author  Andreas Lochbihler 
Tue, 14 Apr 2015 13:56:34 +0200  
changeset 60061  279472fa0b1d 
parent 60057  86fa63ce8156 
child 60758  d8d85a8172b5 
permissions  rwrr 
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(* Title: HOL/Complete_Partial_Order.thy 
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Author: Brian Huffman, Portland State University 
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Author: Alexander Krauss, TU Muenchen 
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*) 
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58889  6 
section {* Chaincomplete partial orders and their fixpoints *} 
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theory Complete_Partial_Order 
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imports Product_Type 
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begin 
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subsection {* Monotone functions *} 
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text {* Dictionarypassing version of @{const Orderings.mono}. *} 
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definition monotone :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool" 
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where "monotone orda ordb f \<longleftrightarrow> (\<forall>x y. orda x y \<longrightarrow> ordb (f x) (f y))" 
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lemma monotoneI[intro?]: "(\<And>x y. orda x y \<Longrightarrow> ordb (f x) (f y)) 
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\<Longrightarrow> monotone orda ordb f" 
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unfolding monotone_def by iprover 
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lemma monotoneD[dest?]: "monotone orda ordb f \<Longrightarrow> orda x y \<Longrightarrow> ordb (f x) (f y)" 
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unfolding monotone_def by iprover 
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subsection {* Chains *} 
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text {* A chain is a totallyordered set. Chains are parameterized over 
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the order for maximal flexibility, since type classes are not enough. 
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*} 
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definition 
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chain :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> bool" 
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where 
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"chain ord S \<longleftrightarrow> (\<forall>x\<in>S. \<forall>y\<in>S. ord x y \<or> ord y x)" 
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lemma chainI: 
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assumes "\<And>x y. x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> ord x y \<or> ord y x" 
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shows "chain ord S" 
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using assms unfolding chain_def by fast 
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lemma chainD: 
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assumes "chain ord S" and "x \<in> S" and "y \<in> S" 
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shows "ord x y \<or> ord y x" 
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using assms unfolding chain_def by fast 
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lemma chainE: 
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assumes "chain ord S" and "x \<in> S" and "y \<in> S" 
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obtains "ord x y"  "ord y x" 
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using assms unfolding chain_def by fast 
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lemma chain_empty: "chain ord {}" 
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by(simp add: chain_def) 
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lemma chain_equality: "chain op = A \<longleftrightarrow> (\<forall>x\<in>A. \<forall>y\<in>A. x = y)" 
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by(auto simp add: chain_def) 

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lemma chain_subset: 
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"\<lbrakk> chain ord A; B \<subseteq> A \<rbrakk> 

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\<Longrightarrow> chain ord B" 

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by(rule chainI)(blast dest: chainD) 

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lemma chain_imageI: 

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assumes chain: "chain le_a Y" 

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and mono: "\<And>x y. \<lbrakk> x \<in> Y; y \<in> Y; le_a x y \<rbrakk> \<Longrightarrow> le_b (f x) (f y)" 

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shows "chain le_b (f ` Y)" 

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by(blast intro: chainI dest: chainD[OF chain] mono) 

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subsection {* Chaincomplete partial orders *} 
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text {* 
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A ccpo has a least upper bound for any chain. In particular, the 
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empty set is a chain, so every ccpo must have a bottom element. 
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*} 
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class ccpo = order + Sup + 
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assumes ccpo_Sup_upper: "\<lbrakk>chain (op \<le>) A; x \<in> A\<rbrakk> \<Longrightarrow> x \<le> Sup A" 
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assumes ccpo_Sup_least: "\<lbrakk>chain (op \<le>) A; \<And>x. x \<in> A \<Longrightarrow> x \<le> z\<rbrakk> \<Longrightarrow> Sup A \<le> z" 
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begin 
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lemma chain_singleton: "Complete_Partial_Order.chain op \<le> {x}" 
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by(rule chainI) simp 

