--- a/src/HOL/Hyperreal/HyperNat.thy Fri Sep 09 17:47:37 2005 +0200
+++ b/src/HOL/Hyperreal/HyperNat.thy Fri Sep 09 19:34:22 2005 +0200
@@ -12,214 +12,48 @@
begin
types hypnat = "nat 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)
+syntax hypnat_of_nat :: "nat => nat star"
+translations "hypnat_of_nat" => "star_of :: nat => nat star"
-instance hypnat :: "{ord, zero, one, plus, times, minus}" ..
-*)
consts whn :: hypnat
-
defs
(* omega is in fact an infinite hypernatural number = [<1,2,3,...>] *)
- hypnat_omega_def: "whn == Abs_star(starrel``{%n::nat. n})"
-
-lemma hypnat_zero_def: "0 == Abs_star(starrel``{%n::nat. 0})"
-by (simp only: star_zero_def star_of_def star_n_def)
-
-lemma hypnat_one_def: "1 == Abs_star(starrel``{%n::nat. 1})"
-by (simp only: star_one_def star_of_def star_n_def)
-
- (** hypernatural arithmetic **)
-(*
- hypnat_zero_def: "0 == Abs_hypnat(hypnatrel``{%n::nat. 0})"
- hypnat_one_def: "1 == Abs_hypnat(hypnatrel``{%n::nat. 1})"
-*)
-(*
- 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})"
-*)
-
-(*
-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)
-
-declare Abs_hypnat_inject [simp] Abs_hypnat_inverse [simp]
-declare equiv_hypnatrel [THEN eq_equiv_class_iff, simp]
-declare hypnatrel_iff [iff]
+ hypnat_omega_def: "whn == star_n (%n::nat. n)"
-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:
- "(%X Y. hypnatrel``{%n. X n + Y n}) respects2 hypnatrel"
-by (simp add: congruent2_def, auto, ultra)
-*)
-lemma hypnat_add:
- "Abs_star(starrel``{%n. X n}) + Abs_star(starrel``{%n. Y n}) =
- Abs_star(starrel``{%n. X n + Y n})"
-by (rule hypreal_add)
-
-lemma hypnat_add_commute: "(z::hypnat) + w = w + z"
-by (rule add_commute)
-
-lemma hypnat_add_assoc: "((z1::hypnat) + z2) + z3 = z1 + (z2 + z3)"
-by (rule add_assoc)
-
-lemma hypnat_add_zero_left: "(0::hypnat) + z = z"
-by (rule comm_monoid_add_class.add_0)
-
-(*
-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:
- "(%X Y. starrel``{%n. X n - Y n}) respects2 starrel"
-by (simp add: congruent2_def, auto, ultra)
-*)
-lemma hypnat_minus:
- "Abs_star(starrel``{%n. X n}) - Abs_star(starrel``{%n. Y n}) =
- Abs_star(starrel``{%n. X n - Y n})"
-by (rule hypreal_diff)
-
-lemma hypnat_minus_zero: "!!z. z - z = (0::hypnat)"
+lemma hypnat_minus_zero [simp]: "!!z. z - z = (0::hypnat)"
by transfer (rule diff_self_eq_0)
-lemma hypnat_diff_0_eq_0: "!!n. (0::hypnat) - n = 0"
+lemma hypnat_diff_0_eq_0 [simp]: "!!n. (0::hypnat) - n = 0"
by transfer (rule diff_0_eq_0)
-declare hypnat_minus_zero [simp] hypnat_diff_0_eq_0 [simp]
-
-lemma hypnat_add_is_0: "!!m n. (m+n = (0::hypnat)) = (m=0 & n=0)"
+lemma hypnat_add_is_0 [iff]: "!!m n. (m+n = (0::hypnat)) = (m=0 & n=0)"
by transfer (rule add_is_0)
-declare hypnat_add_is_0 [iff]
-
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: "!!m n. ((n::hypnat) + m) - n = m"
+lemma hypnat_diff_add_inverse [simp]: "!!m n. ((n::hypnat) + m) - n = m"
by transfer (rule diff_add_inverse)
-declare hypnat_diff_add_inverse [simp]
-lemma hypnat_diff_add_inverse2: "!!m n. ((m::hypnat) + n) - n = m"
+lemma hypnat_diff_add_inverse2 [simp]: "!!m n. ((m::hypnat) + n) - n = m"
by transfer (rule diff_add_inverse2)
-declare hypnat_diff_add_inverse2 [simp]
