_less: "[| abs x < r; abs y < s |] ==> abs(x+y) < r + (s::hypreal)" by (simp add: abs_if split: split_if_asm) lemma hrabs_less_gt_zero: "abs x < r ==> (0::hypreal) < r" by (blast intro!: order_le_less_trans abs_ge_zero) lemma hrabs_disj: "abs x = (x::'a::abs_if) | abs x = -x" by (simp add: abs_if) lemma hrabs_add_lemma_disj: "(y::hypreal) + - x + (y + - z) = abs (x + - z) ==> y = z | x = y" by (simp add: abs_if split add: split_if_asm) subsection{*Embedding the Naturals into the Hyperreals*} abbreviation hypreal_of_nat :: "nat => hypreal" where "hypreal_of_nat == of_nat" lemma SNat_eq: "Nats = {n. ∃N. n = hypreal_of_nat N}" by (simp add: Nats_def image_def) (*------------------------------------------------------------*) (* naturals embedded in hyperreals *) (* is a hyperreal c.f. NS extension *) (*------------------------------------------------------------*) lemma hypreal_of_nat_eq: "hypreal_of_nat (n::nat) = hypreal_of_real (real n)" by (simp add: real_of_nat_def) lemma hypreal_of_nat: "hypreal_of_nat m = star_n (%n. real m)" apply (fold star_of_def) apply (simp add: real_of_nat_def) done (* FIXME: we should declare this, as for type int, but many proofs would break. It replaces x+-y by x-y. Addsimps [symmetric hypreal_diff_def] *) use "hypreal_arith.ML" declaration {* K hypreal_arith_setup *} subsection {* Exponentials on the Hyperreals *} lemma hpowr_0 [simp]: "r ^ 0 = (1::hypreal)" by (rule power_0) lemma hpowr_Suc [simp]: "r ^ (Suc n) = (r::hypreal) * (r ^ n)" by (rule power_Suc) lemma hrealpow_two: "(r::hypreal) ^ Suc (Suc 0) = r * r" by simp lemma hrealpow_two_le [simp]: "(0::hypreal) ≤ r ^ Suc (Suc 0)" by (auto simp add: zero_le_mult_iff) lemma hrealpow_two_le_add_order [simp]: "(0::hypreal) ≤ u ^ Suc (Suc 0) + v ^ Suc (Suc 0)" by (simp only: hrealpow_two_le add_nonneg_nonneg) lemma hrealpow_two_le_add_order2 [simp]: "(0::hypreal) ≤ u ^ Suc (Suc 0) + v ^ Suc (Suc 0) + w ^ Suc (Suc 0)" by (simp only: hrealpow_two_le add_nonneg_nonneg) lemma hypreal_add_nonneg_eq_0_iff: "[| 0 ≤ x; 0 ≤ y |] ==> (x+y = 0) = (x = 0 & y = (0::hypreal))" by arith text{*FIXME: DELETE THESE*} lemma hypreal_three_squares_add_zero_iff: "(x*x + y*y + z*z = 0) = (x = 0 & y = 0 & z = (0::hypreal))" apply (simp only: zero_le_square add_nonneg_nonneg hypreal_add_nonneg_eq_0_iff, auto) done lemma hrealpow_three_squares_add_zero_iff [simp]: "(x ^ Suc (Suc 0) + y ^ Suc (Suc 0) + z ^ Suc (Suc 0) = (0::hypreal)) = (x = 0 & y = 0 & z = 0)" by (simp only: hypreal_three_squares_add_zero_iff hrealpow_two) (*FIXME: This and RealPow.abs_realpow_two should be replaced by an abstract result proved in Ring_and_Field*) lemma hrabs_hrealpow_two [simp]: "abs(x ^ Suc (Suc 0)) = (x::hypreal) ^ Suc (Suc 0)" by (simp add: abs_mult) lemma two_hrealpow_ge_one [simp]: "(1::hypreal) ≤ 2 ^ n" by (insert power_increasing [of 0 n "2::hypreal"], simp) lemma two_hrealpow_gt [simp]: "hypreal_of_nat n < 2 ^ n" apply (induct_tac "n") apply (auto simp add: left_distrib) apply (cut_tac n = n in two_hrealpow_ge_one, arith) done lemma hrealpow: "star_n X ^ m = star_n (%n. (X n::real) ^ m)" apply (induct_tac "m") apply (auto simp add: star_n_one_num star_n_mult power_0) done lemma hrealpow_sum_square_expand: "(x + (y::hypreal)) ^ Suc (Suc 0) = x ^ Suc (Suc 0) + y ^ Suc (Suc 0) + (hypreal_of_nat (Suc (Suc 0)))*x*y" by (simp add: right_distrib left_distrib) lemma power_hypreal_of_real_number_of: "(number_of v :: hypreal) ^ n = hypreal_of_real ((number_of v) ^ n)" by simp declare power_hypreal_of_real_number_of [of _ "number_of w", standard, simp] (* lemma hrealpow_HFinite: fixes x :: "'a::{real_normed_algebra,recpower} star" shows "x ∈ HFinite ==> x ^ n ∈ HFinite" apply (induct_tac "n") apply (auto simp add: power_Suc intro: HFinite_mult) done *) subsection{*Powers with Hypernatural Exponents*} definition (* hypernatural powers of hyperreals *) pow :: "['a::power star, nat star] => 'a star" (infixr "pow" 80) where hyperpow_def [transfer_unfold]: "R pow N = ( *f2* op ^) R N" lemma Standard_hyperpow [simp]: "[|r ∈ Standard; n ∈ Standard|] ==> r pow n ∈ Standard" unfolding hyperpow_def by simp lemma hyperpow: "star_n X pow star_n Y = star_n (%n. X n ^ Y n)" by (simp add: hyperpow_def starfun2_star_n) lemma hyperpow_zero [simp]: "!!n. (0::'a::{recpower,semiring_0} star) pow (n + (1::hypnat)) = 0" by transfer simp lemma hyperpow_not_zero: "!!r n. r ≠ (0::'a::{recpower,field} star) ==> r pow n ≠ 0" by transfer (rule field_power_not_zero) lemma hyperpow_inverse: "!!r n. r ≠ (0::'a::{recpower,division_by_zero,field} star) ==> inverse (r pow n) = (inverse r) pow n" by transfer (rule power_inverse) lemma hyperpow_hrabs: "!!r n. abs (r::'a::{recpower,ordered_idom} star) pow n = abs (r pow n)" by transfer (rule power_abs [symmetric]) lemma hyperpow_add: "!!r n m. (r::'a::recpower star) pow (n + m) = (r pow n) * (r pow m)" by transfer (rule power_add) lemma hyperpow_one [simp]: "!!r. (r::'a::recpower star) pow (1::hypnat) = r" by transfer (rule power_one_right) lemma hyperpow_two: "!!r. (r::'a::recpower star) pow ((1::hypnat) + (1::hypnat)) = r * r" by transfer (simp add: power_Suc) lemma hyperpow_gt_zero: "!!r n. (0::'a::{recpower,ordered_semidom} star) < r ==> 0 < r pow n" by transfer (rule zero_less_power) lemma hyperpow_ge_zero: "!!r n. (0::'a::{recpower,ordered_semidom} star) ≤ r ==> 0 ≤ r pow n" by transfer (rule zero_le_power) lemma hyperpow_le: "!!x y n. [|(0::'a::{recpower,ordered_semidom} star) < x; x ≤ y|] ==> x pow n ≤ y pow n" by transfer (rule power_mono [OF _ order_less_imp_le]) lemma hyperpow_eq_one [simp]: "!!n. 1 pow n = (1::'a::recpower star)" by transfer (rule power_one) lemma hrabs_hyperpow_minus_one [simp]: "!!n. abs(-1 pow n) = (1::'a::{number_ring,recpower,ordered_idom} star)" by transfer (rule abs_power_minus_one) lemma hyperpow_mult: "!!r s n. (r * s::'a::{comm_monoid_mult,recpower} star) pow