# Theory NatStar

```(*  Title:      HOL/Nonstandard_Analysis/NatStar.thy
Author:     Jacques D. Fleuriot

Converted to Isar and polished by lcp
*)

section ‹Star-transforms for the Hypernaturals›

theory NatStar
imports Star
begin

lemma star_n_eq_starfun_whn: "star_n X = ( *f* X) whn"
by (simp add: hypnat_omega_def starfun_def star_of_def Ifun_star_n)

lemma starset_n_Un: "*sn* (λn. (A n) ∪ (B n)) = *sn* A ∪ *sn* B"
proof -
have "⋀N. Iset ((*f* (λn. {x. x ∈ A n ∨ x ∈ B n})) N) =
{x. x ∈ Iset ((*f* A) N) ∨ x ∈ Iset ((*f* B) N)}"
by transfer simp
then show ?thesis
by (simp add: starset_n_def star_n_eq_starfun_whn Un_def)
qed

lemma InternalSets_Un: "X ∈ InternalSets ⟹ Y ∈ InternalSets ⟹ X ∪ Y ∈ InternalSets"
by (auto simp add: InternalSets_def starset_n_Un [symmetric])

lemma starset_n_Int: "*sn* (λn. A n ∩ B n) = *sn* A ∩ *sn* B"
proof -
have "⋀N. Iset ((*f* (λn. {x. x ∈ A n ∧ x ∈ B n})) N) =
{x. x ∈ Iset ((*f* A) N) ∧ x ∈ Iset ((*f* B) N)}"
by transfer simp
then show ?thesis
by (simp add: starset_n_def star_n_eq_starfun_whn Int_def)
qed

lemma InternalSets_Int: "X ∈ InternalSets ⟹ Y ∈ InternalSets ⟹ X ∩ Y ∈ InternalSets"
by (auto simp add: InternalSets_def starset_n_Int [symmetric])

lemma starset_n_Compl: "*sn* ((λn. - A n)) = - ( *sn* A)"
proof -
have "⋀N. Iset ((*f* (λn. {x. x ∉ A n})) N) =
{x. x ∉ Iset ((*f* A) N)}"
by transfer simp
then show ?thesis
by (simp add: starset_n_def star_n_eq_starfun_whn Compl_eq)
qed

lemma InternalSets_Compl: "X ∈ InternalSets ⟹ - X ∈ InternalSets"
by (auto simp add: InternalSets_def starset_n_Compl [symmetric])

lemma starset_n_diff: "*sn* (λn. (A n) - (B n)) = *sn* A - *sn* B"
proof -
have "⋀N. Iset ((*f* (λn. {x. x ∈ A n ∧ x ∉ B n})) N) =
{x. x ∈ Iset ((*f* A) N) ∧ x ∉ Iset ((*f* B) N)}"
by transfer simp
then show ?thesis
by (simp add: starset_n_def star_n_eq_starfun_whn set_diff_eq)
qed

lemma InternalSets_diff: "X ∈ InternalSets ⟹ Y ∈ InternalSets ⟹ X - Y ∈ InternalSets"
by (auto simp add: InternalSets_def starset_n_diff [symmetric])

lemma NatStar_SHNat_subset: "Nats ≤ *s* (UNIV:: nat set)"
by simp

lemma NatStar_hypreal_of_real_Int: "*s* X Int Nats = hypnat_of_nat ` X"

lemma starset_starset_n_eq: "*s* X = *sn* (λn. X)"

lemma InternalSets_starset_n [simp]: "( *s* X) ∈ InternalSets"
by (auto simp add: InternalSets_def starset_starset_n_eq)

lemma InternalSets_UNIV_diff: "X ∈ InternalSets ⟹ UNIV - X ∈ InternalSets"

subsection ‹Nonstandard Extensions of Functions›

text ‹Example of transfer of a property from reals to hyperreals
--- used for limit comparison of sequences.›

lemma starfun_le_mono: "∀n. N ≤ n ⟶ f n ≤ g n ⟹
∀n. hypnat_of_nat N ≤ n ⟶ ( *f* f) n ≤ ( *f* g) n"
by transfer

text ‹And another:›
lemma starfun_less_mono:
"∀n. N ≤ n ⟶ f n < g n ⟹ ∀n. hypnat_of_nat N ≤ n ⟶ ( *f* f) n < ( *f* g) n"
by transfer

text ‹Nonstandard extension when we increment the argument by one.›

lemma starfun_shift_one: "⋀N. ( *f* (λn. f (Suc n))) N = ( *f* f) (N + (1::hypnat))"
by transfer simp

text ‹Nonstandard extension with absolute value.›
lemma starfun_abs: "⋀N. ( *f* (λn. ¦f n¦)) N = ¦( *f* f) N¦"
by transfer (rule refl)

text ‹The ‹hyperpow› function as a nonstandard extension of ‹realpow›.›
lemma starfun_pow: "⋀N. ( *f* (λn. r ^ n)) N = hypreal_of_real r pow N"
by transfer (rule refl)

