(* Title: HOL/Nonstandard_Analysis/StarDef.thy
Author: Jacques D. Fleuriot and Brian Huffman
*)
section \<open>Construction of Star Types Using Ultrafilters\<close>
theory StarDef
imports Free_Ultrafilter
begin
subsection \<open>A Free Ultrafilter over the Naturals\<close>
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 \<open>Definition of \<open>star\<close> type constructor\<close>
definition
starrel :: "((nat \<Rightarrow> 'a) \<times> (nat \<Rightarrow> 'a)) set" where
"starrel = {(X,Y). eventually (\<lambda>n. X n = Y n) \<U>}"
definition "star = (UNIV :: (nat \<Rightarrow> 'a) set) // starrel"
typedef 'a star = "star :: (nat \<Rightarrow> 'a) set set"
unfolding star_def by (auto intro: quotientI)
definition
star_n :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a star" where
"star_n X = Abs_star (starrel `` {X})"
theorem star_cases [case_names star_n, cases type: star]:
"(\<And>X. x = star_n X \<Longrightarrow> P) \<Longrightarrow> P"
by (cases x, unfold star_n_def star_def, erule quotientE, fast)
lemma all_star_eq: "(\<forall>x. P x) = (\<forall>X. P (star_n X))"
by (auto, rule_tac x=x in star_cases, simp)
lemma ex_star_eq: "(\<exists>x. P x) = (\<exists>X. P (star_n X))"
by (auto, rule_tac x=x in star_cases, auto)
text \<open>Proving that @{term starrel} is an equivalence relation\<close>
lemma starrel_iff [iff]: "((X,Y) \<in> starrel) = (eventually (\<lambda>n. X n = Y n) \<U>)"
by (simp add: starrel_def)
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} \<in> star"
by (simp add: star_def quotientI)
lemma star_n_eq_iff: "(star_n X = star_n Y) = (eventually (\<lambda>n. X n = Y n) \<U>)"
by (simp add: star_n_def Abs_star_inject starrel_in_star equiv_starrel_iff)
subsection \<open>Transfer principle\<close>
text \<open>This introduction rule starts each transfer proof.\<close>
lemma transfer_start:
"P \<equiv> eventually (\<lambda>n. Q) \<U> \<Longrightarrow> Trueprop P \<equiv> Trueprop Q"
by (simp add: FreeUltrafilterNat.proper)
text \<open>Initialize transfer tactic.\<close>
ML_file "transfer.ML"
method_setup transfer = \<open>
Attrib.thms >> (fn ths => fn ctxt =>
SIMPLE_METHOD' (Transfer_Principle.transfer_tac ctxt ths))
\<close> "transfer principle"
text \<open>Transfer introduction rules.\<close>
lemma transfer_ex [transfer_intro]:
"\<lbrakk>\<And>X. p (star_n X) \<equiv> eventually (\<lambda>n. P n (X n)) \<U>\<rbrakk>
\<Longrightarrow> \<exists>x::'a star. p x \<equiv> eventually (\<lambda>n. \<exists>x. P n x) \<U>"
by (simp only: ex_star_eq eventually_ex)
lemma transfer_all [transfer_intro]:
"\<lbrakk>\<And>X. p (star_n X) \<equiv> eventually (\<lambda>n. P n (X n)) \<U>\<rbrakk>
\<Longrightarrow> \<forall>x::'a star. p x \<equiv> eventually (\<lambda>n. \<forall>x. P n x) \<U>"
by (simp only: all_star_eq FreeUltrafilterNat.eventually_all_iff)
lemma transfer_not [transfer_intro]:
"\<lbrakk>p \<equiv> eventually P \<U>\<rbrakk> \<Longrightarrow> \<not> p \<equiv> eventually (\<lambda>n. \<not> P n) \<U>"
