starfun, starset, and other functions on NS types are now polymorphic;
many similar theorems have been generalized and merged;
(star_n X) replaces (Abs_star(starrel `` {X}));
many proofs have been simplified with the transfer tactic.
(* Title : HyperNat.thy
Author : Jacques D. Fleuriot
Copyright : 1998 University of Cambridge
Converted to Isar and polished by lcp
*)
header{*Construction of Hypernaturals using Ultrafilters*}
theory HyperNat
imports Star
begin
types hypnat = "nat star"
syntax hypnat_of_nat :: "nat => nat star"
translations "hypnat_of_nat" => "star_of :: nat => nat star"
consts whn :: hypnat
defs
(* omega is in fact an infinite hypernatural number = [<1,2,3,...>] *)
hypnat_omega_def: "whn == star_n (%n::nat. n)"
lemma hypnat_minus_zero [simp]: "!!z. z - z = (0::hypnat)"
by transfer (rule diff_self_eq_0)
lemma hypnat_diff_0_eq_0 [simp]: "!!n. (0::hypnat) - n = 0"
by transfer (rule diff_0_eq_0)
lemma hypnat_add_is_0 [iff]: "!!m n. (m+n = (0::hypnat)) = (m=0 & n=0)"
by transfer (rule add_is_0)
lemma hypnat_diff_diff_left: "!!i j k. (i::hypnat) - j - k = i - (j+k)"
by transfer (rule diff_diff_left)
lemma hypnat_diff_commute: "!!i j k. (i::hypnat) - j - k = i-k-j"
by transfer (rule diff_commute)
lemma hypnat_diff_add_inverse [simp]: "!!m n. ((n::hypnat) + m) - n = m"
by transfer (rule diff_add_inverse)
lemma hypnat_diff_add_inverse2 [simp]: "!!m n. ((m::hypnat) + n) - n = m"
by transfer (rule diff_add_inverse2)
lemma hypnat_diff_cancel [simp]: "!!k m n. ((k::hypnat) + m) - (k+n) = m - n"
by transfer (rule diff_cancel)
lemma hypnat_diff_cancel2 [simp]: "!!k m n. ((m::hypnat) + k) - (n+k) = m - n"
by transfer (rule diff_cancel2)
lemma hypnat_diff_add_0 [simp]: "!!m n. (n::hypnat) - (n+m) = (0::hypnat)"
by transfer (rule diff_add_0)
subsection{*Hyperreal Multiplication*}
lemma hypnat_diff_mult_distrib: "!!k m n. ((m::hypnat) - n) * k = (m * k) - (n * k)"
by transfer (rule diff_mult_distrib)
lemma hypnat_diff_mult_distrib2: "!!k m n. (k::hypnat) * (m - n) = (k * m) - (k * n)"
by transfer (rule diff_mult_distrib2)
subsection{*Properties of The @{text "\<le>"} Relation*}
lemma hypnat_le_zero_cancel [iff]: "!!n. (n \<le> (0::hypnat)) = (n = 0)"
by transfer (rule le_0_eq)
lemma hypnat_mult_is_0 [simp]: "!!m n. (m*n = (0::hypnat)) = (m=0 | n=0)"
by transfer (rule mult_is_0)
lemma hypnat_diff_is_0_eq [simp]: "!!m n. ((m::hypnat) - n = 0) = (m \<le> n)"
by transfer (rule diff_is_0_eq)
subsection{*Theorems for Ordering*}
lemma hypnat_not_less0 [iff]: "!!n. ~ n < (0::hypnat)"
by transfer (rule not_less0)
lemma hypnat_less_one [iff]:
"!!n. (n < (1::hypnat)) = (n=0)"
by transfer (rule less_one)
lemma hypnat_add_diff_inverse: "!!m n. ~ m<n ==> n+(m-n) = (m::hypnat)"
by transfer (rule add_diff_inverse)
lemma hypnat_le_add_diff_inverse [simp]: "!!m n. n \<le> m ==> n+(m-n) = (m::hypnat)"
by transfer (rule le_add_diff_inverse)
