author  huffman 
Fri, 16 May 2008 23:25:37 +0200  
changeset 26927  8684b5240f11 
parent 26806  40b411ec05aa 
child 27267  5ebfb7f25ebb 
permissions  rwrr 
25904  1 
(* Title: HOLCF/CompactBasis.thy 
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ID: $Id$ 

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Author: Brian Huffman 

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*) 

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header {* Compact bases of domains *} 

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theory CompactBasis 

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imports Bifinite SetPcpo 

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begin 

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subsection {* Ideals over a preorder *} 

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context preorder 
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begin 
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definition 

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ideal :: "'a set \<Rightarrow> bool" where 

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"ideal A = ((\<exists>x. x \<in> A) \<and> (\<forall>x\<in>A. \<forall>y\<in>A. \<exists>z\<in>A. x \<sqsubseteq> z \<and> y \<sqsubseteq> z) \<and> 
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(\<forall>x y. x \<sqsubseteq> y \<longrightarrow> y \<in> A \<longrightarrow> x \<in> A))" 
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lemma idealI: 

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assumes "\<exists>x. x \<in> A" 

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assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> \<exists>z\<in>A. x \<sqsubseteq> z \<and> y \<sqsubseteq> z" 
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assumes "\<And>x y. \<lbrakk>x \<sqsubseteq> y; y \<in> A\<rbrakk> \<Longrightarrow> x \<in> A" 
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shows "ideal A" 
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unfolding ideal_def using prems by fast 

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lemma idealD1: 

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"ideal A \<Longrightarrow> \<exists>x. x \<in> A" 

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unfolding ideal_def by fast 

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lemma idealD2: 

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"\<lbrakk>ideal A; x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> \<exists>z\<in>A. x \<sqsubseteq> z \<and> y \<sqsubseteq> z" 
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unfolding ideal_def by fast 
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lemma idealD3: 

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"\<lbrakk>ideal A; x \<sqsubseteq> y; y \<in> A\<rbrakk> \<Longrightarrow> x \<in> A" 
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unfolding ideal_def by fast 
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lemma ideal_directed_finite: 

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assumes A: "ideal A" 

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shows "\<lbrakk>finite U; U \<subseteq> A\<rbrakk> \<Longrightarrow> \<exists>z\<in>A. \<forall>x\<in>U. x \<sqsubseteq> z" 
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apply (induct U set: finite) 
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apply (simp add: idealD1 [OF A]) 

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apply (simp, clarify, rename_tac y) 

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apply (drule (1) idealD2 [OF A]) 

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apply (clarify, erule_tac x=z in rev_bexI) 

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apply (fast intro: trans_less) 
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done 
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lemma ideal_principal: "ideal {x. x \<sqsubseteq> z}" 
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apply (rule idealI) 
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apply (rule_tac x=z in exI) 

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apply fast 
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apply (rule_tac x=z in bexI, fast) 
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apply fast 
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apply (fast intro: trans_less) 
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done 
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lemma directed_image_ideal: 

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assumes A: "ideal A" 

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assumes f: "\<And>x y. x \<sqsubseteq> y \<Longrightarrow> f x \<sqsubseteq> f y" 
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shows "directed (f ` A)" 
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apply (rule directedI) 

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apply (cut_tac idealD1 [OF A], fast) 

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apply (clarify, rename_tac a b) 

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apply (drule (1) idealD2 [OF A]) 

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apply (clarify, rename_tac c) 

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apply (rule_tac x="f c" in rev_bexI) 

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apply (erule imageI) 

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apply (simp add: f) 

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done 

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lemma adm_ideal: "adm (\<lambda>A. ideal A)" 

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unfolding ideal_def by (intro adm_lemmas adm_set_lemmas) 

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lemma lub_image_principal: 
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assumes f: "\<And>x y. x \<sqsubseteq> y \<Longrightarrow> f x \<sqsubseteq> f y" 
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shows "(\<Squnion>x\<in>{x. x \<sqsubseteq> y}. f x) = f y" 
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apply (rule thelubI) 
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apply (rule is_lub_maximal) 

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apply (rule ub_imageI) 

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apply (simp add: f) 

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apply (rule imageI) 

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apply simp 
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done 
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end 
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subsection {* Defining functions in terms of basis elements *} 
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lemma finite_directed_contains_lub: 
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"\<lbrakk>finite S; directed S\<rbrakk> \<Longrightarrow> \<exists>u\<in>S. S << u" 

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apply (drule (1) directed_finiteD, rule subset_refl) 
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apply (erule bexE) 

26034  97 
apply (rule rev_bexI, assumption) 
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apply (erule (1) is_lub_maximal) 
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done 

