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theory StarDef(* Title : HOL/Hyperreal/StarDef.thy ID : $Id: StarDef.thy,v 1.14 2008/05/07 08:57:26 berghofe Exp $ Author : Jacques D. Fleuriot and Brian Huffman *) header {* Construction of Star Types Using Ultrafilters *} theory StarDef imports Filter uses ("transfer.ML") begin subsection {* A Free Ultrafilter over the Naturals *} definition FreeUltrafilterNat :: "nat set set" ("\<U>") where "\<U> = (SOME U. freeultrafilter U)" lemma freeultrafilter_FreeUltrafilterNat: "freeultrafilter \<U>" apply (unfold FreeUltrafilterNat_def) apply (rule someI_ex [where P=freeultrafilter]) apply (rule freeultrafilter_Ex) apply (rule nat_infinite) done interpretation FreeUltrafilterNat: freeultrafilter [FreeUltrafilterNat] by (rule freeultrafilter_FreeUltrafilterNat) text {* This rule takes the place of the old ultra tactic *} lemma ultra: "[|{n. P n} ∈ \<U>; {n. P n --> Q n} ∈ \<U>|] ==> {n. Q n} ∈ \<U>" by (simp add: Collect_imp_eq FreeUltrafilterNat.Un_iff FreeUltrafilterNat.Compl_iff) subsection {* Definition of @{text star} type constructor *} definition starrel :: "((nat => 'a) × (nat => 'a)) set" where "starrel = {(X,Y). {n. X n = Y n} ∈ \<U>}" typedef 'a star = "(UNIV :: (nat => 'a) set) // starrel" by (auto intro: quotientI) definition star_n :: "(nat => 'a) => 'a star" where "star_n X = Abs_star (starrel `` {X})" theorem star_cases [case_names star_n, cases type: star]: "(!!X. x = star_n X ==> P) ==> P" by (cases x, unfold star_n_def star_def, erule quotientE, fast) lemma all_star_eq: "(∀x. P x) = (∀X. P (star_n X))" by (auto, rule_tac x=x in star_cases, simp) lemma ex_star_eq: "(∃x. P x) = (∃X. P (star_n X))" by (auto, rule_tac x=x in star_cases, auto) text {* Proving that @{term starrel} is an equivalence relation *} lemma starrel_iff [iff]: "((X,Y) ∈ starrel) = ({n. X n = Y n} ∈ \<U>)" by (simp add: starrel_def) lemma equiv_starrel: "equiv UNIV starrel" proof (rule equiv.intro) show "reflexive starrel" by (simp add: refl_def) show "sym starrel" by (simp add: sym_def eq_commute) show "trans starrel" by (auto intro: transI elim!: ultra) qed lemmas equiv_starrel_iff = eq_equiv_class_iff [OF equiv_starrel UNIV_I UNIV_I] lemma starrel_in_star: "starrel``{x} ∈ star" by (simp add: star_def quotientI) lemma star_n_eq_iff: "(star_n X = star_n Y) = ({n. X n = Y n} ∈ \<U>)" by (simp add: star_n_def Abs_star_inject starrel_in_star equiv_starrel_iff) subsection {* Transfer principle *} text {* This introduction rule starts each transfer proof. *} lemma transfer_start: "P ≡ {n. Q} ∈ \<U> ==> Trueprop P ≡ Trueprop Q" by (subgoal_tac "P ≡ Q", simp, simp add: atomize_eq) text {*Initialize transfer tactic.*} use "transfer.ML" setup Transfer.setup text {* Transfer introduction rules. *} lemma transfer_ex [transfer_intro]: "[|!!X. p (star_n X) ≡ {n. P n (X n)} ∈ \<U>|] ==> ∃x::'a star. p x ≡ {n. ∃x. P n x} ∈ \<U>" by (simp only: ex_star_eq FreeUltrafilterNat.Collect_ex) lemma transfer_all [transfer_intro]: "[|!!X. p (star_n X) ≡ {n. P n (X n)} ∈ \<U>|] ==> ∀x::'a star. p x ≡ {n. ∀x. P n x} ∈ \<U>" by (simp only: all_star_eq FreeUltrafilterNat.Collect_all) lemma transfer_not [transfer_intro]: "[|p ≡ {n. P n} ∈ \<U>|] ==> ¬ p ≡ {n. ¬ P n} ∈ \<U>" by (simp only: FreeUltrafilterNat.Collect_not) lemma transfer_conj [transfer_intro]: "[|p ≡ {n. P n} ∈ \<U>; q ≡ {n. Q n} ∈ \<U>|] ==> p ∧ q ≡ {n. P n ∧ Q n} ∈ \<U>" by (simp only: FreeUltrafilterNat.Collect_conj) lemma transfer_disj [transfer_intro]: "[|p ≡ {n. P n} ∈ \<U>; q ≡ {n. Q n} ∈ \<U>|] ==> p ∨ q ≡ {n. P n ∨ Q n} ∈ \<U>" by (simp only: FreeUltrafilterNat.Collect_disj) lemma transfer_imp [transfer_intro]: "[|p ≡ {n. P n} ∈ \<U>; q ≡ {n. Q n} ∈ \<U>|] ==> p --> q ≡ {n. P n --> Q n} ∈ \<U>" by (simp only: imp_conv_disj transfer_disj transfer_not) lemma transfer_iff [transfer_intro]: "[|p ≡ {n. P n} ∈ \<U>; q ≡ {n. Q n} ∈ \<U>|] ==> p = q ≡ {n. P n = Q n} ∈ \<U>" by (simp only: iff_conv_conj_imp transfer_conj transfer_imp) lemma transfer_if_bool [transfer_intro]: "[|p ≡ {n. P n} ∈ \<U>; x ≡ {n. X n} ∈ \<U>; y ≡ {n. Y n} ∈ \<U>|] ==> (if p then x else y) ≡ {n. if P n then X n else Y n} ∈ \<U>" by (simp only: if_bool_eq_conj transfer_conj transfer_imp transfer_not) lemma transfer_eq [transfer_intro]: "[|x ≡ star_n X; y ≡ star_n Y|] ==> x = y ≡ {n. X n = Y n} ∈ \<U>" by (simp only: star_n_eq_iff) lemma transfer_if [transfer_intro]: "[|p ≡ {n. P n} ∈ \<U>; x ≡ star_n X; y ≡ star_n Y|] ==> (if p then x else y) ≡ star_n (λn. if P n then X n else Y n)" apply (rule eq_reflection) apply (auto simp add: star_n_eq_iff transfer_not elim!: ultra) done lemma transfer_fun_eq [transfer_intro]: "[|!!X. f (star_n X) = g (star_n X) ≡ {n. F n (X n) = G n (X n)} ∈ \<U>|] ==> f = g ≡ {n. F n = G n} ∈ \<U>" by (simp only: expand_fun_eq transfer_all) lemma transfer_star_n [transfer_intro]: "star_n X ≡ star_n (λn. X n)" by (rule reflexive) lemma transfer_bool [transfer_intro]: "p ≡ {n. p} ∈ \<U>" by (simp add: atomize_eq) subsection {* Standard elements *} definition star_of :: "'a => 'a star" where "star_of x == star_n (λn. x)" definition Standard :: "'a star set" where "Standard = range star_of" text {* Transfer tactic should remove occurrences of @{term star_of} *} setup {* Transfer.add_const "StarDef.star_of" *} declare star_of_def [transfer_intro] lemma star_of_inject: "(star_of x = star_of y) = (x = y)" by (transfer, rule refl) lemma Standard_star_of [simp]: "star_of x ∈ Standard" by (simp add: Standard_def) subsection {* Internal functions *} definition Ifun :: "('a => 'b) star => 'a star => 'b star" ("_ ∗ _" [300,301] 300) where "Ifun f ≡ λx. Abs_star (\<Union>F∈Rep_star f. \<Union>X∈Rep_star x. starrel``{λn. F n (X n)})" lemma Ifun_congruent2: "congruent2 starrel starrel (λF X. starrel``{λn. F n (X n)})" by (auto simp add: congruent2_def equiv_starrel_iff elim!: ultra) lemma Ifun_star_n: "star_n F ∗ star_n X = star_n (λn. F n (X n))" by (simp add: Ifun_def star_n_def Abs_star_inverse starrel_in_star UN_equiv_class2 [OF equiv_starrel equiv_starrel Ifun_congruent2]) text {* Transfer tactic should remove occurrences of @{term Ifun} *} setup {* Transfer.add_const "StarDef.Ifun" *} lemma transfer_Ifun [transfer_intro]: "[|f ≡ star_n F; x ≡ star_n X|] ==> f ∗ x ≡ star_n (λn. F n (X n))" by (simp only: Ifun_star_n) lemma Ifun_star_of [simp]: "star_of f ∗ star_of x = star_of (f x)" by (transfer, rule refl) lemma Standard_Ifun [simp]: "[|f ∈ Standard; x ∈ Standard|] ==> f ∗ x ∈ Standard" by (auto simp add: Standard_def) text {* Nonstandard extensions of functions *} definition starfun :: "('a => 'b) => ('a star => 'b star)" ("*f* _" [80] 80) where "starfun f == λx. star_of f ∗ x" definition starfun2 :: "('a => 'b => 'c) => ('a star => 'b star => 'c star)" ("*f2* _" [80] 80) where "starfun2 f == λx y. star_of f ∗ x ∗ y" declare starfun_def [transfer_unfold] declare starfun2_def [transfer_unfold] lemma starfun_star_n: "( *f* f) (star_n X) = star_n (λn. f (X n))" by (simp only: starfun_def star_of_def Ifun_star_n) lemma starfun2_star_n: "( *f2* f) (star_n X) (star_n Y) = star_n (λn. f (X n) (Y n))" by (simp only: starfun2_def star_of_def Ifun_star_n) lemma starfun_star_of [simp]: "( *f* f) (star_of x) = star_of (f x)" by (transfer, rule refl) lemma starfun2_star_of [simp]: "( *f2* f) (star_of x) = *f* f x" by (transfer, rule refl) lemma Standard_starfun [simp]: "x ∈ Standard ==> starfun f x ∈ Standard" by (simp add: starfun_def) lemma Standard_starfun2 [simp]: "[|x ∈ Standard; y ∈ Standard|] ==> starfun2 f x y ∈ Standard" by (simp add: starfun2_def) lemma Standard_starfun_iff: assumes inj: "!!x y. f x = f y ==> x = y" shows "(starfun f x ∈ Standard) = (x ∈ Standard)" proof assume "x ∈ Standard" thus "starfun f x ∈ Standard" by simp next have inj': "!!x y. starfun f x = starfun f y ==> x = y" using inj by transfer assume "starfun f x ∈ Standard" then obtain b where b: "starfun f x = star_of b" unfolding Standard_def .. hence "∃x. starfun f x = star_of b" .. hence "∃a. f a = b" by transfer then obtain a where "f a = b" .. hence "starfun f (star_of a) = star_of b" by transfer with b have "starfun f x = starfun f (star_of a)" by simp hence "x = star_of a" by (rule inj') thus "x ∈ Standard" unfolding Standard_def by auto qed lemma Standard_starfun2_iff: assumes inj: "!!a b a' b'. f a b = f a' b' ==> a = a' ∧ b = b'" shows "(starfun2 f x y ∈ Standard) = (x ∈ Standard ∧ y ∈ Standard)" proof assume "x ∈ Standard ∧ y ∈ Standard" thus "starfun2 f x y ∈ Standard" by simp next have inj': "!!x y z w. starfun2 f x y = starfun2 f z w ==> x = z ∧ y = w" using inj by transfer assume "starfun2 f x y ∈ Standard" then obtain c where c: "starfun2 f x y = star_of c" unfolding Standard_def .. hence "∃x y. starfun2 f x y = star_of c" by auto hence "∃a b. f a b = c" by transfer then obtain a b where "f a b = c" by auto hence "starfun2 f (star_of a) (star_of b) = star_of c" by transfer with c have "starfun2 f x y = starfun2 f (star_of a) (star_of b)" by simp hence "x = star_of a ∧ y = star_of b" by (rule inj') thus "x ∈ Standard ∧ y ∈ Standard" unfolding Standard_def by auto qed subsection {* Internal predicates *} definition unstar :: "bool star => bool" where "unstar b = (b = star_of True)" lemma unstar_star_n: "unstar (star_n P) = ({n. P n} ∈ \<U>)" by (simp add: unstar_def star_of_def star_n_eq_iff) lemma unstar_star_of [simp]: "unstar (star_of p) = p" by (simp add: unstar_def star_of_inject) text {* Transfer tactic should remove occurrences of @{term unstar} *} setup {* Transfer.add_const "StarDef.unstar" *} lemma transfer_unstar [transfer_intro]: "p ≡ star_n P ==> unstar p ≡ {n. P n} ∈ \<U>" by (simp only: unstar_star_n) definition starP :: "('a => bool) => 'a star => bool" ("*p* _" [80] 80) where "*p* P = (λx. unstar (star_of P ∗ x))" definition starP2 :: "('a => 'b => bool) => 'a star => 'b star => bool" ("*p2* _" [80] 80) where "*p2* P = (λx y. unstar (star_of P ∗ x ∗ y))" declare starP_def [transfer_unfold] declare starP2_def [transfer_unfold] lemma starP_star_n: "( *p* P) (star_n X) = ({n. P (X n)} ∈ \<U>)" by (simp