lemma hypnat_le_add_diff_inverse2 [simp]: "!!m n. n\<le>m ==> (m-n)+n = (m::hypnat)"
by transfer (rule le_add_diff_inverse2)
declare hypnat_le_add_diff_inverse2 [OF order_less_imp_le]
lemma hypnat_le0 [iff]: "!!n. (0::hypnat) \<le> n"
by transfer (rule le0)
lemma hypnat_add_self_le [simp]: "!!x n. (x::hypnat) \<le> n + x"
by transfer (rule le_add2)
lemma hypnat_add_one_self_less [simp]: "(x::hypnat) < x + (1::hypnat)"
by (insert add_strict_left_mono [OF zero_less_one], auto)
lemma hypnat_neq0_conv [iff]: "(n \<noteq> 0) = (0 < (n::hypnat))"
by (simp add: order_less_le)
lemma hypnat_gt_zero_iff: "((0::hypnat) < n) = ((1::hypnat) \<le> n)"
by (auto simp add: linorder_not_less [symmetric])
lemma hypnat_gt_zero_iff2: "(0 < n) = (\<exists>m. n = m + (1::hypnat))"
apply safe
apply (rule_tac x = "n - (1::hypnat) " in exI)
apply (simp add: hypnat_gt_zero_iff)
apply (insert add_le_less_mono [OF _ zero_less_one, of 0], auto)
done
lemma hypnat_add_self_not_less: "~ (x + y < (x::hypnat))"
by (simp add: linorder_not_le [symmetric] add_commute [of x])
lemma hypnat_diff_split:
"P(a - b::hypnat) = ((a<b --> P 0) & (ALL d. a = b + d --> P d))"
-- {* elimination of @{text -} on @{text hypnat} *}
proof (cases "a<b" rule: case_split)
case True
thus ?thesis
by (auto simp add: hypnat_add_self_not_less order_less_imp_le
hypnat_diff_is_0_eq [THEN iffD2])
next
case False
thus ?thesis
by (auto simp add: linorder_not_less dest: order_le_less_trans)
qed
subsection{*The Embedding @{term hypnat_of_nat} Preserves @{text
comm_ring_1} and Order Properties*}
constdefs
(* the set of infinite hypernatural numbers *)
HNatInfinite :: "hypnat set"
"HNatInfinite == {n. n \<notin> Nats}"
lemma hypnat_of_nat_def: "hypnat_of_nat m == of_nat m"
by (transfer star_of_nat_def) simp
lemma hypnat_of_nat_add:
"hypnat_of_nat ((z::nat) + w) = hypnat_of_nat z + hypnat_of_nat w"
by simp
lemma hypnat_of_nat_mult:
"hypnat_of_nat (z * w) = hypnat_of_nat z * hypnat_of_nat w"
by simp
lemma hypnat_of_nat_less_iff:
"(hypnat_of_nat z < hypnat_of_nat w) = (z < w)"
by simp
lemma hypnat_of_nat_le_iff:
"(hypnat_of_nat z \<le> hypnat_of_nat w) = (z \<le> w)"
by simp
lemma hypnat_of_nat_eq_iff:
"(hypnat_of_nat z = hypnat_of_nat w) = (z = w)"
by simp
lemma hypnat_of_nat_one [simp]: "hypnat_of_nat (Suc 0) = (1::hypnat)"
by simp
lemma hypnat_of_nat_zero: "hypnat_of_nat 0 = 0"
by simp
lemma hypnat_of_nat_zero_iff: "(hypnat_of_nat n = 0) = (n = 0)"
by simp
lemma hypnat_of_nat_Suc [simp]:
"hypnat_of_nat (Suc n) = hypnat_of_nat n + (1::hypnat)"
by (simp add: hypnat_of_nat_def)
lemma hypnat_of_nat_minus:
"hypnat_of_nat ((j::nat) - k) = hypnat_of_nat j - hypnat_of_nat k"
by simp
subsection{*Existence of an infinite hypernatural number*}
text{*Existence of infinite number not corresponding to any natural number
follows because member @{term FreeUltrafilterNat} is not finite.