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26034  101 
lemma lub_finite_directed_in_self: 
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"\<lbrakk>finite S; directed S\<rbrakk> \<Longrightarrow> lub S \<in> S" 

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apply (drule (1) finite_directed_contains_lub, clarify) 

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apply (drule thelubI, simp) 

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done 

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lemma finite_directed_has_lub: "\<lbrakk>finite S; directed S\<rbrakk> \<Longrightarrow> \<exists>u. S << u" 

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by (drule (1) finite_directed_contains_lub, fast) 

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lemma is_ub_thelub0: "\<lbrakk>\<exists>u. S << u; x \<in> S\<rbrakk> \<Longrightarrow> x \<sqsubseteq> lub S" 
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apply (erule exE, drule lubI) 

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apply (drule is_lubD1) 

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apply (erule (1) is_ubD) 

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done 

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lemma is_lub_thelub0: "\<lbrakk>\<exists>u. S << u; S < x\<rbrakk> \<Longrightarrow> lub S \<sqsubseteq> x" 

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by (erule exE, drule lubI, erule is_lub_lub) 

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26927  119 
locale basis_take = preorder r + 
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fixes take :: "nat \<Rightarrow> 'a::type \<Rightarrow> 'a" 
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assumes take_less: "r (take n a) a" 

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assumes take_take: "take n (take n a) = take n a" 

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assumes take_mono: "r a b \<Longrightarrow> r (take n a) (take n b)" 

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assumes take_chain: "r (take n a) (take (Suc n) a)" 

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assumes finite_range_take: "finite (range (take n))" 

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assumes take_covers: "\<exists>n. take n a = a" 

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locale ideal_completion = basis_take r + 
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fixes principal :: "'a::type \<Rightarrow> 'b::cpo" 
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fixes rep :: "'b::cpo \<Rightarrow> 'a::type set" 
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assumes ideal_rep: "\<And>x. preorder.ideal r (rep x)" 

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assumes cont_rep: "cont rep" 

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assumes rep_principal: "\<And>a. rep (principal a) = {b. r b a}" 

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assumes subset_repD: "\<And>x y. rep x \<subseteq> rep y \<Longrightarrow> x \<sqsubseteq> y" 

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begin 
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lemma finite_take_rep: "finite (take n ` rep x)" 
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by (rule finite_subset [OF image_mono [OF subset_UNIV] finite_range_take]) 
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lemma basis_fun_lemma0: 

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fixes f :: "'a::type \<Rightarrow> 'c::cpo" 

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assumes f_mono: "\<And>a b. r a b \<Longrightarrow> f a \<sqsubseteq> f b" 

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shows "\<exists>u. f ` take i ` rep x << u" 
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apply (rule finite_directed_has_lub) 
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apply (rule finite_imageI) 

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apply (rule finite_take_rep) 
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apply (subst image_image) 
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apply (rule directed_image_ideal) 

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apply (rule ideal_rep) 
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apply (rule f_mono) 
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apply (erule take_mono) 

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done 

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lemma basis_fun_lemma1: 

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fixes f :: "'a::type \<Rightarrow> 'c::cpo" 

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assumes f_mono: "\<And>a b. r a b \<Longrightarrow> f a \<sqsubseteq> f b" 

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shows "chain (\<lambda>i. lub (f ` take i ` rep x))" 
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apply (rule chainI) 
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apply (rule is_lub_thelub0) 

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apply (rule basis_fun_lemma0, erule f_mono) 

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apply (rule is_ubI, clarsimp, rename_tac a) 

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apply (rule sq_le.trans_less [OF f_mono [OF take_chain]]) 
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apply (rule is_ub_thelub0) 
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apply (rule basis_fun_lemma0, erule f_mono) 

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apply simp 

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done 

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lemma basis_fun_lemma2: 

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fixes f :: "'a::type \<Rightarrow> 'c::cpo" 

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assumes f_mono: "\<And>a b. r a b \<Longrightarrow> f a \<sqsubseteq> f b" 

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shows "f ` rep x << (\<Squnion>i. lub (f ` take i ` rep x))" 
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apply (rule is_lubI) 
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apply (rule ub_imageI, rename_tac a) 

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apply (cut_tac a=a in take_covers, erule exE, rename_tac i) 

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apply (erule subst) 

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apply (rule rev_trans_less) 

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apply (rule_tac x=i in is_ub_thelub) 

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apply (rule basis_fun_lemma1, erule f_mono) 

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apply (rule is_ub_thelub0) 

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apply (rule basis_fun_lemma0, erule f_mono) 

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apply simp 

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apply (rule is_lub_thelub [OF _ ub_rangeI]) 

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apply (rule basis_fun_lemma1, erule f_mono) 

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apply (rule is_lub_thelub0) 

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apply (rule basis_fun_lemma0, erule f_mono) 

