src/HOL/NSA/StarDef.thy
author haftmann
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(*  Title       : HOL/Hyperreal/StarDef.thy
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    Author      : Jacques D. Fleuriot and Brian Huffman
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*)
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header {* Construction of Star Types Using Ultrafilters *}
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theory StarDef
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imports Filter
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uses ("transfer.ML")
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begin
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subsection {* A Free Ultrafilter over the Naturals *}
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definition
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  FreeUltrafilterNat :: "nat set set"  ("\<U>") where
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  "\<U> = (SOME U. freeultrafilter U)"
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lemma freeultrafilter_FreeUltrafilterNat: "freeultrafilter \<U>"
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apply (unfold FreeUltrafilterNat_def)
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apply (rule someI_ex [where P=freeultrafilter])
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apply (rule freeultrafilter_Ex)
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apply (rule nat_infinite)
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done
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interpretation FreeUltrafilterNat: freeultrafilter FreeUltrafilterNat
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by (rule freeultrafilter_FreeUltrafilterNat)
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text {* This rule takes the place of the old ultra tactic *}
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lemma ultra:
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  "\<lbrakk>{n. P n} \<in> \<U>; {n. P n \<longrightarrow> Q n} \<in> \<U>\<rbrakk> \<Longrightarrow> {n. Q n} \<in> \<U>"
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by (simp add: Collect_imp_eq
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    FreeUltrafilterNat.Un_iff FreeUltrafilterNat.Compl_iff)
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subsection {* Definition of @{text star} type constructor *}
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definition
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  starrel :: "((nat \<Rightarrow> 'a) \<times> (nat \<Rightarrow> 'a)) set" where
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  "starrel = {(X,Y). {n. X n = Y n} \<in> \<U>}"
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typedef 'a star = "(UNIV :: (nat \<Rightarrow> 'a) set) // starrel"
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by (auto intro: quotientI)
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definition
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  star_n :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a star" where
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  "star_n X = Abs_star (starrel `` {X})"
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theorem star_cases [case_names star_n, cases type: star]:
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  "(\<And>X. x = star_n X \<Longrightarrow> P) \<Longrightarrow> P"
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by (cases x, unfold star_n_def star_def, erule quotientE, fast)
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lemma all_star_eq: "(\<forall>x. P x) = (\<forall>X. P (star_n X))"
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by (auto, rule_tac x=x in star_cases, simp)
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lemma ex_star_eq: "(\<exists>x. P x) = (\<exists>X. P (star_n X))"
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by (auto, rule_tac x=x in star_cases, auto)
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text {* Proving that @{term starrel} is an equivalence relation *}
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lemma starrel_iff [iff]: "((X,Y) \<in> starrel) = ({n. X n = Y n} \<in> \<U>)"
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by (simp add: starrel_def)
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lemma equiv_starrel: "equiv UNIV starrel"
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proof (rule equiv.intro)
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  show "refl starrel" by (simp add: refl_on_def)
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  show "sym starrel" by (simp add: sym_def eq_commute)
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  show "trans starrel" by (auto intro: transI elim!: ultra)
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qed
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lemmas equiv_starrel_iff =
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  eq_equiv_class_iff [OF equiv_starrel UNIV_I UNIV_I]
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lemma starrel_in_star: "starrel``{x} \<in> star"
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by (simp add: star_def quotientI)
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lemma star_n_eq_iff: "(star_n X = star_n Y) = ({n. X n = Y n} \<in> \<U>)"
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by (simp add: star_n_def Abs_star_inject starrel_in_star equiv_starrel_iff)
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subsection {* Transfer principle *}
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text {* This introduction rule starts each transfer proof. *}
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lemma transfer_start:
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  "P \<equiv> {n. Q} \<in> \<U> \<Longrightarrow> Trueprop P \<equiv> Trueprop Q"
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by (subgoal_tac "P \<equiv> Q", simp, simp add: atomize_eq)
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text {*Initialize transfer tactic.*}
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use "transfer.ML"
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setup Transfer.setup
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text {* Transfer introduction rules. *}
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lemma transfer_ex [transfer_intro]:
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  "\<lbrakk>\<And>X. p (star_n X) \<equiv> {n. P n (X n)} \<in> \<U>\<rbrakk>
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    \<Longrightarrow> \<exists>x::'a star. p x \<equiv> {n. \<exists>x. P n x} \<in> \<U>"
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by (simp only: ex_star_eq FreeUltrafilterNat.Collect_ex)
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lemma transfer_all [transfer_intro]:
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  "\<lbrakk>\<And>X. p (star_n X) \<equiv> {n. P n (X n)} \<in> \<U>\<rbrakk>
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    \<Longrightarrow> \<forall>x::'a star. p x \<equiv> {n. \<forall>x. P n x} \<in> \<U>"
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by (simp only: all_star_eq FreeUltrafilterNat.Collect_all)
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lemma transfer_not [transfer_intro]:
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  "\<lbrakk>p \<equiv> {n. P n} \<in> \<U>\<rbrakk> \<Longrightarrow> \<not> p \<equiv> {n. \<not> P n} \<in> \<U>"
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by (simp only: FreeUltrafilterNat.Collect_not)
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lemma transfer_conj [transfer_intro]:
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  "\<lbrakk>p \<equiv> {n. P n} \<in> \<U>; q \<equiv> {n. Q n} \<in> \<U>\<rbrakk>
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    \<Longrightarrow> p \<and> q \<equiv> {n. P n \<and> Q n} \<in> \<U>"
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by (simp only: FreeUltrafilterNat.Collect_conj)
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lemma transfer_disj [transfer_intro]:
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  "\<lbrakk>p \<equiv> {n. P n} \<in> \<U>; q \<equiv> {n. Q n} \<in> \<U>\<rbrakk>
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    \<Longrightarrow> p \<or> q \<equiv> {n. P n \<or> Q n} \<in> \<U>"
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by (simp only: FreeUltrafilterNat.Collect_disj)
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lemma transfer_imp [transfer_intro]:
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  "\<lbrakk>p \<equiv> {n. P n} \<in> \<U>; q \<equiv> {n. Q n} \<in> \<U>\<rbrakk>
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    \<Longrightarrow> p \<longrightarrow> q \<equiv> {n. P n \<longrightarrow> Q n} \<in> \<U>"
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by (simp only: imp_conv_disj transfer_disj transfer_not)
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lemma transfer_iff [transfer_intro]:
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  "\<lbrakk>p \<equiv> {n. P n} \<in> \<U>; q \<equiv> {n. Q n} \<in> \<U>\<rbrakk>
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    \<Longrightarrow> p = q \<equiv> {n. P n = Q n} \<in> \<U>"
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by (simp only: iff_conv_conj_imp transfer_conj transfer_imp)
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lemma transfer_if_bool [transfer_intro]:
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  "\<lbrakk>p \<equiv> {n. P n} \<in> \<U>; x \<equiv> {n. X n} \<in> \<U>; y \<equiv> {n. Y n} \<in> \<U>\<rbrakk>
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    \<Longrightarrow> (if p then x else y) \<equiv> {n. if P n then X n else Y n} \<in> \<U>"