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lemma ccpo_Sup_singleton [simp]: "\<Squnion>{x} = x" 

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by(rule antisym)(auto intro: ccpo_Sup_least ccpo_Sup_upper simp add: chain_singleton) 

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subsection {* Transfinite iteration of a function *} 
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inductive_set iterates :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a set" 
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for f :: "'a \<Rightarrow> 'a" 
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where 
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step: "x \<in> iterates f \<Longrightarrow> f x \<in> iterates f" 
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 Sup: "chain (op \<le>) M \<Longrightarrow> \<forall>x\<in>M. x \<in> iterates f \<Longrightarrow> Sup M \<in> iterates f" 
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lemma iterates_le_f: 
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"x \<in> iterates f \<Longrightarrow> monotone (op \<le>) (op \<le>) f \<Longrightarrow> x \<le> f x" 
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by (induct x rule: iterates.induct) 
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(force dest: monotoneD intro!: ccpo_Sup_upper ccpo_Sup_least)+ 
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lemma chain_iterates: 
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assumes f: "monotone (op \<le>) (op \<le>) f" 
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shows "chain (op \<le>) (iterates f)" (is "chain _ ?C") 
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proof (rule chainI) 
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fix x y assume "x \<in> ?C" "y \<in> ?C" 
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then show "x \<le> y \<or> y \<le> x" 
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proof (induct x arbitrary: y rule: iterates.induct) 
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fix x y assume y: "y \<in> ?C" 
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and IH: "\<And>z. z \<in> ?C \<Longrightarrow> x \<le> z \<or> z \<le> x" 
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from y show "f x \<le> y \<or> y \<le> f x" 
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proof (induct y rule: iterates.induct) 
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case (step y) with IH f show ?case by (auto dest: monotoneD) 
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next 
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case (Sup M) 
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then have chM: "chain (op \<le>) M" 
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and IH': "\<And>z. z \<in> M \<Longrightarrow> f x \<le> z \<or> z \<le> f x" by auto 
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show "f x \<le> Sup M \<or> Sup M \<le> f x" 
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proof (cases "\<exists>z\<in>M. f x \<le> z") 
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case True then have "f x \<le> Sup M" 
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apply rule 
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apply (erule order_trans) 
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by (rule ccpo_Sup_upper[OF chM]) 
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thus ?thesis .. 
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next 
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case False with IH' 
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show ?thesis by (auto intro: ccpo_Sup_least[OF chM]) 
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qed 
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qed 
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next 
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case (Sup M y) 
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show ?case 
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proof (cases "\<exists>x\<in>M. y \<le> x") 
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case True then have "y \<le> Sup M" 
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apply rule 
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apply (erule order_trans) 
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by (rule ccpo_Sup_upper[OF Sup(1)]) 
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thus ?thesis .. 
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next 
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case False with Sup 
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show ?thesis by (auto intro: ccpo_Sup_least) 
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qed 
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qed 
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qed 
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lemma bot_in_iterates: "Sup {} \<in> iterates f" 
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by(auto intro: iterates.Sup simp add: chain_empty) 
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subsection {* Fixpoint combinator *} 
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definition 
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fixp :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a" 
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where 
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"fixp f = Sup (iterates f)" 
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lemma iterates_fixp: 
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assumes f: "monotone (op \<le>) (op \<le>) f" shows "fixp f \<in> iterates f" 
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unfolding fixp_def 
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by (simp add: iterates.Sup chain_iterates f) 
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lemma fixp_unfold: 
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assumes f: "monotone (op \<le>) (op \<le>) f" 
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shows "fixp f = f (fixp f)" 
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proof (rule antisym) 
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show "fixp f \<le> f (fixp f)" 
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by (intro iterates_le_f iterates_fixp f) 
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have "f (fixp f) \<le> Sup (iterates f)" 
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by (intro ccpo_Sup_upper chain_iterates f iterates.step iterates_fixp) 
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thus "f (fixp f) \<le> fixp f" 
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unfolding fixp_def . 
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qed 
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lemma fixp_lowerbound: 
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assumes f: "monotone (op \<le>) (op \<le>) f" and z: "f z \<le> z" shows "fixp f \<le> z" 
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unfolding fixp_def 
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proof (rule ccpo_Sup_least[OF chain_iterates[OF f]]) 
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fix x assume "x \<in> iterates f" 
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thus "x \<le> z" 
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proof (induct x rule: iterates.induct) 
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fix x assume "x \<le> z" with f have "f x \<le> f z" by (rule monotoneD) 
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also note z finally show "f x \<le> z" . 
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qed (auto intro: ccpo_Sup_least) 
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qed 
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183 