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apply (rule is_ubI, clarsimp, rename_tac a) 

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apply (rule sq_le.trans_less [OF f_mono [OF take_less]]) 
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apply (erule (1) ub_imageD) 
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done 

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lemma basis_fun_lemma: 

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fixes f :: "'a::type \<Rightarrow> 'c::cpo" 

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assumes f_mono: "\<And>a b. r a b \<Longrightarrow> f a \<sqsubseteq> f b" 

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shows "\<exists>u. f ` rep x << u" 
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by (rule exI, rule basis_fun_lemma2, erule f_mono) 
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lemma rep_mono: "x \<sqsubseteq> y \<Longrightarrow> rep x \<subseteq> rep y" 
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apply (drule cont_rep [THEN cont2mono, THEN monofunE]) 

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apply (simp add: set_cpo_simps) 
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done 

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lemma rep_contlub: 
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"chain Y \<Longrightarrow> rep (\<Squnion>i. Y i) = (\<Union>i. rep (Y i))" 

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by (simp add: cont2contlubE [OF cont_rep] set_cpo_simps) 

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lemma less_def: "x \<sqsubseteq> y \<longleftrightarrow> rep x \<subseteq> rep y" 
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by (rule iffI [OF rep_mono subset_repD]) 

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lemma rep_eq: "rep x = {a. principal a \<sqsubseteq> x}" 
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unfolding less_def rep_principal 

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apply safe 
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apply (erule (1) idealD3 [OF ideal_rep]) 
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apply (erule subsetD, simp add: refl) 
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done 

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lemma mem_rep_iff_principal_less: "a \<in> rep x \<longleftrightarrow> principal a \<sqsubseteq> x" 
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by (simp add: rep_eq) 

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lemma principal_less_iff_mem_rep: "principal a \<sqsubseteq> x \<longleftrightarrow> a \<in> rep x" 

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by (simp add: rep_eq) 

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lemma principal_less_iff: "principal a \<sqsubseteq> principal b \<longleftrightarrow> r a b" 
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by (simp add: principal_less_iff_mem_rep rep_principal) 

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lemma principal_eq_iff: "principal a = principal b \<longleftrightarrow> r a b \<and> r b a" 
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unfolding po_eq_conv [where 'a='b] principal_less_iff .. 

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lemma repD: "a \<in> rep x \<Longrightarrow> principal a \<sqsubseteq> x" 

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by (simp add: rep_eq) 

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lemma principal_mono: "r a b \<Longrightarrow> principal a \<sqsubseteq> principal b" 

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by (simp add: principal_less_iff) 
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lemma lessI: "(\<And>a. principal a \<sqsubseteq> x \<Longrightarrow> principal a \<sqsubseteq> u) \<Longrightarrow> x \<sqsubseteq> u" 

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unfolding principal_less_iff_mem_rep 
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by (simp add: less_def subset_eq) 
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lemma lub_principal_rep: "principal ` rep x << x" 
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apply (rule is_lubI) 
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apply (rule ub_imageI) 

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apply (erule repD) 
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apply (subst less_def) 
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apply (rule subsetI) 

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apply (drule (1) ub_imageD) 

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apply (simp add: rep_eq) 
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done 
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definition 

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basis_fun :: "('a::type \<Rightarrow> 'c::cpo) \<Rightarrow> 'b \<rightarrow> 'c" where 

26927  250 
"basis_fun = (\<lambda>f. (\<Lambda> x. lub (f ` rep x)))" 
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lemma basis_fun_beta: 

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fixes f :: "'a::type \<Rightarrow> 'c::cpo" 

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assumes f_mono: "\<And>a b. r a b \<Longrightarrow> f a \<sqsubseteq> f b" 

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shows "basis_fun f\<cdot>x = lub (f ` rep x)" 
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unfolding basis_fun_def 
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proof (rule beta_cfun) 

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have lub: "\<And>x. \<exists>u. f ` rep x << u" 
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using f_mono by (rule basis_fun_lemma) 
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show cont: "cont (\<lambda>x. lub (f ` rep x))" 
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apply (rule contI2) 
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apply (rule monofunI) 

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apply (rule is_lub_thelub0 [OF lub ub_imageI]) 

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apply (rule is_ub_thelub0 [OF lub imageI]) 

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apply (erule (1) subsetD [OF rep_mono]) 
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apply (rule is_lub_thelub0 [OF lub ub_imageI]) 
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apply (simp add: rep_contlub, clarify) 
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apply (erule rev_trans_less [OF is_ub_thelub]) 
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apply (erule is_ub_thelub0 [OF lub imageI]) 

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done 

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qed 

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lemma basis_fun_principal: 

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fixes f :: "'a::type \<Rightarrow> 'c::cpo" 

