(* Title: HOL/Nonstandard_Analysis/NatStar.thy
Author: Jacques D. Fleuriot
Copyright: 1998 University of Cambridge
Converted to Isar and polished by lcp
*)
section\<open>Star-transforms for the Hypernaturals\<close>
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* (%n. (A n) Un (B n)) = *sn* A Un *sn* B"
apply (simp add: starset_n_def star_n_eq_starfun_whn Un_def)
apply (rule_tac x=whn in spec, transfer, simp)
done
lemma InternalSets_Un:
"[| X \<in> InternalSets; Y \<in> InternalSets |]
==> (X Un Y) \<in> InternalSets"
by (auto simp add: InternalSets_def starset_n_Un [symmetric])
lemma starset_n_Int:
"*sn* (%n. (A n) Int (B n)) = *sn* A Int *sn* B"
apply (simp add: starset_n_def star_n_eq_starfun_whn Int_def)
apply (rule_tac x=whn in spec, transfer, simp)
done
lemma InternalSets_Int:
"[| X \<in> InternalSets; Y \<in> InternalSets |]
==> (X Int Y) \<in> InternalSets"
by (auto simp add: InternalSets_def starset_n_Int [symmetric])
lemma starset_n_Compl: "*sn* ((%n. - A n)) = -( *sn* A)"
apply (simp add: starset_n_def star_n_eq_starfun_whn Compl_eq)
apply (rule_tac x=whn in spec, transfer, simp)
done
lemma InternalSets_Compl: "X \<in> InternalSets ==> -X \<in> InternalSets"
by (auto simp add: InternalSets_def starset_n_Compl [symmetric])
lemma starset_n_diff: "*sn* (%n. (A n) - (B n)) = *sn* A - *sn* B"
apply (simp add: starset_n_def star_n_eq_starfun_whn set_diff_eq)
apply (rule_tac x=whn in spec, transfer, simp)
done
lemma InternalSets_diff:
"[| X \<in> InternalSets; Y \<in> InternalSets |]
==> (X - Y) \<in> InternalSets"
by (auto simp add: InternalSets_def starset_n_diff [symmetric])
lemma NatStar_SHNat_subset: "Nats \<le> *s* (UNIV:: nat set)"
by simp
lemma NatStar_hypreal_of_real_Int:
"*s* X Int Nats = hypnat_of_nat ` X"
by (auto simp add: SHNat_eq)
lemma starset_starset_n_eq: "*s* X = *sn* (%n. X)"
by (simp add: starset_n_starset)
lemma InternalSets_starset_n [simp]: "( *s* X) \<in> InternalSets"
by (auto simp add: InternalSets_def starset_starset_n_eq)
lemma InternalSets_UNIV_diff:
"X \<in> InternalSets ==> UNIV - X \<in> InternalSets"
apply (subgoal_tac "UNIV - X = - X")
by (auto intro: InternalSets_Compl)
subsection\<open>Nonstandard Extensions of Functions\<close>
text\<open>Example of transfer of a property from reals to hyperreals
--- used for limit comparison of sequences\<close>
lemma starfun_le_mono:
"\<forall>n. N \<le> n --> f n \<le> g n
==> \<forall>n. hypnat_of_nat N \<le> n --> ( *f* f) n \<le> ( *f* g) n"
by transfer
(*****----- and another -----*****)
lemma starfun_less_mono:
"\<forall>n. N \<le> n --> f n < g n
==> \<forall>n. hypnat_of_nat N \<le> n --> ( *f* f) n < ( *f* g) n"
by transfer
text\<open>Nonstandard extension when we increment the argument by one\<close>
lemma starfun_shift_one:
"!!N. ( *f* (%n. f (Suc n))) N = ( *f* f) (N + (1::hypnat))"
by (transfer, simp)
text\<open>Nonstandard extension with absolute value\<close>
lemma starfun_abs: "!!N. ( *f* (%n. \<bar>f n\<bar>)) N = \<bar>( *f* f) N\<bar>"
by (transfer, rule refl)
text\<open>The hyperpow function as a nonstandard extension of realpow\<close>
lemma starfun_pow: "!!N. ( *f* (%n. r ^ n)) N = (hypreal_of_real r) pow N"
by (transfer, rule refl)
lemma starfun_pow2:
"!!N. ( *f* (%n. (X n) ^ m)) N = ( *f* X) N pow hypnat_of_nat m"
by (transfer, rule refl)
lemma starfun_pow3: "!!R. ( *f* (%r. r ^ n)) R = (R) pow hypnat_of_nat n"
by (transfer, rule refl)
text\<open>The @{term hypreal_of_hypnat} function as a nonstandard extension of
@{term real_of_nat}\<close>
