# Theory HyperNat

theory HyperNat
imports StarDef
```(*  Title:      HOL/Nonstandard_Analysis/HyperNat.thy
Author:     Jacques D. Fleuriot
Copyright:  1998  University of Cambridge

Converted to Isar and polished by lcp
*)

section ‹Hypernatural numbers›

theory HyperNat
imports StarDef
begin

type_synonym hypnat = "nat star"

abbreviation hypnat_of_nat :: "nat ⇒ nat star"
where "hypnat_of_nat ≡ star_of"

definition hSuc :: "hypnat ⇒ hypnat"
where hSuc_def [transfer_unfold]: "hSuc = *f* Suc"

subsection ‹Properties Transferred from Naturals›

lemma hSuc_not_zero [iff]: "⋀m. hSuc m ≠ 0"
by transfer (rule Suc_not_Zero)

lemma zero_not_hSuc [iff]: "⋀m. 0 ≠ hSuc m"
by transfer (rule Zero_not_Suc)

lemma hSuc_hSuc_eq [iff]: "⋀m n. hSuc m = hSuc n ⟷ m = n"
by transfer (rule nat.inject)

lemma zero_less_hSuc [iff]: "⋀n. 0 < hSuc n"
by transfer (rule zero_less_Suc)

lemma hypnat_minus_zero [simp]: "⋀z::hypnat. z - z = 0"
by transfer (rule diff_self_eq_0)

lemma hypnat_diff_0_eq_0 [simp]: "⋀n::hypnat. 0 - n = 0"
by transfer (rule diff_0_eq_0)

lemma hypnat_add_is_0 [iff]: "⋀m n::hypnat. m + n = 0 ⟷ m = 0 ∧ n = 0"
by transfer (rule add_is_0)

lemma hypnat_diff_diff_left: "⋀i j k::hypnat. i - j - k = i - (j + k)"
by transfer (rule diff_diff_left)

lemma hypnat_diff_commute: "⋀i j k::hypnat. i - j - k = i - k - j"
by transfer (rule diff_commute)

lemma hypnat_diff_add_inverse [simp]: "⋀m n::hypnat. n + m - n = m"
by transfer (rule diff_add_inverse)

lemma hypnat_diff_add_inverse2 [simp]:  "⋀m n::hypnat. m + n - n = m"
by transfer (rule diff_add_inverse2)

lemma hypnat_diff_cancel [simp]: "⋀k m n::hypnat. (k + m) - (k + n) = m - n"
by transfer (rule diff_cancel)

lemma hypnat_diff_cancel2 [simp]: "⋀k m n::hypnat. (m + k) - (n + k) = m - n"
by transfer (rule diff_cancel2)

lemma hypnat_diff_add_0 [simp]: "⋀m n::hypnat. n - (n + m) = 0"
by transfer (rule diff_add_0)

lemma hypnat_diff_mult_distrib: "⋀k m n::hypnat. (m - n) * k = (m * k) - (n * k)"
by transfer (rule diff_mult_distrib)

lemma hypnat_diff_mult_distrib2: "⋀k m n::hypnat. k * (m - n) = (k * m) - (k * n)"
by transfer (rule diff_mult_distrib2)

lemma hypnat_le_zero_cancel [iff]: "⋀n::hypnat. n ≤ 0 ⟷ n = 0"
by transfer (rule le_0_eq)

lemma hypnat_mult_is_0 [simp]: "⋀m n::hypnat. m * n = 0 ⟷ m = 0 ∨ n = 0"
by transfer (rule mult_is_0)

lemma hypnat_diff_is_0_eq [simp]: "⋀m n::hypnat. m - n = 0 ⟷ m ≤ n"
by transfer (rule diff_is_0_eq)

lemma hypnat_not_less0 [iff]: "⋀n::hypnat. ¬ n < 0"
by transfer (rule not_less0)

lemma hypnat_less_one [iff]: "⋀n::hypnat. n < 1 ⟷ n = 0"
by transfer (rule less_one)

lemma hypnat_add_diff_inverse: "⋀m n::hypnat. ¬ m < n ⟹ n + (m - n) = m"
by transfer (rule add_diff_inverse)

lemma hypnat_le_add_diff_inverse [simp]: "⋀m n::hypnat. n ≤ m ⟹ n + (m - n) = m"
by transfer (rule le_add_diff_inverse)

lemma hypnat_le_add_diff_inverse2 [simp]: "⋀m n::hypnat. n ≤ m ⟹ (m - n) + n = m"
by transfer (rule le_add_diff_inverse2)

declare hypnat_le_add_diff_inverse2 [OF order_less_imp_le]

lemma hypnat_le0 [iff]: "⋀n::hypnat. 0 ≤ n"
by transfer (rule le0)