by (simp only: FreeUltrafilterNat.eventually_not_iff)
lemma transfer_conj [transfer_intro]:
"\<lbrakk>p \<equiv> eventually P \<U>; q \<equiv> eventually Q \<U>\<rbrakk>
\<Longrightarrow> p \<and> q \<equiv> eventually (\<lambda>n. P n \<and> Q n) \<U>"
by (simp only: eventually_conj_iff)
lemma transfer_disj [transfer_intro]:
"\<lbrakk>p \<equiv> eventually P \<U>; q \<equiv> eventually Q \<U>\<rbrakk>
\<Longrightarrow> p \<or> q \<equiv> eventually (\<lambda>n. P n \<or> Q n) \<U>"
by (simp only: FreeUltrafilterNat.eventually_disj_iff)
lemma transfer_imp [transfer_intro]:
"\<lbrakk>p \<equiv> eventually P \<U>; q \<equiv> eventually Q \<U>\<rbrakk>
\<Longrightarrow> p \<longrightarrow> q \<equiv> eventually (\<lambda>n. P n \<longrightarrow> Q n) \<U>"
by (simp only: FreeUltrafilterNat.eventually_imp_iff)
lemma transfer_iff [transfer_intro]:
"\<lbrakk>p \<equiv> eventually P \<U>; q \<equiv> eventually Q \<U>\<rbrakk>
\<Longrightarrow> p = q \<equiv> eventually (\<lambda>n. P n = Q n) \<U>"
by (simp only: FreeUltrafilterNat.eventually_iff_iff)
lemma transfer_if_bool [transfer_intro]:
"\<lbrakk>p \<equiv> eventually P \<U>; x \<equiv> eventually X \<U>; y \<equiv> eventually Y \<U>\<rbrakk>
\<Longrightarrow> (if p then x else y) \<equiv> eventually (\<lambda>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]:
"\<lbrakk>x \<equiv> star_n X; y \<equiv> star_n Y\<rbrakk> \<Longrightarrow> x = y \<equiv> eventually (\<lambda>n. X n = Y n) \<U>"
by (simp only: star_n_eq_iff)
lemma transfer_if [transfer_intro]:
"\<lbrakk>p \<equiv> eventually (\<lambda>n. P n) \<U>; x \<equiv> star_n X; y \<equiv> star_n Y\<rbrakk>
\<Longrightarrow> (if p then x else y) \<equiv> star_n (\<lambda>n. 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]:
"\<lbrakk>\<And>X. f (star_n X) = g (star_n X)
\<equiv> eventually (\<lambda>n. F n (X n) = G n (X n)) \<U>\<rbrakk>
\<Longrightarrow> f = g \<equiv> eventually (\<lambda>n. F n = G n) \<U>"
by (simp only: fun_eq_iff transfer_all)
lemma transfer_star_n [transfer_intro]: "star_n X \<equiv> star_n (\<lambda>n. X n)"
by (rule reflexive)
lemma transfer_bool [transfer_intro]: "p \<equiv> eventually (\<lambda>n. p) \<U>"
by (simp add: FreeUltrafilterNat.proper)
subsection \<open>Standard elements\<close>
definition
star_of :: "'a \<Rightarrow> 'a star" where
"star_of x == star_n (\<lambda>n. x)"
definition
Standard :: "'a star set" where
"Standard = range star_of"
text \<open>Transfer tactic should remove occurrences of @{term star_of}\<close>
setup \<open>Transfer_Principle.add_const @{const_name star_of}\<close>
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 \<in> Standard"
by (simp add: Standard_def)
subsection \<open>Internal functions\<close>
definition
Ifun :: "('a \<Rightarrow> 'b) star \<Rightarrow> 'a star \<Rightarrow> 'b star" ("_ \<star> _" [300,301] 300) where
"Ifun f \<equiv> \<lambda>x. Abs_star
