# Theory StarDef

theory StarDef
imports Free_Ultrafilter
```(*  Title       : HOL/Hyperreal/StarDef.thy
Author      : Jacques D. Fleuriot and Brian Huffman
*)

section ‹Construction of Star Types Using Ultrafilters›

theory StarDef
imports Free_Ultrafilter
begin

subsection ‹A Free Ultrafilter over the Naturals›

definition
FreeUltrafilterNat :: "nat filter"  ("𝒰") where
"𝒰 = (SOME U. freeultrafilter U)"

lemma freeultrafilter_FreeUltrafilterNat: "freeultrafilter 𝒰"
apply (unfold FreeUltrafilterNat_def)
apply (rule someI_ex)
apply (rule freeultrafilter_Ex)
apply (rule infinite_UNIV_nat)
done

interpretation FreeUltrafilterNat: freeultrafilter FreeUltrafilterNat
by (rule freeultrafilter_FreeUltrafilterNat)

subsection ‹Definition of ‹star› type constructor›

definition
starrel :: "((nat ⇒ 'a) × (nat ⇒ 'a)) set" where
"starrel = {(X,Y). eventually (λn. X n = Y n) 𝒰}"

definition "star = (UNIV :: (nat ⇒ 'a) set) // starrel"

typedef 'a star = "star :: (nat ⇒ 'a) set set"
unfolding star_def 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) = (eventually (λn. X n = Y n) 𝒰)"

lemma equiv_starrel: "equiv UNIV starrel"
proof (rule equivI)
show "refl starrel" by (simp add: refl_on_def)
show "sym starrel" by (simp add: sym_def eq_commute)
show "trans starrel" by (intro transI) (auto elim: eventually_elim2)
qed

lemmas equiv_starrel_iff =
eq_equiv_class_iff [OF equiv_starrel UNIV_I UNIV_I]

lemma starrel_in_star: "starrel``{x} ∈ star"

lemma star_n_eq_iff: "(star_n X = star_n Y) = (eventually (λn. 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 ≡ eventually (λn. Q) 𝒰 ⟹ Trueprop P ≡ Trueprop Q"

text ‹Initialize transfer tactic.›
ML_file "transfer.ML"

method_setup transfer = ‹
Attrib.thms >> (fn ths => fn ctxt =>
SIMPLE_METHOD' (Transfer_Principle.transfer_tac ctxt ths))
› "transfer principle"

text ‹Transfer introduction rules.›

lemma transfer_ex [transfer_intro]:
"⟦⋀X. p (star_n X) ≡ eventually (λn. P n (X n)) 𝒰⟧
⟹ ∃x::'a star. p x ≡ eventually (λn. ∃x. P n x) 𝒰"
by (simp only: ex_star_eq eventually_ex)

lemma transfer_all [transfer_intro]:
"⟦⋀X. p (star_n X) ≡ eventually (λn. P n (X n)) 𝒰⟧
⟹ ∀x::'a star. p x ≡ eventually (λn. ∀x. P n x) 𝒰"
by (simp only: all_star_eq FreeUltrafilterNat.eventually_all_iff)

lemma transfer_not [transfer_intro]:
"⟦p ≡ eventually P 𝒰⟧ ⟹ ¬ p ≡ eventually (λn. ¬ P n) 𝒰"
by (simp only: FreeUltrafilterNat.eventually_not_iff)

lemma transfer_conj [transfer_intro]:
"⟦p ≡ eventually P 𝒰; q ≡ eventually Q 𝒰⟧
⟹ p ∧ q ≡ eventually (λn. P n ∧ Q n) 𝒰"
by (simp only: eventually_conj_iff)

lemma transfer_disj [transfer_intro]:
"⟦p ≡ eventually P 𝒰; q ≡ eventually Q 𝒰⟧
⟹ p ∨ q ≡ eventually (λn. P n ∨ Q n) 𝒰"
by (simp only: FreeUltrafilterNat.eventually_disj_iff)

lemma transfer_imp [transfer_intro]:
"⟦p ≡ eventually P 𝒰; q ≡ eventually Q 𝒰⟧
⟹ p ⟶ q ≡ eventually (λn. P n ⟶ Q n) 𝒰"
by (simp only: FreeUltrafilterNat.eventually_imp_iff)

lemma transfer_iff [transfer_intro]:
"⟦p ≡ eventually P 𝒰; q ≡ eventually Q 𝒰⟧
⟹ p = q ≡ eventually (λn. P n = Q n) 𝒰"
by (simp only: FreeUltrafilterNat.eventually_iff_iff)

lemma transfer_if_bool [transfer_intro]:
"⟦p ≡ eventually P 𝒰; x ≡ eventually X 𝒰; y ≡ eventually Y 𝒰⟧
⟹ (if p then x else y) ≡ eventually (λn. 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 ≡ eventually (λn. X n = Y n) 𝒰"
by (simp only: star_n_eq_iff)

lemma transfer_if [transfer_intro]:
"⟦p ≡ eventually (λn. 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!: eventually_mono)
done

lemma transfer_fun_eq [transfer_intro]:
"⟦⋀X. f (star_n X) = g (star_n X)
≡ eventually (λn. F n (X n) = G n (X n)) 𝒰⟧
⟹ f = g ≡ eventually (λn. F n = G n) 𝒰"
by (simp only: fun_eq_iff 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 ≡ eventually (λn. p) 𝒰"

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}›

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"

subsection ‹Internal functions›

definition