-lemma hypnat_diff_cancel: "!!k m n. ((k::hypnat) + m) - (k+n) = m - n"
+lemma hypnat_diff_cancel [simp]: "!!k m n. ((k::hypnat) + m) - (k+n) = m - n"
by transfer (rule diff_cancel)
-declare hypnat_diff_cancel [simp]
-lemma hypnat_diff_cancel2: "!!k m n. ((m::hypnat) + k) - (n+k) = m - n"
+lemma hypnat_diff_cancel2 [simp]: "!!k m n. ((m::hypnat) + k) - (n+k) = m - n"
by transfer (rule diff_cancel2)
-declare hypnat_diff_cancel2 [simp]
-lemma hypnat_diff_add_0: "!!m n. (n::hypnat) - (n+m) = (0::hypnat)"
+lemma hypnat_diff_add_0 [simp]: "!!m n. (n::hypnat) - (n+m) = (0::hypnat)"
by transfer (rule diff_add_0)
-declare hypnat_diff_add_0 [simp]
subsection{*Hyperreal Multiplication*}
-(*
-lemma hypnat_mult_congruent2:
- "(%X Y. starrel``{%n. X n * Y n}) respects2 starrel"
-by (simp add: congruent2_def, auto, ultra)
-*)
-lemma hypnat_mult:
- "Abs_star(starrel``{%n. X n}) * Abs_star(starrel``{%n. Y n}) =
- Abs_star(starrel``{%n. X n * Y n})"
-by (rule hypreal_mult)
-
-lemma hypnat_mult_commute: "(z::hypnat) * w = w * z"
-by (rule mult_commute)
-
-lemma hypnat_mult_assoc: "((z1::hypnat) * z2) * z3 = z1 * (z2 * z3)"
-by (rule mult_assoc)
-
-lemma hypnat_mult_1: "(1::hypnat) * z = z"
-by (rule mult_1_left)
lemma hypnat_diff_mult_distrib: "!!k m n. ((m::hypnat) - n) * k = (m * k) - (n * k)"
by transfer (rule diff_mult_distrib)
@@ -227,87 +61,8 @@
lemma hypnat_diff_mult_distrib2: "!!k m n. (k::hypnat) * (m - n) = (k * m) - (k * n)"
by transfer (rule diff_mult_distrib2)
-lemma hypnat_add_mult_distrib: "((z1::hypnat) + z2) * w = (z1 * w) + (z2 * w)"
-by (rule left_distrib)
-
-lemma hypnat_add_mult_distrib2: "(w::hypnat) * (z1 + z2) = (w * z1) + (w * z2)"
-by (rule right_distrib)
-
-text{*one and zero are distinct*}
-lemma hypnat_zero_not_eq_one: "(0::hypnat) \<noteq> (1::hypnat)"
-by (rule zero_neq_one)
-declare hypnat_zero_not_eq_one [THEN not_sym, simp]
-
-(*
-text{*The hypernaturals form a @{text 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_star(starrel``{%n. X n}) \<le> Abs_star(starrel``{%n. Y n})) =
- ({n. X n \<le> Y n} \<in> FreeUltrafilterNat)"
-by (rule hypreal_le)
-
-lemma hypnat_le_refl: "w \<le> (w::hypnat)"
-by (rule order_refl)
-
-lemma hypnat_le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::hypnat)"
-by (rule order_trans)
-
-lemma hypnat_le_anti_sym: "[| z \<le> w; w \<le> z |] ==> z = (w::hypnat)"
-by (rule order_antisym)
-
-(* Axiom 'order_less_le' of class 'order': *)
-lemma hypnat_less_le: "((w::hypnat) < z) = (w \<le> z & w \<noteq> z)"
-by (rule order_less_le)
-
-(*
-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"
-by (rule linorder_linear)
-(*
-instance hypnat :: linorder
- by intro_classes (rule hypnat_le_linear)
-*)
-lemma hypnat_add_left_mono: "x \<le> y ==> z + x \<le> z + (y::hypnat)"
-by (rule add_left_mono)
-
-lemma hypnat_mult_less_mono2: "[| (0::hypnat)<z; x<y |] ==> z*x<z*y"
-by (rule mult_strict_left_mono)
-
-
-subsection{*The Hypernaturals Form an Ordered @{text 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. (n \<le> (0::hypnat)) = (n = 0)"
by transfer (rule le_0_eq)
@@ -321,11 +76,6 @@
subsection{*Theorems for Ordering*}
-lemma hypnat_less:
- "(Abs_star(starrel``{%n. X n}) < Abs_star(starrel``{%n. Y n})) =