(\<Union>F\<in>Rep_star f. \<Union>X\<in>Rep_star x. starrel``{\<lambda>n. F n (X n)})"
lemma Ifun_congruent2:
"congruent2 starrel starrel (\<lambda>F X. starrel``{\<lambda>n. 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> star_n X = star_n (\<lambda>n. 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 \<open>Transfer tactic should remove occurrences of @{term Ifun}\<close>
setup \<open>Transfer_Principle.add_const @{const_name Ifun}\<close>
lemma transfer_Ifun [transfer_intro]:
"\<lbrakk>f \<equiv> star_n F; x \<equiv> star_n X\<rbrakk> \<Longrightarrow> f \<star> x \<equiv> star_n (\<lambda>n. F n (X n))"
by (simp only: Ifun_star_n)
lemma Ifun_star_of [simp]: "star_of f \<star> star_of x = star_of (f x)"
by (transfer, rule refl)
lemma Standard_Ifun [simp]:
"\<lbrakk>f \<in> Standard; x \<in> Standard\<rbrakk> \<Longrightarrow> f \<star> x \<in> Standard"
by (auto simp add: Standard_def)
text \<open>Nonstandard extensions of functions\<close>
definition
starfun :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a star \<Rightarrow> 'b star)" ("*f* _" [80] 80) where
"starfun f == \<lambda>x. star_of f \<star> x"
definition
starfun2 :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> ('a star \<Rightarrow> 'b star \<Rightarrow> 'c star)"
("*f2* _" [80] 80) where
"starfun2 f == \<lambda>x y. star_of f \<star> x \<star> y"
declare starfun_def [transfer_unfold]
declare starfun2_def [transfer_unfold]
lemma starfun_star_n: "( *f* f) (star_n X) = star_n (\<lambda>n. 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 (\<lambda>n. 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 \<in> Standard \<Longrightarrow> starfun f x \<in> Standard"
by (simp add: starfun_def)
lemma Standard_starfun2 [simp]:
"\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> starfun2 f x y \<in> Standard"
by (simp add: starfun2_def)
lemma Standard_starfun_iff:
assumes inj: "\<And>x y. f x = f y \<Longrightarrow> x = y"
shows "(starfun f x \<in> Standard) = (x \<in> Standard)"
proof
assume "x \<in> Standard"
thus "starfun f x \<in> Standard" by simp
next
have inj': "\<And>x y. starfun f x = starfun f y \<Longrightarrow> x = y"
using inj by transfer
assume "starfun f x \<in> Standard"
then obtain b where b: "starfun f x = star_of b"
unfolding Standard_def ..
hence "\<exists>x. starfun f x = star_of b" ..
hence "\<exists>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 \<in> Standard"
unfolding Standard_def by auto
qed
lemma Standard_starfun2_iff:
assumes inj: "\<And>a b a' b'. f a b = f a' b' \<Longrightarrow> a = a' \<and> b = b'"
shows "(starfun2 f x y \<in> Standard) = (x \<in> Standard \<and> y \<in> Standard)"
proof
assume "x \<in> Standard \<and> y \<in> Standard"
thus "starfun2 f x y \<in> Standard" by simp
next
have inj': "\<And>x y z w. starfun2 f x y = starfun2 f z w \<Longrightarrow> x = z \<and> y = w"
using inj by transfer
assume "starfun2 f x y \<in> Standard"
then obtain c where c: "starfun2 f x y = star_of c"
unfolding Standard_def ..