lemma starfunNat_real_of_nat: "( *f* real) = hypreal_of_hypnat"
by transfer (simp add: fun_eq_iff)
lemma starfun_inverse_real_of_nat_eq:
"N \<in> HNatInfinite
==> ( *f* (%x::nat. inverse(real x))) N = inverse(hypreal_of_hypnat N)"
apply (rule_tac f1 = inverse in starfun_o2 [THEN subst])
apply (subgoal_tac "hypreal_of_hypnat N ~= 0")
apply (simp_all add: zero_less_HNatInfinite starfunNat_real_of_nat starfun_inverse_inverse)
done
text\<open>Internal functions - some redundancy with *f* now\<close>
lemma starfun_n: "( *fn* f) (star_n X) = star_n (%n. f n (X n))"
by (simp add: starfun_n_def Ifun_star_n)
text\<open>Multiplication: \<open>( *fn) x ( *gn) = *(fn x gn)\<close>\<close>
lemma starfun_n_mult:
"( *fn* f) z * ( *fn* g) z = ( *fn* (% i x. f i x * g i x)) z"
apply (cases z)
apply (simp add: starfun_n star_n_mult)
done
text\<open>Addition: \<open>( *fn) + ( *gn) = *(fn + gn)\<close>\<close>
lemma starfun_n_add:
"( *fn* f) z + ( *fn* g) z = ( *fn* (%i x. f i x + g i x)) z"
apply (cases z)
apply (simp add: starfun_n star_n_add)
done
text\<open>Subtraction: \<open>( *fn) - ( *gn) = *(fn + - gn)\<close>\<close>
lemma starfun_n_add_minus:
"( *fn* f) z + -( *fn* g) z = ( *fn* (%i x. f i x + -g i x)) z"
apply (cases z)
apply (simp add: starfun_n star_n_minus star_n_add)
done
text\<open>Composition: \<open>( *fn) o ( *gn) = *(fn o gn)\<close>\<close>
lemma starfun_n_const_fun [simp]:
"( *fn* (%i x. k)) z = star_of k"
apply (cases z)
apply (simp add: starfun_n star_of_def)
done
lemma starfun_n_minus: "- ( *fn* f) x = ( *fn* (%i x. - (f i) x)) x"
apply (cases x)
apply (simp add: starfun_n star_n_minus)
done
lemma starfun_n_eq [simp]:
"( *fn* f) (star_of n) = star_n (%i. f i n)"
by (simp add: starfun_n star_of_def)
lemma starfun_eq_iff: "(( *f* f) = ( *f* g)) = (f = g)"
by (transfer, rule refl)
lemma starfunNat_inverse_real_of_nat_Infinitesimal [simp]:
"N \<in> HNatInfinite ==> ( *f* (%x. inverse (real x))) N \<in> Infinitesimal"
apply (rule_tac f1 = inverse in starfun_o2 [THEN subst])
apply (subgoal_tac "hypreal_of_hypnat N ~= 0")
apply (simp_all add: zero_less_HNatInfinite starfunNat_real_of_nat)
done
subsection\<open>Nonstandard Characterization of Induction\<close>
lemma hypnat_induct_obj:
"!!n. (( *p* P) (0::hypnat) &
(\<forall>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* (op =)) = (op =)"
by transfer (rule refl)
lemma starP2_eq_iff2: "( *p2* (%x y. x = y)) X Y = (X = Y)"
by (simp add: starP2_eq_iff)
lemma nonempty_nat_set_Least_mem:
"c \<in> (S :: nat set) ==> (LEAST n. n \<in> S) \<in> S"
by (erule LeastI)
lemma nonempty_set_star_has_least:
"!!S::nat set star. Iset S \<noteq> {} ==> \<exists>n \<in> Iset S. \<forall>m \<in> Iset S. n \<le> m"
apply (transfer empty_def)
apply (rule_tac x="LEAST n. n \<in> S" in bexI)
apply (simp add: Least_le)
apply (rule LeastI_ex, auto)
done
lemma nonempty_InternalNatSet_has_least:
"[| (S::hypnat set) \<in> InternalSets; S \<noteq> {} |] ==> \<exists>n \<in> S. \<forall>m \<in> S. n \<le> m"
apply (clarsimp simp add: InternalSets_def starset_n_def)
apply (erule nonempty_set_star_has_least)
done
text\<open>Goldblatt page 129 Thm 11.3.2\<close>
lemma internal_induct_lemma:
"!!X::nat set star. [| (0::hypnat) \<in> Iset X; \<forall>n. n \<in> Iset X --> n + 1 \<in> 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 \<in> InternalSets; (0::hypnat) \<in> X; \<forall>n. n \<in> X --> n + 1 \<in> X |]
==> X = (UNIV:: hypnat set)"
apply (clarsimp simp add: InternalSets_def starset_n_def)
apply (erule (1) internal_induct_lemma)
done
end