(* Title : HOL/Hyperreal/HyperDef.thy
ID : $Id$
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 \<Rightarrow> hypreal" where
"hypreal_of_hypnat \<equiv> of_hypnat"
definition
omega :: hypreal where
-- {*an infinite number @{text "= [<1,2,3,...>]"} *}
"omega = star_n (\<lambda>n. real (Suc n))"
definition
epsilon :: hypreal where
-- {*an infinitesimal number @{text "= [<1,1/2,1/3,...>]"} *}
"epsilon = star_n (\<lambda>n. inverse (real (Suc n)))"
notation (xsymbols)
omega ("\<omega>") and
epsilon ("\<epsilon>")
notation (HTML output)
omega ("\<omega>") and
epsilon ("\<epsilon>")
subsection {* Real vector class instances *}
instantiation star :: (scaleR) scaleR
begin
definition
star_scaleR_def [transfer_unfold, code func del]: "scaleR r \<equiv> *f* (scaleR r)"
instance ..
end
lemma Standard_scaleR [simp]: "x \<in> Standard \<Longrightarrow> scaleR r x \<in> 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 "\<And>x y::'a star. scaleR a (x + y) = scaleR a x + scaleR a y"
by transfer (rule scaleR_right_distrib)
show "\<And>x::'a star. scaleR (a + b) x = scaleR a x + scaleR b x"
by transfer (rule scaleR_left_distrib)
show "\<And>x::'a star. scaleR a (scaleR b x) = scaleR (a * b) x"
by transfer (rule scaleR_scaleR)
show "\<And>x::'a star. scaleR 1 x = x"
by transfer (rule scaleR_one)
qed
instance star :: (real_algebra) real_algebra
proof
fix a :: real
show "\<And>x y::'a star. scaleR a x * y = scaleR a (x * y)"
by transfer (rule mult_scaleR_left)
show "\<And>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 :: (field_char_0) field_char_0 ..
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 \<in> 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 \<Rightarrow> 'a::real_algebra_1 star" where
[transfer_unfold, code func del]: "of_hypreal = *f* of_real"
lemma Standard_of_hypreal [simp]:
"r \<in> Standard \<Longrightarrow> of_hypreal r \<in> 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]:
"\<And>x y. of_hypreal (x + y) = of_hypreal x + of_hypreal y"
by transfer (rule of_real_add)
lemma of_hypreal_minus [simp]: "\<And>x. of_hypreal (- x) = - of_hypreal x"
by transfer (rule of_real_minus)
lemma of_hypreal_diff [simp]:
"\<And>x y. of_hypreal (x - y) = of_hypreal x - of_hypreal y"
by transfer (rule of_real_diff)
lemma of_hypreal_mult [simp]:
"\<And>x y. of_hypreal (x * y) = of_hypreal x * of_hypreal y"
by transfer (rule of_real_mult)
lemma of_hypreal_inverse [simp]:
"\<And>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]:
"\<And>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]:
"\<And>x y. (of_hypreal x = of_hypreal y) = (x = y)"
by transfer (rule of_real_eq_iff)
lemma of_hypreal_eq_0_iff [simp]:
"\<And>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 \<in> 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 \<in> Rep_star x) = (x = star_n X)"
by (cases x, simp add: star_n_def)
lemma Rep_star_star_n_iff [simp]:
"(X \<in> Rep_star (star_n Y)) = ({n. Y n = X n} \<in> \<U>)"
by (simp add: star_n_def)
lemma Rep_star_star_n: "X \<in> 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 \<le> star_n Y =
({n. X n \<le> Y n} \<in> 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} \<in> 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 \<noteq> y"
by (auto)
lemma hypreal_eq_minus_iff: "((x::hypreal) = y) = (x + - y = 0)"
by auto
lemma hypreal_mult_left_cancel: "(c::hypreal) \<noteq> 0 ==> (c*a=c*b) = (a=b)"
by auto
lemma hypreal_mult_right_cancel: "(c::hypreal) \<noteq> 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} = {} |
(\<exists>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:
"~ (\<exists>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 \<noteq> 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))} = {} |
(\<exists>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: "~ (\<exists>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 \<noteq> epsilon"
by (insert not_ex_hypreal_of_real_eq_epsilon, auto)
lemma hypreal_epsilon_not_zero: "epsilon \<noteq> 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. \<exists>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) \<le> r ^ Suc (Suc 0)"
