(* Title : NatStar.thy Author : Jacques D. Fleuriot Copyright : 1998 University of Cambridge Converted to Isar and polished by lcp *) header{*Star-transforms for the Hypernaturals*} theory NatStar imports HyperPow 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 ∈ InternalSets; Y ∈ InternalSets |] ==> (X Un Y) ∈ 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 ∈ InternalSets; Y ∈ InternalSets |] ==> (X Int Y) ∈ 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_def) apply (rule_tac x=whn in spec, transfer, simp) done lemma InternalSets_Compl: "X ∈ InternalSets ==> -X ∈ 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_def) apply (rule_tac x=whn in spec, transfer, simp) done 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" by (auto simp add: SHNat_eq STAR_mem_iff) lemma starset_starset_n_eq: "*s* X = *sn* (%n. X)" by (simp add: starset_n_starset) 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" apply (subgoal_tac "UNIV - X = - X") by (auto intro: InternalSets_Compl) 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 (*****----- 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* (%n. f (Suc n))) N = ( *f* f) (N + (1::hypnat))" by (transfer, simp) text{*Nonstandard extension with absolute value*} lemma starfun_abs: "!!N. ( *f* (%n. abs (f n))) N = abs(( *f* f) N)" by (transfer, rule refl) text{*The hyperpow function as a nonstandard extension of realpow*} 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{*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" by (transfer, rule refl) lemma starfun_inverse_real_of_nat_eq: "N ∈ 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: HNatInfinite_not_eq_zero starfunNat_real_of_nat starfun_inverse_inverse) done text{*Internal functions - some redundancy with *f* now*} 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{*Multiplication: @{text "( *fn) x ( *gn) = *(fn x gn)"}*} 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{*Addition: @{text "( *fn) + ( *gn) = *(fn + gn)"}*} 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{*Subtraction: @{text "( *fn) - ( *gn) = *(fn + - gn)"}*} 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{*Composition: @{text "( *fn) o ( *gn) = *(fn o gn)"}*} 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 ∈ HNatInfinite ==> ( *f* (%x. inverse (real x))) N ∈ Infinitesimal" apply (rule_tac f1 = inverse in starfun_o2 [THEN subst]) apply (subgoal_tac "hypreal_of_hypnat N ~= 0") apply (simp_all add: HNatInfinite_not_eq_zero starfunNat_real_of_nat) done ML {* val starset_n_Un = thm "starset_n_Un"; val InternalSets_Un = thm "InternalSets_Un"; val starset_n_Int = thm "starset_n_Int"; val InternalSets_Int = thm "InternalSets_Int"; val starset_n_Compl = thm "starset_n_Compl"; val InternalSets_Compl = thm "InternalSets_Compl"; val starset_n_diff = thm "starset_n_diff"; val InternalSets_diff = thm "InternalSets_diff"; val NatStar_SHNat_subset = thm "NatStar_SHNat_subset"; val NatStar_hypreal_of_real_Int = thm "NatStar_hypreal_of_real_Int"; val starset_starset_n_eq = thm "starset_starset_n_eq"; val InternalSets_starset_n = thm "InternalSets_starset_n"; val InternalSets_UNIV_diff = thm "InternalSets_UNIV_diff"; val starset_n_starset = thm "starset_n_starset"; val starfun_const_fun = thm "starfun_const_fun"; val starfun_le_mono = thm "starfun_le_mono"; val starfun_less_mono = thm "starfun_less_mono"; val starfun_shift_one = thm "starfun_shift_one"; val starfun_abs = thm "starfun_abs"; val starfun_pow = thm "starfun_pow"; val starfun_pow2 = thm "starfun_pow2"; val starfun_pow3 = thm "starfun_pow3"; val starfunNat_real_of_nat = thm "starfunNat_real_of_nat"; val starfun_inverse_real_of_nat_eq = thm "starfun_inverse_real_of_nat_eq"; val starfun_n = thm "starfun_n"; val starfun_n_mult = thm "starfun_n_mult"; val starfun_n_add = thm "starfun_n_add"; val starfun_n_add_minus = thm "starfun_n_add_minus"; val starfun_n_const_fun = thm "starfun_n_const_fun"; val starfun_n_minus = thm "starfun_n_minus"; val starfun_n_eq = thm "starfun_n_eq"; val starfun_eq_iff = thm "starfun_eq_iff"; val starfunNat_inverse_real_of_nat_Infinitesimal = thm "starfunNat_inverse_real_of_nat_Infinitesimal"; *} subsection{*Nonstandard Characterization of Induction*} constdefs hSuc :: "hypnat => hypnat" "hSuc n == n + 1" lemma starP: "(( *p* P) (star_n X)) = ({n. P (X n)} ∈ FreeUltrafilterNat)" by (rule starP_star_n) 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: "(( *p2* P) (star_n X) (star_n Y)) = ({n. P (X n) (Y n)} ∈ FreeUltrafilterNat)" by (rule starP2_star_n) 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 hSuc_not_zero [iff]: "hSuc m ≠ 0" by (simp add: hSuc_def) lemmas zero_not_hSuc = hSuc_not_zero [THEN not_sym, standard, iff] lemma hSuc_hSuc_eq [iff]: "(hSuc m = hSuc n) = (m = n)" by (simp add: hSuc_def star_n_one_num) lemma nonempty_nat_set_Least_mem: "c ∈ (S :: nat set) ==> (LEAST n. n ∈ S) ∈ S" by (erule LeastI) lemma nonempty_set_star_has_least: "!!S::nat set star. Iset S ≠ {} ==> ∃n ∈ Iset S. ∀m ∈ Iset S. n ≤ m" apply (transfer empty_def) apply (rule_tac x="LEAST n. n ∈ S" in bexI) apply (simp add: Least_le) apply (rule LeastI_ex, auto) done lemma nonempty_InternalNatSet_has_least: "[| (S::hypnat set) ∈ InternalSets; S ≠ {} |] ==> ∃n ∈ S. ∀m ∈ S. n ≤ m" apply (clarsimp simp add: InternalSets_def starset_n_def) apply (erule nonempty_set_star_has_least) done 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
lemma star_n_eq_starfun_whn:
star_n X = (*f* X) whn
lemma starset_n_Un:
*sn* (%n. A n ∪ B n) = *sn* A ∪ *sn* B
lemma InternalSets_Un:
[| X ∈ InternalSets; Y ∈ InternalSets |] ==> X ∪ Y ∈ InternalSets
lemma starset_n_Int:
*sn* (%n. A n ∩ B n) = *sn* A ∩ *sn* B
lemma InternalSets_Int:
[| X ∈ InternalSets; Y ∈ InternalSets |] ==> X ∩ Y ∈ InternalSets
lemma starset_n_Compl:
*sn* (%n. - A n) = - (*sn* A)
lemma InternalSets_Compl:
X ∈ InternalSets ==> - X ∈ InternalSets
lemma starset_n_diff:
*sn* (%n. A n - B n) = *sn* A - *sn* B
lemma InternalSets_diff:
[| X ∈ InternalSets; Y ∈ InternalSets |] ==> X - Y ∈ InternalSets
lemma NatStar_SHNat_subset:
Nats ⊆ *s* UNIV
lemma NatStar_hypreal_of_real_Int:
*s* X ∩ Nats = star_of ` X
lemma starset_starset_n_eq:
*s* X = *sn* (%n. X)
lemma InternalSets_starset_n:
*s* X ∈ InternalSets
lemma InternalSets_UNIV_diff:
X ∈ InternalSets ==> UNIV - X ∈ InternalSets
lemma starfun_le_mono:
∀n≥N. f n ≤ g n ==> ∀n≥star_of N. (*f* f) n ≤ (*f* g) n
lemma starfun_less_mono:
∀n≥N. f n < g n ==> ∀n≥star_of N. (*f* f) n < (*f* g) n
lemma starfun_shift_one:
(*f* (%n. f (Suc n))) N = (*f* f) (N + 1)
lemma starfun_abs:
(*f* (%n. ¦f n¦)) N = ¦(*f* f) N¦
lemma starfun_pow:
(*f* op ^ r) N = star_of r pow N
lemma starfun_pow2:
(*f* (%n. X n ^ m)) N = (*f* X) N pow star_of m
lemma starfun_pow3:
(*f* (%r. r ^ n)) R = R pow star_of n
lemma starfunNat_real_of_nat:
*f* real = hypreal_of_hypnat
lemma starfun_inverse_real_of_nat_eq:
N ∈ HNatInfinite ==> (*f* (%x. inverse (real x))) N = inverse (hypreal_of_hypnat N)
lemma starfun_n:
(*fn* f) (star_n X) = star_n (%n. f n (X n))
lemma starfun_n_mult:
(*fn* f) z * (*fn* g) z = (*fn* (%i x. f i x * g i x)) z
lemma starfun_n_add:
(*fn* f) z + (*fn* g) z = (*fn* (%i x. f i x + g i x)) z
lemma starfun_n_add_minus:
(*fn* f) z + - (*fn* g) z = (*fn* (%i x. f i x + - g i x)) z
lemma starfun_n_const_fun:
(*fn* (%i x. k)) z = star_of k
lemma starfun_n_minus:
- (*fn* f) x = (*fn* (%i x. - f i x)) x
lemma starfun_n_eq:
(*fn* f) (star_of n) = star_n (%i. f i n)
lemma starfun_eq_iff:
(*f* f = *f* g) = (f = g)
lemma starfunNat_inverse_real_of_nat_Infinitesimal:
N ∈ HNatInfinite ==> (*f* (%x. inverse (real x))) N ∈ Infinitesimal
lemma starP:
(*p* P) (star_n X) = ({n. P (X n)} ∈ \<U>)
lemma hypnat_induct_obj:
(*p* P) 0 ∧ (∀n. (*p* P) n --> (*p* P) (n + 1)) --> (*p* P) n
lemma hypnat_induct:
[| (*p* P) 0; !!n. (*p* P) n ==> (*p* P) (n + 1) |] ==> (*p* P) n
lemma starP2:
(*p2* P) (star_n X) (star_n Y) = ({n. P (X n) (Y n)} ∈ \<U>)
lemma starP2_eq_iff:
*p2* op = = op =
lemma starP2_eq_iff2:
(*p2* op =) X Y = (X = Y)
lemma hSuc_not_zero:
hSuc m ≠ 0
lemmas zero_not_hSuc:
0 ≠ hSuc m
lemmas zero_not_hSuc:
0 ≠ hSuc m
lemma hSuc_hSuc_eq:
(hSuc m = hSuc n) = (m = n)
lemma nonempty_nat_set_Least_mem:
c ∈ S ==> (LEAST n. n ∈ S) ∈ S
lemma nonempty_set_star_has_least:
Iset S ≠ {} ==> ∃n∈Iset S. ∀m∈Iset S. n ≤ m
lemma nonempty_InternalNatSet_has_least:
[| S ∈ InternalSets; S ≠ {} |] ==> ∃n∈S. ∀m∈S. n ≤ m
lemma internal_induct_lemma:
[| 0 ∈ Iset X; ∀n. n ∈ Iset X --> n + 1 ∈ Iset X |] ==> Iset X = UNIV
lemma internal_induct:
[| X ∈ InternalSets; 0 ∈ X; ∀n. n ∈ X --> n + 1 ∈ X |] ==> X = UNIV