(*  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 = Star:

constdefs

    hypnatrel :: "((nat=>nat)*(nat=>nat)) set"

    "hypnatrel == {p. \<exists>X Y. p = ((X::nat=>nat),Y) &

                       {n::nat. X(n) = Y(n)} \<in> FreeUltrafilterNat}"

typedef hypnat = "UNIV//hypnatrel"

    by (auto simp add: quotient_def)

instance hypnat :: "{ord, zero, one, plus, times, minus}" ..

consts whn :: hypnat

defs (overloaded)

  (** hypernatural arithmetic **)

  hypnat_zero_def:  "0 == Abs_hypnat(hypnatrel``{%n::nat. 0})"

  hypnat_one_def:   "1 == Abs_hypnat(hypnatrel``{%n::nat. 1})"

  (* omega is in fact an infinite hypernatural number = [<1,2,3,...>] *)

  hypnat_omega_def:  "whn == Abs_hypnat(hypnatrel``{%n::nat. n})"

  hypnat_add_def:

  "P + Q == Abs_hypnat(\<Union>X \<in> Rep_hypnat(P). \<Union>Y \<in> Rep_hypnat(Q).

                hypnatrel``{%n::nat. X n + Y n})"

  hypnat_mult_def:

  "P * Q == Abs_hypnat(\<Union>X \<in> Rep_hypnat(P). \<Union>Y \<in> Rep_hypnat(Q).

                hypnatrel``{%n::nat. X n * Y n})"

  hypnat_minus_def:

  "P - Q == Abs_hypnat(\<Union>X \<in> Rep_hypnat(P). \<Union>Y \<in> Rep_hypnat(Q).

                hypnatrel``{%n::nat. X n - Y n})"

  hypnat_le_def:

  "P \<le> (Q::hypnat) == \<exists>X Y. X \<in> Rep_hypnat(P) & Y \<in> Rep_hypnat(Q) &

                            {n::nat. X n \<le> Y n} \<in> FreeUltrafilterNat"

  hypnat_less_def: "(x < (y::hypnat)) == (x \<le> y & x \<noteq> y)"

subsection{*Properties of @{term hypnatrel}*}

text{*Proving that @{term hypnatrel} is an equivalence relation*}

lemma hypnatrel_iff:

     "((X,Y) \<in> hypnatrel) = ({n. X n = Y n}: FreeUltrafilterNat)"

apply (simp add: hypnatrel_def)

done

lemma hypnatrel_refl: "(x,x) \<in> hypnatrel"

by (simp add: hypnatrel_def)

lemma hypnatrel_sym: "(x,y) \<in> hypnatrel ==> (y,x) \<in> hypnatrel"

by (auto simp add: hypnatrel_def eq_commute)

lemma hypnatrel_trans [rule_format (no_asm)]:

     "(x,y) \<in> hypnatrel --> (y,z) \<in> hypnatrel --> (x,z) \<in> hypnatrel"

by (auto simp add: hypnatrel_def, ultra)

lemma equiv_hypnatrel:

     "equiv UNIV hypnatrel"

apply (simp add: equiv_def refl_def sym_def trans_def hypnatrel_refl)

apply (blast intro: hypnatrel_sym hypnatrel_trans)

done

(* (hypnatrel `` {x} = hypnatrel `` {y}) = ((x,y) \<in> hypnatrel) *)

lemmas equiv_hypnatrel_iff =

    eq_equiv_class_iff [OF equiv_hypnatrel UNIV_I UNIV_I, simp]

lemma hypnatrel_in_hypnat [simp]: "hypnatrel``{x}:hypnat"

by (simp add: hypnat_def hypnatrel_def quotient_def, blast)

lemma inj_on_Abs_hypnat: "inj_on Abs_hypnat hypnat"

apply (rule inj_on_inverseI)

apply (erule Abs_hypnat_inverse)

done

declare inj_on_Abs_hypnat [THEN inj_on_iff, simp]