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by (simp only: if_bool_eq_conj transfer_conj transfer_imp transfer_not)
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lemma transfer_eq [transfer_intro]:
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  "\<lbrakk>x \<equiv> star_n X; y \<equiv> star_n Y\<rbrakk> \<Longrightarrow> x = y \<equiv> {n. X n = Y n} \<in> \<U>"
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by (simp only: star_n_eq_iff)
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lemma transfer_if [transfer_intro]:
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  "\<lbrakk>p \<equiv> {n. P n} \<in> \<U>; x \<equiv> star_n X; y \<equiv> star_n Y\<rbrakk>
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    \<Longrightarrow> (if p then x else y) \<equiv> star_n (\<lambda>n. if P n then X n else Y n)"
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apply (rule eq_reflection)
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apply (auto simp add: star_n_eq_iff transfer_not elim!: ultra)
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done
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lemma transfer_fun_eq [transfer_intro]:
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  "\<lbrakk>\<And>X. f (star_n X) = g (star_n X) 
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    \<equiv> {n. F n (X n) = G n (X n)} \<in> \<U>\<rbrakk>
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      \<Longrightarrow> f = g \<equiv> {n. F n = G n} \<in> \<U>"
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by (simp only: expand_fun_eq transfer_all)
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lemma transfer_star_n [transfer_intro]: "star_n X \<equiv> star_n (\<lambda>n. X n)"
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by (rule reflexive)
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lemma transfer_bool [transfer_intro]: "p \<equiv> {n. p} \<in> \<U>"
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by (simp add: atomize_eq)
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subsection {* Standard elements *}
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definition
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  star_of :: "'a \<Rightarrow> 'a star" where
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  "star_of x == star_n (\<lambda>n. x)"
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definition
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  Standard :: "'a star set" where
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  "Standard = range star_of"
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text {* Transfer tactic should remove occurrences of @{term star_of} *}
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setup {* Transfer.add_const "StarDef.star_of" *}
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declare star_of_def [transfer_intro]
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lemma star_of_inject: "(star_of x = star_of y) = (x = y)"
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by (transfer, rule refl)
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lemma Standard_star_of [simp]: "star_of x \<in> Standard"
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by (simp add: Standard_def)
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subsection {* Internal functions *}
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definition
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  Ifun :: "('a \<Rightarrow> 'b) star \<Rightarrow> 'a star \<Rightarrow> 'b star" ("_ \<star> _" [300,301] 300) where
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  "Ifun f \<equiv> \<lambda>x. Abs_star
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       (\<Union>F\<in>Rep_star f. \<Union>X\<in>Rep_star x. starrel``{\<lambda>n. F n (X n)})"
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lemma Ifun_congruent2:
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  "congruent2 starrel starrel (\<lambda>F X. starrel``{\<lambda>n. F n (X n)})"
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by (auto simp add: congruent2_def equiv_starrel_iff elim!: ultra)
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lemma Ifun_star_n: "star_n F \<star> star_n X = star_n (\<lambda>n. F n (X n))"
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by (simp add: Ifun_def star_n_def Abs_star_inverse starrel_in_star
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    UN_equiv_class2 [OF equiv_starrel equiv_starrel Ifun_congruent2])
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text {* Transfer tactic should remove occurrences of @{term Ifun} *}
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setup {* Transfer.add_const "StarDef.Ifun" *}
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lemma transfer_Ifun [transfer_intro]:
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  "\<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))"
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by (simp only: Ifun_star_n)
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lemma Ifun_star_of [simp]: "star_of f \<star> star_of x = star_of (f x)"
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by (transfer, rule refl)
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lemma Standard_Ifun [simp]:
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  "\<lbrakk>f \<in> Standard; x \<in> Standard\<rbrakk> \<Longrightarrow> f \<star> x \<in> Standard"
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by (auto simp add: Standard_def)
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text {* Nonstandard extensions of functions *}
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definition
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  starfun :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a star \<Rightarrow> 'b star)"  ("*f* _" [80] 80) where
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  "starfun f == \<lambda>x. star_of f \<star> x"
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definition
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  starfun2 :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> ('a star \<Rightarrow> 'b star \<Rightarrow> 'c star)"
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   216
    ("*f2* _" [80] 80) where
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  "starfun2 f == \<lambda>x y. star_of f \<star> x \<star> y"
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declare starfun_def [transfer_unfold]
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declare starfun2_def [transfer_unfold]
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lemma starfun_star_n: "( *f* f) (star_n X) = star_n (\<lambda>n. f (X n))"
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by (simp only: starfun_def star_of_def Ifun_star_n)
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   224
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lemma starfun2_star_n:
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  "( *f2* f) (star_n X) (star_n Y) = star_n (\<lambda>n. f (X n) (Y n))"
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by (simp only: starfun2_def star_of_def Ifun_star_n)
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   228
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lemma starfun_star_of [simp]: "( *f* f) (star_of x) = star_of (f x)"
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   230
by (transfer, rule refl)
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   231
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lemma starfun2_star_of [simp]: "( *f2* f) (star_of x) = *f* f x"
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   233
by (transfer, rule refl)
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   234
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lemma Standard_starfun [simp]: "x \<in> Standard \<Longrightarrow> starfun f x \<in> Standard"
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   236
by (simp add: starfun_def)
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   237
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lemma Standard_starfun2 [simp]:
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  "\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> starfun2 f x y \<in> Standard"
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   240
by (simp add: starfun2_def)
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   241
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lemma Standard_starfun_iff:
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   243
  assumes inj: "\<And>x y. f x = f y \<Longrightarrow> x = y"
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   244
  shows "(starfun f x \<in> Standard) = (x \<in> Standard)"
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   245
proof
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   246
  assume "x \<in> Standard"
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  thus "starfun f x \<in> Standard" by simp
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   248
next
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   249
  have inj': "\<And>x y. starfun f x = starfun f y \<Longrightarrow> x = y"
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   250
    using inj by transfer
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   251
  assume "starfun f x \<in> Standard"
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   252
  then obtain b where b: "starfun f x = star_of b"
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   253
    unfolding Standard_def ..
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   254
  hence "\<exists>x. starfun f x = star_of b" ..
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   255
  hence "\<exists>a. f a = b" by transfer
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   256
  then obtain a where "f a = b" ..