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assumes f_mono: "\<And>a b. r a b \<Longrightarrow> f a \<sqsubseteq> f b" 

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shows "basis_fun f\<cdot>(principal a) = f a" 

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apply (subst basis_fun_beta, erule f_mono) 

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apply (subst rep_principal) 
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apply (rule lub_image_principal, erule f_mono) 
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done 

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lemma basis_fun_mono: 

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assumes f_mono: "\<And>a b. r a b \<Longrightarrow> f a \<sqsubseteq> f b" 

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assumes g_mono: "\<And>a b. r a b \<Longrightarrow> g a \<sqsubseteq> g b" 

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assumes less: "\<And>a. f a \<sqsubseteq> g a" 

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shows "basis_fun f \<sqsubseteq> basis_fun g" 

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apply (rule less_cfun_ext) 

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apply (simp only: basis_fun_beta f_mono g_mono) 

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apply (rule is_lub_thelub0) 

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apply (rule basis_fun_lemma, erule f_mono) 

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apply (rule ub_imageI, rename_tac a) 

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apply (rule sq_le.trans_less [OF less]) 
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apply (rule is_ub_thelub0) 
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apply (rule basis_fun_lemma, erule g_mono) 

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apply (erule imageI) 

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done 

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lemma compact_principal: "compact (principal a)" 

26927  299 
by (rule compactI2, simp add: principal_less_iff_mem_rep rep_contlub) 
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definition 

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completion_approx :: "nat \<Rightarrow> 'b \<rightarrow> 'b" where 

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"completion_approx = (\<lambda>i. basis_fun (\<lambda>a. principal (take i a)))" 

25904  304 

26927  305 
lemma completion_approx_beta: 
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"completion_approx i\<cdot>x = (\<Squnion>a\<in>rep x. principal (take i a))" 

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unfolding completion_approx_def 

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by (simp add: basis_fun_beta principal_mono take_mono) 

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lemma completion_approx_principal: 

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"completion_approx i\<cdot>(principal a) = principal (take i a)" 

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unfolding completion_approx_def 

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by (simp add: basis_fun_principal principal_mono take_mono) 

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lemma chain_completion_approx: "chain completion_approx" 

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unfolding completion_approx_def 

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apply (rule chainI) 
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apply (rule basis_fun_mono) 

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apply (erule principal_mono [OF take_mono]) 

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apply (erule principal_mono [OF take_mono]) 

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apply (rule principal_mono [OF take_chain]) 

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done 

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26927  324 
lemma lub_completion_approx: "(\<Squnion>i. completion_approx i\<cdot>x) = x" 
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unfolding completion_approx_beta 

25904  326 
apply (subst image_image [where f=principal, symmetric]) 
26927  327 
apply (rule unique_lub [OF _ lub_principal_rep]) 
25904  328 
apply (rule basis_fun_lemma2, erule principal_mono) 
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done 

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26927  331 
lemma completion_approx_eq_principal: 
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"\<exists>a\<in>rep x. completion_approx i\<cdot>x = principal (take i a)" 

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unfolding completion_approx_beta 

25904  334 
apply (subst image_image [where f=principal, symmetric]) 
26927  335 
apply (subgoal_tac "finite (principal ` take i ` rep x)") 
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apply (subgoal_tac "directed (principal ` take i ` rep x)") 

25904  337 
apply (drule (1) lub_finite_directed_in_self, fast) 
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apply (subst image_image) 

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apply (rule directed_image_ideal) 

26927  340 
apply (rule ideal_rep) 
25904  341 
apply (erule principal_mono [OF take_mono]) 
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apply (rule finite_imageI) 

26927  343 
apply (rule finite_take_rep) 
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done 

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lemma completion_approx_idem: 

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"completion_approx i\<cdot>(completion_approx i\<cdot>x) = completion_approx i\<cdot>x" 

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using completion_approx_eq_principal [where i=i and x=x] 

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by (auto simp add: completion_approx_principal take_take) 

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lemma finite_fixes_completion_approx: 

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"finite {x. completion_approx i\<cdot>x = x}" (is "finite ?S") 

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apply (subgoal_tac "?S \<subseteq> principal ` range (take i)") 

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apply (erule finite_subset) 

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apply (rule finite_imageI) 

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apply (rule finite_range_take) 

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apply (clarify, erule subst) 

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apply (cut_tac x=x and i=i in completion_approx_eq_principal) 

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apply fast 

25904  360 
done 
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lemma principal_induct: 

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assumes adm: "adm P" 

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assumes P: "\<And>a. P (principal a)" 

365 
shows "P x" 

26927  366 
apply (subgoal_tac "P (\<Squnion>i. completion_approx i\<cdot>x)") 
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apply (simp add: lub_completion_approx) 