        Abs_hypnat_inverse [simp]

declare equiv_hypnatrel [THEN eq_equiv_class_iff, simp]

declare hypnatrel_iff [iff]

lemma inj_Rep_hypnat: "inj(Rep_hypnat)"

apply (rule inj_on_inverseI)

apply (rule Rep_hypnat_inverse)

done

lemma lemma_hypnatrel_refl: "x \<in> hypnatrel `` {x}"

by (simp add: hypnatrel_def)

declare lemma_hypnatrel_refl [simp]

lemma hypnat_empty_not_mem: "{} \<notin> hypnat"

apply (simp add: hypnat_def)

apply (auto elim!: quotientE equalityCE)

done

declare hypnat_empty_not_mem [simp]

lemma Rep_hypnat_nonempty: "Rep_hypnat x \<noteq> {}"

by (cut_tac x = x in Rep_hypnat, auto)

declare Rep_hypnat_nonempty [simp]

lemma eq_Abs_hypnat:

    "(!!x. z = Abs_hypnat(hypnatrel``{x}) ==> P) ==> P"

apply (rule_tac x1=z in Rep_hypnat [unfolded hypnat_def, THEN quotientE])

apply (drule_tac f = Abs_hypnat in arg_cong)

apply (force simp add: Rep_hypnat_inverse)

done

theorem hypnat_cases [case_names Abs_hypnat, cases type: hypnat]:

    "(!!x. z = Abs_hypnat(hypnatrel``{x}) ==> P) ==> P"

by (rule eq_Abs_hypnat [of z], blast)

subsection{*Hypernat Addition*}

lemma hypnat_add_congruent2:

     "congruent2 hypnatrel hypnatrel (%X Y. hypnatrel``{%n. X n + Y n})"

by (simp add: congruent2_def, auto, ultra)

lemma hypnat_add:

  "Abs_hypnat(hypnatrel``{%n. X n}) + Abs_hypnat(hypnatrel``{%n. Y n}) =

   Abs_hypnat(hypnatrel``{%n. X n + Y n})"

by (simp add: hypnat_add_def 

    UN_equiv_class2 [OF equiv_hypnatrel equiv_hypnatrel hypnat_add_congruent2])

lemma hypnat_add_commute: "(z::hypnat) + w = w + z"

apply (cases z, cases w)

apply (simp add: add_ac hypnat_add)

done

lemma hypnat_add_assoc: "((z1::hypnat) + z2) + z3 = z1 + (z2 + z3)"

apply (cases z1, cases z2, cases z3)

apply (simp add: hypnat_add nat_add_assoc)

done

lemma hypnat_add_zero_left: "(0::hypnat) + z = z"

apply (cases z)

apply (simp add: hypnat_zero_def hypnat_add)

done

instance hypnat :: comm_monoid_add

  by intro_classes

    (assumption |

      rule hypnat_add_commute hypnat_add_assoc hypnat_add_zero_left)+

subsection{*Subtraction inverse on @{typ hypreal}*}

lemma hypnat_minus_congruent2:

    "congruent2 hypnatrel hypnatrel (%X Y. hypnatrel``{%n. X n - Y n})"

by (simp add: congruent2_def, auto, ultra)

lemma hypnat_minus:

  "Abs_hypnat(hypnatrel``{%n. X n}) - Abs_hypnat(hypnatrel``{%n. Y n}) =

   Abs_hypnat(hypnatrel``{%n. X n - Y n})"

by (simp add: hypnat_minus_def 

  UN_equiv_class2 [OF equiv_hypnatrel equiv_hypnatrel hypnat_minus_congruent2])

lemma hypnat_minus_zero: "z - z = (0::hypnat)"

apply (cases z)

apply (simp add: hypnat_zero_def hypnat_minus)

done

lemma hypnat_diff_0_eq_0: "(0::hypnat) - n = 0"

apply (cases n)

apply (simp add: hypnat_minus hypnat_zero_def)

