(* Title : HyperNat.thy Author : Jacques D. Fleuriot Copyright : 1998 University of Cambridge Converted to Isar and polished by lcp *) header{*Hypernatural numbers*} 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" subsection{*Properties Transferred from Naturals*} 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) 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) lemma hypnat_le_zero_cancel [iff]: "!!n. (n ≤ (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 ≤ n)" by transfer (rule diff_is_0_eq) 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 ≤ m ==> n+(m-n) = (m::hypnat)" by transfer (rule le_add_diff_inverse) lemma hypnat_le_add_diff_inverse2 [simp]: "!!m n. n≤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) ≤ n" by transfer (rule le0) lemma hypnat_add_self_le [simp]: "!!x n. (x::hypnat) ≤ 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. (n ≠ 0) = (0 < (n::hypnat))" by transfer (rule neq0_conv) lemma hypnat_gt_zero_iff: "((0::hypnat) < n) = ((1::hypnat) ≤ n)" by (auto simp add: linorder_not_less [symmetric]) lemma hypnat_gt_zero_iff2: "(0 < n) = (∃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{*Properties of the set of embedded natural numbers*} lemma hypnat_of_nat_def: "hypnat_of_nat m == of_nat m" by (transfer, simp) lemma hypnat_of_nat_one [simp]: "hypnat_of_nat (Suc 0) = (1::hypnat)" 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 of_nat_eq_add [rule_format]: "∀d::hypnat. of_nat m = of_nat n + d --> d ∈ 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 ∈ Nats; b ∈ Nats|] ==> (a-b :: hypnat) ∈ Nats" by (auto simp add: of_nat_eq_add Nats_def split: hypnat_diff_split) subsection{*Existence of an infinite hypernatural number*} consts whn :: hypnat defs (* omega is in fact an infinite hypernatural number = [<1,2,3,...>] *) hypnat_omega_def: "whn == star_n (%n::nat. n)" 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.*} lemma lemma_unbounded_set [simp]: "{n::nat. m < n} ∈ 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 ≤ 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)" by (simp add: star_of_def) lemma SHNat_eq: "Nats = {n. ∃N. n = hypnat_of_nat N}" by (force simp add: hypnat_of_nat_def Nats_def) lemma hypnat_omega_gt_SHNat: "n ∈ 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 ∉ 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 ≤ 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}*} constdefs (* the set of infinite hypernatural numbers *) HNatInfinite :: "hypnat set" "HNatInfinite == {n. n ∉ Nats}" lemma HNatInfinite_whn [simp]: "whn ∈ HNatInfinite" by (simp add: HNatInfinite_def) lemma Nats_not_HNatInfinite_iff: "(x ∈ Nats) = (x ∉ HNatInfinite)" by (simp add: HNatInfinite_def) lemma HNatInfinite_not_Nats_iff: "(x ∈ HNatInfinite) = (x ∉ Nats)" by (simp add: HNatInfinite_def) subsubsection{*Alternative characterization of the set of infinite hypernaturals*} text{* @{term "HNatInfinite = {N. ∀n ∈ Nats. n < N}"}*} (*??delete? similar reasoning in hypnat_omega_gt_SHNat above*) lemma HNatInfinite_FreeUltrafilterNat_lemma: "∀N::nat. {n. f n ≠ N} ∈ FreeUltrafilterNat ==> {n. N < f n} ∈ 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. ∀n ∈ 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 subsubsection{*Alternative Characterization of @{term HNatInfinite} using Free Ultrafilter*} lemma