See @{text HyperDef.thy} for similar argument.*}
subsection{*Properties of the set of embedded natural numbers*}
lemma of_nat_eq_add [rule_format]:
"\<forall>d::hypnat. of_nat m = of_nat n + d --> d \<in> range of_nat"
apply (induct n)
apply (auto simp add: add_assoc)
apply (case_tac x)
apply (auto simp add: add_commute [of 1])
done
lemma Nats_diff [simp]: "[|a \<in> Nats; b \<in> Nats|] ==> (a-b :: hypnat) \<in> Nats"
by (auto simp add: of_nat_eq_add Nats_def split: hypnat_diff_split)
lemma lemma_unbounded_set [simp]: "{n::nat. m < n} \<in> FreeUltrafilterNat"
apply (insert finite_atMost [of m])
apply (simp add: atMost_def)
apply (drule FreeUltrafilterNat_finite)
apply (drule FreeUltrafilterNat_Compl_mem, ultra)
done
lemma Compl_Collect_le: "- {n::nat. N \<le> n} = {n. n < N}"
by (simp add: Collect_neg_eq [symmetric] linorder_not_le)
lemma hypnat_of_nat_eq:
"hypnat_of_nat m = star_n (%n::nat. m)"
apply (induct m)
apply (simp_all add: star_n_zero_num star_n_one_num star_n_add)
done
lemma SHNat_eq: "Nats = {n. \<exists>N. n = hypnat_of_nat N}"
by (force simp add: hypnat_of_nat_def Nats_def)
lemma hypnat_omega_gt_SHNat:
"n \<in> Nats ==> n < whn"
by (auto simp add: hypnat_of_nat_eq star_n_less hypnat_omega_def SHNat_eq)
(* Infinite hypernatural not in embedded Nats *)
lemma SHNAT_omega_not_mem [simp]: "whn \<notin> Nats"
by (blast dest: hypnat_omega_gt_SHNat)
lemma hypnat_of_nat_less_whn [simp]: "hypnat_of_nat n < whn"
apply (insert hypnat_omega_gt_SHNat [of "hypnat_of_nat n"])
apply (simp add: hypnat_of_nat_def)
done
lemma hypnat_of_nat_le_whn [simp]: "hypnat_of_nat n \<le> whn"
by (rule hypnat_of_nat_less_whn [THEN order_less_imp_le])
lemma hypnat_zero_less_hypnat_omega [simp]: "0 < whn"
by (simp add: hypnat_omega_gt_SHNat)
lemma hypnat_one_less_hypnat_omega [simp]: "(1::hypnat) < whn"
by (simp add: hypnat_omega_gt_SHNat)
subsection{*Infinite Hypernatural Numbers -- @{term HNatInfinite}*}
lemma HNatInfinite_whn [simp]: "whn \<in> HNatInfinite"
by (simp add: HNatInfinite_def)
lemma Nats_not_HNatInfinite_iff: "(x \<in> Nats) = (x \<notin> HNatInfinite)"
by (simp add: HNatInfinite_def)
lemma HNatInfinite_not_Nats_iff: "(x \<in> HNatInfinite) = (x \<notin> Nats)"
by (simp add: HNatInfinite_def)
subsection{*Alternative characterization of the set of infinite hypernaturals*}
text{* @{term "HNatInfinite = {N. \<forall>n \<in> Nats. n < N}"}*}
(*??delete? similar reasoning in hypnat_omega_gt_SHNat above*)
lemma HNatInfinite_FreeUltrafilterNat_lemma:
"\<forall>N::nat. {n. f n \<noteq> N} \<in> FreeUltrafilterNat
==> {n. N < f n} \<in> FreeUltrafilterNat"
apply (induct_tac N)
apply (drule_tac x = 0 in spec)
apply (rule ccontr, drule FreeUltrafilterNat_Compl_mem, drule FreeUltrafilterNat_Int, assumption, simp)
apply (drule_tac x = "Suc n" in spec, ultra)
done
lemma HNatInfinite_iff: "HNatInfinite = {N. \<forall>n \<in> Nats. n < N}"
apply (auto simp add: HNatInfinite_def SHNat_eq hypnat_of_nat_eq)
apply (rule_tac x = x in star_cases)
apply (auto elim: HNatInfinite_FreeUltrafilterNat_lemma