by (auto simp add: zero_le_mult_iff)
lemma hrealpow_two_le_add_order [simp]:
"(0::hypreal) \<le> 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) \<le> 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 \<le> x; 0 \<le> 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) \<le> 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 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 \<in> HFinite ==> x ^ n \<in> HFinite"
apply (induct_tac "n")
apply (auto simp add: power_Suc intro: HFinite_mult)
done
*)
subsection{*Powers with Hypernatural Exponents*}
definition pow :: "['a::power star, nat star] \<Rightarrow> 'a star" (infixr "pow" 80) where
hyperpow_def [transfer_unfold, code func del]: "R pow N = ( *f2* op ^) R N"
(* hypernatural powers of hyperreals *)
lemma Standard_hyperpow [simp]:
"\<lbrakk>r \<in> Standard; n \<in> Standard\<rbrakk> \<Longrightarrow> r pow n \<in> 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]:
"\<And>n. (0::'a::{recpower,semiring_0} star) pow (n + (1::hypnat)) = 0"
by transfer simp
lemma hyperpow_not_zero:
"\<And>r n. r \<noteq> (0::'a::{recpower,field} star) ==> r pow n \<noteq> 0"
by transfer (rule field_power_not_zero)
lemma hyperpow_inverse:
"\<And>r n. r \<noteq> (0::'a::{recpower,division_by_zero,field} star)
\<Longrightarrow> inverse (r pow n) = (inverse r) pow n"
by transfer (rule power_inverse)
lemma hyperpow_hrabs:
"\<And>r n. abs (r::'a::{recpower,ordered_idom} star) pow n = abs (r pow n)"
by transfer (rule power_abs [symmetric])
lemma hyperpow_add:
"\<And>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]:
"\<And>r. (r::'a::recpower star) pow (1::hypnat) = r"
by transfer (rule power_one_right)
lemma hyperpow_two:
"\<And>r. (r::'a::recpower star) pow ((1::hypnat) + (1::hypnat)) = r * r"
by transfer (simp add: power_Suc)
lemma hyperpow_gt_zero:
"\<And>r n. (0::'a::{recpower,ordered_semidom} star) < r \<Longrightarrow> 0 < r pow n"
by transfer (rule zero_less_power)
lemma hyperpow_ge_zero:
"\<And>r n. (0::'a::{recpower,ordered_semidom} star) \<le> r \<Longrightarrow> 0 \<le> r pow n"
by transfer (rule zero_le_power)
lemma hyperpow_le:
"\<And>x y n. \<lbrakk>(0::'a::{recpower,ordered_semidom} star) < x; x \<le> y\<rbrakk>
\<Longrightarrow> x pow n \<le> y pow n"
by transfer (rule power_mono [OF _ order_less_imp_le])
lemma hyperpow_eq_one [simp]:
"\<And>n. 1 pow n = (1::'a::recpower star)"
by transfer (rule power_one)
lemma hrabs_hyperpow_minus_one [simp]:
"\<And>n. abs(-1 pow n) = (1::'a::{number_ring,recpower,ordered_idom} star)"
by transfer (rule abs_power_minus_one)
lemma hyperpow_mult:
"\<And>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) \<le> 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\<le>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:
"\<And>r::'a::{recpower,ordered_semidom} star. 1 < r \<Longrightarrow> 1 < r pow (1 + 1)"
by transfer (simp add: power_gt1)
lemma hyperpow_two_ge_one:
"\<And>r::'a::{recpower,ordered_semidom} star. 1 \<le> r \<Longrightarrow> 1 \<le> r pow (1 + 1)"
by transfer (simp add: one_le_power)
lemma two_hyperpow_ge_one [simp]: "(1::hypreal) \<le> 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) \<le> r; r \<le> 1; n < N|] ==> r pow N \<le> r pow n"
by transfer (rule power_decreasing [OF order_less_imp_le])
lemma hyperpow_SHNat_le:
"[| 0 \<le> r; r \<le> (1::hypreal); N \<in> HNatInfinite |]
==> ALL n: Nats. r pow N \<le> 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) \<in> Reals"
by (simp add: Reals_eq_Standard)
lemma hyperpow_zero_HNatInfinite [simp]:
"N \<in> HNatInfinite ==> (0::hypreal) pow N = 0"
by (drule HNatInfinite_is_Suc, auto)
lemma hyperpow_le_le:
"[| (0::hypreal) \<le> r; r \<le> 1; n \<le> N |] ==> r pow N \<le> 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) \<le> r; r < 1 |] ==> r pow (n + (1::hypnat)) \<le> r"
apply (drule_tac n = " (1::hypnat) " in hyperpow_le_le)
apply auto
done
lemma hyperpow_hypnat_of_nat: "\<And>x. x pow hypnat_of_nat n = x ^ n"
by transfer (rule refl)
lemma of_hypreal_hyperpow:
"\<And>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