starfun, starset, and other functions on NS types are now polymorphic;
many similar theorems have been generalized and merged;
(star_n X) replaces (Abs_star(starrel `` {X}));
many proofs have been simplified with the transfer tactic.
(* Title : HOL/Hyperreal/StarType.thy
ID : $Id$
Author : Jacques D. Fleuriot and Brian Huffman
*)
header {* Construction of Star Types Using Ultrafilters *}
theory StarType
imports Filter
begin
subsection {* A Free Ultrafilter over the Naturals *}
constdefs
FreeUltrafilterNat :: "nat set set" ("\<U>")
"\<U> \<equiv> SOME U. freeultrafilter U"
lemma freeultrafilter_FUFNat: "freeultrafilter \<U>"
apply (unfold FreeUltrafilterNat_def)
apply (rule someI_ex)
apply (rule freeultrafilter_Ex)
apply (rule nat_infinite)
done
lemmas ultrafilter_FUFNat =
freeultrafilter_FUFNat [THEN freeultrafilter.ultrafilter]
lemmas filter_FUFNat =
freeultrafilter_FUFNat [THEN freeultrafilter.filter]
lemmas FUFNat_empty [iff] =
filter_FUFNat [THEN filter.empty]
lemmas FUFNat_UNIV [iff] =
filter_FUFNat [THEN filter.UNIV]
text {* This rule takes the place of the old ultra tactic *}
lemma ultra:
"\<lbrakk>{n. P n} \<in> \<U>; {n. P n \<longrightarrow> Q n} \<in> \<U>\<rbrakk> \<Longrightarrow> {n. Q n} \<in> \<U>"
by (simp add: Collect_imp_eq
ultrafilter_FUFNat [THEN ultrafilter.Un_iff]
ultrafilter_FUFNat [THEN ultrafilter.Compl_iff])
subsection {* Definition of @{text star} type constructor *}
constdefs
starrel :: "((nat \<Rightarrow> 'a) \<times> (nat \<Rightarrow> 'a)) set"
"starrel \<equiv> {(X,Y). {n. X n = Y n} \<in> \<U>}"
typedef 'a star = "(UNIV :: (nat \<Rightarrow> 'a) set) // starrel"
by (auto intro: quotientI)
text {* Proving that @{term starrel} is an equivalence relation *}
lemma starrel_iff [iff]: "((X,Y) \<in> starrel) = ({n. X n = Y n} \<in> \<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]
-- {* @{term "(starrel `` {x} = starrel `` {y}) = ((x,y) \<in> starrel)"} *}
lemma starrel_in_star: "starrel``{x} \<in> star"
by (simp add: star_def starrel_def quotient_def, fast)
lemma eq_Abs_star:
"(\<And>x. z = Abs_star (starrel``{x}) \<Longrightarrow> P) \<Longrightarrow> P"
apply (rule_tac x=z in Abs_star_cases)
apply (unfold star_def)
apply (erule quotientE)
apply simp
done
subsection {* Constructors for type @{typ "'a star"} *}
constdefs
star_n :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a star"
"star_n X \<equiv> Abs_star (starrel `` {X})"
star_of :: "'a \<Rightarrow> 'a star"
"star_of x \<equiv> star_n (\<lambda>n. x)"
theorem star_cases [case_names star_n, cases type: star]:
"(\<And>X. x = star_n X \<Longrightarrow> P) \<Longrightarrow> P"
by (unfold star_n_def, rule eq_Abs_star[of x], blast)
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)
lemma star_n_eq_iff: "(star_n X = star_n Y) = ({n. X n = Y n} \<in> \<U>)"
apply (unfold star_n_def)
apply (simp add: Abs_star_inject starrel_in_star equiv_starrel_iff)
done
lemma star_of_inject: "(star_of x = star_of y) = (x = y)"
by (simp add: star_of_def star_n_eq_iff)
subsection {* Internal functions *}
constdefs