- ({n. X n < Y n} \<in> FreeUltrafilterNat)"
-by (rule hypreal_less)
-
lemma hypnat_not_less0 [iff]: "!!n. ~ n < (0::hypnat)"
by transfer (rule not_less0)
@@ -389,42 +139,42 @@
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_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 add: hypnat_of_nat_def)
+by simp
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)
+by simp
-lemma hypnat_of_nat_less_iff [simp]:
+lemma hypnat_of_nat_less_iff:
"(hypnat_of_nat z < hypnat_of_nat w) = (z < w)"
-by (simp add: hypnat_of_nat_def)
+by simp
-lemma hypnat_of_nat_le_iff [simp]:
+lemma hypnat_of_nat_le_iff:
"(hypnat_of_nat z \<le> hypnat_of_nat w) = (z \<le> w)"
-by (simp add: hypnat_of_nat_def)
+by simp
-lemma hypnat_of_nat_eq_iff [simp]:
+lemma hypnat_of_nat_eq_iff:
"(hypnat_of_nat z = hypnat_of_nat w) = (z = w)"
-by (simp add: hypnat_of_nat_def)
+by simp
lemma hypnat_of_nat_one [simp]: "hypnat_of_nat (Suc 0) = (1::hypnat)"
-by (simp add: hypnat_of_nat_def)
+by simp
-lemma hypnat_of_nat_zero [simp]: "hypnat_of_nat 0 = 0"
-by (simp add: hypnat_of_nat_def)
+lemma hypnat_of_nat_zero: "hypnat_of_nat 0 = 0"
+by simp
-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_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)"
@@ -432,17 +182,11 @@
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)
+by simp
subsection{*Existence of an infinite hypernatural number*}
-lemma hypnat_omega: "starrel``{%n::nat. n} \<in> star"
-by auto
-
-lemma Rep_star_omega: "Rep_star(whn) \<in> star"
-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.*}
@@ -471,11 +215,10 @@
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_star(starrel``{%n::nat. m})"
+ "hypnat_of_nat m = star_n (%n::nat. m)"
apply (induct m)
-apply (simp_all add: hypnat_zero_def hypnat_one_def hypnat_add)
+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}"
@@ -483,7 +226,7 @@
lemma hypnat_omega_gt_SHNat:
"n \<in> Nats ==> n < whn"
-by (auto simp add: hypnat_of_nat_eq hypnat_less hypnat_omega_def SHNat_eq)
+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"
@@ -532,10 +275,10 @@
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_star)
+apply (rule_tac x = x in star_cases)
apply (auto elim: HNatInfinite_FreeUltrafilterNat_lemma
- simp add: hypnat_less FreeUltrafilterNat_Compl_iff1
- Collect_neg_eq [symmetric])
+ simp add: star_n_less FreeUltrafilterNat_Compl_iff1
+ star_n_eq_iff Collect_neg_eq [symmetric])
done
@@ -545,17 +288,17 @@
lemma HNatInfinite_FreeUltrafilterNat:
"x \<in> HNatInfinite
==> \<exists>X \<in> Rep_star x. \<forall>u. {n. u < X n}: FreeUltrafilterNat"
-apply (rule_tac z=x in eq_Abs_star)
+apply (cases x)
apply (auto simp add: HNatInfinite_iff SHNat_eq hypnat_of_nat_eq)
-apply (rule bexI [OF _ lemma_starrel_refl], clarify)
-apply (auto simp add: hypnat_of_nat_def hypnat_less)
+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 (rule_tac z=x in eq_Abs_star)
-apply (auto simp add: hypnat_less HNatInfinite_iff SHNat_eq hypnat_of_nat_eq)
+apply (cases x)
+apply (auto simp add: star_n_less HNatInfinite_iff SHNat_eq hypnat_of_nat_eq)
apply (drule spec, ultra, auto)
done
@@ -628,72 +371,51 @@
constdefs
hypreal_of_hypnat :: "hypnat => hypreal"
- "hypreal_of_hypnat N ==