hence "\<exists>x y. starfun2 f x y = star_of c" by auto
hence "\<exists>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 \<and> y = star_of b"
by (rule inj')
thus "x \<in> Standard \<and> y \<in> Standard"
unfolding Standard_def by auto
qed
subsection \<open>Internal predicates\<close>
definition unstar :: "bool star \<Rightarrow> bool" where
"unstar b \<longleftrightarrow> 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"
by (simp add: unstar_def star_of_inject)
text \<open>Transfer tactic should remove occurrences of @{term unstar}\<close>
setup \<open>Transfer_Principle.add_const @{const_name unstar}\<close>
lemma transfer_unstar [transfer_intro]:
"p \<equiv> star_n P \<Longrightarrow> unstar p \<equiv> eventually P \<U>"
by (simp only: unstar_star_n)
definition
starP :: "('a \<Rightarrow> bool) \<Rightarrow> 'a star \<Rightarrow> bool" ("*p* _" [80] 80) where
"*p* P = (\<lambda>x. unstar (star_of P \<star> x))"
definition
starP2 :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a star \<Rightarrow> 'b star \<Rightarrow> bool" ("*p2* _" [80] 80) where
"*p2* P = (\<lambda>x y. unstar (star_of P \<star> x \<star> y))"
declare starP_def [transfer_unfold]
declare starP2_def [transfer_unfold]
lemma starP_star_n: "( *p* P) (star_n X) = (eventually (\<lambda>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) = (eventually (\<lambda>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 \<open>Internal sets\<close>
definition
Iset :: "'a set star \<Rightarrow> 'a star set" where
"Iset A = {x. ( *p2* op \<in>) x A}"
lemma Iset_star_n:
"(star_n X \<in> Iset (star_n A)) = (eventually (\<lambda>n. X n \<in> A n) \<U>)"
by (simp add: Iset_def starP2_star_n)
text \<open>Transfer tactic should remove occurrences of @{term Iset}\<close>
setup \<open>Transfer_Principle.add_const @{const_name Iset}\<close>
lemma transfer_mem [transfer_intro]:
"\<lbrakk>x \<equiv> star_n X; a \<equiv> Iset (star_n A)\<rbrakk>
\<Longrightarrow> x \<in> a \<equiv> eventually (\<lambda>n. X n \<in> A n) \<U>"
by (simp only: Iset_star_n)
lemma transfer_Collect [transfer_intro]:
"\<lbrakk>\<And>X. p (star_n X) \<equiv> eventually (\<lambda>n. P n (X n)) \<U>\<rbrakk>
\<Longrightarrow> Collect p \<equiv> Iset (star_n (\<lambda>n. Collect (P n)))"
by (simp add: atomize_eq set_eq_iff all_star_eq Iset_star_n)
lemma transfer_set_eq [transfer_intro]:
"\<lbrakk>a \<equiv> Iset (star_n A); b \<equiv> Iset (star_n B)\<rbrakk>
\<Longrightarrow> a = b \<equiv> eventually (\<lambda>n. A n = B n) \<U>"
by (simp only: set_eq_iff transfer_all transfer_iff transfer_mem)
lemma transfer_ball [transfer_intro]:
"\<lbrakk>a \<equiv> Iset (star_n A); \<And>X. p (star_n X) \<equiv> eventually (\<lambda>n. P n (X n)) \<U>\<rbrakk>
\<Longrightarrow> \<forall>x\<in>a. p x \<equiv> eventually (\<lambda>n. \<forall>x\<in>A n. P n x) \<U>"
by (simp only: Ball_def transfer_all transfer_imp transfer_mem)
lemma transfer_bex [transfer_intro]:
"\<lbrakk>a \<equiv> Iset (star_n A); \<And>X. p (star_n X) \<equiv> eventually (\<lambda>n. P n (X n)) \<U>\<rbrakk>
\<Longrightarrow> \<exists>x\<in>a. p x \<equiv> eventually (\<lambda>n. \<exists>x\<in>A n. P n x) \<U>"
by (simp only: Bex_def transfer_ex transfer_conj transfer_mem)
lemma transfer_Iset [transfer_intro]:
"\<lbrakk>a \<equiv> star_n A\<rbrakk> \<Longrightarrow> Iset a \<equiv> Iset (star_n (\<lambda>n. A n))"
by simp
text \<open>Nonstandard extensions of sets.\<close>
definition
starset :: "'a set \<Rightarrow> 'a star set" ("*s* _" [80] 80) where
"starset A = Iset (star_of A)"
declare starset_def [transfer_unfold]
lemma starset_mem: "(star_of x \<in> *s* A) = (x \<in> 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 \<union> B) = *s* A \<union> *s* B"
by (transfer Un_def, rule refl)
lemma starset_Int: "*s* (A \<inter> B) = *s* A \<inter> *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 \<subseteq> *s* B) = (A \<subseteq> 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 \<open>Syntactic classes\<close>
instantiation star :: (zero) zero
begin
definition
star_zero_def: "0 \<equiv> star_of 0"
instance ..