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   257
  hence "starfun f (star_of a) = star_of b" by transfer
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parents:
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   258
  with b have "starfun f x = starfun f (star_of a)" by simp
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   259
  hence "x = star_of a" by (rule inj')
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   260
  thus "x \<in> Standard"
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   261
    unfolding Standard_def by auto
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   262
qed
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   263
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   264
lemma Standard_starfun2_iff:
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   265
  assumes inj: "\<And>a b a' b'. f a b = f a' b' \<Longrightarrow> a = a' \<and> b = b'"
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   266
  shows "(starfun2 f x y \<in> Standard) = (x \<in> Standard \<and> y \<in> Standard)"
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parents:
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   267
proof
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   268
  assume "x \<in> Standard \<and> y \<in> Standard"
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   269
  thus "starfun2 f x y \<in> Standard" by simp
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   270
next
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parents:
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   271
  have inj': "\<And>x y z w. starfun2 f x y = starfun2 f z w \<Longrightarrow> x = z \<and> y = w"
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parents:
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   272
    using inj by transfer
0783dd1dc13d move nonstandard analysis theories to NSA directory
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   273
  assume "starfun2 f x y \<in> Standard"
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   274
  then obtain c where c: "starfun2 f x y = star_of c"
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   275
    unfolding Standard_def ..
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   276
  hence "\<exists>x y. starfun2 f x y = star_of c" by auto
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   277
  hence "\<exists>a b. f a b = c" by transfer
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   278
  then obtain a b where "f a b = c" by auto
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   279
  hence "starfun2 f (star_of a) (star_of b) = star_of c"
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   280
    by transfer
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   281
  with c have "starfun2 f x y = starfun2 f (star_of a) (star_of b)"
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   282
    by simp
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   283
  hence "x = star_of a \<and> y = star_of b"
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   284
    by (rule inj')
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   285
  thus "x \<in> Standard \<and> y \<in> Standard"
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   286
    unfolding Standard_def by auto
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   287
qed
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   288
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   289
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   290
subsection {* Internal predicates *}
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   291
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   292
definition unstar :: "bool star \<Rightarrow> bool" where
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parents: 28059
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   293
  [code del]: "unstar b \<longleftrightarrow> b = star_of True"
27468
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   294
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   295
lemma unstar_star_n: "unstar (star_n P) = ({n. P n} \<in> \<U>)"
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   296
by (simp add: unstar_def star_of_def star_n_eq_iff)
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   297
0783dd1dc13d move nonstandard analysis theories to NSA directory
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   298
lemma unstar_star_of [simp]: "unstar (star_of p) = p"
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   299
by (simp add: unstar_def star_of_inject)
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   300
0783dd1dc13d move nonstandard analysis theories to NSA directory
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   301
text {* Transfer tactic should remove occurrences of @{term unstar} *}
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   302
setup {* Transfer.add_const "StarDef.unstar" *}
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diff changeset
   303
0783dd1dc13d move nonstandard analysis theories to NSA directory
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   304
lemma transfer_unstar [transfer_intro]:
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parents:
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   305
  "p \<equiv> star_n P \<Longrightarrow> unstar p \<equiv> {n. P n} \<in> \<U>"
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parents:
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   306
by (simp only: unstar_star_n)
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   307
0783dd1dc13d move nonstandard analysis theories to NSA directory
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   308
definition
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   309
  starP :: "('a \<Rightarrow> bool) \<Rightarrow> 'a star \<Rightarrow> bool"  ("*p* _" [80] 80) where
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   310
  "*p* P = (\<lambda>x. unstar (star_of P \<star> x))"
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diff changeset
   311
0783dd1dc13d move nonstandard analysis theories to NSA directory
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   312
definition
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parents:
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   313
  starP2 :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a star \<Rightarrow> 'b star \<Rightarrow> bool"  ("*p2* _" [80] 80) where
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   314
  "*p2* P = (\<lambda>x y. unstar (star_of P \<star> x \<star> y))"
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   315
0783dd1dc13d move nonstandard analysis theories to NSA directory
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   316
declare starP_def [transfer_unfold]
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   317
declare starP2_def [transfer_unfold]
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   318
0783dd1dc13d move nonstandard analysis theories to NSA directory
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   319
lemma starP_star_n: "( *p* P) (star_n X) = ({n. P (X n)} \<in> \<U>)"
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parents:
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   320
by (simp only: starP_def star_of_def Ifun_star_n unstar_star_n)
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parents:
diff changeset
   321
0783dd1dc13d move nonstandard analysis theories to NSA directory
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parents:
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   322
lemma starP2_star_n:
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parents:
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   323
  "( *p2* P) (star_n X) (star_n Y) = ({n. P (X n) (Y n)} \<in> \<U>)"
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parents:
diff changeset
   324
by (simp only: starP2_def star_of_def Ifun_star_n unstar_star_n)
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huffman
parents:
diff changeset
   325
0783dd1dc13d move nonstandard analysis theories to NSA directory
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parents:
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   326
lemma starP_star_of [simp]: "( *p* P) (star_of x) = P x"
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huffman
parents:
diff changeset
   327
by (transfer, rule refl)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   328
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   329
lemma starP2_star_of [simp]: "( *p2* P) (star_of x) = *p* P x"
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huffman
parents:
diff changeset
   330
by (transfer, rule refl)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   331
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   332
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
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   333
subsection {* Internal sets *}
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parents:
diff changeset
   334
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
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   335
definition
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huffman
parents:
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   336
  Iset :: "'a set star \<Rightarrow> 'a star set" where
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huffman
parents:
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   337
  "Iset A = {x. ( *p2* op \<in>) x A}"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   338
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
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   339
lemma Iset_star_n:
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huffman
parents:
diff changeset
   340
  "(star_n X \<in> Iset (star_n A)) = ({n. X n \<in> A n} \<in> \<U>)"
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huffman
parents:
diff changeset
   341
by (simp add: Iset_def starP2_star_n)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   342
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
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   343
text {* Transfer tactic should remove occurrences of @{term Iset} *}
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
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   344
setup {* Transfer.add_const "StarDef.Iset" *}
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huffman
parents:
diff changeset
   345
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
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   346
lemma transfer_mem [transfer_intro]:
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huffman
parents:
diff changeset
   347
  "\<lbrakk>x \<equiv> star_n X; a \<equiv> Iset (star_n A)\<rbrakk>
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huffman
parents:
diff changeset
   348
    \<Longrightarrow> x \<in> a \<equiv> {n. X n \<in> A n} \<in> \<U>"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   349
by (simp only: Iset_star_n)
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huffman
parents:
diff changeset
   350
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
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   351
lemma transfer_Collect [transfer_intro]:
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huffman
parents:
diff changeset
   352
  "\<lbrakk>\<And>X. p (star_n X) \<equiv> {n. P n (X n)} \<in> \<U>\<rbrakk>
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   353
    \<Longrightarrow> Collect p \<equiv> Iset (star_n (\<lambda>n. Collect (P n)))"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   354
by (simp add: atomize_eq expand_set_eq all_star_eq Iset_star_n)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   355
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
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   356