done

declare hypnat_minus_zero [simp] hypnat_diff_0_eq_0 [simp]

lemma hypnat_add_is_0: "(m+n = (0::hypnat)) = (m=0 & n=0)"

apply (cases m, cases n)

apply (auto intro: FreeUltrafilterNat_subset dest: FreeUltrafilterNat_Int simp add: hypnat_zero_def hypnat_add)

done

declare hypnat_add_is_0 [iff]

lemma hypnat_diff_diff_left: "(i::hypnat) - j - k = i - (j+k)"

apply (cases i, cases j, cases k)

apply (simp add: hypnat_minus hypnat_add diff_diff_left)

done

lemma hypnat_diff_commute: "(i::hypnat) - j - k = i-k-j"

by (simp add: hypnat_diff_diff_left hypnat_add_commute)

lemma hypnat_diff_add_inverse: "((n::hypnat) + m) - n = m"

apply (cases m, cases n)

apply (simp add: hypnat_minus hypnat_add)

done

declare hypnat_diff_add_inverse [simp]

lemma hypnat_diff_add_inverse2:  "((m::hypnat) + n) - n = m"

apply (cases m, cases n)

apply (simp add: hypnat_minus hypnat_add)

done

declare hypnat_diff_add_inverse2 [simp]

lemma hypnat_diff_cancel: "((k::hypnat) + m) - (k+n) = m - n"

apply (cases k, cases m, cases n)

apply (simp add: hypnat_minus hypnat_add)

done

declare hypnat_diff_cancel [simp]

lemma hypnat_diff_cancel2: "((m::hypnat) + k) - (n+k) = m - n"

by (simp add: hypnat_add_commute [of _ k])

declare hypnat_diff_cancel2 [simp]

lemma hypnat_diff_add_0: "(n::hypnat) - (n+m) = (0::hypnat)"

apply (cases m, cases n)

apply (simp add: hypnat_zero_def hypnat_minus hypnat_add)

done

declare hypnat_diff_add_0 [simp]

subsection{*Hyperreal Multiplication*}

lemma hypnat_mult_congruent2:

    "congruent2 hypnatrel hypnatrel (%X Y. hypnatrel``{%n. X n * Y n})"

by (simp add: congruent2_def, auto, ultra)

lemma hypnat_mult:

  "Abs_hypnat(hypnatrel``{%n. X n}) * Abs_hypnat(hypnatrel``{%n. Y n}) =

   Abs_hypnat(hypnatrel``{%n. X n * Y n})"

by (simp add: hypnat_mult_def

   UN_equiv_class2 [OF equiv_hypnatrel equiv_hypnatrel hypnat_mult_congruent2])

lemma hypnat_mult_commute: "(z::hypnat) * w = w * z"

by (cases z, cases w, simp add: hypnat_mult mult_ac)

lemma hypnat_mult_assoc: "((z1::hypnat) * z2) * z3 = z1 * (z2 * z3)"

apply (cases z1, cases z2, cases z3)

apply (simp add: hypnat_mult mult_assoc)

done

lemma hypnat_mult_1: "(1::hypnat) * z = z"

apply (cases z)

apply (simp add: hypnat_mult hypnat_one_def)

done

lemma hypnat_diff_mult_distrib: "((m::hypnat) - n) * k = (m * k) - (n * k)"

apply (cases k, cases m, cases n)

apply (simp add: hypnat_mult hypnat_minus diff_mult_distrib)

done

lemma hypnat_diff_mult_distrib2: "(k::hypnat) * (m - n) = (k * m) - (k * n)"

by (simp add: hypnat_diff_mult_distrib hypnat_mult_commute [of k])

lemma hypnat_add_mult_distrib: "((z1::hypnat) + z2) * w = (z1 * w) + (z2 * w)"

apply (cases z1, cases z2, cases w)

apply (simp add: hypnat_mult hypnat_add add_mult_distrib)