Ifun :: "('a \<Rightarrow> 'b) star \<Rightarrow> 'a star \<Rightarrow> 'b star" ("_ \<star> _" [300,301] 300)
"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_star_n: "star_n F \<star> star_n X = star_n (\<lambda>n. F n (X n))"
apply (unfold Ifun_def star_n_def)
apply (simp add: Abs_star_inverse starrel_in_star)
apply (rule_tac f=Abs_star in arg_cong)
apply safe
apply (erule ultra)+
apply simp
apply force
done
lemma Ifun [simp]: "star_of f \<star> star_of x = star_of (f x)"
by (simp only: star_of_def Ifun_star_n)
subsection {* Testing lifted booleans *}
constdefs
unstar :: "bool star \<Rightarrow> bool"
"unstar b \<equiv> b = star_of True"
lemma unstar_star_n: "unstar (star_n P) = ({n. P n} \<in> \<U>)"
by (simp add: unstar_def star_of_def star_n_eq_iff)
lemma unstar [simp]: "unstar (star_of p) = p"
by (simp add: unstar_def star_of_inject)
subsection {* Internal functions and predicates *}
constdefs
Ifun_of :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a star \<Rightarrow> 'b star)"
"Ifun_of f \<equiv> Ifun (star_of f)"
Ifun2 :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) star \<Rightarrow> ('a star \<Rightarrow> 'b star \<Rightarrow> 'c star)"
"Ifun2 f \<equiv> \<lambda>x y. f \<star> x \<star> y"
Ifun2_of :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> ('a star \<Rightarrow> 'b star \<Rightarrow> 'c star)"
"Ifun2_of f \<equiv> \<lambda>x y. star_of f \<star> x \<star> y"
Ipred :: "('a \<Rightarrow> bool) star \<Rightarrow> ('a star \<Rightarrow> bool)"
"Ipred P \<equiv> \<lambda>x. unstar (P \<star> x)"
Ipred_of :: "('a \<Rightarrow> bool) \<Rightarrow> ('a star \<Rightarrow> bool)"
"Ipred_of P \<equiv> \<lambda>x. unstar (star_of P \<star> x)"
Ipred2 :: "('a \<Rightarrow> 'b \<Rightarrow> bool) star \<Rightarrow> ('a star \<Rightarrow> 'b star \<Rightarrow> bool)"
"Ipred2 P \<equiv> \<lambda>x y. unstar (P \<star> x \<star> y)"
Ipred2_of :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a star \<Rightarrow> 'b star \<Rightarrow> bool)"
"Ipred2_of P \<equiv> \<lambda>x y. unstar (star_of P \<star> x \<star> y)"
lemma Ifun_of [simp]:
"Ifun_of f (star_of x) = star_of (f x)"
by (simp only: Ifun_of_def Ifun)
lemma Ifun2_of [simp]:
"Ifun2_of f (star_of x) (star_of y) = star_of (f x y)"
by (simp only: Ifun2_of_def Ifun)
lemma Ipred_of [simp]:
"Ipred_of P (star_of x) = P x"
by (simp only: Ipred_of_def Ifun unstar)
lemma Ipred2_of [simp]:
"Ipred2_of P (star_of x) (star_of y) = P x y"
by (simp only: Ipred2_of_def Ifun unstar)
lemmas Ifun_defs =
star_of_def Ifun_of_def Ifun2_def Ifun2_of_def
Ipred_def Ipred_of_def Ipred2_def Ipred2_of_def
subsection {* Internal sets *}
constdefs
Iset :: "'a set star \<Rightarrow> 'a star set"
"Iset A \<equiv> {x. Ipred2_of (op \<in>) x A}"
Iset_of :: "'a set \<Rightarrow> 'a star set"
"Iset_of A \<equiv> Iset (star_of A)"
lemma Iset_star_n:
"(star_n X \<in> Iset (star_n A)) = ({n. X n \<in> A n} \<in> \<U>)"
by (simp add: Iset_def Ipred2_of_def star_of_def Ifun_star_n unstar_star_n)
subsection {* Class constants *}
instance star :: (ord) ord ..
instance star :: (zero) zero ..
instance star :: (one) one ..