lemma transfer_set_eq [transfer_intro]:
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huffman
parents:
diff changeset
   357
  "\<lbrakk>a \<equiv> Iset (star_n A); b \<equiv> Iset (star_n B)\<rbrakk>
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   358
    \<Longrightarrow> a = b \<equiv> {n. A n = B n} \<in> \<U>"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   359
by (simp only: expand_set_eq transfer_all transfer_iff transfer_mem)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   360
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   361
lemma transfer_ball [transfer_intro]:
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   362
  "\<lbrakk>a \<equiv> Iset (star_n A); \<And>X. p (star_n X) \<equiv> {n. P n (X n)} \<in> \<U>\<rbrakk>
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   363
    \<Longrightarrow> \<forall>x\<in>a. p x \<equiv> {n. \<forall>x\<in>A n. P n x} \<in> \<U>"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   364
by (simp only: Ball_def transfer_all transfer_imp transfer_mem)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   365
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   366
lemma transfer_bex [transfer_intro]:
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   367
  "\<lbrakk>a \<equiv> Iset (star_n A); \<And>X. p (star_n X) \<equiv> {n. P n (X n)} \<in> \<U>\<rbrakk>
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   368
    \<Longrightarrow> \<exists>x\<in>a. p x \<equiv> {n. \<exists>x\<in>A n. P n x} \<in> \<U>"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   369
by (simp only: Bex_def transfer_ex transfer_conj transfer_mem)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   370
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   371
lemma transfer_Iset [transfer_intro]:
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huffman
parents:
diff changeset
   372
  "\<lbrakk>a \<equiv> star_n A\<rbrakk> \<Longrightarrow> Iset a \<equiv> Iset (star_n (\<lambda>n. A n))"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   373
by simp
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   374
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   375
text {* Nonstandard extensions of sets. *}
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   376
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   377
definition
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   378
  starset :: "'a set \<Rightarrow> 'a star set" ("*s* _" [80] 80) where
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   379
  "starset A = Iset (star_of A)"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   380
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   381
declare starset_def [transfer_unfold]
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   382
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   383
lemma starset_mem: "(star_of x \<in> *s* A) = (x \<in> A)"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   384
by (transfer, rule refl)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   385
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   386
lemma starset_UNIV: "*s* (UNIV::'a set) = (UNIV::'a star set)"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   387
by (transfer UNIV_def, rule refl)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   388
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   389
lemma starset_empty: "*s* {} = {}"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   390
by (transfer empty_def, rule refl)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   391
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   392
lemma starset_insert: "*s* (insert x A) = insert (star_of x) ( *s* A)"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   393
by (transfer insert_def Un_def, rule refl)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   394
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   395
lemma starset_Un: "*s* (A \<union> B) = *s* A \<union> *s* B"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   396
by (transfer Un_def, rule refl)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   397
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   398
lemma starset_Int: "*s* (A \<inter> B) = *s* A \<inter> *s* B"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   399
by (transfer Int_def, rule refl)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   400
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   401
lemma starset_Compl: "*s* -A = -( *s* A)"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   402
by (transfer Compl_eq, rule refl)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   403
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   404
lemma starset_diff: "*s* (A - B) = *s* A - *s* B"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   405
by (transfer set_diff_eq, rule refl)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   406
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   407
lemma starset_image: "*s* (f ` A) = ( *f* f) ` ( *s* A)"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   408
by (transfer image_def, rule refl)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   409
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   410
lemma starset_vimage: "*s* (f -` A) = ( *f* f) -` ( *s* A)"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   411
by (transfer vimage_def, rule refl)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   412
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   413
lemma starset_subset: "( *s* A \<subseteq> *s* B) = (A \<subseteq> B)"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   414
by (transfer subset_eq, rule refl)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   415
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   416
lemma starset_eq: "( *s* A = *s* B) = (A = B)"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   417
by (transfer, rule refl)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   418
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   419
lemmas starset_simps [simp] =
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   420
  starset_mem     starset_UNIV
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   421
  starset_empty   starset_insert
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   422
  starset_Un      starset_Int
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   423
  starset_Compl   starset_diff
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   424
  starset_image   starset_vimage
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   425
  starset_subset  starset_eq
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   426
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   427
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   428
subsection {* Syntactic classes *}
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   429
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   430
instantiation star :: (zero) zero
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   431
begin
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   432
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   433
definition
28562
4e74209f113e `code func` now just `code`
haftmann
parents: 28059
diff changeset
   434
  star_zero_def [code del]:    "0 \<equiv> star_of 0"
27468
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   435
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   436
instance ..
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   437
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   438
end
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   439
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   440
instantiation star :: (one) one
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   441
begin
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   442
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   443
definition
28562
4e74209f113e `code func` now just `code`
haftmann
parents: 28059
diff changeset
   444
  star_one_def [code del]:     "1 \<equiv> star_of 1"
27468
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   445
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   446
instance ..
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   447
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   448
end
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   449
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   450
instantiation star :: (plus) plus
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   451
begin
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   452
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   453
definition
28562
4e74209f113e `code func` now just `code`
haftmann
parents: 28059
diff changeset
   454
  star_add_def [code del]:     "(op +) \<equiv> *f2* (op +)"
27468
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   455
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   456
instance ..
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   457
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   458
end
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   459
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   460
instantiation star :: (times) times
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   461
begin
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   462
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   463
definition
28562
4e74209f113e `code func` now just `code`
haftmann
parents: 28059
diff changeset
   464
  star_mult_def [code del]:    "(op *) \<equiv> *f2* (op *)"
27468
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   465
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   466
instance ..
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   467
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   468
end
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   469
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   470
instantiation star :: (uminus) uminus
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   471
begin
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   472
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   473
definition
28562
4e74209f113e `code func` now just `code`
haftmann
parents: 28059
diff changeset
   474
  star_minus_def [code del]:   "uminus \<equiv> *f* uminus"
27468
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   475
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   476
instance ..
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   477
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   478
end
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   479
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   480
instantiation star :: (minus) minus
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   481
begin
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   482
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   483
definition
28562
4e74209f113e `code func` now just `code`
haftmann
parents: 28059
diff changeset
   484
  star_diff_def [code del]:    "(op -) \<equiv> *f2* (op -)"
27468
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   485
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   486
instance ..
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   487
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   488
end
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   489
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   490
instantiation star :: (abs) abs
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   491
begin
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   492
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   493
definition
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   494
  star_abs_def:     "abs \<equiv> *f* abs"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   495
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   496
instance ..