done

lemma hypnat_add_mult_distrib2: "(w::hypnat) * (z1 + z2) = (w * z1) + (w * z2)"

by (simp add: hypnat_mult_commute [of w] hypnat_add_mult_distrib)

text{*one and zero are distinct*}

lemma hypnat_zero_not_eq_one: "(0::hypnat) \<noteq> (1::hypnat)"

by (auto simp add: hypnat_zero_def hypnat_one_def)

declare hypnat_zero_not_eq_one [THEN not_sym, simp]

text{*The Hypernaturals Form A comm_semiring_1_cancel*}

instance hypnat :: comm_semiring_1_cancel

proof

  fix i j k :: hypnat

  show "(i * j) * k = i * (j * k)" by (rule hypnat_mult_assoc)

  show "i * j = j * i" by (rule hypnat_mult_commute)

  show "1 * i = i" by (rule hypnat_mult_1)

  show "(i + j) * k = i * k + j * k" by (simp add: hypnat_add_mult_distrib)

  show "0 \<noteq> (1::hypnat)" by (rule hypnat_zero_not_eq_one)

  assume "k+i = k+j"

  hence "(k+i) - k = (k+j) - k" by simp

  thus "i=j" by simp

qed

subsection{*Properties of The @{text "\<le>"} Relation*}

lemma hypnat_le:

      "(Abs_hypnat(hypnatrel``{%n. X n}) \<le> Abs_hypnat(hypnatrel``{%n. Y n})) =

       ({n. X n \<le> Y n} \<in> FreeUltrafilterNat)"

apply (simp add: hypnat_le_def)

apply (auto intro!: lemma_hypnatrel_refl, ultra)

done

lemma hypnat_le_refl: "w \<le> (w::hypnat)"

apply (cases w)

apply (simp add: hypnat_le)

done

lemma hypnat_le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::hypnat)"

apply (cases i, cases j, cases k)

apply (simp add: hypnat_le, ultra)

done

lemma hypnat_le_anti_sym: "[| z \<le> w; w \<le> z |] ==> z = (w::hypnat)"

apply (cases z, cases w)

apply (simp add: hypnat_le, ultra)

done

(* Axiom 'order_less_le' of class 'order': *)

lemma hypnat_less_le: "((w::hypnat) < z) = (w \<le> z & w \<noteq> z)"

by (simp add: hypnat_less_def)

instance hypnat :: order

  by intro_classes

    (assumption |

      rule hypnat_le_refl hypnat_le_trans hypnat_le_anti_sym hypnat_less_le)+

(* Axiom 'linorder_linear' of class 'linorder': *)

lemma hypnat_le_linear: "(z::hypnat) \<le> w | w \<le> z"

apply (cases z, cases w)

apply (auto simp add: hypnat_le, ultra)

done

instance hypnat :: linorder

  by intro_classes (rule hypnat_le_linear)

lemma hypnat_add_left_mono: "x \<le> y ==> z + x \<le> z + (y::hypnat)"

apply (cases x, cases y, cases z)

apply (auto simp add: hypnat_le hypnat_add)

done

lemma hypnat_mult_less_mono2: "[| (0::hypnat)<z; x<y |] ==> z*x<z*y"

apply (cases x, cases y, cases z)

apply (simp add: hypnat_zero_def  hypnat_mult linorder_not_le [symmetric])

apply (auto simp add: hypnat_le, ultra)

done

subsection{*The Hypernaturals Form an Ordered comm_semiring_1_cancel*}

instance hypnat :: ordered_semidom

proof

  fix x y z :: hypnat

  show "0 < (1::hypnat)"

    by (simp add: hypnat_zero_def hypnat_one_def linorder_not_le [symmetric],

        simp add: hypnat_le)

  show "x \<le> y ==> z + x \<le> z + y"

    by (rule hypnat_add_left_mono)

  show "x < y ==> 0 < z ==> z * x < z * y"

    by (simp add: hypnat_mult_less_mono2)

