# Theory HyperNat

Up to index of Isabelle/HOL/HOL-Complex

theory HyperNat
imports Star
begin

    (*  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

### Properties Transferred from Naturals

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))

### Properties of the set of embedded natural numbers

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

### Existence of an infinite hypernatural number

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

### Infinite Hypernatural Numbers -- @{term HNatInfinite}

lemma HNatInfinite_whn:

      whn ∈ HNatInfinite

lemma Nats_not_HNatInfinite_iff:

      (x ∈ Nats) = (x ∉ HNatInfinite)

lemma HNatInfinite_not_Nats_iff:

      (x ∈ HNatInfinite) = (x ∉ Nats)

#### Alternative characterization of the set of infinite hypernaturals

lemma HNatInfinite_FreeUltrafilterNat_lemma:

      ∀N. {n. f n ≠ N} ∈ \<U> ==> {n. N < f n} ∈ \<U>

lemma HNatInfinite_iff:

      HNatInfinite = {N. ∀n∈Nats. n < N}

#### Alternative Characterization of @{term HNatInfinite} using Free Ultrafilter

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

#### Closure Rules

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

### Embedding of the Hypernaturals into the Hyperreals

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