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   497
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   498
end
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   499
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   500
instantiation star :: (sgn) sgn
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   501
begin
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   502
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   503
definition
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   504
  star_sgn_def:     "sgn \<equiv> *f* sgn"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   505
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   506
instance ..
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   507
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   508
end
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   509
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   510
instantiation star :: (inverse) inverse
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   511
begin
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   512
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   513
definition
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   514
  star_divide_def:  "(op /) \<equiv> *f2* (op /)"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   515
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   516
definition
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   517
  star_inverse_def: "inverse \<equiv> *f* inverse"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   518
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   519
instance ..
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   520
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   521
end
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   522
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   523
instantiation star :: (number) number
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   524
begin
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   525
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   526
definition
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   527
  star_number_def:  "number_of b \<equiv> star_of (number_of b)"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   528
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   529
instance ..
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   530
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   531
end
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   532
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27468
diff changeset
   533
instance star :: (Ring_and_Field.dvd) Ring_and_Field.dvd ..
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27468
diff changeset
   534
27468
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   535
instantiation star :: (Divides.div) Divides.div
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   536
begin
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   537
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   538
definition
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   539
  star_div_def:     "(op div) \<equiv> *f2* (op div)"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   540
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   541
definition
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   542
  star_mod_def:     "(op mod) \<equiv> *f2* (op mod)"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   543
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   544
instance ..
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   545
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   546
end
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   547
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   548
instantiation star :: (ord) ord
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   549
begin
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   550
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   551
definition
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   552
  star_le_def:      "(op \<le>) \<equiv> *p2* (op \<le>)"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   553
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   554
definition
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   555
  star_less_def:    "(op <) \<equiv> *p2* (op <)"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   556
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   557
instance ..
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   558
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   559
end
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   560
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   561
lemmas star_class_defs [transfer_unfold] =
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   562
  star_zero_def     star_one_def      star_number_def
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   563
  star_add_def      star_diff_def     star_minus_def
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   564
  star_mult_def     star_divide_def   star_inverse_def
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   565
  star_le_def       star_less_def     star_abs_def       star_sgn_def
30968
10fef94f40fc adaptions due to rearrangment of power operation
haftmann
parents: 30729
diff changeset
   566
  star_div_def      star_mod_def
27468
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   567
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   568
text {* Class operations preserve standard elements *}
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   569
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   570
lemma Standard_zero: "0 \<in> Standard"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   571
by (simp add: star_zero_def)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   572
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   573
lemma Standard_one: "1 \<in> Standard"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   574
by (simp add: star_one_def)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   575
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   576
lemma Standard_number_of: "number_of b \<in> Standard"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   577
by (simp add: star_number_def)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   578
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   579
lemma Standard_add: "\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> x + y \<in> Standard"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   580
by (simp add: star_add_def)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   581
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   582
lemma Standard_diff: "\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> x - y \<in> Standard"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   583
by (simp add: star_diff_def)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   584
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   585
lemma Standard_minus: "x \<in> Standard \<Longrightarrow> - x \<in> Standard"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   586
by (simp add: star_minus_def)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   587
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   588
lemma Standard_mult: "\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> x * y \<in> Standard"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   589
by (simp add: star_mult_def)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   590
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   591
lemma Standard_divide: "\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> x / y \<in> Standard"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   592
by (simp add: star_divide_def)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   593
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   594
lemma Standard_inverse: "x \<in> Standard \<Longrightarrow> inverse x \<in> Standard"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   595
by (simp add: star_inverse_def)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   596
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   597
lemma Standard_abs: "x \<in> Standard \<Longrightarrow> abs x \<in> Standard"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   598
by (simp add: star_abs_def)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   599
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   600
lemma Standard_div: "\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> x div y \<in> Standard"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   601
by (simp add: star_div_def)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   602
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   603
lemma Standard_mod: "\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> x mod y \<in> Standard"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   604
by (simp add: star_mod_def)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   605
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   606
lemmas Standard_simps [simp] =
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   607
  Standard_zero  Standard_one  Standard_number_of
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   608
  Standard_add  Standard_diff  Standard_minus
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   609
  Standard_mult  Standard_divide  Standard_inverse
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   610
  Standard_abs  Standard_div  Standard_mod
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   611
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   612
text {* @{term star_of} preserves class operations *}
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   613
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   614
lemma star_of_add: "star_of (x + y) = star_of x + star_of y"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   615
by transfer (rule refl)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   616
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   617
lemma star_of_diff: "star_of (x - y) = star_of x - star_of y"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   618
by transfer (rule refl)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   619
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   620
lemma star_of_minus: "star_of (-x) = - star_of x"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   621
by transfer (rule refl)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   622
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   623
lemma star_of_mult: "star_of (x * y) = star_of x * star_of y"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   624
by transfer (rule refl)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   625
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   626
lemma star_of_divide: "star_of (x / y) = star_of x / star_of y"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   627
by transfer (rule refl)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   628
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   629
lemma star_of_inverse: "star_of (inverse x) = inverse (star_of x)"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   630
by transfer (rule refl)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   631
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   632
lemma star_of_div: "star_of (x div y) = star_of x div star_of y"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   633
by transfer (rule refl)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   634
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   635
lemma star_of_mod: "star_of (x mod y) = star_of x mod star_of y"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   636
by transfer (rule refl)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   637
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   638
lemma star_of_abs: "star_of (abs x) = abs (star_of x)"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   639
by transfer (rule refl)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   640
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   641
text {* @{term star_of} preserves numerals *}
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   642
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   643
lemma star_of_zero: "star_of 0 = 0"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   644
by transfer (rule refl)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   645
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   646
lemma star_of_one: "star_of 1 = 1"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   647
by transfer (rule refl)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   648
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   649
lemma star_of_number_of: "star_of (number_of x) = number_of x"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   650
by transfer (rule refl)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   651
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   652
text {* @{term star_of} preserves orderings *}
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   653
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   654
lemma star_of_less: "(star_of x < star_of y) = (x < y)"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   655
by transfer (rule refl)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   656
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   657
lemma star_of_le: "(star_of x \<le> star_of y) = (x \<le> y)"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   658
by transfer (rule refl)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   659
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   660
lemma star_of_eq: "(star_of x = star_of y) = (x = y)"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   661
by transfer (rule refl)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   662
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   663
text{*As above, for 0*}
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   664
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   665
lemmas star_of_0_less = star_of_less [of 0, simplified star_of_zero]
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   666
lemmas star_of_0_le   = star_of_le   [of 0, simplified star_of_zero]
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   667
lemmas star_of_0_eq   = star_of_eq   [of 0, simplified star_of_zero]
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   668
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   669