25925  368 
apply (rule admD [OF adm]) 
26927  369 
apply (simp add: chain_completion_approx) 
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apply (cut_tac x=x and i=i in completion_approx_eq_principal) 

25904  371 
apply (clarify, simp add: P) 
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done 

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374 
end 

375 

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subsection {* Compact bases of bifinite domains *} 

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defaultsort profinite 
25904  380 

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typedef (open) 'a compact_basis = "{x::'a::profinite. compact x}" 
25904  382 
by (fast intro: compact_approx) 
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lemma compact_Rep_compact_basis [simp]: "compact (Rep_compact_basis a)" 

26927  385 
by (rule Rep_compact_basis [unfolded mem_Collect_eq]) 
25904  386 

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lemma Rep_Abs_compact_basis_approx [simp]: 

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"Rep_compact_basis (Abs_compact_basis (approx n\<cdot>x)) = approx n\<cdot>x" 

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by (rule Abs_compact_basis_inverse, simp) 

390 

391 
lemma compact_imp_Rep_compact_basis: 

392 
"compact x \<Longrightarrow> \<exists>y. x = Rep_compact_basis y" 

393 
by (rule exI, rule Abs_compact_basis_inverse [symmetric], simp) 

394 

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395 
instantiation compact_basis :: (profinite) sq_ord 
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396 
begin 
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397 

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398 
definition 
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399 
compact_le_def: 
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400 
"(op \<sqsubseteq>) \<equiv> (\<lambda>x y. Rep_compact_basis x \<sqsubseteq> Rep_compact_basis y)" 
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401 

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402 
instance .. 
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403 

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404 
end 
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405 

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406 
instance compact_basis :: (profinite) po 
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407 
by (rule typedef_po 
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[OF type_definition_compact_basis compact_le_def]) 
25904  409 

410 
text {* minimal compact element *} 

411 

412 
definition 

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413 
compact_bot :: "'a::bifinite compact_basis" where 
25904  414 
"compact_bot = Abs_compact_basis \<bottom>" 
415 

416 
lemma Rep_compact_bot: "Rep_compact_basis compact_bot = \<bottom>" 

417 
unfolding compact_bot_def by (simp add: Abs_compact_basis_inverse) 

418 

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lemma compact_minimal [simp]: "compact_bot \<sqsubseteq> a" 
25904  420 
unfolding compact_le_def Rep_compact_bot by simp 
421 

422 
text {* compacts *} 

423 

424 
definition 

425 
compacts :: "'a \<Rightarrow> 'a compact_basis set" where 

426 
"compacts = (\<lambda>x. {a. Rep_compact_basis a \<sqsubseteq> x})" 

427 

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lemma ideal_compacts: "preorder.ideal sq_le (compacts w)" 
25904  429 
unfolding compacts_def 
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430 
apply (rule preorder.idealI) 
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431 
apply (rule preorder_class.axioms) 
25904  432 
apply (rule_tac x="Abs_compact_basis (approx 0\<cdot>w)" in exI) 
433 
apply (simp add: approx_less) 

434 
apply simp 

435 
apply (cut_tac a=x in compact_Rep_compact_basis) 

436 
apply (cut_tac a=y in compact_Rep_compact_basis) 

437 
apply (drule bifinite_compact_eq_approx) 

438 
apply (drule bifinite_compact_eq_approx) 

439 
apply (clarify, rename_tac i j) 

440 
apply (rule_tac x="Abs_compact_basis (approx (max i j)\<cdot>w)" in exI) 

441 
apply (simp add: approx_less compact_le_def) 

442 
apply (erule subst, erule subst) 

25922
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443 
apply (simp add: monofun_cfun chain_mono [OF chain_approx]) 
25904  444 
apply (simp add: compact_le_def) 
445 
apply (erule (1) trans_less) 

446 
done 

447 

448 
lemma compacts_Rep_compact_basis: 

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449 
"compacts (Rep_compact_basis b) = {a. a \<sqsubseteq> b}" 
25904  450 
unfolding compacts_def compact_le_def .. 
451 

452 
lemma cont_compacts: "cont compacts" 

453 
unfolding compacts_def 

454 
apply (rule contI2) 

455 
apply (rule monofunI) 

456 
apply (simp add: set_cpo_simps) 

457 
apply (fast intro: trans_less) 

458 
apply (simp add: set_cpo_simps) 

459 
apply clarify 

460 
apply simp 

461 
apply (erule (1) compactD2 [OF compact_Rep_compact_basis]) 

462 
done 

463 

464 
lemma compacts_lessD: "compacts x \<subseteq> compacts y \<Longrightarrow> x \<sqsubseteq> y" 