instance star :: (plus) plus ..
instance star :: (times) times ..
instance star :: (minus) minus ..
instance star :: (inverse) inverse ..
instance star :: (number) number ..
instance star :: ("Divides.div") "Divides.div" ..
instance star :: (power) power ..
defs (overloaded)
star_zero_def: "0 \<equiv> star_of 0"
star_one_def: "1 \<equiv> star_of 1"
star_number_def: "number_of b \<equiv> star_of (number_of b)"
star_add_def: "(op +) \<equiv> Ifun2_of (op +)"
star_diff_def: "(op -) \<equiv> Ifun2_of (op -)"
star_minus_def: "uminus \<equiv> Ifun_of uminus"
star_mult_def: "(op *) \<equiv> Ifun2_of (op *)"
star_divide_def: "(op /) \<equiv> Ifun2_of (op /)"
star_inverse_def: "inverse \<equiv> Ifun_of inverse"
star_le_def: "(op \<le>) \<equiv> Ipred2_of (op \<le>)"
star_less_def: "(op <) \<equiv> Ipred2_of (op <)"
star_abs_def: "abs \<equiv> Ifun_of abs"
star_div_def: "(op div) \<equiv> Ifun2_of (op div)"
star_mod_def: "(op mod) \<equiv> Ifun2_of (op mod)"
star_power_def: "(op ^) \<equiv> \<lambda>x n. Ifun_of (\<lambda>x. x ^ n) x"
lemmas star_class_defs =
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_div_def star_mod_def star_power_def
text {* @{term star_of} preserves class operations *}
lemma star_of_add: "star_of (x + y) = star_of x + star_of y"
by (simp add: star_add_def)
lemma star_of_diff: "star_of (x - y) = star_of x - star_of y"
by (simp add: star_diff_def)
lemma star_of_minus: "star_of (-x) = - star_of x"
by (simp add: star_minus_def)
lemma star_of_mult: "star_of (x * y) = star_of x * star_of y"
by (simp add: star_mult_def)
lemma star_of_divide: "star_of (x / y) = star_of x / star_of y"
by (simp add: star_divide_def)
lemma star_of_inverse: "star_of (inverse x) = inverse (star_of x)"
by (simp add: star_inverse_def)
lemma star_of_div: "star_of (x div y) = star_of x div star_of y"
by (simp add: star_div_def)
lemma star_of_mod: "star_of (x mod y) = star_of x mod star_of y"
by (simp add: star_mod_def)
lemma star_of_power: "star_of (x ^ n) = star_of x ^ n"
by (simp add: star_power_def)
lemma star_of_abs: "star_of (abs x) = abs (star_of x)"
by (simp add: star_abs_def)
text {* @{term star_of} preserves numerals *}
lemma star_of_zero: "star_of 0 = 0"
by (simp add: star_zero_def)
lemma star_of_one: "star_of 1 = 1"
by (simp add: star_one_def)
lemma star_of_number_of: "star_of (number_of x) = number_of x"
by (simp add: star_number_def)
text {* @{term star_of} preserves orderings *}
lemma star_of_less: "(star_of x < star_of y) = (x < y)"
by (simp add: star_less_def)
lemma star_of_le: "(star_of x \<le> star_of y) = (x \<le> y)"
by (simp add: star_le_def)
lemma star_of_eq: "(star_of x = star_of y) = (x = y)"
by (rule star_of_inject)
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 b", simplified star_of_number_of, standard]
lemmas star_of_number_le =
star_of_le [of "number_of b", simplified star_of_number_of, standard]
lemmas star_of_number_eq =
star_of_eq [of "number_of b", simplified star_of_number_of, standard]
lemmas star_of_less_number =
star_of_less [of _ "number_of b", simplified star_of_number_of, standard]
lemmas star_of_le_number =
star_of_le [of _ "number_of b", simplified star_of_number_of, standard]
lemmas star_of_eq_number =
star_of_eq [of _ "number_of b", simplified star_of_number_of, standard]
lemmas star_of_simps =
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
declare star_of_simps [simp]
end