lemmas star_of_less_0 = star_of_less [of _ 0, simplified star_of_zero]
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   670
lemmas star_of_le_0   = star_of_le   [of _ 0, simplified star_of_zero]
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   671
lemmas star_of_eq_0   = star_of_eq   [of _ 0, simplified star_of_zero]
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   672
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   673
text{*As above, for 1*}
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   674
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   675
lemmas star_of_1_less = star_of_less [of 1, simplified star_of_one]
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   676
lemmas star_of_1_le   = star_of_le   [of 1, simplified star_of_one]
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   677
lemmas star_of_1_eq   = star_of_eq   [of 1, simplified star_of_one]
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   678
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   679
lemmas star_of_less_1 = star_of_less [of _ 1, simplified star_of_one]
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   680
lemmas star_of_le_1   = star_of_le   [of _ 1, simplified star_of_one]
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   681
lemmas star_of_eq_1   = star_of_eq   [of _ 1, simplified star_of_one]
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   682
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   683
text{*As above, for numerals*}
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   684
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   685
lemmas star_of_number_less =
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   686
  star_of_less [of "number_of w", standard, simplified star_of_number_of]
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   687
lemmas star_of_number_le   =
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   688
  star_of_le   [of "number_of w", standard, simplified star_of_number_of]
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   689
lemmas star_of_number_eq   =
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   690
  star_of_eq   [of "number_of w", standard, simplified star_of_number_of]
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   691
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   692
lemmas star_of_less_number =
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   693
  star_of_less [of _ "number_of w", standard, simplified star_of_number_of]
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   694
lemmas star_of_le_number   =
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   695
  star_of_le   [of _ "number_of w", standard, simplified star_of_number_of]
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   696
lemmas star_of_eq_number   =
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   697
  star_of_eq   [of _ "number_of w", standard, simplified star_of_number_of]
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   698
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   699
lemmas star_of_simps [simp] =
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   700
  star_of_add     star_of_diff    star_of_minus
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   701
  star_of_mult    star_of_divide  star_of_inverse
30968
10fef94f40fc adaptions due to rearrangment of power operation
haftmann
parents: 30729
diff changeset
   702
  star_of_div     star_of_mod     star_of_abs
27468
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   703
  star_of_zero    star_of_one     star_of_number_of
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   704
  star_of_less    star_of_le      star_of_eq
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   705
  star_of_0_less  star_of_0_le    star_of_0_eq
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   706
  star_of_less_0  star_of_le_0    star_of_eq_0
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   707
  star_of_1_less  star_of_1_le    star_of_1_eq
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   708
  star_of_less_1  star_of_le_1    star_of_eq_1
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   709
  star_of_number_less star_of_number_le star_of_number_eq
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   710
  star_of_less_number star_of_le_number star_of_eq_number
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   711
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   712
subsection {* Ordering and lattice classes *}
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   713
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   714
instance star :: (order) order
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   715
apply (intro_classes)
27682
25aceefd4786 added class preorder
haftmann
parents: 27651
diff changeset
   716
apply (transfer, rule less_le_not_le)
27468
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   717
apply (transfer, rule order_refl)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   718
apply (transfer, erule (1) order_trans)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   719
apply (transfer, erule (1) order_antisym)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   720
done
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   721
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 31021
diff changeset
   722
instantiation star :: (semilattice_inf) semilattice_inf
27468
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   723
begin
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   724
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   725
definition
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   726
  star_inf_def [transfer_unfold]: "inf \<equiv> *f2* inf"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   727
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   728
instance
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   729
  by default (transfer star_inf_def, auto)+
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   730
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   731
end
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   732
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 31021
diff changeset
   733
instantiation star :: (semilattice_sup) semilattice_sup
27468
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   734
begin
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   735
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   736
definition
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   737
  star_sup_def [transfer_unfold]: "sup \<equiv> *f2* sup"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   738
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   739
instance
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   740
  by default (transfer star_sup_def, auto)+
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   741
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   742
end
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   743
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   744
instance star :: (lattice) lattice ..
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   745
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   746
instance star :: (distrib_lattice) distrib_lattice
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   747
  by default (transfer, auto simp add: sup_inf_distrib1)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   748
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   749
lemma Standard_inf [simp]:
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   750
  "\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> inf x y \<in> Standard"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   751
by (simp add: star_inf_def)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   752
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   753
lemma Standard_sup [simp]:
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   754
  "\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> sup x y \<in> Standard"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   755
by (simp add: star_sup_def)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   756
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   757
lemma star_of_inf [simp]: "star_of (inf x y) = inf (star_of x) (star_of y)"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   758
by transfer (rule refl)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   759
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   760
lemma star_of_sup [simp]: "star_of (sup x y) = sup (star_of x) (star_of y)"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   761
by transfer (rule refl)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   762
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   763
instance star :: (linorder) linorder
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   764
by (intro_classes, transfer, rule linorder_linear)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   765
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   766
lemma star_max_def [transfer_unfold]: "max = *f2* max"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   767
apply (rule ext, rule ext)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   768
apply (unfold max_def, transfer, fold max_def)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   769
apply (rule refl)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   770
done
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   771
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   772
lemma star_min_def [transfer_unfold]: "min = *f2* min"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   773
apply (rule ext, rule ext)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   774
apply (unfold min_def, transfer, fold min_def)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   775
apply (rule refl)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   776
done
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   777
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   778
lemma Standard_max [simp]:
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   779
  "\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> max x y \<in> Standard"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   780
by (simp add: star_max_def)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   781
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   782
lemma Standard_min [simp]:
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   783
  "\<lbrakk>x \<in> Standard; y \<in> Standard\<rbrakk> \<Longrightarrow> min x y \<in> Standard"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   784
by (simp add: star_min_def)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   785
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   786
lemma star_of_max [simp]: "star_of (max x y) = max (star_of x) (star_of y)"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   787
by transfer (rule refl)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   788
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   789
lemma star_of_min [simp]: "star_of (min x y) = min (star_of x) (star_of y)"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   790
by transfer (rule refl)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   791
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   792
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   793
subsection {* Ordered group classes *}
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   794
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   795
instance star :: (semigroup_add) semigroup_add
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   796
by (intro_classes, transfer, rule add_assoc)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   797
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   798
instance star :: (ab_semigroup_add) ab_semigroup_add
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   799
by (intro_classes, transfer, rule add_commute)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   800
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   801
instance star :: (semigroup_mult) semigroup_mult
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   802
by (intro_classes, transfer, rule mult_assoc)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   803
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   804
instance star :: (ab_semigroup_mult) ab_semigroup_mult
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   805
by (intro_classes, transfer, rule mult_commute)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   806
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   807
instance star :: (comm_monoid_add) comm_monoid_add
28059
295a8fc92684 fixed names of class assumptions
haftmann
parents: 27682
diff changeset
   808
by (intro_classes, transfer, rule comm_monoid_add_class.add_0)
27468
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   809
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   810
instance star :: (monoid_mult) monoid_mult
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   811
apply (intro_classes)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   812
apply (transfer, rule mult_1_left)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   813
apply (transfer, rule mult_1_right)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   814
done
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   815
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   816
instance star :: (comm_monoid_mult) comm_monoid_mult
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   817
by (intro_classes, transfer, rule mult_1)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   818
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   819
instance star :: (cancel_semigroup_add) cancel_semigroup_add
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   820
apply (intro_classes)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   821
apply (transfer, erule add_left_imp_eq)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   822
apply (transfer, erule add_right_imp_eq)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   823
done
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   824
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   825
instance star :: (cancel_ab_semigroup_add) cancel_ab_semigroup_add
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   826
by (intro_classes, transfer, rule add_imp_eq)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   827
29904
856f16a3b436 add class cancel_comm_monoid_add
huffman
parents: 29234
diff changeset
   828
instance star :: (cancel_comm_monoid_add) cancel_comm_monoid_add ..