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end 
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subsection {* Fixpoint induction *} 
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187 

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setup {* Sign.map_naming (Name_Space.mandatory_path "ccpo") *} 
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189 

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definition admissible :: "('a set \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" 
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where "admissible lub ord P = (\<forall>A. chain ord A \<longrightarrow> (A \<noteq> {}) \<longrightarrow> (\<forall>x\<in>A. P x) \<longrightarrow> P (lub A))" 
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lemma admissibleI: 
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assumes "\<And>A. chain ord A \<Longrightarrow> A \<noteq> {} \<Longrightarrow> \<forall>x\<in>A. P x \<Longrightarrow> P (lub A)" 
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shows "ccpo.admissible lub ord P" 
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using assms unfolding ccpo.admissible_def by fast 
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197 

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lemma admissibleD: 
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assumes "ccpo.admissible lub ord P" 
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assumes "chain ord A" 
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assumes "A \<noteq> {}" 
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assumes "\<And>x. x \<in> A \<Longrightarrow> P x" 
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shows "P (lub A)" 
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using assms by (auto simp: ccpo.admissible_def) 
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setup {* Sign.map_naming Name_Space.parent_path *} 
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207 

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lemma (in ccpo) fixp_induct: 
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assumes adm: "ccpo.admissible Sup (op \<le>) P" 
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assumes mono: "monotone (op \<le>) (op \<le>) f" 
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assumes bot: "P (Sup {})" 
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assumes step: "\<And>x. P x \<Longrightarrow> P (f x)" 
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shows "P (fixp f)" 
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unfolding fixp_def using adm chain_iterates[OF mono] 
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proof (rule ccpo.admissibleD) 
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show "iterates f \<noteq> {}" using bot_in_iterates by auto 
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fix x assume "x \<in> iterates f" 
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thus "P x" 
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by (induct rule: iterates.induct) 
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(case_tac "M = {}", auto intro: step bot ccpo.admissibleD adm) 
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qed 
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222 

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lemma admissible_True: "ccpo.admissible lub ord (\<lambda>x. True)" 
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224 
unfolding ccpo.admissible_def by simp 
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225 

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(*lemma admissible_False: "\<not> ccpo.admissible lub ord (\<lambda>x. False)" 
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227 
unfolding ccpo.admissible_def chain_def by simp 
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*) 
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lemma admissible_const: "ccpo.admissible lub ord (\<lambda>x. t)" 
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by(auto intro: ccpo.admissibleI) 
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231 

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lemma admissible_conj: 
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assumes "ccpo.admissible lub ord (\<lambda>x. P x)" 
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assumes "ccpo.admissible lub ord (\<lambda>x. Q x)" 
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shows "ccpo.admissible lub ord (\<lambda>x. P x \<and> Q x)" 
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using assms unfolding ccpo.admissible_def by simp 
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237 

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lemma admissible_all: 
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assumes "\<And>y. ccpo.admissible lub ord (\<lambda>x. P x y)" 
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240 
shows "ccpo.admissible lub ord (\<lambda>x. \<forall>y. P x y)" 
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using assms unfolding ccpo.admissible_def by fast 
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242 