lemma hypnat_le_add1 [simp]: "⋀x n::hypnat. x ≤ x + n"
by transfer (rule le_add1)

lemma hypnat_add_self_le [simp]: "⋀x n::hypnat. x ≤ n + x"
by transfer (rule le_add2)

lemma hypnat_add_one_self_less [simp]: "x < x + 1" for x :: hypnat

lemma hypnat_neq0_conv [iff]: "⋀n::hypnat. n ≠ 0 ⟷ 0 < n"
by transfer (rule neq0_conv)

lemma hypnat_gt_zero_iff: "0 < n ⟷ 1 ≤ n" for n :: hypnat
by (auto simp add: linorder_not_less [symmetric])

lemma hypnat_gt_zero_iff2: "0 < n ⟷ (∃m. n = m + 1)" for n :: hypnat
by (auto intro!: add_nonneg_pos exI[of _ "n - 1"] simp: hypnat_gt_zero_iff)

lemma hypnat_add_self_not_less: "¬ x + y < x" for x y :: hypnat
by (simp add: linorder_not_le [symmetric] add.commute [of x])

lemma hypnat_diff_split: "P (a - b) ⟷ (a < b ⟶ P 0) ∧ (∀d. a = b + d ⟶ P d)"
for a b :: hypnat
― ‹elimination of ‹-› on ‹hypnat››
proof (cases "a < b" rule: case_split)
case True
then show ?thesis
by (auto simp add: hypnat_add_self_not_less order_less_imp_le hypnat_diff_is_0_eq [THEN iffD2])
next
case False
then show ?thesis
by (auto simp add: linorder_not_less dest: order_le_less_trans)
qed

subsection ‹Properties of the set of embedded natural numbers›

lemma of_nat_eq_star_of [simp]: "of_nat = star_of"
proof
show "of_nat n = star_of n" for n
by transfer simp
qed

lemma Nats_eq_Standard: "(Nats :: nat star set) = Standard"
by (auto simp: Nats_def Standard_def)

lemma hypnat_of_nat_mem_Nats [simp]: "hypnat_of_nat n ∈ Nats"
by (simp add: Nats_eq_Standard)

lemma hypnat_of_nat_one [simp]: "hypnat_of_nat (Suc 0) = 1"
by transfer simp

lemma hypnat_of_nat_Suc [simp]: "hypnat_of_nat (Suc n) = hypnat_of_nat n + 1"
by transfer simp

lemma of_nat_eq_add [rule_format]: "∀d::hypnat. of_nat m = of_nat n + d --> d ∈ range of_nat"
apply (induct n)
apply (case_tac x)
apply (auto simp add: add.commute [of 1])
done

lemma Nats_diff [simp]: "a ∈ Nats ⟹ b ∈ Nats ⟹ a - b ∈ Nats" for a b :: hypnat
by (simp add: Nats_eq_Standard)

subsection ‹Infinite Hypernatural Numbers -- @{term HNatInfinite}›

text ‹The set of infinite hypernatural numbers.›
definition HNatInfinite :: "hypnat set"
where "HNatInfinite = {n. n ∉ Nats}"

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)

lemma star_of_neq_HNatInfinite: "N ∈ HNatInfinite ⟹ star_of n ≠ N"
by (auto simp add: HNatInfinite_def Nats_eq_Standard)

lemma star_of_Suc_lessI: "⋀N. star_of n < N ⟹ star_of (Suc n) ≠ N ⟹ star_of (Suc n) < N"
by transfer (rule Suc_lessI)

lemma star_of_less_HNatInfinite:
assumes N: "N ∈ HNatInfinite"
shows "star_of n < N"
proof (induct n)
case 0
from N have "star_of 0 ≠ N"
by (rule star_of_neq_HNatInfinite)
then show ?case by simp
next
case (Suc n)
from N have "star_of (Suc n) ≠ N"
by (rule star_of_neq_HNatInfinite)
with Suc show ?case
by (rule star_of_Suc_lessI)
qed

lemma star_of_le_HNatInfinite: "N ∈ HNatInfinite ⟹ star_of n ≤ N"
by (rule star_of_less_HNatInfinite [THEN order_less_imp_le])

subsubsection ‹Closure Rules›

lemma Nats_less_HNatInfinite: "x ∈ Nats ⟹ y ∈ HNatInfinite ⟹ x < y"
by (auto simp add: Nats_def star_of_less_HNatInfinite)