856f16a3b436 add class cancel_comm_monoid_add
huffman
parents: 29234
diff changeset
   829
27468
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   830
instance star :: (ab_group_add) ab_group_add
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   831
apply (intro_classes)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   832
apply (transfer, rule left_minus)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   833
apply (transfer, rule diff_minus)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   834
done
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   835
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 31021
diff changeset
   836
instance star :: (ordered_ab_semigroup_add) ordered_ab_semigroup_add
27468
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   837
by (intro_classes, transfer, rule add_left_mono)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   838
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 31021
diff changeset
   839
instance star :: (ordered_cancel_ab_semigroup_add) ordered_cancel_ab_semigroup_add ..
27468
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   840
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 31021
diff changeset
   841
instance star :: (ordered_ab_semigroup_add_imp_le) ordered_ab_semigroup_add_imp_le
27468
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   842
by (intro_classes, transfer, rule add_le_imp_le_left)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   843
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 31021
diff changeset
   844
instance star :: (ordered_comm_monoid_add) ordered_comm_monoid_add ..
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 31021
diff changeset
   845
instance star :: (ordered_ab_group_add) ordered_ab_group_add ..
27468
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   846
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 31021
diff changeset
   847
instance star :: (ordered_ab_group_add_abs) ordered_ab_group_add_abs 
27468
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   848
  by intro_classes (transfer,
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   849
    simp add: abs_ge_self abs_leI abs_triangle_ineq)+
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   850
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 31021
diff changeset
   851
instance star :: (linordered_cancel_ab_semigroup_add) linordered_cancel_ab_semigroup_add ..
27468
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   852
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   853
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   854
subsection {* Ring and field classes *}
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   855
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   856
instance star :: (semiring) semiring
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   857
apply (intro_classes)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   858
apply (transfer, rule left_distrib)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   859
apply (transfer, rule right_distrib)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   860
done
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   861
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   862
instance star :: (semiring_0) semiring_0 
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   863
by intro_classes (transfer, simp)+
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   864
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   865
instance star :: (semiring_0_cancel) semiring_0_cancel ..
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   866
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   867
instance star :: (comm_semiring) comm_semiring 
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   868
by (intro_classes, transfer, rule left_distrib)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   869
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   870
instance star :: (comm_semiring_0) comm_semiring_0 ..
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   871
instance star :: (comm_semiring_0_cancel) comm_semiring_0_cancel ..
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   872
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   873
instance star :: (zero_neq_one) zero_neq_one
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   874
by (intro_classes, transfer, rule zero_neq_one)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   875
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   876
instance star :: (semiring_1) semiring_1 ..
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   877
instance star :: (comm_semiring_1) comm_semiring_1 ..
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   878
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   879
instance star :: (no_zero_divisors) no_zero_divisors
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   880
by (intro_classes, transfer, rule no_zero_divisors)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   881
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   882
instance star :: (semiring_1_cancel) semiring_1_cancel ..
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   883
instance star :: (comm_semiring_1_cancel) comm_semiring_1_cancel ..
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   884
instance star :: (ring) ring ..
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   885
instance star :: (comm_ring) comm_ring ..
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   886
instance star :: (ring_1) ring_1 ..
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   887
instance star :: (comm_ring_1) comm_ring_1 ..
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   888
instance star :: (ring_no_zero_divisors) ring_no_zero_divisors ..
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   889
instance star :: (ring_1_no_zero_divisors) ring_1_no_zero_divisors ..
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   890
instance star :: (idom) idom .. 
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   891
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   892
instance star :: (division_ring) division_ring
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   893
apply (intro_classes)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   894
apply (transfer, erule left_inverse)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   895
apply (transfer, erule right_inverse)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   896
done
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   897
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   898
instance star :: (field) field
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   899
apply (intro_classes)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   900
apply (transfer, erule left_inverse)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   901
apply (transfer, rule divide_inverse)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   902
done
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   903
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   904
instance star :: (division_by_zero) division_by_zero
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   905
by (intro_classes, transfer, rule inverse_zero)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   906
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 31021
diff changeset
   907
instance star :: (ordered_semiring) ordered_semiring
27468
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   908
apply (intro_classes)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   909
apply (transfer, erule (1) mult_left_mono)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   910
apply (transfer, erule (1) mult_right_mono)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   911
done
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   912
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 31021
diff changeset
   913
instance star :: (ordered_cancel_semiring) ordered_cancel_semiring ..
27468
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   914
35043
07dbdf60d5ad dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents: 35035
diff changeset
   915
instance star :: (linordered_semiring_strict) linordered_semiring_strict
27468
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   916
apply (intro_classes)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   917
apply (transfer, erule (1) mult_strict_left_mono)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   918
apply (transfer, erule (1) mult_strict_right_mono)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   919
done
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   920
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 31021
diff changeset
   921
instance star :: (ordered_comm_semiring) ordered_comm_semiring
28059
295a8fc92684 fixed names of class assumptions
haftmann
parents: 27682
diff changeset
   922
by (intro_classes, transfer, rule mult_mono1_class.mult_mono1)
27468
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   923
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 31021
diff changeset
   924
instance star :: (ordered_cancel_comm_semiring) ordered_cancel_comm_semiring ..
27468
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   925
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 31021
diff changeset
   926
instance star :: (linordered_comm_semiring_strict) linordered_comm_semiring_strict
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 31021
diff changeset
   927
by (intro_classes, transfer, rule mult_strict_left_mono_comm)
27468
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   928
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 31021
diff changeset
   929
instance star :: (ordered_ring) ordered_ring ..
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 31021
diff changeset
   930
instance star :: (ordered_ring_abs) ordered_ring_abs
27468
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   931
  by intro_classes  (transfer, rule abs_eq_mult)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   932
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   933
instance star :: (abs_if) abs_if
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   934
by (intro_classes, transfer, rule abs_if)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   935
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   936
instance star :: (sgn_if) sgn_if
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   937
by (intro_classes, transfer, rule sgn_if)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   938
35043
07dbdf60d5ad dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents: 35035
diff changeset
   939
instance star :: (linordered_ring_strict) linordered_ring_strict ..