only: starP_def star_of_def Ifun_star_n unstar_star_n) lemma starP2_star_n: "( *p2* P) (star_n X) (star_n Y) = ({n. P (X n) (Y n)} ∈ \<U>)" by (simp only: starP2_def star_of_def Ifun_star_n unstar_star_n) lemma starP_star_of [simp]: "( *p* P) (star_of x) = P x" by (transfer, rule refl) lemma starP2_star_of [simp]: "( *p2* P) (star_of x) = *p* P x" by (transfer, rule refl) subsection {* Internal sets *} definition Iset :: "'a set star => 'a star set" where "Iset A = {x. ( *p2* op ∈) x A}" lemma Iset_star_n: "(star_n X ∈ Iset (star_n A)) = ({n. X n ∈ A n} ∈ \<U>)" by (simp add: Iset_def starP2_star_n) text {* Transfer tactic should remove occurrences of @{term Iset} *} setup {* Transfer.add_const "StarDef.Iset" *} lemma transfer_mem [transfer_intro]: "[|x ≡ star_n X; a ≡ Iset (star_n A)|] ==> x ∈ a ≡ {n. X n ∈ A n} ∈ \<U>" by (simp only: Iset_star_n) lemma transfer_Collect [transfer_intro]: "[|!!X. p (star_n X) ≡ {n. P n (X n)} ∈ \<U>|] ==> Collect p ≡ Iset (star_n (λn. Collect (P n)))" by (simp add: atomize_eq expand_set_eq all_star_eq Iset_star_n) lemma transfer_set_eq [transfer_intro]: "[|a ≡ Iset (star_n A); b ≡ Iset (star_n B)|] ==> a = b ≡ {n. A n = B n} ∈ \<U>" by (simp only: expand_set_eq transfer_all transfer_iff transfer_mem) lemma transfer_ball [transfer_intro]: "[|a ≡ Iset (star_n A); !!X. p (star_n X) ≡ {n. P n (X n)} ∈ \<U>|] ==> ∀x∈a. p x ≡ {n. ∀x∈A n. P n x} ∈ \<U>" by (simp only: Ball_def transfer_all transfer_imp transfer_mem) lemma transfer_bex [transfer_intro]: "[|a ≡ Iset (star_n A); !!X. p (star_n X) ≡ {n. P n (X n)} ∈ \<U>|] ==> ∃x∈a. p x ≡ {n. ∃x∈A n. P n x} ∈ \<U>" by (simp only: Bex_def transfer_ex transfer_conj transfer_mem) lemma transfer_Iset [transfer_intro]: "[|a ≡ star_n A|] ==> Iset a ≡ Iset (star_n (λn. A n))" by simp text {* Nonstandard extensions of sets. *} definition starset :: "'a set => 'a star set" ("*s* _" [80] 80) where "starset A = Iset (star_of A)" declare starset_def [transfer_unfold] lemma starset_mem: "(star_of x ∈ *s* A) = (x ∈ A)" by (transfer, rule refl) lemma starset_UNIV: "*s* (UNIV::'a set) = (UNIV::'a star set)" by (transfer UNIV_def, rule refl) lemma starset_empty: "*s* {} = {}" by (transfer empty_def, rule refl) lemma starset_insert: "*s* (insert x A) = insert (star_of x) ( *s* A)" by (transfer insert_def Un_def, rule refl) lemma starset_Un: "*s* (A ∪ B) = *s* A ∪ *s* B" by (transfer Un_def, rule refl) lemma starset_Int: "*s* (A ∩ B) = *s* A ∩ *s* B" by (transfer Int_def, rule refl) lemma starset_Compl: "*s* -A = -( *s* A)" by (transfer Compl_eq, rule refl) lemma starset_diff: "*s* (A - B) = *s* A - *s* B" by (transfer set_diff_eq, rule refl) lemma starset_image: "*s* (f ` A) = ( *f* f) ` ( *s* A)" by (transfer image_def, rule refl) lemma starset_vimage: "*s* (f -` A) = ( *f* f) -` ( *s* A)" by (transfer vimage_def, rule refl) lemma starset_subset: "( *s* A ⊆ *s* B) = (A ⊆ B)" by (transfer subset_eq, rule refl) lemma starset_eq: "( *s* A = *s* B) = (A = B)" by (transfer, rule refl) lemmas starset_simps [simp] = starset_mem starset_UNIV starset_empty starset_insert starset_Un starset_Int starset_Compl starset_diff starset_image starset_vimage starset_subset starset_eq subsection {* Syntactic classes *} instantiation star :: (zero) zero begin definition star_zero_def: "0 ≡ star_of 0" instance .. end instantiation star :: (one) one begin definition star_one_def: "1 ≡ star_of 1" instance .. end instantiation star :: (plus) plus begin definition star_add_def: "(op +) ≡ *f2* (op +)" instance .. end instantiation star :: (times) times begin definition star_mult_def: "(op *) ≡ *f2* (op *)" instance .. end instantiation star :: (uminus) uminus begin definition star_minus_def: "uminus ≡ *f* uminus" instance .. end instantiation star :: (minus) minus begin definition star_diff_def: "(op -) ≡ *f2* (op -)" instance .. end instantiation star :: (abs) abs begin definition star_abs_def: "abs ≡ *f* abs" instance .. end instantiation star :: (sgn) sgn begin definition star_sgn_def: "sgn ≡ *f* sgn" instance .. end instantiation star :: (inverse) inverse begin definition star_divide_def: "(op /) ≡ *f2* (op /)" definition star_inverse_def: "inverse ≡ *f* inverse" instance .. end instantiation star :: (number) number begin definition star_number_def: "number_of b ≡ star_of (number_of b)" instance .. end instantiation star :: (Divides.div) Divides.div begin definition star_div_def: "(op div) ≡ *f2* (op div)" definition star_mod_def: "(op mod) ≡ *f2* (op mod)" instance .. end instantiation star :: (power) power begin definition star_power_def: "(op ^) ≡ λx n. ( *f* (λx. x ^ n)) x" instance .. end instantiation star :: (ord) ord begin definition star_le_def: "(op ≤) ≡ *p2* (op ≤)" definition star_less_def: "(op <) ≡ *p2* (op <)" instance .. end lemmas star_class_defs [transfer_unfold] = star_zero_def star_one_def star_number_def star_add_def star_diff_def star_minus_def star_mult_def star_divide_def star_inverse_def star_le_def star_less_def star_abs_def star_sgn_def star_div_def star_mod_def