simp add: star_n_less FreeUltrafilterNat_Compl_iff1
star_n_eq_iff Collect_neg_eq [symmetric])
done
subsection{*Alternative Characterization of @{term HNatInfinite} using
Free Ultrafilter*}
lemma HNatInfinite_FreeUltrafilterNat:
"x \<in> HNatInfinite
==> \<exists>X \<in> Rep_star x. \<forall>u. {n. u < X n}: FreeUltrafilterNat"
apply (cases x)
apply (auto simp add: HNatInfinite_iff SHNat_eq hypnat_of_nat_eq)
apply (rule bexI [OF _ Rep_star_star_n], clarify)
apply (auto simp add: hypnat_of_nat_def star_n_less)
done
lemma FreeUltrafilterNat_HNatInfinite:
"\<exists>X \<in> Rep_star x. \<forall>u. {n. u < X n}: FreeUltrafilterNat
==> x \<in> HNatInfinite"
apply (cases x)
apply (auto simp add: star_n_less HNatInfinite_iff SHNat_eq hypnat_of_nat_eq)
apply (drule spec, ultra, auto)
done
lemma HNatInfinite_FreeUltrafilterNat_iff:
"(x \<in> HNatInfinite) =
(\<exists>X \<in> Rep_star x. \<forall>u. {n. u < X n}: FreeUltrafilterNat)"
by (blast intro: HNatInfinite_FreeUltrafilterNat
FreeUltrafilterNat_HNatInfinite)
lemma HNatInfinite_gt_one [simp]: "x \<in> HNatInfinite ==> (1::hypnat) < x"
by (auto simp add: HNatInfinite_iff)
lemma zero_not_mem_HNatInfinite [simp]: "0 \<notin> HNatInfinite"
apply (auto simp add: HNatInfinite_iff)
apply (drule_tac a = " (1::hypnat) " in equals0D)
apply simp
done
lemma HNatInfinite_not_eq_zero: "x \<in> HNatInfinite ==> 0 < x"
apply (drule HNatInfinite_gt_one)
apply (auto simp add: order_less_trans [OF zero_less_one])
done
lemma HNatInfinite_ge_one [simp]: "x \<in> HNatInfinite ==> (1::hypnat) \<le> x"
by (blast intro: order_less_imp_le HNatInfinite_gt_one)
subsection{*Closure Rules*}
lemma HNatInfinite_add:
"[| x \<in> HNatInfinite; y \<in> HNatInfinite |] ==> x + y \<in> HNatInfinite"
apply (auto simp add: HNatInfinite_iff)
apply (drule bspec, assumption)
apply (drule bspec [OF _ Nats_0])
apply (drule add_strict_mono, assumption, simp)
done
lemma HNatInfinite_SHNat_add:
"[| x \<in> HNatInfinite; y \<in> Nats |] ==> x + y \<in> HNatInfinite"
apply (auto simp add: HNatInfinite_not_Nats_iff)
apply (drule_tac a = "x + y" in Nats_diff, auto)
done
lemma HNatInfinite_Nats_imp_less: "[| x \<in> HNatInfinite; y \<in> Nats |] ==> y < x"
by (simp add: HNatInfinite_iff)
lemma HNatInfinite_SHNat_diff:
assumes x: "x \<in> HNatInfinite" and y: "y \<in> Nats"
shows "x - y \<in> HNatInfinite"
proof -
have "y < x" by (simp add: HNatInfinite_Nats_imp_less prems)
hence "x - y + y = x" by (simp add: order_less_imp_le)
with x show ?thesis
by (force simp add: HNatInfinite_not_Nats_iff
dest: Nats_add [of "x-y", OF _ y])
qed
lemma HNatInfinite_add_one:
"x \<in> HNatInfinite ==> x + (1::hypnat) \<in> HNatInfinite"
by (auto intro: HNatInfinite_SHNat_add)
lemma HNatInfinite_is_Suc: "x \<in> HNatInfinite ==> \<exists>y. x = y + (1::hypnat)"
apply (rule_tac x = "x - (1::hypnat) " in exI)
apply auto
done
subsection{*Embedding of the Hypernaturals into the Hyperreals*}
text{*Obtained using the nonstandard extension of the naturals*}
constdefs