n = (r pow n) * (s pow n)" by transfer (rule power_mult_distrib) lemma hyperpow_two_le [simp]: "(0::'a::{recpower,ordered_ring_strict} star) ≤ r pow (1 + 1)" by (auto simp add: hyperpow_two zero_le_mult_iff) lemma hrabs_hyperpow_two [simp]: "abs(x pow (1 + 1)) = (x::'a::{recpower,ordered_ring_strict} star) pow (1 + 1)" by (simp only: abs_of_nonneg hyperpow_two_le) lemma hyperpow_two_hrabs [simp]: "abs(x::'a::{recpower,ordered_idom} star) pow (1 + 1) = x pow (1 + 1)" by (simp add: hyperpow_hrabs) text{*The precondition could be weakened to @{term "0≤x"}*} lemma hypreal_mult_less_mono: "[| u<v; x<y; (0::hypreal) < v; 0 < x |] ==> u*x < v* y" by (simp add: Ring_and_Field.mult_strict_mono order_less_imp_le) lemma hyperpow_two_gt_one: "!!r::'a::{recpower,ordered_semidom} star. 1 < r ==> 1 < r pow (1 + 1)" by transfer (simp add: power_gt1) lemma hyperpow_two_ge_one: "!!r::'a::{recpower,ordered_semidom} star. 1 ≤ r ==> 1 ≤ r pow (1 + 1)" by transfer (simp add: one_le_power) lemma two_hyperpow_ge_one [simp]: "(1::hypreal) ≤ 2 pow n" apply (rule_tac y = "1 pow n" in order_trans) apply (rule_tac [2] hyperpow_le, auto) done lemma hyperpow_minus_one2 [simp]: "!!n. -1 pow ((1 + 1)*n) = (1::hypreal)" by transfer (subst power_mult, simp) lemma hyperpow_less_le: "!!r n N. [|(0::hypreal) ≤ r; r ≤ 1; n < N|] ==> r pow N ≤ r pow n" by transfer (rule power_decreasing [OF order_less_imp_le]) lemma hyperpow_SHNat_le: "[| 0 ≤ r; r ≤ (1::hypreal); N ∈ HNatInfinite |] ==> ALL n: Nats. r pow N ≤ r pow n" by (auto intro!: hyperpow_less_le simp add: HNatInfinite_iff) lemma hyperpow_realpow: "(hypreal_of_real r) pow (hypnat_of_nat n) = hypreal_of_real (r ^ n)" by transfer (rule refl) lemma hyperpow_SReal [simp]: "(hypreal_of_real r) pow (hypnat_of_nat n) ∈ Reals" by (simp add: Reals_eq_Standard) lemma hyperpow_zero_HNatInfinite [simp]: "N ∈ HNatInfinite ==> (0::hypreal) pow N = 0" by (drule HNatInfinite_is_Suc, auto) lemma hyperpow_le_le: "[| (0::hypreal) ≤ r; r ≤ 1; n ≤ N |] ==> r pow N ≤ r pow n" apply (drule order_le_less [of n, THEN iffD1]) apply (auto intro: hyperpow_less_le) done lemma hyperpow_Suc_le_self2: "[| (0::hypreal) ≤ r; r < 1 |] ==> r pow (n + (1::hypnat)) ≤ r" apply (drule_tac n = " (1::hypnat) " in hyperpow_le_le) apply auto done lemma hyperpow_hypnat_of_nat: "!!x. x pow hypnat_of_nat n = x ^ n" by transfer (rule refl) lemma of_hypreal_hyperpow: "!!x n. of_hypreal (x pow n) = (of_hypreal x::'a::{real_algebra_1,recpower} star) pow n" by transfer (rule of_real_power) end
lemma Standard_scaleR:
x ∈ Standard ==> r *R x ∈ Standard
lemma star_of_scaleR:
star_of (r *R x) = r *R star_of x
lemma star_of_real_def:
of_real r = star_of (of_real r)
lemma Standard_of_real:
of_real r ∈ Standard
lemma star_of_of_real:
star_of (of_real r) = of_real r
lemma of_real_eq_star_of:
of_real = hypreal_of_real
lemma Reals_eq_Standard:
Reals = Standard
lemma Standard_of_hypreal:
r ∈ Standard ==> of_hypreal r ∈ Standard
lemma of_hypreal_0:
of_hypreal 0 = 0
lemma of_hypreal_1:
of_hypreal 1 = 1
lemma of_hypreal_add:
of_hypreal (x + y) = of_hypreal x + of_hypreal y
lemma of_hypreal_minus:
of_hypreal (- x) = - of_hypreal x
lemma of_hypreal_diff:
of_hypreal (x - y) = of_hypreal x - of_hypreal y
lemma of_hypreal_mult:
of_hypreal (x * y) = of_hypreal x * of_hypreal y
lemma of_hypreal_inverse:
of_hypreal (inverse x) = inverse (of_hypreal x)
lemma of_hypreal_divide:
of_hypreal (x / y) = of_hypreal x / of_hypreal y
lemma of_hypreal_eq_iff:
(of_hypreal x = of_hypreal y) = (x = y)
lemma of_hypreal_eq_0_iff:
(of_hypreal x = 0) = (x = 0)
lemma lemma_starrel_refl:
x ∈ starrel `` {x}
lemma starrel_in_hypreal:
starrel `` {x} ∈ star
lemma inj_star_of:
inj star_of
lemma mem_Rep_star_iff:
(X ∈ Rep_star x) = (x = star_n X)
lemma Rep_star_star_n_iff:
(X ∈ Rep_star (star_n Y)) = ({n. Y n = X n} ∈ \<U>)
lemma Rep_star_star_n:
X ∈ Rep_star (star_n X)
lemma star_n_add:
star_n X + star_n Y = star_n (λn. X n + Y n)
lemma star_n_minus:
- star_n X = star_n (λn. - X n)
lemma star_n_diff:
star_n X - star_n Y = star_n (λn. X n - Y n)
lemma star_n_mult:
star_n X * star_n Y = star_n (λn. X n * Y n)
lemma star_n_inverse:
inverse (star_n X) = star_n (λn. inverse (X n))
lemma star_n_le:
(star_n X ≤ star_n Y) = ({n. X n ≤ Y n} ∈ \<U>)
lemma star_n_less:
(star_n X < star_n Y) = ({n. X n < Y n} ∈ \<U>)
lemma star_n_zero_num:
0 = star_n (λn. 0::'a)
lemma star_n_one_num:
1 = star_n (λn. 1::'a)
lemma star_n_abs:
¦star_n X¦ = star_n (λn. ¦X n¦)
lemma hypreal_not_refl2:
x < y ==> x ≠ y
lemma hypreal_eq_minus_iff:
(x = y) = (x + - y = 0)
lemma hypreal_mult_left_cancel:
c ≠ 0 ==> (c * a = c * b) = (a = b)
lemma hypreal_mult_right_cancel:
c ≠ 0 ==> (a * c = b * c) = (a = b)
lemma hypreal_omega_gt_zero:
0 < ω
lemma lemma_omega_empty_singleton_disj:
{n. x = real n} = {} ∨ (∃y. {n. x = real n} = {y})
lemma lemma_finite_omega_set:
finite {n. x = real n}
lemma not_ex_hypreal_of_real_eq_omega:
¬ (∃x. hypreal_of_real x = ω)
lemma hypreal_of_real_not_eq_omega:
hypreal_of_real x ≠ ω
lemma lemma_epsilon_empty_singleton_disj:
{n. x = inverse (real (Suc n))} = {} ∨
(∃y. {n. x = inverse (real (Suc n))} = {y})
lemma lemma_finite_epsilon_set:
finite {n. x = inverse (real (Suc n))}
lemma not_ex_hypreal_of_real_eq_epsilon:
¬ (∃x. hypreal_of_real x = ε)
lemma hypreal_of_real_not_eq_epsilon:
hypreal_of_real x ≠ ε
lemma hypreal_epsilon_not_zero:
ε ≠ 0
lemma hypreal_epsilon_inverse_omega:
ε = inverse ω
lemma hypreal_epsilon_gt_zero:
0 < ε
lemma hrabs_add_less:
[| ¦x¦ < r; ¦y¦ < s |] ==> ¦x + y¦ < r + s
lemma hrabs_less_gt_zero:
¦x¦ < r ==> 0 < r
lemma hrabs_disj:
¦x¦ = x ∨ ¦x¦ = - x
lemma hrabs_add_lemma_disj:
y + - x + (y + - z) = ¦x + - z¦ ==> y = z ∨ x = y
lemma SNat_eq:
Nats = {n. ∃N. n = hypreal_of_nat N}
lemma hypreal_of_nat_eq:
hypreal_of_nat n = hypreal_of_real (real n)
lemma hypreal_of_nat:
hypreal_of_nat m = star_n (λn. real m)