Ifun :: "('a ⇒ 'b) star ⇒ 'a star ⇒ 'b star" ("_ ⋆ _" [300,301] 300) where
"Ifun f ≡ λx. Abs_star
(⋃F∈Rep_star f. ⋃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!: eventually_rev_mp)

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}›

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"

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"

lemma Standard_starfun2 [simp]:
"⟦x ∈ Standard; y ∈ Standard⟧ ⟹ starfun2 f x y ∈ Standard"

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) = (eventually P 𝒰)"
by (simp add: unstar_def star_of_def star_n_eq_iff)

lemma unstar_star_of [simp]: "unstar (star_of p) = p"

text ‹Transfer tactic should remove occurrences of @{term unstar}›

lemma transfer_unstar [transfer_intro]:
"p ≡ star_n P ⟹ unstar p ≡ eventually P 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) = (eventually (λn. 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) = (eventually (λn. 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)) = (eventually (λn. X n ∈ A n) 𝒰)"

text ‹Transfer tactic should remove occurrences of @{term Iset}›

lemma transfer_mem [transfer_intro]:
"⟦x ≡ star_n X; a ≡ Iset (star_n A)⟧
⟹ x ∈ a ≡ eventually (λn. X n ∈ A n) 𝒰"
by (simp only: Iset_star_n)

lemma transfer_Collect [transfer_intro]:
"⟦⋀X. p (star_n X) ≡ eventually (λn. P n (X n)) 𝒰⟧
⟹ Collect p ≡ Iset (star_n (λn. Collect (P n)))"
by (simp add: atomize_eq set_eq_iff all_star_eq Iset_star_n)

lemma transfer_set_eq [transfer_intro]:
"⟦a ≡ Iset (star_n A); b ≡ Iset (star_n B)⟧
⟹ a = b ≡ eventually (λn. A n = B n) 𝒰"
by (simp only: set_eq_iff transfer_all transfer_iff transfer_mem)

lemma transfer_ball [transfer_intro]:
"⟦a ≡ Iset (star_n A); ⋀X. p (star_n X) ≡ eventually (λn. P n (X n)) 𝒰⟧
⟹ ∀x∈a. p x ≡ eventually (λn. ∀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) ≡ eventually (λn. P n (X n)) 𝒰⟧
⟹ ∃x∈a. p x ≡ eventually (λn. ∃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 :: (divide) divide
begin

definition
star_divide_def:  "divide ≡ *f2* divide"

instance ..

end

instantiation star :: (inverse) inverse
begin

definition
star_inverse_def: "inverse ≡ *f* inverse"

instance ..

end

instance star :: (Rings.dvd) Rings.dvd ..

instantiation star :: (Divides.div) Divides.div
begin

definition
star_mod_def:     "(op mod) ≡ *f2* (op mod)"

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_mult_def     star_divide_def   star_inverse_def
star_le_def       star_less_def     star_abs_def       star_sgn_def
star_mod_def

text ‹Class operations preserve standard elements›

lemma Standard_zero: "0 ∈ Standard"

lemma Standard_one: "1 ∈ 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_mod: "⟦x ∈ Standard; y ∈ Standard⟧ ⟹ x mod y ∈ Standard"

lemmas Standard_simps [simp] =
Standard_zero  Standard_one
Standard_mult  Standard_divide  Standard_inverse
Standard_abs   Standard_mod

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_mod: "star_of (x mod y) = star_of x mod star_of y"
by transfer (rule refl)

lemma star_of_abs: "star_of ¦x¦ = ¦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)

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]

lemmas star_of_simps [simp] =
star_of_mult    star_of_divide  star_of_inverse
star_of_mod     star_of_abs
star_of_zero    star_of_one
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

subsection ‹Ordering and lattice classes›

instance star :: (order) order
apply (intro_classes)
apply (transfer, rule less_le_not_le)
apply (transfer, rule order_refl)
apply (transfer, erule (1) order_trans)
apply (transfer, erule (1) order_antisym)
done

instantiation star :: (semilattice_inf) semilattice_inf
begin

definition
star_inf_def [transfer_unfold]: "inf ≡ *f2* inf"

instance
by (standard; transfer) auto

end

instantiation star :: (semilattice_sup) semilattice_sup
begin

definition
star_sup_def [transfer_unfold]: "sup ≡ *f2* sup"

instance
by (standard; transfer) auto

end

instance star :: (lattice) lattice ..