lemma starfun_pow2: "⋀N. ( *f* (λn. X n ^ m)) N = ( *f* X) N pow hypnat_of_nat m"
by transfer (rule refl)

lemma starfun_pow3: "⋀R. ( *f* (λr. r ^ n)) R = R pow hypnat_of_nat n"
by transfer (rule refl)

text ‹The \<^term>‹hypreal_of_hypnat› function as a nonstandard extension of
\<^term>‹real_of_nat›.›
lemma starfunNat_real_of_nat: "( *f* real) = hypreal_of_hypnat"

lemma starfun_inverse_real_of_nat_eq:
"N ∈ HNatInfinite ⟹ ( *f* (λx::nat. inverse (real x))) N = inverse (hypreal_of_hypnat N)"
by (metis of_hypnat_def starfun_inverse2)

text ‹Internal functions -- some redundancy with ‹*f*› now.›

lemma starfun_n: "( *fn* f) (star_n X) = star_n (λn. f n (X n))"

text ‹Multiplication: ‹( *fn) x ( *gn) = *(fn x gn)››

lemma starfun_n_mult: "( *fn* f) z * ( *fn* g) z = ( *fn* (λi x. f i x * g i x)) z"
by (cases z) (simp add: starfun_n star_n_mult)

text ‹Addition: ‹( *fn) + ( *gn) = *(fn + gn)››
lemma starfun_n_add: "( *fn* f) z + ( *fn* g) z = ( *fn* (λi x. f i x + g i x)) z"

text ‹Subtraction: ‹( *fn) - ( *gn) = *(fn + - gn)››
lemma starfun_n_add_minus: "( *fn* f) z + -( *fn* g) z = ( *fn* (λi x. f i x + -g i x)) z"

text ‹Composition: ‹( *fn) ∘ ( *gn) = *(fn ∘ gn)››

lemma starfun_n_const_fun [simp]: "( *fn* (λi x. k)) z = star_of k"
by (cases z) (simp add: starfun_n star_of_def)

lemma starfun_n_minus: "- ( *fn* f) x = ( *fn* (λi x. - (f i) x)) x"
by (cases x) (simp add: starfun_n star_n_minus)

lemma starfun_n_eq [simp]: "( *fn* f) (star_of n) = star_n (λi. f i n)"

lemma starfun_eq_iff: "(( *f* f) = ( *f* g)) ⟷ f = g"
by transfer (rule refl)

lemma starfunNat_inverse_real_of_nat_Infinitesimal [simp]:
"N ∈ HNatInfinite ⟹ ( *f* (λx. inverse (real x))) N ∈ Infinitesimal"
using starfun_inverse_real_of_nat_eq by auto

subsection ‹Nonstandard Characterization of Induction›

lemma hypnat_induct_obj:
"⋀n. (( *p* P) (0::hypnat) ∧ (∀n. ( *p* P) n ⟶ ( *p* P) (n + 1))) ⟶ ( *p* P) n"
by transfer (induct_tac n, auto)

lemma hypnat_induct:
"⋀n. ( *p* P) (0::hypnat) ⟹ (⋀n. ( *p* P) n ⟹ ( *p* P) (n + 1)) ⟹ ( *p* P) n"
by transfer (induct_tac n, auto)

lemma starP2_eq_iff: "( *p2* (=)) = (=)"
by transfer (rule refl)

lemma starP2_eq_iff2: "( *p2* (λx y. x = y)) X Y ⟷ X = Y"

lemma nonempty_set_star_has_least_lemma:
"∃n∈S. ∀m∈S. n ≤ m" if "S ≠ {}" for S :: "nat set"
proof
show "∀m∈S. (LEAST n. n ∈ S) ≤ m"
show "(LEAST n. n ∈ S) ∈ S"
by (meson that LeastI_ex equals0I)
qed

lemma nonempty_set_star_has_least:
"⋀S::nat set star. Iset S ≠ {} ⟹ ∃n ∈ Iset S. ∀m ∈ Iset S. n ≤ m"
using nonempty_set_star_has_least_lemma by (transfer empty_def)

lemma nonempty_InternalNatSet_has_least: "S ∈ InternalSets ⟹ S ≠ {} ⟹ ∃n ∈ S. ∀m ∈ S. n ≤ m"
for S :: "hypnat set"
by (force simp add: InternalSets_def starset_n_def dest!: nonempty_set_star_has_least)

text ‹Goldblatt, page 129 Thm 11.3.2.›
lemma internal_induct_lemma:
"⋀X::nat set star.
(0::hypnat) ∈ Iset X ⟹ ∀n. n ∈ Iset X ⟶ n + 1 ∈ Iset X ⟹ Iset X = (UNIV:: hypnat set)"
apply (transfer UNIV_def)
apply (rule equalityI [OF subset_UNIV subsetI])
apply (induct_tac x, auto)
done

lemma internal_induct:
"X ∈ InternalSets ⟹ (0::hypnat) ∈ X ⟹ ∀n. n ∈ X ⟶ n + 1 ∈ X ⟹ X = (UNIV:: hypnat set)"
apply (clarsimp simp add: InternalSets_def starset_n_def)
apply (erule (1) internal_induct_lemma)
done

end
```