end
instantiation star :: (one) one
begin
definition
star_one_def: "1 \<equiv> star_of 1"
instance ..
end
instantiation star :: (plus) plus
begin
definition
star_add_def: "(op +) \<equiv> *f2* (op +)"
instance ..
end
instantiation star :: (times) times
begin
definition
star_mult_def: "(op *) \<equiv> *f2* (op *)"
instance ..
end
instantiation star :: (uminus) uminus
begin
definition
star_minus_def: "uminus \<equiv> *f* uminus"
instance ..
end
instantiation star :: (minus) minus
begin
definition
star_diff_def: "(op -) \<equiv> *f2* (op -)"
instance ..
end
instantiation star :: (abs) abs
begin
definition
star_abs_def: "abs \<equiv> *f* abs"
instance ..
end
instantiation star :: (sgn) sgn
begin
definition
star_sgn_def: "sgn \<equiv> *f* sgn"
instance ..
end
instantiation star :: (divide) divide
begin
definition
star_divide_def: "divide \<equiv> *f2* divide"
instance ..
end
instantiation star :: (inverse) inverse
begin
definition
star_inverse_def: "inverse \<equiv> *f* inverse"
instance ..
end
instance star :: (Rings.dvd) Rings.dvd ..
instantiation star :: (Divides.div) Divides.div
begin
definition
star_mod_def: "(op mod) \<equiv> *f2* (op mod)"
instance ..
end
instantiation star :: (ord) ord
begin
definition
star_le_def: "(op \<le>) \<equiv> *p2* (op \<le>)"
definition
star_less_def: "(op <) \<equiv> *p2* (op <)"
instance ..
end
lemmas star_class_defs [transfer_unfold] =
star_zero_def star_one_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_mod_def
text \<open>Class operations preserve standard elements\<close>
lemma Standard_zero: "0 \<in> Standard"
by (simp add: star_zero_def)
lemma Standard_one: "1 \<in> Standard"
by (simp add: star_one_def)
lemma Standard_add: "\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> x + y \<in> Standard"
by (simp add: star_add_def)
lemma Standard_diff: "\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> x - y \<in> Standard"
by (simp add: star_diff_def)
lemma Standard_minus: "x \<in> Standard \<Longrightarrow> - x \<in> Standard"
by (simp add: star_minus_def)
lemma Standard_mult: "\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> x * y \<in> Standard"
by (simp add: star_mult_def)
lemma Standard_divide: "\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> x / y \<in> Standard"
by (simp add: star_divide_def)
lemma Standard_inverse: "x \<in> Standard \<Longrightarrow> inverse x \<in> Standard"
by (simp add: star_inverse_def)
lemma Standard_abs: "x \<in> Standard \<Longrightarrow> \<bar>x\<bar> \<in> Standard"
by (simp add: star_abs_def)
lemma Standard_mod: "\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> x mod y \<in> Standard"
by (simp add: star_mod_def)
lemmas Standard_simps [simp] =
Standard_zero Standard_one
Standard_add Standard_diff Standard_minus
Standard_mult Standard_divide Standard_inverse
Standard_abs Standard_mod
text \<open>@{term star_of} preserves class operations\<close>
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 \<bar>x\<bar> = \<bar>star_of x\<bar>"
by transfer (rule refl)
text \<open>@{term star_of} preserves numerals\<close>
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 \<open>@{term star_of} preserves orderings\<close>
lemma star_of_less: "(star_of x < star_of y) = (x < y)"
by transfer (rule refl)
lemma star_of_le: "(star_of x \<le> star_of y) = (x \<le> y)"
by transfer (rule refl)
lemma star_of_eq: "(star_of x = star_of y) = (x = y)"
by transfer (rule refl)
text\<open>As above, for 0\<close>
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\<open>As above, for 1\<close>
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_add star_of_diff star_of_minus
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 \<open>Ordering and lattice classes\<close>
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 \<equiv> *f2* inf"
instance
by (standard; transfer) auto
end
instantiation star :: (semilattice_sup) semilattice_sup
begin
definition
star_sup_def [transfer_unfold]: "sup \<equiv> *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]:
"\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> inf x y \<in> Standard"
by (simp add: star_inf_def)
lemma Standard_sup [simp]:
"\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> sup x y \<in> 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]:
"\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> max x y \<in> Standard"
by (simp add: star_max_def)
lemma Standard_min [simp]:
"\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> min x y \<in> 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 \<open>Ordered group classes\<close>
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.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 :: (power) power ..
