src/HOL/Hyperreal/StarType.thy

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