465 
apply (subgoal_tac "(\<Squnion>i. approx i\<cdot>x) \<sqsubseteq> y", simp) 

25925  466 
apply (rule admD, simp, simp) 
25904  467 
apply (drule_tac c="Abs_compact_basis (approx i\<cdot>x)" in subsetD) 
468 
apply (simp add: compacts_def Abs_compact_basis_inverse approx_less) 

469 
apply (simp add: compacts_def Abs_compact_basis_inverse) 

470 
done 

471 

472 
lemma compacts_mono: "x \<sqsubseteq> y \<Longrightarrow> compacts x \<subseteq> compacts y" 

473 
unfolding compacts_def by (fast intro: trans_less) 

474 

475 
lemma less_compact_basis_iff: "(x \<sqsubseteq> y) = (compacts x \<subseteq> compacts y)" 

476 
by (rule iffI [OF compacts_mono compacts_lessD]) 

477 

478 
lemma compact_basis_induct: 

479 
"\<lbrakk>adm P; \<And>a. P (Rep_compact_basis a)\<rbrakk> \<Longrightarrow> P x" 

480 
apply (erule approx_induct) 

481 
apply (drule_tac x="Abs_compact_basis (approx n\<cdot>x)" in meta_spec) 

482 
apply (simp add: Abs_compact_basis_inverse) 

483 
done 

484 

485 
text {* approximation on compact bases *} 

486 

487 
definition 

488 
compact_approx :: "nat \<Rightarrow> 'a compact_basis \<Rightarrow> 'a compact_basis" where 

489 
"compact_approx = (\<lambda>n a. Abs_compact_basis (approx n\<cdot>(Rep_compact_basis a)))" 

490 

491 
lemma Rep_compact_approx: 

492 
"Rep_compact_basis (compact_approx n a) = approx n\<cdot>(Rep_compact_basis a)" 

493 
unfolding compact_approx_def 

494 
by (simp add: Abs_compact_basis_inverse) 

495 

496 
lemmas approx_Rep_compact_basis = Rep_compact_approx [symmetric] 

497 

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lemma compact_approx_le: "compact_approx n a \<sqsubseteq> a" 
25904  499 
unfolding compact_le_def 
500 
by (simp add: Rep_compact_approx approx_less) 

501 

502 
lemma compact_approx_mono1: 

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503 
"i \<le> j \<Longrightarrow> compact_approx i a \<sqsubseteq> compact_approx j a" 
25904  504 
unfolding compact_le_def 
505 
apply (simp add: Rep_compact_approx) 

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506 
apply (rule chain_mono, simp, assumption) 
25904  507 
done 
508 

509 
lemma compact_approx_mono: 

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"a \<sqsubseteq> b \<Longrightarrow> compact_approx n a \<sqsubseteq> compact_approx n b" 
25904  511 
unfolding compact_le_def 
512 
apply (simp add: Rep_compact_approx) 

513 
apply (erule monofun_cfun_arg) 

514 
done 

515 

516 
lemma ex_compact_approx_eq: "\<exists>n. compact_approx n a = a" 

517 
apply (simp add: Rep_compact_basis_inject [symmetric]) 

518 
apply (simp add: Rep_compact_approx) 

519 
apply (rule bifinite_compact_eq_approx) 

520 
apply (rule compact_Rep_compact_basis) 

521 
done 

522 

523 
lemma compact_approx_idem: 

524 
"compact_approx n (compact_approx n a) = compact_approx n a" 

525 
apply (rule Rep_compact_basis_inject [THEN iffD1]) 

526 
apply (simp add: Rep_compact_approx) 

527 
done 

528 

529 
lemma finite_fixes_compact_approx: "finite {a. compact_approx n a = a}" 

530 
apply (subgoal_tac "finite (Rep_compact_basis ` {a. compact_approx n a = a})") 

531 
apply (erule finite_imageD) 

532 
apply (rule inj_onI, simp add: Rep_compact_basis_inject) 

533 
apply (rule finite_subset [OF _ finite_fixes_approx [where i=n]]) 

534 
apply (rule subsetI, clarify, rename_tac a) 

535 
apply (simp add: Rep_compact_basis_inject [symmetric]) 

536 
apply (simp add: Rep_compact_approx) 

537 
done 

538 

539 
lemma finite_range_compact_approx: "finite (range (compact_approx n))" 

540 
apply (cut_tac n=n in finite_fixes_compact_approx) 

541 
apply (simp add: idem_fixes_eq_range compact_approx_idem) 

542 
apply (simp add: image_def) 

543 
done 

544 

545 
interpretation compact_basis: 