- Abs_star(\<Union>X \<in> Rep_star(N). starrel``{%n::nat. real (X n)})"
+ "hypreal_of_hypnat == *f* real"
lemma HNat_hypreal_of_nat [simp]: "hypreal_of_nat N \<in> Nats"
by (simp add: hypreal_of_nat_def)
-(*WARNING: FRAGILE!*)
-lemma lemma_starrel_FUFN:
- "(Ya \<in> starrel ``{%n. f(n)}) = ({n. f n = Ya n} \<in> FreeUltrafilterNat)"
-by force
-
lemma hypreal_of_hypnat:
- "hypreal_of_hypnat (Abs_star(starrel``{%n. X n})) =
- Abs_star(starrel `` {%n. real (X n)})"
-apply (simp add: hypreal_of_hypnat_def)
-apply (rule_tac f = Abs_star in arg_cong)
-apply (auto elim: FreeUltrafilterNat_Int [THEN FreeUltrafilterNat_subset]
- simp add: lemma_starrel_FUFN)
-done
+ "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]:
- "(hypreal_of_hypnat m = hypreal_of_hypnat n) = (m=n)"
-apply (rule_tac z=m in eq_Abs_star, rule_tac z=n in eq_Abs_star)
-apply (auto simp add: hypreal_of_hypnat)
-done
+ "!!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: hypnat_zero_def hypreal_zero_def hypreal_of_hypnat)
+by (simp add: star_n_zero_num 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)
+by (simp add: star_n_one_num hypreal_of_hypnat)
lemma hypreal_of_hypnat_add [simp]:
- "hypreal_of_hypnat (m + n) = hypreal_of_hypnat m + hypreal_of_hypnat n"
-apply (rule_tac z=m in eq_Abs_star, rule_tac z=n in eq_Abs_star)
-apply (simp add: hypreal_of_hypnat hypreal_add hypnat_add real_of_nat_add)
-done
+ "!!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]:
- "hypreal_of_hypnat (m * n) = hypreal_of_hypnat m * hypreal_of_hypnat n"
-apply (rule_tac z=m in eq_Abs_star, rule_tac z=n in eq_Abs_star)
-apply (simp add: hypreal_of_hypnat hypreal_mult hypnat_mult real_of_nat_mult)
-done
+ "!!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]:
- "(hypreal_of_hypnat n < hypreal_of_hypnat m) = (n < m)"
-apply (rule_tac z=m in eq_Abs_star, rule_tac z=n in eq_Abs_star)
-apply (simp add: hypreal_of_hypnat hypreal_less hypnat_less)
-done
+ "!!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]: "0 \<le> hypreal_of_hypnat n"
-apply (rule_tac z=n in eq_Abs_star)
-apply (simp add: hypreal_of_hypnat hypreal_zero_num hypreal_le)
-done
+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 (rule_tac z=n in eq_Abs_star)
-apply (auto simp add: hypreal_of_hypnat hypreal_inverse
+apply (cases n)
+apply (auto simp add: hypreal_of_hypnat star_n_inverse
HNatInfinite_FreeUltrafilterNat_iff Infinitesimal_FreeUltrafilterNat_iff2)
-apply (rule bexI, rule_tac [2] lemma_starrel_refl, auto)
+apply (rule bexI [OF _ Rep_star_star_n], auto)
apply (drule_tac x = "m + 1" in spec, ultra)
done
@@ -709,22 +431,11 @@
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 starrel_iff = thm "starrel_iff";
-(* val starrel_in_hypnat = thm "starrel_in_hypnat"; *)
val lemma_starrel_refl = thm "lemma_starrel_refl";
-(* val hypnat_empty_not_mem = thm "hypnat_empty_not_mem"; *)
-(* val Rep_star_nonempty = thm "Rep_star_nonempty"; *)
val eq_Abs_star = thm "eq_Abs_star";
-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";
@@ -735,26 +446,9 @@
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";
@@ -777,8 +471,6 @@
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_star_omega = thm "Rep_star_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";