lemma Nats_le_HNatInfinite: "x ∈ Nats ⟹ y ∈ HNatInfinite ⟹ x ≤ y"
by (rule Nats_less_HNatInfinite [THEN order_less_imp_le])

lemma zero_less_HNatInfinite: "x ∈ HNatInfinite ⟹ 0 < x"
by (simp add: Nats_less_HNatInfinite)

lemma one_less_HNatInfinite: "x ∈ HNatInfinite ⟹ 1 < x"
by (simp add: Nats_less_HNatInfinite)

lemma one_le_HNatInfinite: "x ∈ HNatInfinite ⟹ 1 ≤ x"
by (simp add: Nats_le_HNatInfinite)

lemma zero_not_mem_HNatInfinite [simp]: "0 ∉ HNatInfinite"
by (simp add: HNatInfinite_def)

lemma Nats_downward_closed: "x ∈ Nats ⟹ y ≤ x ⟹ y ∈ Nats" for x y :: hypnat
apply (simp only: linorder_not_less [symmetric])
apply (erule contrapos_np)
apply (drule HNatInfinite_not_Nats_iff [THEN iffD2])
apply (erule (1) Nats_less_HNatInfinite)
done

lemma HNatInfinite_upward_closed: "x ∈ HNatInfinite ⟹ x ≤ y ⟹ y ∈ HNatInfinite"
apply (simp only: HNatInfinite_not_Nats_iff)
apply (erule contrapos_nn)
apply (erule (1) Nats_downward_closed)
done

lemma HNatInfinite_add: "x ∈ HNatInfinite ⟹ x + y ∈ HNatInfinite"
apply (erule HNatInfinite_upward_closed)
done

lemma HNatInfinite_add_one: "x ∈ HNatInfinite ⟹ x + 1 ∈ HNatInfinite"

lemma HNatInfinite_diff: "x ∈ HNatInfinite ⟹ y ∈ Nats ⟹ x - y ∈ HNatInfinite"
apply (frule (1) Nats_le_HNatInfinite)
apply (simp only: HNatInfinite_not_Nats_iff)
apply (erule contrapos_nn)
apply (drule (1) Nats_add, simp)
done

lemma HNatInfinite_is_Suc: "x ∈ HNatInfinite ⟹ ∃y. x = y + 1" for x :: hypnat
apply (rule_tac x = "x - (1::hypnat) " in exI)
apply (simp add: Nats_le_HNatInfinite)
done

subsection ‹Existence of an infinite hypernatural number›

text ‹‹ω› is in fact an infinite hypernatural number = ‹[<1, 2, 3, …>]››
definition whn :: hypnat
where hypnat_omega_def: "whn = star_n (λn::nat. n)"

lemma hypnat_of_nat_neq_whn: "hypnat_of_nat n ≠ whn"
by (simp add: FreeUltrafilterNat.singleton' hypnat_omega_def star_of_def star_n_eq_iff)

lemma whn_neq_hypnat_of_nat: "whn ≠ hypnat_of_nat n"
by (simp add: FreeUltrafilterNat.singleton hypnat_omega_def star_of_def star_n_eq_iff)

lemma whn_not_Nats [simp]: "whn ∉ Nats"
by (simp add: Nats_def image_def whn_neq_hypnat_of_nat)

lemma HNatInfinite_whn [simp]: "whn ∈ HNatInfinite"
by (simp add: HNatInfinite_def)

lemma lemma_unbounded_set [simp]: "eventually (λn::nat. m < n) 𝒰"
by (rule filter_leD[OF FreeUltrafilterNat.le_cofinite])
(auto simp add: cofinite_eq_sequentially eventually_at_top_dense)

lemma hypnat_of_nat_eq: "hypnat_of_nat m  = star_n (λn::nat. m)"
by (simp add: star_of_def)

lemma SHNat_eq: "Nats = {n. ∃N. n = hypnat_of_nat N}"
by (simp add: Nats_def image_def)

lemma Nats_less_whn: "n ∈ Nats ⟹ n < whn"
by (simp add: Nats_less_HNatInfinite)

lemma Nats_le_whn: "n ∈ Nats ⟹ n ≤ whn"
by (simp add: Nats_le_HNatInfinite)

lemma hypnat_of_nat_less_whn [simp]: "hypnat_of_nat n < whn"
by (simp add: Nats_less_whn)

lemma hypnat_of_nat_le_whn [simp]: "hypnat_of_nat n ≤ whn"
by (simp add: Nats_le_whn)

lemma hypnat_zero_less_hypnat_omega [simp]: "0 < whn"
by (simp add: Nats_less_whn)

lemma hypnat_one_less_hypnat_omega [simp]: "1 < whn"
by (simp add: Nats_less_whn)