star_power_def text {* Class operations preserve standard elements *} lemma Standard_zero: "0 ∈ Standard" by (simp add: star_zero_def) lemma Standard_one: "1 ∈ Standard" by (simp add: star_one_def) lemma Standard_number_of: "number_of b ∈ Standard" by (simp add: star_number_def) lemma Standard_add: "[|x ∈ Standard; y ∈ Standard|] ==> x + y ∈ Standard" by (simp add: star_add_def) lemma Standard_diff: "[|x ∈ Standard; y ∈ Standard|] ==> x - y ∈ Standard" by (simp add: star_diff_def) lemma Standard_minus: "x ∈ Standard ==> - x ∈ Standard" by (simp add: star_minus_def) lemma Standard_mult: "[|x ∈ Standard; y ∈ Standard|] ==> x * y ∈ Standard" by (simp add: star_mult_def) lemma Standard_divide: "[|x ∈ Standard; y ∈ Standard|] ==> x / y ∈ Standard" by (simp add: star_divide_def) lemma Standard_inverse: "x ∈ Standard ==> inverse x ∈ Standard" by (simp add: star_inverse_def) lemma Standard_abs: "x ∈ Standard ==> abs x ∈ Standard" by (simp add: star_abs_def) lemma Standard_div: "[|x ∈ Standard; y ∈ Standard|] ==> x div y ∈ Standard" by (simp add: star_div_def) lemma Standard_mod: "[|x ∈ Standard; y ∈ Standard|] ==> x mod y ∈ Standard" by (simp add: star_mod_def) lemma Standard_power: "x ∈ Standard ==> x ^ n ∈ Standard" by (simp add: star_power_def) lemmas Standard_simps [simp] = Standard_zero Standard_one Standard_number_of Standard_add Standard_diff Standard_minus Standard_mult Standard_divide Standard_inverse Standard_abs Standard_div Standard_mod Standard_power text {* @{term star_of} preserves class operations *} lemma star_of_add: "star_of (x + y) = star_of x + star_of y" by transfer (rule refl) lemma star_of_diff: "star_of (x - y) = star_of x - star_of y" by transfer (rule refl) lemma star_of_minus: "star_of (-x) = - star_of x" by transfer (rule refl) lemma star_of_mult: "star_of (x * y) = star_of x * star_of y" by transfer (rule refl) lemma star_of_divide: "star_of (x / y) = star_of x / star_of y" by transfer (rule refl) lemma star_of_inverse: "star_of (inverse x) = inverse (star_of x)" by transfer (rule refl) lemma star_of_div: "star_of (x div y) = star_of x div star_of y" by transfer (rule refl) lemma star_of_mod: "star_of (x mod y) = star_of x mod star_of y" by transfer (rule refl) lemma star_of_power: "star_of (x ^ n) = star_of x ^ n" by transfer (rule refl) lemma star_of_abs: "star_of (abs x) = abs (star_of x)" by transfer (rule refl) text {* @{term star_of} preserves numerals *} lemma star_of_zero: "star_of 0 = 0" by transfer (rule refl) lemma star_of_one: "star_of 1 = 1" by transfer (rule refl) lemma star_of_number_of: "star_of (number_of x) = number_of x" by transfer (rule refl) text {* @{term star_of} preserves orderings *} lemma star_of_less: "(star_of x < star_of y) = (x < y)" by transfer (rule refl) lemma star_of_le: "(star_of x ≤ star_of y) = (x ≤ y)" by transfer (rule refl) lemma star_of_eq: "(star_of x = star_of y) = (x = y)" by transfer (rule refl) text{*As above, for 0*} lemmas star_of_0_less = star_of_less [of 0, simplified star_of_zero] lemmas star_of_0_le = star_of_le [of 0, simplified star_of_zero] lemmas star_of_0_eq = star_of_eq [of 0, simplified star_of_zero] lemmas star_of_less_0 = star_of_less [of _ 0, simplified star_of_zero] lemmas star_of_le_0 = star_of_le [of _ 0, simplified star_of_zero] lemmas star_of_eq_0 = star_of_eq [of _ 0, simplified star_of_zero] text{*As above, for 1*} lemmas star_of_1_less = star_of_less [of 1, simplified star_of_one] lemmas star_of_1_le = star_of_le [of 1, simplified star_of_one] lemmas star_of_1_eq = star_of_eq [of 1, simplified star_of_one] lemmas star_of_less_1 = star_of_less [of _ 1, simplified star_of_one] lemmas star_of_le_1 = star_of_le [of _ 1, simplified star_of_one] lemmas star_of_eq_1 = star_of_eq [of _ 1, simplified star_of_one] text{*As above, for numerals*} lemmas star_of_number_less = star_of_less [of "number_of w", standard, simplified star_of_number_of] lemmas star_of_number_le = star_of_le [of "number_of w", standard, simplified star_of_number_of] lemmas star_of_number_eq = star_of_eq [of "number_of w", standard, simplified star_of_number_of] lemmas star_of_less_number = star_of_less [of _ "number_of w", standard, simplified star_of_number_of] lemmas star_of_le_number = star_of_le [of _ "number_of w", standard, simplified star_of_number_of] lemmas star_of_eq_number = star_of_eq [of _ "number_of w", standard, simplified star_of_number_of] lemmas star_of_simps [simp] = star_of_add star_of_diff star_of_minus star_of_mult star_of_divide star_of_inverse star_of_div star_of_mod star_of_power star_of_abs star_of_zero star_of_one star_of_number_of star_of_less star_of_le star_of_eq star_of_0_less star_of_0_le star_of_0_eq star_of_less_0 star_of_le_0 star_of_eq_0 star_of_1_less star_of_1_le star_of_1_eq star_of_less_1 star_of_le_1 star_of_eq_1 star_of_number_less star_of_number_le star_of_number_eq star_of_less_number star_of_le_number star_of_eq_number subsection {* Ordering and lattice classes *} instance star :: (order) order apply (intro_classes) apply (transfer, rule