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lemma admissible_ball: 
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assumes "\<And>y. y \<in> A \<Longrightarrow> ccpo.admissible lub ord (\<lambda>x. P x y)" 
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shows "ccpo.admissible lub ord (\<lambda>x. \<forall>y\<in>A. P x y)" 
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using assms unfolding ccpo.admissible_def by fast 
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247 

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lemma chain_compr: "chain ord A \<Longrightarrow> chain ord {x \<in> A. P x}" 
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unfolding chain_def by fast 
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250 

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context ccpo begin 
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252 

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lemma admissible_disj_lemma: 
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assumes A: "chain (op \<le>)A" 
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assumes P: "\<forall>x\<in>A. \<exists>y\<in>A. x \<le> y \<and> P y" 
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256 
shows "Sup A = Sup {x \<in> A. P x}" 
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proof (rule antisym) 
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have *: "chain (op \<le>) {x \<in> A. P x}" 
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by (rule chain_compr [OF A]) 
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show "Sup A \<le> Sup {x \<in> A. P x}" 
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apply (rule ccpo_Sup_least [OF A]) 
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apply (drule P [rule_format], clarify) 
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apply (erule order_trans) 
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264 
apply (simp add: ccpo_Sup_upper [OF *]) 
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done 
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show "Sup {x \<in> A. P x} \<le> Sup A" 
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apply (rule ccpo_Sup_least [OF *]) 
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apply (simp add: ccpo_Sup_upper [OF A]) 
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done 
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qed 
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lemma admissible_disj: 
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fixes P Q :: "'a \<Rightarrow> bool" 
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assumes P: "ccpo.admissible Sup (op \<le>) (\<lambda>x. P x)" 
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assumes Q: "ccpo.admissible Sup (op \<le>) (\<lambda>x. Q x)" 
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shows "ccpo.admissible Sup (op \<le>) (\<lambda>x. P x \<or> Q x)" 
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proof (rule ccpo.admissibleI) 
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fix A :: "'a set" assume A: "chain (op \<le>) A" 
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assume "A \<noteq> {}" 
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and "\<forall>x\<in>A. P x \<or> Q x" 
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hence "(\<exists>x\<in>A. P x) \<and> (\<forall>x\<in>A. \<exists>y\<in>A. x \<le> y \<and> P y) \<or> (\<exists>x\<in>A. Q x) \<and> (\<forall>x\<in>A. \<exists>y\<in>A. x \<le> y \<and> Q y)" 
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using chainD[OF A] by blast 
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hence "(\<exists>x. x \<in> A \<and> P x) \<and> Sup A = Sup {x \<in> A. P x} \<or> (\<exists>x. x \<in> A \<and> Q x) \<and> Sup A = Sup {x \<in> A. Q x}" 
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using admissible_disj_lemma [OF A] by blast 
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thus "P (Sup A) \<or> Q (Sup A)" 
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apply (rule disjE, simp_all) 
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apply (rule disjI1, rule ccpo.admissibleD [OF P chain_compr [OF A]], simp, simp) 
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apply (rule disjI2, rule ccpo.admissibleD [OF Q chain_compr [OF A]], simp, simp) 
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done 
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qed 
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end 
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instance complete_lattice \<subseteq> ccpo 
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by default (fast intro: Sup_upper Sup_least)+ 
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lemma lfp_eq_fixp: 
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assumes f: "mono f" shows "lfp f = fixp f" 
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proof (rule antisym) 
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from f have f': "monotone (op \<le>) (op \<le>) f" 
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unfolding mono_def monotone_def . 
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show "lfp f \<le> fixp f" 
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by (rule lfp_lowerbound, subst fixp_unfold [OF f'], rule order_refl) 
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show "fixp f \<le> lfp f" 
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by (rule fixp_lowerbound [OF f'], subst lfp_unfold [OF f], rule order_refl) 
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qed 
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hide_const (open) iterates fixp 
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end 