hypreal_of_hypnat :: "hypnat => hypreal"
"hypreal_of_hypnat == *f* real"
lemma HNat_hypreal_of_nat [simp]: "hypreal_of_nat N \<in> Nats"
by (simp add: hypreal_of_nat_def)
lemma hypreal_of_hypnat:
"hypreal_of_hypnat (star_n X) = star_n (%n. real (X n))"
by (simp add: hypreal_of_hypnat_def starfun)
lemma hypreal_of_hypnat_inject [simp]:
"!!m n. (hypreal_of_hypnat m = hypreal_of_hypnat n) = (m=n)"
by (unfold hypreal_of_hypnat_def, transfer, simp)
lemma hypreal_of_hypnat_zero: "hypreal_of_hypnat 0 = 0"
by (simp add: star_n_zero_num hypreal_of_hypnat)
lemma hypreal_of_hypnat_one: "hypreal_of_hypnat (1::hypnat) = 1"
by (simp add: star_n_one_num hypreal_of_hypnat)
lemma hypreal_of_hypnat_add [simp]:
"!!m n. hypreal_of_hypnat (m + n) = hypreal_of_hypnat m + hypreal_of_hypnat n"
by (unfold hypreal_of_hypnat_def, transfer, rule real_of_nat_add)
lemma hypreal_of_hypnat_mult [simp]:
"!!m n. hypreal_of_hypnat (m * n) = hypreal_of_hypnat m * hypreal_of_hypnat n"
by (unfold hypreal_of_hypnat_def, transfer, rule real_of_nat_mult)
lemma hypreal_of_hypnat_less_iff [simp]:
"!!m n. (hypreal_of_hypnat n < hypreal_of_hypnat m) = (n < m)"
by (unfold hypreal_of_hypnat_def, transfer, simp)
lemma hypreal_of_hypnat_eq_zero_iff: "(hypreal_of_hypnat N = 0) = (N = 0)"
by (simp add: hypreal_of_hypnat_zero [symmetric])
declare hypreal_of_hypnat_eq_zero_iff [simp]
lemma hypreal_of_hypnat_ge_zero [simp]: "!!n. 0 \<le> hypreal_of_hypnat n"
by (unfold hypreal_of_hypnat_def, transfer, simp)
lemma HNatInfinite_inverse_Infinitesimal [simp]:
"n \<in> HNatInfinite ==> inverse (hypreal_of_hypnat n) \<in> Infinitesimal"
apply (cases n)
apply (auto simp add: hypreal_of_hypnat star_n_inverse
HNatInfinite_FreeUltrafilterNat_iff Infinitesimal_FreeUltrafilterNat_iff2)
apply (rule bexI [OF _ Rep_star_star_n], auto)
apply (drule_tac x = "m + 1" in spec, ultra)
done
lemma HNatInfinite_hypreal_of_hypnat_gt_zero:
"N \<in> HNatInfinite ==> 0 < hypreal_of_hypnat N"
apply (rule ccontr)
apply (simp add: hypreal_of_hypnat_zero [symmetric] linorder_not_less)
done
ML
{*
val hypnat_of_nat_def = thm"hypnat_of_nat_def";
val HNatInfinite_def = thm"HNatInfinite_def";
val hypreal_of_hypnat_def = thm"hypreal_of_hypnat_def";
val hypnat_omega_def = thm"hypnat_omega_def";
val starrel_iff = thm "starrel_iff";
val lemma_starrel_refl = thm "lemma_starrel_refl";
val eq_Abs_star = thm "eq_Abs_star";
val hypnat_minus_zero = thm "hypnat_minus_zero";
val hypnat_diff_0_eq_0 = thm "hypnat_diff_0_eq_0";
val hypnat_add_is_0 = thm "hypnat_add_is_0";
val hypnat_diff_diff_left = thm "hypnat_diff_diff_left";
val hypnat_diff_commute = thm "hypnat_diff_commute";
val hypnat_diff_add_inverse = thm "hypnat_diff_add_inverse";
val hypnat_diff_add_inverse2 = thm "hypnat_diff_add_inverse2";
val hypnat_diff_cancel = thm "hypnat_diff_cancel";
val hypnat_diff_cancel2 = thm "hypnat_diff_cancel2";
val hypnat_diff_add_0 = thm "hypnat_diff_add_0";
val hypnat_diff_mult_distrib = thm "hypnat_diff_mult_distrib";
val hypnat_diff_mult_distrib2 = thm "hypnat_diff_mult_distrib2";