instance star :: (distrib_lattice) distrib_lattice
by (standard; transfer) (auto simp add: sup_inf_distrib1)

lemma Standard_inf [simp]:
"⟦x ∈ Standard; y ∈ Standard⟧ ⟹ inf x y ∈ Standard"

lemma Standard_sup [simp]:
"⟦x ∈ Standard; y ∈ Standard⟧ ⟹ sup x y ∈ Standard"

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"

lemma Standard_min [simp]:
"⟦x ∈ Standard; y ∈ Standard⟧ ⟹ min x y ∈ Standard"

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_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 :: (monoid_mult) monoid_mult
apply (intro_classes)
apply (transfer, rule mult_1_left)
apply (transfer, rule mult_1_right)
done

instance star :: (power) power ..

instance star :: (comm_monoid_mult) comm_monoid_mult
by (intro_classes, transfer, rule mult_1)

apply (intro_classes)
done

by intro_classes (transfer, simp add: diff_diff_eq)+

apply (intro_classes)
apply (transfer, rule left_minus)
done

by intro_classes (transfer,

subsection ‹Ring and field classes›

instance star :: (semiring) semiring
by (intro_classes; transfer) (fact distrib_right distrib_left)+

instance star :: (semiring_0) semiring_0
by (intro_classes; transfer) simp_all

instance star :: (semiring_0_cancel) semiring_0_cancel ..

instance star :: (comm_semiring) comm_semiring
by (intro_classes; transfer) (fact distrib_right)

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) (fact zero_neq_one)

instance star :: (semiring_1) semiring_1 ..
instance star :: (comm_semiring_1) comm_semiring_1 ..

declare dvd_def [transfer_refold]

instance star :: (comm_semiring_1_cancel) comm_semiring_1_cancel
by (intro_classes; transfer) (fact right_diff_distrib')

instance star :: (semiring_no_zero_divisors) semiring_no_zero_divisors
by (intro_classes; transfer) (fact no_zero_divisors)

instance star :: (semiring_1_no_zero_divisors) semiring_1_no_zero_divisors ..

instance star :: (semiring_no_zero_divisors_cancel) semiring_no_zero_divisors_cancel
by (intro_classes; transfer) simp_all

instance star :: (semiring_1_cancel) 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 :: (semidom) semidom ..

instance star :: (semidom_divide) semidom_divide
by (intro_classes; transfer) simp_all

instance star :: (semiring_div) semiring_div
by (intro_classes; transfer) (simp_all add: mod_div_equality)

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 :: (idom_divide) idom_divide ..

instance star :: (division_ring) division_ring
by (intro_classes; transfer) (simp_all add: divide_inverse)

instance star :: (field) field
by (intro_classes; transfer) (simp_all add: divide_inverse)

instance star :: (ordered_semiring) ordered_semiring
by (intro_classes; transfer) (fact mult_left_mono mult_right_mono)+

instance star :: (ordered_cancel_semiring) ordered_cancel_semiring ..

instance star :: (linordered_semiring_strict) linordered_semiring_strict
by (intro_classes; transfer) (fact mult_strict_left_mono mult_strict_right_mono)+

instance star :: (ordered_comm_semiring) ordered_comm_semiring
by (intro_classes; transfer) (fact mult_left_mono)

instance star :: (ordered_cancel_comm_semiring) ordered_cancel_comm_semiring ..

instance star :: (linordered_comm_semiring_strict) linordered_comm_semiring_strict
by (intro_classes; transfer) (fact mult_strict_left_mono)

instance star :: (ordered_ring) ordered_ring ..

instance star :: (ordered_ring_abs) ordered_ring_abs
by (intro_classes; transfer) (fact abs_eq_mult)

instance star :: (abs_if) abs_if
by (intro_classes; transfer) (fact abs_if)

instance star :: (sgn_if) sgn_if
by (intro_classes; transfer) (fact sgn_if)

instance star :: (linordered_ring_strict) linordered_ring_strict ..
instance star :: (ordered_comm_ring) ordered_comm_ring ..

instance star :: (linordered_semidom) linordered_semidom
apply intro_classes
apply(transfer, fact zero_less_one)
done

instance star :: (linordered_idom) linordered_idom ..
instance star :: (linordered_field) linordered_field ..