HNatInfinite_FreeUltrafilterNat: "x ∈ HNatInfinite ==> ∃X ∈ Rep_star x. ∀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: "∃X ∈ Rep_star x. ∀u. {n. u < X n}: FreeUltrafilterNat ==> x ∈ 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 ∈ HNatInfinite) = (∃X ∈ Rep_star x. ∀u. {n. u < X n}: FreeUltrafilterNat)" by (blast intro: HNatInfinite_FreeUltrafilterNat FreeUltrafilterNat_HNatInfinite) lemma HNatInfinite_gt_one [simp]: "x ∈ HNatInfinite ==> (1::hypnat) < x" by (auto simp add: HNatInfinite_iff) lemma zero_not_mem_HNatInfinite [simp]: "0 ∉ HNatInfinite" apply (auto simp add: HNatInfinite_iff) apply (drule_tac a = " (1::hypnat) " in equals0D) apply simp done lemma HNatInfinite_not_eq_zero: "x ∈ 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 ∈ HNatInfinite ==> (1::hypnat) ≤ x" by (blast intro: order_less_imp_le HNatInfinite_gt_one) subsubsection{*Closure Rules*} lemma HNatInfinite_add: "[| x ∈ HNatInfinite; y ∈ HNatInfinite |] ==> x + y ∈ 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 ∈ HNatInfinite; y ∈ Nats |] ==> x + y ∈ 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 ∈ HNatInfinite; y ∈ Nats |] ==> y < x" by (simp add: HNatInfinite_iff) lemma HNatInfinite_SHNat_diff: assumes x: "x ∈ HNatInfinite" and y: "y ∈ Nats" shows "x - y ∈ 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 ∈ HNatInfinite ==> x + (1::hypnat) ∈ HNatInfinite" by (auto intro: HNatInfinite_SHNat_add) lemma HNatInfinite_is_Suc: "x ∈ HNatInfinite ==> ∃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" declare hypreal_of_hypnat_def [transfer_unfold] lemma HNat_hypreal_of_nat [simp]: "hypreal_of_nat N ∈ 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 (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 (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 (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 (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 ≤ hypreal_of_hypnat n" by (transfer, simp) lemma HNatInfinite_inverse_Infinitesimal [simp]: "n ∈ HNatInfinite ==> inverse (hypreal_of_hypnat n) ∈ 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 ∈ 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 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 SHNat_eq = thm"SHNat_eq" val hypnat_of_nat_one = thm "hypnat_of_nat_one"; 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
lemma hypnat_minus_zero:
z - z = 0
lemma hypnat_diff_0_eq_0:
0 - n = 0
lemma hypnat_add_is_0:
(m + n = 0) = (m = 0 ∧ n = 0)
lemma hypnat_diff_diff_left:
i - j - k = i - (j + k)
lemma hypnat_diff_commute:
i - j - k = i - k - j
lemma hypnat_diff_add_inverse:
n + m - n = m
lemma hypnat_diff_add_inverse2:
m + n - n = m
lemma hypnat_diff_cancel:
k + m - (k + n) = m - n
lemma hypnat_diff_cancel2:
m + k - (n + k) = m - n
lemma hypnat_diff_add_0:
n - (n + m) = 0
lemma hypnat_diff_mult_distrib:
(m - n) * k = m * k - n * k
lemma hypnat_diff_mult_distrib2:
k * (m - n) = k * m - k * n
lemma hypnat_le_zero_cancel:
(n ≤ 0) = (n = 0)
lemma hypnat_mult_is_0:
(m * n = 0) = (m = 0 ∨ n = 0)
lemma hypnat_diff_is_0_eq:
(m - n = 0) = (m ≤ n)
lemma hypnat_not_less0:
¬ n < 0
lemma hypnat_less_one:
(n < 1) = (n = 0)
lemma hypnat_add_diff_inverse:
¬ m < n ==> n + (m - n) = m
lemma hypnat_le_add_diff_inverse:
n ≤ m ==> n + (m - n) = m
lemma hypnat_le_add_diff_inverse2:
n ≤ m ==> m - n + n = m
lemma hypnat_le0:
0 ≤ n
lemma hypnat_add_self_le:
x ≤ n + x
lemma hypnat_add_one_self_less:
x < x + 1
lemma hypnat_neq0_conv:
(n ≠ 0) = (0 < n)
lemma hypnat_gt_zero_iff:
(0 < n) = (1 ≤ n)
lemma hypnat_gt_zero_iff2:
(0 < n) = (∃m. n = m + 1)
lemma hypnat_add_self_not_less:
¬ x + y < x
lemma hypnat_diff_split:
P (a - b) = ((a < b --> P 0) ∧ (∀d. a = b + d --> P d))
lemma hypnat_of_nat_def:
star_of m == of_nat m
lemma hypnat_of_nat_one:
star_of (Suc 0) = 1
lemma hypnat_of_nat_Suc:
star_of (Suc n) = star_of n + 1
lemma of_nat_eq_add:
of_nat m = of_nat n + d ==> d ∈ range of_nat
lemma Nats_diff:
[| a ∈ Nats; b ∈ Nats |] ==> a - b ∈ Nats
lemma lemma_unbounded_set:
{n. m < n} ∈ \<U>
lemma Compl_Collect_le:
- {n. N ≤ n} = {n. n < N}
lemma hypnat_of_nat_eq:
star_of m = star_n (%n. m)
lemma SHNat_eq:
Nats = {n. ∃N. n = star_of N}
lemma hypnat_omega_gt_SHNat:
n ∈ Nats ==> n < whn
lemma SHNAT_omega_not_mem:
whn ∉ Nats
lemma hypnat_of_nat_less_whn:
star_of n < whn
lemma hypnat_of_nat_le_whn:
star_of n ≤ whn
lemma hypnat_zero_less_hypnat_omega:
0 < whn
lemma hypnat_one_less_hypnat_omega:
1 < whn
lemma HNatInfinite_whn:
whn ∈ HNatInfinite
lemma Nats_not_HNatInfinite_iff:
(x ∈ Nats) = (x ∉ HNatInfinite)
lemma HNatInfinite_not_Nats_iff:
(x ∈ HNatInfinite) = (x ∉ Nats)
lemma HNatInfinite_FreeUltrafilterNat_lemma:
∀N. {n. f n ≠ N} ∈ \<U> ==> {n. N < f n} ∈ \<U>
lemma HNatInfinite_iff:
HNatInfinite = {N. ∀n∈Nats. n < N}
lemma HNatInfinite_FreeUltrafilterNat:
x ∈ HNatInfinite ==> ∃X∈Rep_star x. ∀u. {n. u < X n} ∈ \<U>
lemma FreeUltrafilterNat_HNatInfinite:
∃X∈Rep_star x. ∀u. {n. u < X n} ∈ \<U> ==> x ∈ HNatInfinite
lemma HNatInfinite_FreeUltrafilterNat_iff:
(x ∈ HNatInfinite) = (∃X∈Rep_star x. ∀u. {n. u < X n} ∈ \<U>)
lemma HNatInfinite_gt_one:
x ∈ HNatInfinite ==> 1 < x
lemma zero_not_mem_HNatInfinite:
0 ∉ HNatInfinite
lemma HNatInfinite_not_eq_zero:
x ∈ HNatInfinite ==> 0 < x
lemma HNatInfinite_ge_one:
x ∈ HNatInfinite ==> 1 ≤ x
lemma HNatInfinite_add:
[| x ∈ HNatInfinite; y ∈ HNatInfinite |] ==> x + y ∈ HNatInfinite
lemma HNatInfinite_SHNat_add:
[| x ∈ HNatInfinite; y ∈ Nats |] ==> x + y ∈ HNatInfinite
lemma HNatInfinite_Nats_imp_less:
[| x ∈ HNatInfinite; y ∈ Nats |] ==> y < x
lemma HNatInfinite_SHNat_diff:
[| x ∈ HNatInfinite; y ∈ Nats |] ==> x - y ∈ HNatInfinite
lemma HNatInfinite_add_one:
x ∈ HNatInfinite ==> x + 1 ∈ HNatInfinite
lemma HNatInfinite_is_Suc:
x ∈ HNatInfinite ==> ∃y. x = y + 1
lemma HNat_hypreal_of_nat:
hypreal_of_nat N ∈ Nats
lemma hypreal_of_hypnat:
hypreal_of_hypnat (star_n X) = star_n (%n. real (X n))
lemma hypreal_of_hypnat_inject:
(hypreal_of_hypnat m = hypreal_of_hypnat n) = (m = n)
lemma hypreal_of_hypnat_zero:
hypreal_of_hypnat 0 = 0
lemma hypreal_of_hypnat_one:
hypreal_of_hypnat 1 = 1
lemma hypreal_of_hypnat_add:
hypreal_of_hypnat (m + n) = hypreal_of_hypnat m + hypreal_of_hypnat n
lemma hypreal_of_hypnat_mult:
hypreal_of_hypnat (m * n) = hypreal_of_hypnat m * hypreal_of_hypnat n
lemma hypreal_of_hypnat_less_iff:
(hypreal_of_hypnat n < hypreal_of_hypnat m) = (n < m)
lemma hypreal_of_hypnat_eq_zero_iff:
(hypreal_of_hypnat N = 0) = (N = 0)
lemma hypreal_of_hypnat_ge_zero:
0 ≤ hypreal_of_hypnat n
lemma HNatInfinite_inverse_Infinitesimal:
n ∈ HNatInfinite ==> inverse (hypreal_of_hypnat n) ∈ Infinitesimal
lemma HNatInfinite_hypreal_of_hypnat_gt_zero:
N ∈ HNatInfinite ==> 0 < hypreal_of_hypnat N