order_less_le) apply (transfer, rule order_refl) apply (transfer, erule (1) order_trans) apply (transfer, erule (1) order_antisym) done instantiation star :: (lower_semilattice) lower_semilattice begin definition star_inf_def [transfer_unfold]: "inf ≡ *f2* inf" instance by default (transfer star_inf_def, auto)+ end instantiation star :: (upper_semilattice) upper_semilattice begin definition star_sup_def [transfer_unfold]: "sup ≡ *f2* sup" instance by default (transfer star_sup_def, auto)+ end instance star :: (lattice) lattice .. instance star :: (distrib_lattice) distrib_lattice by default (transfer, auto simp add: sup_inf_distrib1) lemma Standard_inf [simp]: "[|x ∈ Standard; y ∈ Standard|] ==> inf x y ∈ Standard" by (simp add: star_inf_def) lemma Standard_sup [simp]: "[|x ∈ Standard; y ∈ Standard|] ==> sup x y ∈ Standard" by (simp add: star_sup_def) lemma star_of_inf [simp]: "star_of (inf x y) = inf (star_of x) (star_of y)" by transfer (rule refl) lemma star_of_sup [simp]: "star_of (sup x y) = sup (star_of x) (star_of y)" by transfer (rule refl) instance star :: (linorder) linorder by (intro_classes, transfer, rule linorder_linear) lemma star_max_def [transfer_unfold]: "max = *f2* max" apply (rule ext, rule ext) apply (unfold max_def, transfer, fold max_def) apply (rule refl) done lemma star_min_def [transfer_unfold]: "min = *f2* min" apply (rule ext, rule ext) apply (unfold min_def, transfer, fold min_def) apply (rule refl) done lemma Standard_max [simp]: "[|x ∈ Standard; y ∈ Standard|] ==> max x y ∈ Standard" by (simp add: star_max_def) lemma Standard_min [simp]: "[|x ∈ Standard; y ∈ Standard|] ==> min x y ∈ Standard" by (simp add: star_min_def) lemma star_of_max [simp]: "star_of (max x y) = max (star_of x) (star_of y)" by transfer (rule refl) lemma star_of_min [simp]: "star_of (min x y) = min (star_of x) (star_of y)" by transfer (rule refl) subsection {* Ordered group classes *} instance star :: (semigroup_add) semigroup_add by (intro_classes, transfer, rule add_assoc) instance star :: (ab_semigroup_add) ab_semigroup_add by (intro_classes, transfer, rule add_commute) instance star :: (semigroup_mult) semigroup_mult by (intro_classes, transfer, rule mult_assoc) instance star :: (ab_semigroup_mult) ab_semigroup_mult by (intro_classes, transfer, rule mult_commute) instance star :: (comm_monoid_add) comm_monoid_add by (intro_classes, transfer, rule comm_monoid_add_class.zero_plus.add_0) instance star :: (monoid_mult) monoid_mult apply (intro_classes) apply (transfer, rule mult_1_left) apply (transfer, rule mult_1_right) done instance star :: (comm_monoid_mult) comm_monoid_mult by (intro_classes, transfer, rule mult_1) instance star :: (cancel_semigroup_add) cancel_semigroup_add apply (intro_classes) apply (transfer, erule add_left_imp_eq) apply (transfer, erule add_right_imp_eq) done instance star :: (cancel_ab_semigroup_add) cancel_ab_semigroup_add by (intro_classes, transfer, rule add_imp_eq) instance star :: (ab_group_add) ab_group_add apply (intro_classes) apply (transfer, rule left_minus) apply (transfer, rule diff_minus) done instance star :: (pordered_ab_semigroup_add) pordered_ab_semigroup_add by (intro_classes, transfer, rule add_left_mono) instance star :: (pordered_cancel_ab_semigroup_add) pordered_cancel_ab_semigroup_add .. instance star :: (pordered_ab_semigroup_add_imp_le) pordered_ab_semigroup_add_imp_le by (intro_classes, transfer, rule add_le_imp_le_left) instance star :: (pordered_comm_monoid_add) pordered_comm_monoid_add .. instance star :: (pordered_ab_group_add) pordered_ab_group_add .. instance star :: (pordered_ab_group_add_abs) pordered_ab_group_add_abs by intro_classes (transfer, simp add: abs_ge_self abs_leI abs_triangle_ineq)+ instance star :: (ordered_cancel_ab_semigroup_add) ordered_cancel_ab_semigroup_add .. instance star :: (lordered_ab_group_add_meet) lordered_ab_group_add_meet .. instance star :: (lordered_ab_group_add_meet) lordered_ab_group_add_meet .. instance star :: (lordered_ab_group_add) lordered_ab_group_add .. instance star :: (lordered_ab_group_add_abs) lordered_ab_group_add_abs by (intro_classes, transfer, rule abs_lattice) subsection {* Ring and field classes *} instance star :: (semiring) semiring apply (intro_classes) apply (transfer, rule left_distrib) apply (transfer, rule right_distrib) done instance star :: (semiring_0) semiring_0 by intro_classes (transfer, simp)+ instance star :: (semiring_0_cancel) semiring_0_cancel .. instance star :: (comm_semiring) comm_semiring by (intro_classes, transfer, rule left_distrib) instance star :: (comm_semiring_0) comm_semiring_0 .. instance star :: (comm_semiring_0_cancel) comm_semiring_0_cancel .. instance star :: (zero_neq_one) zero_neq_one by (intro_classes, transfer, rule zero_neq_one) instance star :: (semiring_1) semiring_1 .. instance star :: (comm_semiring_1) comm_semiring_1 .. instance star :: (no_zero_divisors) no_zero_divisors by (intro_classes, transfer, rule no_zero_divisors) instance star :: (semiring_1_cancel) semiring_1_cancel .. instance star :: (comm_semiring_1_cancel) comm_semiring_1_cancel .. instance star :: (ring) ring .. instance star :: (comm_ring) comm_ring .. instance star :: (ring_1) ring_1 .. instance star :: (comm_ring_1) comm_ring_1 .. instance star :: (ring_no_zero_divisors) ring_no_zero_divisors .. instance star :: (ring_1_no_zero_divisors) ring_1_no_zero_divisors .. instance star :: (idom) idom .. instance star :: (division_ring) division_ring apply (intro_classes) apply (transfer, erule left_inverse) apply (transfer, erule right_inverse) done instance star :: (field) field apply (intro_classes) apply (transfer, erule left_inverse) apply (transfer, rule divide_inverse) done instance star :: (division_by_zero) division_by_zero by (intro_classes, transfer, rule inverse_zero) instance star :: (pordered_semiring) pordered_semiring apply (intro_classes) apply (transfer, erule (1) mult_left_mono) apply (transfer, erule (1) mult_right_mono) done instance star :: (pordered_cancel_semiring) pordered_cancel_semiring .. instance star :: (ordered_semiring_strict) ordered_semiring_strict apply (intro_classes) apply (transfer, erule (1) mult_strict_left_mono) apply (transfer, erule (1) mult_strict_right_mono) done instance star :: (pordered_comm_semiring) pordered_comm_semiring by (intro_classes, transfer, rule mult_mono1_class.less_eq_less_times_zero.mult_mono1) instance star :: (pordered_cancel_comm_semiring) pordered_cancel_comm_semiring .. instance star :: (ordered_comm_semiring_strict) ordered_comm_semiring_strict by (intro_classes, transfer, rule ordered_comm_semiring_strict_class.plus_less_eq_less_zero_times.mult_strict_left_mono_comm) instance star :: (pordered_ring) pordered_ring .. instance star :: (pordered_ring_abs) pordered_ring_abs by intro_classes (transfer, rule abs_eq_mult) instance star :: (lordered_ring) lordered_ring .. instance star :: (abs_if) abs_if by (intro_classes, transfer, rule abs_if) instance star :: (sgn_if) sgn_if by (intro_classes, transfer, rule sgn_if) instance star :: (ordered_ring_strict) ordered_ring_strict .. instance star :: (pordered_comm_ring) pordered_comm_ring .. instance star :: (ordered_semidom) ordered_semidom by (intro_classes, transfer, rule zero_less_one) instance star :: (ordered_idom) ordered_idom .. instance star :: (ordered_field) ordered_field .. subsection {* Power classes *} text {* Proving the class axiom @{thm [source] power_Suc} for type @{typ "'a star"} is a little tricky, because it quantifies over values of type @{typ nat}. The transfer principle does not handle quantification over non-star types in general, but we can work around this by fixing an arbitrary @{typ nat} value, and then applying the transfer principle. *} instance star :: (recpower) recpower proof show "!!a::'a star. a ^ 0 = 1" by transfer (rule power_0) next fix n show "!!a::'a star. a ^ Suc n = a * a ^ n" by transfer (rule power_Suc) qed subsection {* Number classes *} lemma star_of_nat_def [transfer_unfold]: "of_nat n = star_of (of_nat n)" by (induct n, simp_all) lemma Standard_of_nat [simp]: "of_nat n ∈ Standard" by (simp add: star_of_nat_def) lemma star_of_of_nat [simp]: "star_of (of_nat n) = of_nat n" by transfer (rule refl) lemma star_of_int_def [transfer_unfold]: "of_int z = star_of (of_int z)" by (rule_tac z=z in int_diff_cases, simp) lemma Standard_of_int [simp]: "of_int z ∈ Standard" by (simp add: star_of_int_def) lemma star_of_of_int [simp]: "star_of (of_int z) = of_int z" by transfer (rule refl) instance star :: (semiring_char_0) semiring_char_0 by intro_classes (simp only: star_of_nat_def star_of_eq of_nat_eq_iff) instance star :: (ring_char_0) ring_char_0 .. instance star :: (number_ring) number_ring by (intro_classes, simp only: star_number_def star_of_int_def number_of_eq) subsection {* Finite class *} lemma starset_finite: "finite A ==> *s* A = star_of ` A" by (erule finite_induct, simp_all) instance star :: (finite) finite apply (intro_classes) apply (subst starset_UNIV [symmetric]) apply (subst starset_finite [OF finite]) apply (rule finite_imageI [OF finite]) done end
lemma freeultrafilter_FreeUltrafilterNat:
freeultrafilter \<U>
lemma ultra:
[| {n. P n} ∈ \<U>; {n. P n --> Q n} ∈ \<U> |] ==> {n. Q n} ∈ \<U>
theorem star_cases:
(!!X. x = star_n X ==> P) ==> P
lemma all_star_eq:
(∀x. P x) = (∀X. P (star_n X))
lemma ex_star_eq:
(∃x. P x) = (∃X. P (star_n X))
lemma starrel_iff:
((X, Y) ∈ starrel) = ({n. X n = Y n} ∈ \<U>)
lemma equiv_starrel:
equiv UNIV starrel
lemma equiv_starrel_iff:
(starrel `` {x} = starrel `` {y}) = ((x, y) ∈ starrel)
lemma starrel_in_star:
starrel `` {x} ∈ star
lemma star_n_eq_iff:
(star_n X = star_n Y) = ({n. X n = Y n} ∈ \<U>)
lemma transfer_start:
P == {n. Q} ∈ \<U> ==> P == Q
lemma transfer_ex:
(!!X. p (star_n X) == {n. P n (X n)} ∈ \<U>)
==> ∃x. p x == {n. ∃x. P n x} ∈ \<U>
lemma transfer_all:
(!!X. p (star_n X) == {n. P n (X n)} ∈ \<U>)
==> ∀x. p x == {n. ∀x. P n x} ∈ \<U>
lemma transfer_not:
p == {n. P n} ∈ \<U> ==> ¬ p == {n. ¬ P n} ∈ \<U>
lemma transfer_conj:
[| p == {n. P n} ∈ \<U>; q == {n. Q n} ∈ \<U> |]
==> p ∧ q == {n. P n ∧ Q n} ∈ \<U>
lemma transfer_disj:
[| p == {n. P n} ∈ \<U>; q == {n. Q n} ∈ \<U> |]
==> p ∨ q == {n. P n ∨ Q n} ∈ \<U>
lemma transfer_imp:
[| p == {n. P n} ∈ \<U>; q == {n. Q n} ∈ \<U> |]
==> p --> q == {n. P n --> Q n} ∈ \<U>
lemma transfer_iff:
[| p == {n. P n} ∈ \<U>; q == {n. Q n} ∈ \<U> |]
==> p = q == {n. P n = Q n} ∈ \<U>
lemma transfer_if_bool:
[| p == {n. P n} ∈ \<U>; x == {n. X n} ∈ \<U>; y == {n. Y n} ∈ \<U> |]
==> if p then x else y == {n. if P n then X n else Y n} ∈ \<U>
lemma transfer_eq:
[| x == star_n X; y == star_n Y |] ==> x = y == {n. X n = Y n} ∈ \<U>
lemma transfer_if:
[| p == {n. P n} ∈ \<U>; x == star_n X; y == star_n Y |]
==> if p then x else y == star_n (λn. if P n then X n else Y n)
lemma transfer_fun_eq:
(!!X. f (star_n X) = g (star_n X) == {n. F n (X n) = G n (X n)} ∈ \<U>)
==> f = g == {n. F n = G n} ∈ \<U>
lemma transfer_star_n:
star_n X == star_n X
lemma transfer_bool:
p == {n. p} ∈ \<U>
lemma star_of_inject:
(star_of x = star_of y) = (x = y)
lemma Standard_star_of:
star_of x ∈ Standard
lemma Ifun_congruent2:
congruent2 starrel starrel (λF X. starrel `` {λn. F n (X n)})
lemma Ifun_star_n:
star_n F ∗ star_n X = star_n (λn. F n (X n))
lemma transfer_Ifun:
[| f == star_n F; x == star_n X |] ==> f ∗ x == star_n (λn. F n (X n))
lemma Ifun_star_of:
star_of f ∗ star_of x = star_of (f x)
lemma Standard_Ifun:
[| f ∈ Standard; x ∈ Standard |] ==> f ∗ x ∈ Standard
lemma starfun_star_n:
(*f* f) (star_n X) = star_n (λn. f (X n))
lemma starfun2_star_n:
(*f2* f) (star_n X) (star_n Y) = star_n (λn. f (X n) (Y n))
lemma starfun_star_of:
(*f* f) (star_of x) = star_of (f x)
lemma starfun2_star_of:
(*f2* f) (star_of x) = *f* f x
lemma Standard_starfun:
x ∈ Standard ==> (*f* f) x ∈ Standard
lemma Standard_starfun2:
[| x ∈ Standard; y ∈ Standard |] ==> (*f2* f) x y ∈ Standard
lemma Standard_starfun_iff:
(!!x y. f x = f y ==> x = y) ==> ((*f* f) x ∈ Standard) = (x ∈ Standard)
lemma Standard_starfun2_iff:
(!!a b a' b'. f a b = f a' b' ==> a = a' ∧ b = b')
==> ((*f2* f) x y ∈ Standard) = (x ∈ Standard ∧ y ∈ Standard)
lemma unstar_star_n:
unstar (star_n P) = ({n. P n} ∈ \<U>)
lemma unstar_star_of:
unstar (star_of p) = p
lemma transfer_unstar:
p == star_n P ==> unstar p == {n. P n} ∈ \<U>
lemma starP_star_n:
(*p* P) (star_n X) = ({n. P (X n)} ∈ \<U>)
lemma starP2_star_n:
(*p2* P) (star_n X) (star_n Y) = ({n. P (X n) (Y n)} ∈ \<U>)
lemma starP_star_of:
(*p* P) (star_of x) = P x
lemma starP2_star_of:
(*p2* P) (star_of x) = *p* P x
lemma Iset_star_n:
(star_n X ∈ Iset (star_n A)) = ({n. X n ∈ A n} ∈ \<U>)
lemma transfer_mem:
[| x == star_n X; a == Iset (star_n A) |] ==> x ∈ a == {n. X n ∈ A n} ∈ \<U>
lemma transfer_Collect:
(!!X. p (star_n X) == {n. P n (X n)} ∈ \<U>)
==> Collect p == Iset (star_n (λn. Collect (P n)))
lemma transfer_set_eq:
[| a == Iset (star_n A); b == Iset (star_n B) |]
==> a = b == {n. A n = B n} ∈ \<U>
lemma transfer_ball:
[| a == Iset (star_n A); !!X. p (star_n X) == {n. P n (X n)} ∈ \<U> |]
==> ∀x∈a. p x == {n. ∀x∈A n. P n x} ∈ \<U>
lemma transfer_bex:
[| a == Iset (star_n A); !!X. p (star_n X) == {n. P n (X n)} ∈ \<U> |]
==> ∃x∈a. p x == {n. ∃x∈A n. P n x} ∈ \<U>
lemma transfer_Iset:
a == star_n A ==> Iset a == Iset (star_n A)
lemma starset_mem:
(star_of x ∈ *s* A) = (x ∈ A)
lemma starset_UNIV:
*s* UNIV = UNIV
lemma starset_empty:
*s* {} = {}
lemma starset_insert:
*s* insert x A = insert (star_of x) (*s* A)
lemma starset_Un:
*s* (A ∪ B) = *s* A ∪ *s* B
lemma starset_Int:
*s* (A ∩ B) = *s* A ∩ *s* B
lemma starset_Compl:
*s* - A = - (*s* A)
lemma starset_diff:
*s* (A - B) = *s* A - *s* B
lemma starset_image:
*s* f ` A = (*f* f) ` (*s* A)
lemma starset_vimage:
*s* f -` A = (*f* f) -` (*s* A)
lemma starset_subset:
(*s* A ⊆ *s* B) = (A ⊆ B)
lemma starset_eq:
(*s* A = *s* B) = (A = B)
lemma starset_simps:
(star_of x ∈ *s* A) = (x ∈ A)
*s* UNIV = UNIV
*s* {} = {}
*s* insert x A = insert (star_of x) (*s* A)
*s* (A ∪ B) = *s* A ∪ *s* B
*s* (A ∩ B) = *s* A ∩ *s* B
*s* - A = - (*s* A)
*s* (A - B) = *s* A - *s* B
*s* f ` A = (*f* f) ` (*s* A)
*s* f -` A = (*f* f) -` (*s* A)
(*s* A ⊆ *s* B) = (A ⊆ B)
(*s* A = *s* B) = (A = B)
lemma star_class_defs:
0 == star_of (0::'a)
1 == star_of (1::'a)
number_of b == star_of (number_of b)
op + == *f2* op +
op - == *f2* op -
uminus == *f* uminus
op * == *f2* op *
op / == *f2* op /
inverse == *f* inverse
op ≤ == *p2* op ≤
op < == *p2* op <
abs == *f* abs
sgn == *f* sgn
op div == *f2* op div
op mod == *f2* op mod
op ^ == λx n. (*f* (λx. x ^ n)) x
lemma Standard_zero:
0 ∈ Standard
lemma Standard_one:
1 ∈ Standard
lemma Standard_number_of:
number_of b ∈ Standard
lemma Standard_add:
[| x ∈ Standard; y ∈ Standard |] ==> x + y ∈ Standard
lemma Standard_diff:
[| x ∈ Standard; y ∈ Standard |] ==> x - y ∈ Standard
lemma Standard_minus:
x ∈ Standard ==> - x ∈ Standard
lemma Standard_mult:
[| x ∈ Standard; y ∈ Standard |] ==> x * y ∈ Standard
lemma Standard_divide:
[| x ∈ Standard; y ∈ Standard |] ==> x / y ∈ Standard
lemma Standard_inverse:
x ∈ Standard ==> inverse x ∈ Standard
lemma Standard_abs:
x ∈ Standard ==> ¦x¦ ∈ Standard
lemma Standard_div:
[| x ∈ Standard; y ∈ Standard |] ==> x div y ∈ Standard
lemma Standard_mod:
[| x ∈ Standard; y ∈ Standard |] ==> x mod y ∈ Standard
lemma Standard_power:
x ∈ Standard ==> x ^ n ∈ Standard
lemma Standard_simps:
0 ∈ Standard
1 ∈ Standard
number_of b ∈ Standard
[| x ∈ Standard; y ∈ Standard |] ==> x + y ∈ Standard
[| x ∈ Standard; y ∈ Standard |] ==> x - y ∈ Standard
x ∈ Standard ==> - x ∈ Standard
[| x ∈ Standard; y ∈ Standard |] ==> x * y ∈ Standard
[| x ∈ Standard; y ∈ Standard |] ==> x / y ∈ Standard
x ∈ Standard ==> inverse x ∈ Standard
x ∈ Standard ==> ¦x¦ ∈ Standard
[| x ∈ Standard; y ∈ Standard |] ==> x div y ∈ Standard
[| x ∈ Standard; y ∈ Standard |] ==> x mod y ∈ Standard
x ∈ Standard ==> x ^ n ∈ Standard
lemma star_of_add:
star_of (x + y) = star_of x + star_of y
lemma star_of_diff:
star_of (x - y) = star_of x - star_of y
lemma star_of_minus:
star_of (- x) = - star_of x
lemma star_of_mult:
star_of (x * y) = star_of x * star_of y
lemma star_of_divide:
star_of (x / y) = star_of x / star_of y
lemma star_of_inverse:
star_of (inverse x) = inverse (star_of x)
lemma star_of_div:
star_of (x div y) = star_of x div star_of y
lemma star_of_mod:
star_of (x mod y) = star_of x mod star_of y
lemma star_of_power:
star_of (x ^ n) = star_of x ^ n
lemma star_of_abs:
star_of ¦x¦ = ¦star_of x¦
lemma star_of_zero:
star_of (0::'a) = 0
lemma star_of_one:
star_of (1::'a) = 1
lemma star_of_number_of:
star_of (number_of x) = number_of x
lemma star_of_less:
(star_of x < star_of y) = (x < y)
lemma star_of_le:
(star_of x ≤ star_of y) = (x ≤ y)
lemma star_of_eq:
(star_of x = star_of y) = (x = y)
lemma star_of_0_less:
(0 < star_of y) = ((0::'b1) < y)
lemma star_of_0_le:
(0 ≤ star_of y) = ((0::'b1) ≤ y)
lemma star_of_0_eq:
(0 = star_of y) = ((0::'b1) = y)
lemma star_of_less_0:
(star_of x < 0) = (x < (0::'b1))
lemma star_of_le_0:
(star_of x ≤ 0) = (x ≤ (0::'b1))
lemma star_of_eq_0:
(star_of x = 0) = (x = (0::'b1))
lemma star_of_1_less:
(1 < star_of y) = ((1::'b1) < y)
lemma star_of_1_le:
(1 ≤ star_of y) = ((1::'b1) ≤ y)
lemma star_of_1_eq:
(1 = star_of y) = ((1::'b1) = y)
lemma star_of_less_1:
(star_of x < 1) = (x < (1::'b1))
lemma star_of_le_1:
(star_of x ≤ 1) = (x ≤ (1::'b1))
lemma star_of_eq_1:
(star_of x = 1) = (x = (1::'b1))
lemma star_of_number_less:
(number_of w < star_of y) = (number_of w < y)
lemma star_of_number_le:
(number_of w ≤ star_of y) = (number_of w ≤ y)
lemma star_of_number_eq:
(number_of w = star_of y) = (number_of w = y)
lemma star_of_less_number:
(star_of x < number_of w) = (x < number_of w)
lemma star_of_le_number:
(star_of x ≤ number_of w) = (x ≤ number_of w)
lemma star_of_eq_number:
(star_of x = number_of w) = (x = number_of w)
lemma star_of_simps:
star_of (x + y) = star_of x + star_of y
star_of (x - y) = star_of x - star_of y
star_of (- x) = - star_of x
star_of (x * y) = star_of x * star_of y
star_of (x / y) = star_of x / star_of y
star_of (inverse x) = inverse (star_of x)
star_of (x div y) = star_of x div star_of y
star_of (x mod y) = star_of x mod star_of y
star_of (x ^ n) = star_of x ^ n
star_of ¦x¦ = ¦star_of x¦
star_of (0::'a) = 0
star_of (1::'a) = 1
star_of (number_of x) = number_of x
(star_of x < star_of y) = (x < y)
(star_of x ≤ star_of y) = (x ≤ y)
(star_of x = star_of y) = (x = y)
(0 < star_of y) = ((0::'b) < y)
(0 ≤ star_of y) = ((0::'b) ≤ y)
(0 = star_of y) = ((0::'b) = y)
(star_of x < 0) = (x < (0::'b))
(star_of x ≤ 0) = (x ≤ (0::'b))
(star_of x = 0) = (x = (0::'b))
(1 < star_of y) = ((1::'b) < y)
(1 ≤ star_of y) = ((1::'b) ≤ y)
(1 = star_of y) = ((1::'b) = y)
(star_of x < 1) = (x < (1::'b))
(star_of x ≤ 1) = (x ≤ (1::'b))
(star_of x = 1) = (x = (1::'b))
(number_of w < star_of y) = (number_of w < y)
(number_of w ≤ star_of y) = (number_of w ≤ y)
(number_of w = star_of y) = (number_of w = y)
(star_of x < number_of w) = (x < number_of w)
(star_of x ≤ number_of w) = (x ≤ number_of w)
(star_of x = number_of w) = (x = number_of w)
lemma Standard_inf:
[| x ∈ Standard; y ∈ Standard |] ==> inf x y ∈ Standard
lemma Standard_sup:
[| x ∈ Standard; y ∈ Standard |] ==> sup x y ∈ Standard
lemma star_of_inf:
star_of (inf x y) = inf (star_of x) (star_of y)
lemma star_of_sup:
star_of (sup x y) = sup (star_of x) (star_of y)
lemma star_max_def:
max = *f2* max
lemma star_min_def:
min = *f2* min
lemma Standard_max:
[| x ∈ Standard; y ∈ Standard |] ==> max x y ∈ Standard
lemma Standard_min:
[| x ∈ Standard; y ∈ Standard |] ==> min x y ∈ Standard
lemma star_of_max:
star_of (max x y) = max (star_of x) (star_of y)
lemma star_of_min:
star_of (min x y) = min (star_of x) (star_of y)
lemma star_of_nat_def:
of_nat n = star_of (of_nat n)
lemma Standard_of_nat:
of_nat n ∈ Standard
lemma star_of_of_nat:
star_of (of_nat n) = of_nat n
lemma star_of_int_def:
of_int z = star_of (of_int z)
lemma Standard_of_int:
of_int z ∈ Standard
lemma star_of_of_int:
star_of (of_int z) = of_int z
lemma starset_finite:
finite A ==> *s* A = star_of ` A