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, simp add: diff_diff_eq)+
instance star :: (cancel_comm_monoid_add) cancel_comm_monoid_add ..
instance star :: (ab_group_add) ab_group_add
apply (intro_classes)
apply (transfer, rule left_minus)
apply (transfer, rule diff_conv_add_uminus)
done
instance star :: (ordered_ab_semigroup_add) ordered_ab_semigroup_add
by (intro_classes, transfer, rule add_left_mono)
instance star :: (ordered_cancel_ab_semigroup_add) ordered_cancel_ab_semigroup_add ..
instance star :: (ordered_ab_semigroup_add_imp_le) ordered_ab_semigroup_add_imp_le
by (intro_classes, transfer, rule add_le_imp_le_left)
instance star :: (ordered_comm_monoid_add) ordered_comm_monoid_add ..
instance star :: (ordered_cancel_comm_monoid_add) ordered_cancel_comm_monoid_add ..
instance star :: (ordered_ab_group_add) ordered_ab_group_add ..
instance star :: (ordered_ab_group_add_abs) ordered_ab_group_add_abs
by intro_classes (transfer,
simp add: abs_ge_self abs_leI abs_triangle_ineq)+
instance star :: (linordered_cancel_ab_semigroup_add) linordered_cancel_ab_semigroup_add ..
subsection \<open>Ring and field classes\<close>
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)
apply(transfer, fact le_add_diff_inverse2)
done
instance star :: (linordered_idom) linordered_idom ..
instance star :: (linordered_field) linordered_field ..
subsection \<open>Power\<close>
lemma star_power_def [transfer_unfold]:
"(op ^) \<equiv> \<lambda>x n. ( *f* (\<lambda>x. x ^ n)) x"
proof (rule eq_reflection, rule ext, rule ext)
fix n :: nat
show "\<And>x::'a star. x ^ n = ( *f* (\<lambda>x. x ^ n)) x"
proof (induct n)
case 0
have "\<And>x::'a star. ( *f* (\<lambda>x. 1)) x = 1"
by transfer simp
then show ?case by simp
next
case (Suc n)
have "\<And>x::'a star. x * ( *f* (\<lambda>x::'a. x ^ n)) x = ( *f* (\<lambda>x::'a. x * x ^ n)) x"
by transfer simp
with Suc show ?case by simp
qed
qed
lemma Standard_power [simp]: "x \<in> Standard \<Longrightarrow> x ^ n \<in> Standard"
by (simp add: star_power_def)
lemma star_of_power [simp]: "star_of (x ^ n) = star_of x ^ n"
by transfer (rule refl)
subsection \<open>Number classes\<close>
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 \<in> Standard"
by (simp add: star_numeral_def)
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 \<in> 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 \<in> 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
proof
have "inj (star_of :: 'a \<Rightarrow> 'a star)" by (rule injI) simp
then have "inj (star_of \<circ> of_nat :: nat \<Rightarrow> 'a star)" using inj_of_nat by (rule inj_comp)
then show "inj (of_nat :: nat \<Rightarrow> '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 (1) odd_even_add)
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 \<Rightarrow> num \<Rightarrow> 'a star \<times> '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 \<Rightarrow> 'a star \<times> 'a star \<Rightarrow> 'a star \<times> 'a star"
where
"divmod_step_star l qr = (let (q, r) = qr
in if r \<ge> 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 \<ge> numeral l then (2 * q + 1, r - numeral l)
else (2 * q, r))" for l and qr :: "'a star \<times> '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 \<open>Finite class\<close>
lemma starset_finite: "finite A \<Longrightarrow> *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