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:
assumes "∀N::nat. eventually (λn. f n ≠ N) 𝒰"
shows "eventually (λn. N < f n) 𝒰"
apply (induct N)
using assms
apply (drule_tac x = 0 in spec, simp)
using assms
apply (drule_tac x = "Suc N" in spec)
apply (auto elim: eventually_elim2)
done

lemma HNatInfinite_iff: "HNatInfinite = {N. ∀n ∈ Nats. n < N}"
apply (safe intro!: Nats_less_HNatInfinite)
apply (auto simp add: HNatInfinite_def)
done

subsubsection ‹Alternative Characterization of @{term HNatInfinite} using Free Ultrafilter›

lemma HNatInfinite_FreeUltrafilterNat:
"star_n X ∈ HNatInfinite ⟹ ∀u. eventually (λn. u < X n) 𝒰"
apply (auto simp add: HNatInfinite_iff SHNat_eq)
apply (drule_tac x="star_of u" in spec, simp)
apply (simp add: star_of_def star_less_def starP2_star_n)
done

lemma FreeUltrafilterNat_HNatInfinite:
"∀u. eventually (λn. u < X n) 𝒰 ⟹ star_n X ∈ HNatInfinite"
by (auto simp add: star_less_def starP2_star_n HNatInfinite_iff SHNat_eq hypnat_of_nat_eq)

lemma HNatInfinite_FreeUltrafilterNat_iff:
"(star_n X ∈ HNatInfinite) = (∀u. eventually (λn. u < X n) 𝒰)"
by (rule iffI [OF HNatInfinite_FreeUltrafilterNat FreeUltrafilterNat_HNatInfinite])

subsection ‹Embedding of the Hypernaturals into other types›

definition of_hypnat :: "hypnat ⇒ 'a::semiring_1_cancel star"
where of_hypnat_def [transfer_unfold]: "of_hypnat = *f* of_nat"

lemma of_hypnat_0 [simp]: "of_hypnat 0 = 0"
by transfer (rule of_nat_0)

lemma of_hypnat_1 [simp]: "of_hypnat 1 = 1"
by transfer (rule of_nat_1)

lemma of_hypnat_hSuc: "⋀m. of_hypnat (hSuc m) = 1 + of_hypnat m"
by transfer (rule of_nat_Suc)

lemma of_hypnat_add [simp]: "⋀m n. of_hypnat (m + n) = of_hypnat m + of_hypnat n"
by transfer (rule of_nat_add)

lemma of_hypnat_mult [simp]: "⋀m n. of_hypnat (m * n) = of_hypnat m * of_hypnat n"
by transfer (rule of_nat_mult)

lemma of_hypnat_less_iff [simp]:
"⋀m n. of_hypnat m < (of_hypnat n::'a::linordered_semidom star) ⟷ m < n"
by transfer (rule of_nat_less_iff)

lemma of_hypnat_0_less_iff [simp]:
"⋀n. 0 < (of_hypnat n::'a::linordered_semidom star) ⟷ 0 < n"
by transfer (rule of_nat_0_less_iff)

lemma of_hypnat_less_0_iff [simp]: "⋀m. ¬ (of_hypnat m::'a::linordered_semidom star) < 0"
by transfer (rule of_nat_less_0_iff)

lemma of_hypnat_le_iff [simp]:
"⋀m n. of_hypnat m ≤ (of_hypnat n::'a::linordered_semidom star) ⟷ m ≤ n"
by transfer (rule of_nat_le_iff)

lemma of_hypnat_0_le_iff [simp]: "⋀n. 0 ≤ (of_hypnat n::'a::linordered_semidom star)"
by transfer (rule of_nat_0_le_iff)

lemma of_hypnat_le_0_iff [simp]: "⋀m. (of_hypnat m::'a::linordered_semidom star) ≤ 0 ⟷ m = 0"
by transfer (rule of_nat_le_0_iff)

lemma of_hypnat_eq_iff [simp]:
"⋀m n. of_hypnat m = (of_hypnat n::'a::linordered_semidom star) ⟷ m = n"
by transfer (rule of_nat_eq_iff)

lemma of_hypnat_eq_0_iff [simp]: "⋀m. (of_hypnat m::'a::linordered_semidom star) = 0 ⟷ m = 0"
by transfer (rule of_nat_eq_0_iff)

lemma HNatInfinite_of_hypnat_gt_zero:
"N ∈ HNatInfinite ⟹ (0::'a::linordered_semidom star) < of_hypnat N"
by (rule ccontr) (simp add: linorder_not_less)

end
```