qed

lemma hypnat_le_zero_cancel [iff]: "(n \<le> (0::hypnat)) = (n = 0)"

apply (cases n)

apply (simp add: hypnat_zero_def hypnat_le)

done

lemma hypnat_mult_is_0 [simp]: "(m*n = (0::hypnat)) = (m=0 | n=0)"

apply (cases m, cases n)

apply (auto simp add: hypnat_zero_def hypnat_mult, ultra+)

done

lemma hypnat_diff_is_0_eq [simp]: "((m::hypnat) - n = 0) = (m \<le> n)"

apply (cases m, cases n)

apply (simp add: hypnat_le hypnat_minus hypnat_zero_def)

done

subsection{*Theorems for Ordering*}

lemma hypnat_less:

      "(Abs_hypnat(hypnatrel``{%n. X n}) < Abs_hypnat(hypnatrel``{%n. Y n})) =

       ({n. X n < Y n} \<in> FreeUltrafilterNat)"

apply (auto simp add: hypnat_le  linorder_not_le [symmetric])

apply (ultra+)

done

lemma hypnat_not_less0 [iff]: "~ n < (0::hypnat)"

apply (cases n)

apply (auto simp add: hypnat_zero_def hypnat_less)

done

lemma hypnat_less_one [iff]:

      "(n < (1::hypnat)) = (n=0)"

apply (cases n)

apply (auto simp add: hypnat_zero_def hypnat_one_def hypnat_less)

done

lemma hypnat_add_diff_inverse: "~ m<n ==> n+(m-n) = (m::hypnat)"

apply (cases m, cases n)

apply (simp add: hypnat_minus hypnat_add hypnat_less split: nat_diff_split, ultra)

done

lemma hypnat_le_add_diff_inverse [simp]: "n \<le> m ==> n+(m-n) = (m::hypnat)"

by (simp add: hypnat_add_diff_inverse linorder_not_less [symmetric])

lemma hypnat_le_add_diff_inverse2 [simp]: "n\<le>m ==> (m-n)+n = (m::hypnat)"

by (simp add: hypnat_le_add_diff_inverse hypnat_add_commute)

declare hypnat_le_add_diff_inverse2 [OF order_less_imp_le]

lemma hypnat_le0 [iff]: "(0::hypnat) \<le> n"

by (simp add: linorder_not_less [symmetric])

lemma hypnat_add_self_le [simp]: "(x::hypnat) \<le> n + x"

by (insert add_right_mono [of 0 n x], simp)

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 comm_ring_1 and 

      Order Properties*}

constdefs

  hypnat_of_nat   :: "nat => hypnat"

  "hypnat_of_nat m  == of_nat m"

  (* the set of infinite hypernatural numbers *)

  HNatInfinite :: "hypnat set"

  "HNatInfinite == {n. n \<notin> Nats}"

lemma hypnat_of_nat_add:

      "hypnat_of_nat ((z::nat) + w) = hypnat_of_nat z + hypnat_of_nat w"

by (simp add: hypnat_of_nat_def)

lemma hypnat_of_nat_mult:

      "hypnat_of_nat (z * w) = hypnat_of_nat z * hypnat_of_nat w"

by (simp add: hypnat_of_nat_def)

lemma hypnat_of_nat_less_iff [simp]:

      "(hypnat_of_nat z < hypnat_of_nat w) = (z < w)"

by (simp add: hypnat_of_nat_def)

lemma hypnat_of_nat_le_iff [simp]:

      "(hypnat_of_nat z \<le> hypnat_of_nat w) = (z \<le> w)"

by (simp add: hypnat_of_nat_def)

lemma hypnat_of_nat_eq_iff [simp]:

      "(hypnat_of_nat z = hypnat_of_nat w) = (z = w)"

by (simp add: hypnat_of_nat_def)