val hypnat_mult_is_0 = thm "hypnat_mult_is_0";
val hypnat_not_less0 = thm "hypnat_not_less0";
val hypnat_less_one = thm "hypnat_less_one";
val hypnat_add_diff_inverse = thm "hypnat_add_diff_inverse";
val hypnat_le_add_diff_inverse = thm "hypnat_le_add_diff_inverse";
val hypnat_le_add_diff_inverse2 = thm "hypnat_le_add_diff_inverse2";
val hypnat_le0 = thm "hypnat_le0";
val hypnat_add_self_le = thm "hypnat_add_self_le";
val hypnat_add_one_self_less = thm "hypnat_add_one_self_less";
val hypnat_neq0_conv = thm "hypnat_neq0_conv";
val hypnat_gt_zero_iff = thm "hypnat_gt_zero_iff";
val hypnat_gt_zero_iff2 = thm "hypnat_gt_zero_iff2";
val hypnat_of_nat_add = thm "hypnat_of_nat_add";
val hypnat_of_nat_minus = thm "hypnat_of_nat_minus";
val hypnat_of_nat_mult = thm "hypnat_of_nat_mult";
val hypnat_of_nat_less_iff = thm "hypnat_of_nat_less_iff";
val hypnat_of_nat_le_iff = thm "hypnat_of_nat_le_iff";
val hypnat_of_nat_eq = thm"hypnat_of_nat_eq"
val SHNat_eq = thm"SHNat_eq"
val hypnat_of_nat_one = thm "hypnat_of_nat_one";
val hypnat_of_nat_zero = thm "hypnat_of_nat_zero";
val hypnat_of_nat_zero_iff = thm "hypnat_of_nat_zero_iff";
val hypnat_of_nat_Suc = thm "hypnat_of_nat_Suc";
val SHNAT_omega_not_mem = thm "SHNAT_omega_not_mem";
val cofinite_mem_FreeUltrafilterNat = thm "cofinite_mem_FreeUltrafilterNat";
val hypnat_omega_gt_SHNat = thm "hypnat_omega_gt_SHNat";
val hypnat_of_nat_less_whn = thm "hypnat_of_nat_less_whn";
val hypnat_of_nat_le_whn = thm "hypnat_of_nat_le_whn";
val hypnat_zero_less_hypnat_omega = thm "hypnat_zero_less_hypnat_omega";
val hypnat_one_less_hypnat_omega = thm "hypnat_one_less_hypnat_omega";
val HNatInfinite_whn = thm "HNatInfinite_whn";
val HNatInfinite_iff = thm "HNatInfinite_iff";
val HNatInfinite_FreeUltrafilterNat = thm "HNatInfinite_FreeUltrafilterNat";
val FreeUltrafilterNat_HNatInfinite = thm "FreeUltrafilterNat_HNatInfinite";
val HNatInfinite_FreeUltrafilterNat_iff = thm "HNatInfinite_FreeUltrafilterNat_iff";
val HNatInfinite_gt_one = thm "HNatInfinite_gt_one";
val zero_not_mem_HNatInfinite = thm "zero_not_mem_HNatInfinite";
val HNatInfinite_not_eq_zero = thm "HNatInfinite_not_eq_zero";
val HNatInfinite_ge_one = thm "HNatInfinite_ge_one";
val HNatInfinite_add = thm "HNatInfinite_add";
val HNatInfinite_SHNat_add = thm "HNatInfinite_SHNat_add";
val HNatInfinite_SHNat_diff = thm "HNatInfinite_SHNat_diff";
val HNatInfinite_add_one = thm "HNatInfinite_add_one";
val HNatInfinite_is_Suc = thm "HNatInfinite_is_Suc";
val HNat_hypreal_of_nat = thm "HNat_hypreal_of_nat";
val hypreal_of_hypnat = thm "hypreal_of_hypnat";
val hypreal_of_hypnat_zero = thm "hypreal_of_hypnat_zero";
val hypreal_of_hypnat_one = thm "hypreal_of_hypnat_one";
val hypreal_of_hypnat_add = thm "hypreal_of_hypnat_add";
val hypreal_of_hypnat_mult = thm "hypreal_of_hypnat_mult";
val hypreal_of_hypnat_less_iff = thm "hypreal_of_hypnat_less_iff";
val hypreal_of_hypnat_ge_zero = thm "hypreal_of_hypnat_ge_zero";
val HNatInfinite_inverse_Infinitesimal = thm "HNatInfinite_inverse_Infinitesimal";
*}
end