26927  546 
ideal_completion [sq_le compact_approx Rep_compact_basis compacts] 
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547 
proof (unfold_locales) 
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548 
fix n :: nat and a b :: "'a compact_basis" and x :: "'a" 
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549 
show "compact_approx n a \<sqsubseteq> a" 
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550 
by (rule compact_approx_le) 
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551 
show "compact_approx n (compact_approx n a) = compact_approx n a" 
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552 
by (rule compact_approx_idem) 
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553 
show "compact_approx n a \<sqsubseteq> compact_approx (Suc n) a" 
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554 
by (rule compact_approx_mono1, simp) 
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555 
show "finite (range (compact_approx n))" 
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556 
by (rule finite_range_compact_approx) 
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557 
show "\<exists>n\<Colon>nat. compact_approx n a = a" 
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558 
by (rule ex_compact_approx_eq) 
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559 
show "preorder.ideal sq_le (compacts x)" 
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560 
by (rule ideal_compacts) 
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561 
show "cont compacts" 
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562 
by (rule cont_compacts) 
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563 
show "compacts (Rep_compact_basis a) = {b. b \<sqsubseteq> a}" 
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564 
by (rule compacts_Rep_compact_basis) 
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565 
next 
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566 
fix n :: nat and a b :: "'a compact_basis" 
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567 
assume "a \<sqsubseteq> b" 
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568 
thus "compact_approx n a \<sqsubseteq> compact_approx n b" 
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569 
by (rule compact_approx_mono) 
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570 
next 
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571 
fix x y :: "'a" 
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572 
assume "compacts x \<subseteq> compacts y" thus "x \<sqsubseteq> y" 
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573 
by (rule compacts_lessD) 
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574 
qed 
25904  575 

576 

577 
subsection {* A compact basis for powerdomains *} 

578 

579 
typedef 'a pd_basis = 

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580 
"{S::'a::profinite compact_basis set. finite S \<and> S \<noteq> {}}" 
25904  581 
by (rule_tac x="{arbitrary}" in exI, simp) 
582 

583 
lemma finite_Rep_pd_basis [simp]: "finite (Rep_pd_basis u)" 

584 
by (insert Rep_pd_basis [of u, unfolded pd_basis_def]) simp 

585 

586 
lemma Rep_pd_basis_nonempty [simp]: "Rep_pd_basis u \<noteq> {}" 

587 
by (insert Rep_pd_basis [of u, unfolded pd_basis_def]) simp 

588 

589 
text {* unit and plus *} 

590 

591 
definition 

592 
PDUnit :: "'a compact_basis \<Rightarrow> 'a pd_basis" where 

593 
"PDUnit = (\<lambda>x. Abs_pd_basis {x})" 

594 

595 
definition 

596 
PDPlus :: "'a pd_basis \<Rightarrow> 'a pd_basis \<Rightarrow> 'a pd_basis" where 

597 
"PDPlus t u = Abs_pd_basis (Rep_pd_basis t \<union> Rep_pd_basis u)" 

598 

599 
lemma Rep_PDUnit: 

600 
"Rep_pd_basis (PDUnit x) = {x}" 

601 
unfolding PDUnit_def by (rule Abs_pd_basis_inverse) (simp add: pd_basis_def) 

602 

603 
lemma Rep_PDPlus: 

604 
"Rep_pd_basis (PDPlus u v) = Rep_pd_basis u \<union> Rep_pd_basis v" 

605 
unfolding PDPlus_def by (rule Abs_pd_basis_inverse) (simp add: pd_basis_def) 

606 

607 
lemma PDUnit_inject [simp]: "(PDUnit a = PDUnit b) = (a = b)" 

608 
unfolding Rep_pd_basis_inject [symmetric] Rep_PDUnit by simp 

609 

610 
lemma PDPlus_assoc: "PDPlus (PDPlus t u) v = PDPlus t (PDPlus u v)" 

611 
unfolding Rep_pd_basis_inject [symmetric] Rep_PDPlus by (rule Un_assoc) 

612 

613 
lemma PDPlus_commute: "PDPlus t u = PDPlus u t" 

614 
unfolding Rep_pd_basis_inject [symmetric] Rep_PDPlus by (rule Un_commute) 

615 

616 
lemma PDPlus_absorb: "PDPlus t t = t" 

617 
unfolding Rep_pd_basis_inject [symmetric] Rep_PDPlus by (rule Un_absorb) 

618 

619 
lemma pd_basis_induct1: 

620 
assumes PDUnit: "\<And>a. P (PDUnit a)" 

621 
assumes PDPlus: "\<And>a t. P t \<Longrightarrow> P (PDPlus (PDUnit a) t)" 

622 
shows "P x" 

623 
apply (induct x, unfold pd_basis_def, clarify) 

624 
apply (erule (1) finite_ne_induct) 

625 
apply (cut_tac a=x in PDUnit) 

626 
apply (simp add: PDUnit_def) 