lemma hpowr_0:
r ^ 0 = 1
lemma hpowr_Suc:
r ^ Suc n = r * r ^ n
lemma hrealpow_two:
r ^ Suc (Suc 0) = r * r
lemma hrealpow_two_le:
0 ≤ r ^ Suc (Suc 0)
lemma hrealpow_two_le_add_order:
0 ≤ u ^ Suc (Suc 0) + v ^ Suc (Suc 0)
lemma hrealpow_two_le_add_order2:
0 ≤ u ^ Suc (Suc 0) + v ^ Suc (Suc 0) + w ^ Suc (Suc 0)
lemma hypreal_add_nonneg_eq_0_iff:
[| 0 ≤ x; 0 ≤ y |] ==> (x + y = 0) = (x = 0 ∧ y = 0)
lemma hypreal_three_squares_add_zero_iff:
(x * x + y * y + z * z = 0) = (x = 0 ∧ y = 0 ∧ z = 0)
lemma hrealpow_three_squares_add_zero_iff:
(x ^ Suc (Suc 0) + y ^ Suc (Suc 0) + z ^ Suc (Suc 0) = 0) =
(x = 0 ∧ y = 0 ∧ z = 0)
lemma hrabs_hrealpow_two:
¦x ^ Suc (Suc 0)¦ = x ^ Suc (Suc 0)
lemma two_hrealpow_ge_one:
1 ≤ 2 ^ n
lemma two_hrealpow_gt:
hypreal_of_nat n < 2 ^ n
lemma hrealpow:
star_n X ^ m = star_n (λn. X n ^ m)
lemma hrealpow_sum_square_expand:
(x + y) ^ Suc (Suc 0) =
x ^ Suc (Suc 0) + y ^ Suc (Suc 0) + hypreal_of_nat (Suc (Suc 0)) * x * y
lemma power_hypreal_of_real_number_of:
number_of v ^ n = hypreal_of_real (number_of v ^ n)
lemma Standard_hyperpow:
[| r ∈ Standard; n ∈ Standard |] ==> r pow n ∈ Standard
lemma hyperpow:
star_n X pow star_n Y = star_n (λn. X n ^ Y n)
lemma hyperpow_zero:
0 pow (n + 1) = 0
lemma hyperpow_not_zero:
r ≠ 0 ==> r pow n ≠ 0
lemma hyperpow_inverse:
r ≠ 0 ==> inverse (r pow n) = inverse r pow n
lemma hyperpow_hrabs:
¦r¦ pow n = ¦r pow n¦
lemma hyperpow_add:
r pow (n + m) = r pow n * r pow m
lemma hyperpow_one:
r pow 1 = r
lemma hyperpow_two:
r pow (1 + 1) = r * r
lemma hyperpow_gt_zero:
0 < r ==> 0 < r pow n
lemma hyperpow_ge_zero:
0 ≤ r ==> 0 ≤ r pow n
lemma hyperpow_le:
[| 0 < x; x ≤ y |] ==> x pow n ≤ y pow n
lemma hyperpow_eq_one:
1 pow n = 1
lemma hrabs_hyperpow_minus_one:
¦-1 pow n¦ = 1
lemma hyperpow_mult:
(r * s) pow n = r pow n * s pow n
lemma hyperpow_two_le:
0 ≤ r pow (1 + 1)
lemma hrabs_hyperpow_two:
¦x pow (1 + 1)¦ = x pow (1 + 1)
lemma hyperpow_two_hrabs:
¦x¦ pow (1 + 1) = x pow (1 + 1)
lemma hypreal_mult_less_mono:
[| u < v; x < y; 0 < v; 0 < x |] ==> u * x < v * y
lemma hyperpow_two_gt_one:
1 < r ==> 1 < r pow (1 + 1)
lemma hyperpow_two_ge_one:
1 ≤ r ==> 1 ≤ r pow (1 + 1)
lemma two_hyperpow_ge_one:
1 ≤ 2 pow n
lemma hyperpow_minus_one2:
-1 pow ((1 + 1) * n) = 1
lemma hyperpow_less_le:
[| 0 ≤ r; r ≤ 1; n < N |] ==> r pow N ≤ r pow n
lemma hyperpow_SHNat_le:
[| 0 ≤ r; r ≤ 1; N ∈ HNatInfinite |] ==> ∀n∈Nats. r pow N ≤ r pow n
lemma hyperpow_realpow:
hypreal_of_real r pow hypnat_of_nat n = hypreal_of_real (r ^ n)
lemma hyperpow_SReal:
hypreal_of_real r pow hypnat_of_nat n ∈ Reals
lemma hyperpow_zero_HNatInfinite:
N ∈ HNatInfinite ==> 0 pow N = 0
lemma hyperpow_le_le:
[| 0 ≤ r; r ≤ 1; n ≤ N |] ==> r pow N ≤ r pow n
lemma hyperpow_Suc_le_self2:
[| 0 ≤ r; r < 1 |] ==> r pow (n + 1) ≤ r
lemma hyperpow_hypnat_of_nat:
x pow hypnat_of_nat n = x ^ n
lemma of_hypreal_hyperpow:
of_hypreal (x pow n) = of_hypreal x pow n