subsection ‹Power›

lemma star_power_def [transfer_unfold]:
"(op ^) ≡ λx n. ( *f* (λx. x ^ n)) x"
proof (rule eq_reflection, rule ext, rule ext)
fix n :: nat
show "⋀x::'a star. x ^ n = ( *f* (λx. x ^ n)) x"
proof (induct n)
case 0
have "⋀x::'a star. ( *f* (λx. 1)) x = 1"
by transfer simp
then show ?case by simp
next
case (Suc n)
have "⋀x::'a star. x * ( *f* (λx::'a. x ^ n)) x = ( *f* (λx::'a. x * x ^ n)) x"
by transfer simp
with Suc show ?case by simp
qed
qed

lemma Standard_power [simp]: "x ∈ Standard ⟹ x ^ n ∈ Standard"

lemma star_of_power [simp]: "star_of (x ^ n) = star_of x ^ n"
by transfer (rule refl)

subsection ‹Number classes›

instance star :: (numeral) numeral ..

lemma star_numeral_def [transfer_unfold]:
"numeral k = star_of (numeral k)"
by (induct k, simp_all only: numeral.simps star_of_one star_of_add)

lemma Standard_numeral [simp]: "numeral k ∈ Standard"

lemma star_of_numeral [simp]: "star_of (numeral k) = numeral k"
by transfer (rule refl)

lemma star_of_nat_def [transfer_unfold]: "of_nat n = star_of (of_nat n)"
by (induct n, simp_all)

lemmas star_of_compare_numeral [simp] =
star_of_less [of "numeral k", simplified star_of_numeral]
star_of_le   [of "numeral k", simplified star_of_numeral]
star_of_eq   [of "numeral k", simplified star_of_numeral]
star_of_less [of _ "numeral k", simplified star_of_numeral]
star_of_le   [of _ "numeral k", simplified star_of_numeral]
star_of_eq   [of _ "numeral k", simplified star_of_numeral]
star_of_less [of "- numeral k", simplified star_of_numeral]
star_of_le   [of "- numeral k", simplified star_of_numeral]
star_of_eq   [of "- numeral k", simplified star_of_numeral]
star_of_less [of _ "- numeral k", simplified star_of_numeral]
star_of_le   [of _ "- numeral k", simplified star_of_numeral]
star_of_eq   [of _ "- numeral k", simplified star_of_numeral] for k

lemma Standard_of_nat [simp]: "of_nat n ∈ Standard"

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"

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 proof
have "inj (star_of :: 'a ⇒ 'a star)" by (rule injI) simp
then have "inj (star_of ∘ of_nat :: nat ⇒ 'a star)" using inj_of_nat by (rule inj_comp)
then show "inj (of_nat :: nat ⇒ 'a star)" by (simp add: comp_def)
qed

instance star :: (ring_char_0) ring_char_0 ..

instance star :: (semiring_parity) semiring_parity
apply intro_classes
apply(transfer, rule odd_one)
apply(transfer, erule even_multD)
apply(transfer, erule odd_ex_decrement)
done

instance star :: (semiring_div_parity) semiring_div_parity
apply intro_classes
apply(transfer, rule parity)
apply(transfer, rule one_mod_two_eq_one)
apply(transfer, rule zero_not_eq_two)
done

instantiation star :: (semiring_numeral_div) semiring_numeral_div
begin

definition divmod_star :: "num ⇒ num ⇒ 'a star × 'a star"
where
divmod_star_def: "divmod_star m n = (numeral m div numeral n, numeral m mod numeral n)"

definition divmod_step_star :: "num ⇒ 'a star × 'a star ⇒ 'a star × 'a star"
where
"divmod_step_star l qr = (let (q, r) = qr
in if r ≥ numeral l then (2 * q + 1, r - numeral l)
else (2 * q, r))"

instance proof
show "divmod m n = (numeral m div numeral n :: 'a star, numeral m mod numeral n)"
for m n by (fact divmod_star_def)
show "divmod_step l qr = (let (q, r) = qr
in if r ≥ numeral l then (2 * q + 1, r - numeral l)
else (2 * q, r))" for l and qr :: "'a star × 'a star"
by (fact divmod_step_star_def)
qed (transfer,
fact
semiring_numeral_div_class.div_less
semiring_numeral_div_class.mod_less
semiring_numeral_div_class.div_positive
semiring_numeral_div_class.mod_less_eq_dividend
semiring_numeral_div_class.pos_mod_bound
semiring_numeral_div_class.pos_mod_sign
semiring_numeral_div_class.mod_mult2_eq
semiring_numeral_div_class.div_mult2_eq
semiring_numeral_div_class.discrete)+

end

declare divmod_algorithm_code [where ?'a = "'a::semiring_numeral_div star", code]

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
```