627 
apply (drule_tac a=x in PDPlus) 

628 
apply (simp add: PDUnit_def PDPlus_def Abs_pd_basis_inverse [unfolded pd_basis_def]) 

629 
done 

630 

631 
lemma pd_basis_induct: 

632 
assumes PDUnit: "\<And>a. P (PDUnit a)" 

633 
assumes PDPlus: "\<And>t u. \<lbrakk>P t; P u\<rbrakk> \<Longrightarrow> P (PDPlus t u)" 

634 
shows "P x" 

635 
apply (induct x rule: pd_basis_induct1) 

636 
apply (rule PDUnit, erule PDPlus [OF PDUnit]) 

637 
done 

638 

639 
text {* foldpd *} 

640 

641 
definition 

642 
fold_pd :: 

643 
"('a compact_basis \<Rightarrow> 'b::type) \<Rightarrow> ('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a pd_basis \<Rightarrow> 'b" 

644 
where "fold_pd g f t = fold1 f (g ` Rep_pd_basis t)" 

645 

26927  646 
lemma fold_pd_PDUnit: 
647 
includes ab_semigroup_idem_mult f 

648 
shows "fold_pd g f (PDUnit x) = g x" 

25904  649 
unfolding fold_pd_def Rep_PDUnit by simp 
650 

26927  651 
lemma fold_pd_PDPlus: 
652 
includes ab_semigroup_idem_mult f 

653 
shows "fold_pd g f (PDPlus t u) = f (fold_pd g f t) (fold_pd g f u)" 

25904  654 
unfolding fold_pd_def Rep_PDPlus by (simp add: image_Un fold1_Un2) 
655 

656 
text {* approxpd *} 

657 

658 
definition 

659 
approx_pd :: "nat \<Rightarrow> 'a pd_basis \<Rightarrow> 'a pd_basis" where 

660 
"approx_pd n = (\<lambda>t. Abs_pd_basis (compact_approx n ` Rep_pd_basis t))" 

661 

662 
lemma Rep_approx_pd: 

663 
"Rep_pd_basis (approx_pd n t) = compact_approx n ` Rep_pd_basis t" 

664 
unfolding approx_pd_def 

665 
apply (rule Abs_pd_basis_inverse) 

666 
apply (simp add: pd_basis_def) 

667 
done 

668 

669 
lemma approx_pd_simps [simp]: 

670 
"approx_pd n (PDUnit a) = PDUnit (compact_approx n a)" 

671 
"approx_pd n (PDPlus t u) = PDPlus (approx_pd n t) (approx_pd n u)" 

672 
apply (simp_all add: Rep_pd_basis_inject [symmetric]) 

673 
apply (simp_all add: Rep_approx_pd Rep_PDUnit Rep_PDPlus image_Un) 

674 
done 

675 

676 
lemma approx_pd_idem: "approx_pd n (approx_pd n t) = approx_pd n t" 

677 
apply (induct t rule: pd_basis_induct) 

678 
apply (simp add: compact_approx_idem) 

679 
apply simp 

680 
done 

681 

682 
lemma range_image_f: "range (image f) = Pow (range f)" 

683 
apply (safe, fast) 

684 
apply (rule_tac x="f ` x" in range_eqI) 

685 
apply (simp, fast) 

686 
done 

687 

688 
lemma finite_range_approx_pd: "finite (range (approx_pd n))" 

689 
apply (subgoal_tac "finite (Rep_pd_basis ` range (approx_pd n))") 

690 
apply (erule finite_imageD) 

691 
apply (rule inj_onI, simp add: Rep_pd_basis_inject) 

692 
apply (subst image_image) 

693 
apply (subst Rep_approx_pd) 

694 
apply (simp only: range_composition) 

695 
apply (rule finite_subset [OF image_mono [OF subset_UNIV]]) 

696 
apply (simp add: range_image_f) 

697 
apply (rule finite_range_compact_approx) 

698 
done 

699 

700 
lemma ex_approx_pd_eq: "\<exists>n. approx_pd n t = t" 

701 
apply (subgoal_tac "\<exists>n. \<forall>m\<ge>n. approx_pd m t = t", fast) 

702 
apply (induct t rule: pd_basis_induct) 

703 
apply (cut_tac a=a in ex_compact_approx_eq) 

704 
apply (clarify, rule_tac x=n in exI) 

705 
apply (clarify, simp) 

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706 
apply (rule antisym_less) 
25904  707 
apply (rule compact_approx_le) 
708 
apply (drule_tac a=a in compact_approx_mono1) 

709 
apply simp 

710 
apply (clarify, rename_tac i j) 

711 
apply (rule_tac x="max i j" in exI, simp) 

712 
done 

713 

714 
end 