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

by (simp add: hypnat_of_nat_def)

lemma hypnat_of_nat_zero [simp]: "hypnat_of_nat 0 = 0"

by (simp add: hypnat_of_nat_def)

lemma hypnat_of_nat_zero_iff [simp]: "(hypnat_of_nat n = 0) = (n = 0)"

by (simp add: hypnat_of_nat_def)

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 add: hypnat_of_nat_def split: nat_diff_split hypnat_diff_split)

subsection{*Existence of an Infinite Hypernatural Number*}

lemma hypnat_omega: "hypnatrel``{%n::nat. n} \<in> hypnat"

by auto

lemma Rep_hypnat_omega: "Rep_hypnat(whn) \<in> hypnat"

by (simp add: hypnat_omega_def)

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 @{term Nats} 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  = Abs_hypnat(hypnatrel``{%n::nat. m})"

apply (induct m) 

apply (simp_all add: hypnat_zero_def hypnat_one_def hypnat_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"

apply (auto simp add: hypnat_of_nat_eq hypnat_less_def hypnat_le_def

                      hypnat_omega_def SHNat_eq)

 prefer 2 apply (force dest: FreeUltrafilterNat_not_finite)

apply (auto intro!: exI)

apply (rule cofinite_mem_FreeUltrafilterNat)

apply (simp add: Compl_Collect_le finite_nat_segment) 

done

(* 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:

@{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 z = x in eq_Abs_hypnat)

apply (auto elim: HNatInfinite_FreeUltrafilterNat_lemma 

            simp add: hypnat_less FreeUltrafilterNat_Compl_iff1 

                      Collect_neg_eq [symmetric])

done

subsection{*Alternative Characterization of @{term HNatInfinite} using 

Free Ultrafilter*}

lemma HNatInfinite_FreeUltrafilterNat:

     "x \<in> HNatInfinite 

      ==> \<exists>X \<in> Rep_hypnat 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 _ lemma_hypnatrel_refl], clarify) 

apply (auto simp add: hypnat_of_nat_def hypnat_less)

done

lemma FreeUltrafilterNat_HNatInfinite:

     "\<exists>X \<in> Rep_hypnat x. \<forall>u. {n. u < X n}:  FreeUltrafilterNat

      ==> x \<in> HNatInfinite"

apply (cases x)

apply (auto simp add: hypnat_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_hypnat 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 N  == 

      Abs_hypreal(\<Union>X \<in> Rep_hypnat(N). hyprel``{%n::nat. real (X n)})"

lemma HNat_hypreal_of_nat [simp]: "hypreal_of_nat N \<in> Nats"

by (simp add: hypreal_of_nat_def) 

(*WARNING: FRAGILE!*)

lemma lemma_hyprel_FUFN:

     "(Ya \<in> hyprel ``{%n. f(n)}) = ({n. f n = Ya n} \<in> FreeUltrafilterNat)"

by force

lemma hypreal_of_hypnat:

      "hypreal_of_hypnat (Abs_hypnat(hypnatrel``{%n. X n})) =

       Abs_hypreal(hyprel `` {%n. real (X n)})"

apply (simp add: hypreal_of_hypnat_def)

apply (rule_tac f = Abs_hypreal in arg_cong)

apply (auto elim: FreeUltrafilterNat_Int [THEN FreeUltrafilterNat_subset] 

       simp add: lemma_hyprel_FUFN)

done

lemma hypreal_of_hypnat_inject [simp]:

     "(hypreal_of_hypnat m = hypreal_of_hypnat n) = (m=n)"

apply (cases m, cases n)

apply (auto simp add: hypreal_of_hypnat)

done

lemma hypreal_of_hypnat_zero: "hypreal_of_hypnat 0 = 0"

by (simp add: hypnat_zero_def hypreal_zero_def hypreal_of_hypnat)

lemma hypreal_of_hypnat_one: "hypreal_of_hypnat (1::hypnat) = 1"

by (simp add: hypnat_one_def hypreal_one_def hypreal_of_hypnat)

lemma hypreal_of_hypnat_add [simp]:

     "hypreal_of_hypnat (m + n) = hypreal_of_hypnat m + hypreal_of_hypnat n"

apply (cases m, cases n)

apply (simp add: hypreal_of_hypnat hypreal_add hypnat_add real_of_nat_add)

done

lemma hypreal_of_hypnat_mult [simp]:

     "hypreal_of_hypnat (m * n) = hypreal_of_hypnat m * hypreal_of_hypnat n"

apply (cases m, cases n)

apply (simp add: hypreal_of_hypnat hypreal_mult hypnat_mult real_of_nat_mult)

done

lemma hypreal_of_hypnat_less_iff [simp]:

     "(hypreal_of_hypnat n < hypreal_of_hypnat m) = (n < m)"

apply (cases m, cases n)

apply (simp add: hypreal_of_hypnat hypreal_less hypnat_less)

done

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]: "0 \<le> hypreal_of_hypnat n"

apply (cases n)

apply (simp add: hypreal_of_hypnat hypreal_zero_num hypreal_le)

done

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 hypreal_inverse 

      HNatInfinite_FreeUltrafilterNat_iff Infinitesimal_FreeUltrafilterNat_iff2)

apply (rule bexI, rule_tac [2] lemma_hyprel_refl, 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_zero_def = thm"hypnat_zero_def";

val hypnat_one_def = thm"hypnat_one_def";

val hypnat_omega_def = thm"hypnat_omega_def";

val hypnatrel_iff = thm "hypnatrel_iff";

val hypnatrel_in_hypnat = thm "hypnatrel_in_hypnat";

val inj_on_Abs_hypnat = thm "inj_on_Abs_hypnat";

val inj_Rep_hypnat = thm "inj_Rep_hypnat";

val lemma_hypnatrel_refl = thm "lemma_hypnatrel_refl";

val hypnat_empty_not_mem = thm "hypnat_empty_not_mem";

val Rep_hypnat_nonempty = thm "Rep_hypnat_nonempty";

val eq_Abs_hypnat = thm "eq_Abs_hypnat";

val hypnat_add = thm "hypnat_add";

val hypnat_add_commute = thm "hypnat_add_commute";

val hypnat_add_assoc = thm "hypnat_add_assoc";

val hypnat_add_zero_left = thm "hypnat_add_zero_left";

val hypnat_minus_congruent2 = thm "hypnat_minus_congruent2";

val hypnat_minus = thm "hypnat_minus";

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_mult_congruent2 = thm "hypnat_mult_congruent2";

val hypnat_mult = thm "hypnat_mult";

val hypnat_mult_commute = thm "hypnat_mult_commute";

val hypnat_mult_assoc = thm "hypnat_mult_assoc";

val hypnat_mult_1 = thm "hypnat_mult_1";

val hypnat_diff_mult_distrib = thm "hypnat_diff_mult_distrib";

val hypnat_diff_mult_distrib2 = thm "hypnat_diff_mult_distrib2";

val hypnat_add_mult_distrib = thm "hypnat_add_mult_distrib";

val hypnat_add_mult_distrib2 = thm "hypnat_add_mult_distrib2";

val hypnat_zero_not_eq_one = thm "hypnat_zero_not_eq_one";

val hypnat_le = thm "hypnat_le";

val hypnat_le_refl = thm "hypnat_le_refl";

val hypnat_le_trans = thm "hypnat_le_trans";

val hypnat_le_anti_sym = thm "hypnat_le_anti_sym";

val hypnat_less_le = thm "hypnat_less_le";

val hypnat_le_linear = thm "hypnat_le_linear";

val hypnat_add_left_mono = thm "hypnat_add_left_mono";

val hypnat_mult_less_mono2 = thm "hypnat_mult_less_mono2";

val hypnat_mult_is_0 = thm "hypnat_mult_is_0";

val hypnat_less = thm "hypnat_less";

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 hypnat_omega = thm "hypnat_omega";

val Rep_hypnat_omega = thm "Rep_hypnat_omega";

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