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 31021
diff changeset
   940
instance star :: (ordered_comm_ring) ordered_comm_ring ..
27468
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   941
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 31021
diff changeset
   942
instance star :: (linordered_semidom) linordered_semidom
27468
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   943
by (intro_classes, transfer, rule zero_less_one)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   944
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 31021
diff changeset
   945
instance star :: (linordered_idom) linordered_idom ..
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 31021
diff changeset
   946
instance star :: (linordered_field) linordered_field ..
27468
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   947
30968
10fef94f40fc adaptions due to rearrangment of power operation
haftmann
parents: 30729
diff changeset
   948
10fef94f40fc adaptions due to rearrangment of power operation
haftmann
parents: 30729
diff changeset
   949
subsection {* Power *}
10fef94f40fc adaptions due to rearrangment of power operation
haftmann
parents: 30729
diff changeset
   950
10fef94f40fc adaptions due to rearrangment of power operation
haftmann
parents: 30729
diff changeset
   951
lemma star_power_def [transfer_unfold]:
10fef94f40fc adaptions due to rearrangment of power operation
haftmann
parents: 30729
diff changeset
   952
  "(op ^) \<equiv> \<lambda>x n. ( *f* (\<lambda>x. x ^ n)) x"
10fef94f40fc adaptions due to rearrangment of power operation
haftmann
parents: 30729
diff changeset
   953
proof (rule eq_reflection, rule ext, rule ext)
10fef94f40fc adaptions due to rearrangment of power operation
haftmann
parents: 30729
diff changeset
   954
  fix n :: nat
10fef94f40fc adaptions due to rearrangment of power operation
haftmann
parents: 30729
diff changeset
   955
  show "\<And>x::'a star. x ^ n = ( *f* (\<lambda>x. x ^ n)) x" 
10fef94f40fc adaptions due to rearrangment of power operation
haftmann
parents: 30729
diff changeset
   956
  proof (induct n)
10fef94f40fc adaptions due to rearrangment of power operation
haftmann
parents: 30729
diff changeset
   957
    case 0
10fef94f40fc adaptions due to rearrangment of power operation
haftmann
parents: 30729
diff changeset
   958
    have "\<And>x::'a star. ( *f* (\<lambda>x. 1)) x = 1"
10fef94f40fc adaptions due to rearrangment of power operation
haftmann
parents: 30729
diff changeset
   959
      by transfer simp
10fef94f40fc adaptions due to rearrangment of power operation
haftmann
parents: 30729
diff changeset
   960
    then show ?case by simp
10fef94f40fc adaptions due to rearrangment of power operation
haftmann
parents: 30729
diff changeset
   961
  next
10fef94f40fc adaptions due to rearrangment of power operation
haftmann
parents: 30729
diff changeset
   962
    case (Suc n)
10fef94f40fc adaptions due to rearrangment of power operation
haftmann
parents: 30729
diff changeset
   963
    have "\<And>x::'a star. x * ( *f* (\<lambda>x\<Colon>'a. x ^ n)) x = ( *f* (\<lambda>x\<Colon>'a. x * x ^ n)) x"
10fef94f40fc adaptions due to rearrangment of power operation
haftmann
parents: 30729
diff changeset
   964
      by transfer simp
10fef94f40fc adaptions due to rearrangment of power operation
haftmann
parents: 30729
diff changeset
   965
    with Suc show ?case by simp
10fef94f40fc adaptions due to rearrangment of power operation
haftmann
parents: 30729
diff changeset
   966
  qed
10fef94f40fc adaptions due to rearrangment of power operation
haftmann
parents: 30729
diff changeset
   967
qed
27468
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   968
30968
10fef94f40fc adaptions due to rearrangment of power operation
haftmann
parents: 30729
diff changeset
   969
lemma Standard_power [simp]: "x \<in> Standard \<Longrightarrow> x ^ n \<in> Standard"
10fef94f40fc adaptions due to rearrangment of power operation
haftmann
parents: 30729
diff changeset
   970
  by (simp add: star_power_def)
10fef94f40fc adaptions due to rearrangment of power operation
haftmann
parents: 30729
diff changeset
   971
10fef94f40fc adaptions due to rearrangment of power operation
haftmann
parents: 30729
diff changeset
   972
lemma star_of_power [simp]: "star_of (x ^ n) = star_of x ^ n"
10fef94f40fc adaptions due to rearrangment of power operation
haftmann
parents: 30729
diff changeset
   973
  by transfer (rule refl)
10fef94f40fc adaptions due to rearrangment of power operation
haftmann
parents: 30729
diff changeset
   974
27468
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   975
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   976
subsection {* Number classes *}
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   977
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   978
lemma star_of_nat_def [transfer_unfold]: "of_nat n = star_of (of_nat n)"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   979
by (induct n, simp_all)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   980
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   981
lemma Standard_of_nat [simp]: "of_nat n \<in> Standard"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   982
by (simp add: star_of_nat_def)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   983
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   984
lemma star_of_of_nat [simp]: "star_of (of_nat n) = of_nat n"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   985
by transfer (rule refl)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   986
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   987
lemma star_of_int_def [transfer_unfold]: "of_int z = star_of (of_int z)"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   988
by (rule_tac z=z in int_diff_cases, simp)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   989
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   990
lemma Standard_of_int [simp]: "of_int z \<in> Standard"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   991
by (simp add: star_of_int_def)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   992
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   993
lemma star_of_of_int [simp]: "star_of (of_int z) = of_int z"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   994
by transfer (rule refl)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   995
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   996
instance star :: (semiring_char_0) semiring_char_0
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   997
by intro_classes (simp only: star_of_nat_def star_of_eq of_nat_eq_iff)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   998
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
   999
instance star :: (ring_char_0) ring_char_0 ..
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
  1000
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
  1001
instance star :: (number_ring) number_ring
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
  1002
by (intro_classes, simp only: star_number_def star_of_int_def number_of_eq)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
  1003
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
  1004
subsection {* Finite class *}
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
  1005
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
  1006
lemma starset_finite: "finite A \<Longrightarrow> *s* A = star_of ` A"
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
  1007
by (erule finite_induct, simp_all)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
  1008
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
  1009
instance star :: (finite) finite
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
  1010
apply (intro_classes)
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
  1011
apply (subst starset_UNIV [symmetric])
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
  1012
apply (subst starset_finite [OF finite])
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
  1013
apply (rule finite_imageI [OF finite])
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
  1014
done
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
  1015
0783dd1dc13d move nonstandard analysis theories to NSA directory
huffman
parents:
diff changeset
  1016
end