author  wenzelm 
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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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header {* 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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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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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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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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136 
shows "fixp f = f (fixp f)" 
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137 
proof (rule antisym) 
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138 
show "fixp f \<le> f (fixp f)" 
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139 
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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142 
thus "f (fixp f) \<le> fixp f" 
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143 
unfolding fixp_def . 
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144 
qed 
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145 

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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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148 
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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151 
thus "x \<le> z" 
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152 
proof (induct x rule: iterates.induct) 
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153 
fix x assume "x \<le> z" with f have "f x \<le> f z" by (rule monotoneD) 
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154 
also note z finally show "f x \<le> z" . 
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qed (auto intro: ccpo_Sup_least) 
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156 
qed 
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157 

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158 

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subsection {* Fixpoint induction *} 
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definition 
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admissible :: "('a \<Rightarrow> bool) \<Rightarrow> bool" 
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where 
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"admissible P = (\<forall>A. chain (op \<le>) A \<longrightarrow> (\<forall>x\<in>A. P x) \<longrightarrow> P (Sup A))" 
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165 

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lemma admissibleI: 
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assumes "\<And>A. chain (op \<le>) A \<Longrightarrow> \<forall>x\<in>A. P x \<Longrightarrow> P (Sup A)" 
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168 
shows "admissible P" 
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169 
using assms unfolding admissible_def by fast 
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170 

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lemma admissibleD: 
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172 
assumes "admissible P" 
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assumes "chain (op \<le>) A" 
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assumes "\<And>x. x \<in> A \<Longrightarrow> P x" 
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shows "P (Sup A)" 
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176 
using assms by (auto simp: admissible_def) 
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Complete_Partial_Order.thy: complete partial orders over arbitrary chains, with fixpoint theorem
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177 

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lemma fixp_induct: 
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assumes adm: "admissible P" 
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assumes mono: "monotone (op \<le>) (op \<le>) f" 
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181 
assumes step: "\<And>x. P x \<Longrightarrow> P (f x)" 
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182 
shows "P (fixp f)" 
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183 
unfolding fixp_def using adm chain_iterates[OF mono] 
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184 
proof (rule admissibleD) 
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185 
fix x assume "x \<in> iterates f" 
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Complete_Partial_Order.thy: complete partial orders over arbitrary chains, with fixpoint theorem
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186 
thus "P x" 
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by (induct rule: iterates.induct) 
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188 
(auto intro: step admissibleD adm) 
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189 
qed 
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190 

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lemma admissible_True: "admissible (\<lambda>x. True)" 
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192 
unfolding admissible_def by simp 
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193 

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lemma admissible_False: "\<not> admissible (\<lambda>x. False)" 
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195 
unfolding admissible_def chain_def by simp 
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196 

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lemma admissible_const: "admissible (\<lambda>x. t) = t" 
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198 
by (cases t, simp_all add: admissible_True admissible_False) 
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Complete_Partial_Order.thy: complete partial orders over arbitrary chains, with fixpoint theorem
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199 

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200 
lemma admissible_conj: 
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201 
assumes "admissible (\<lambda>x. P x)" 
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202 
assumes "admissible (\<lambda>x. Q x)" 
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203 
shows "admissible (\<lambda>x. P x \<and> Q x)" 
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204 
using assms unfolding admissible_def by simp 
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Complete_Partial_Order.thy: complete partial orders over arbitrary chains, with fixpoint theorem
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205 

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206 
lemma admissible_all: 
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207 
assumes "\<And>y. admissible (\<lambda>x. P x y)" 
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208 
shows "admissible (\<lambda>x. \<forall>y. P x y)" 
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209 
using assms unfolding admissible_def by fast 
c58951943cba
Complete_Partial_Order.thy: complete partial orders over arbitrary chains, with fixpoint theorem
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210 

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211 
lemma admissible_ball: 
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212 
assumes "\<And>y. y \<in> A \<Longrightarrow> admissible (\<lambda>x. P x y)" 
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213 
shows "admissible (\<lambda>x. \<forall>y\<in>A. P x y)" 
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Complete_Partial_Order.thy: complete partial orders over arbitrary chains, with fixpoint theorem
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214 
using assms unfolding admissible_def by fast 
c58951943cba
Complete_Partial_Order.thy: complete partial orders over arbitrary chains, with fixpoint theorem
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215 

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216 
lemma chain_compr: "chain (op \<le>) A \<Longrightarrow> chain (op \<le>) {x \<in> A. P x}" 
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217 
unfolding chain_def by fast 
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Complete_Partial_Order.thy: complete partial orders over arbitrary chains, with fixpoint theorem
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218 

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219 
lemma admissible_disj_lemma: 
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220 
assumes A: "chain (op \<le>)A" 
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221 
assumes P: "\<forall>x\<in>A. \<exists>y\<in>A. x \<le> y \<and> P y" 
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222 
shows "Sup A = Sup {x \<in> A. P x}" 
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223 
proof (rule antisym) 
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224 
have *: "chain (op \<le>) {x \<in> A. P x}" 
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225 
by (rule chain_compr [OF A]) 
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226 
show "Sup A \<le> Sup {x \<in> A. P x}" 
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227 
apply (rule ccpo_Sup_least [OF A]) 
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228 
apply (drule P [rule_format], clarify) 
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229 
apply (erule order_trans) 
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230 
apply (simp add: ccpo_Sup_upper [OF *]) 
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231 
done 
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232 
show "Sup {x \<in> A. P x} \<le> Sup A" 
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233 
apply (rule ccpo_Sup_least [OF *]) 
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234 
apply clarify 
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235 
apply (simp add: ccpo_Sup_upper [OF A]) 
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236 
done 
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Complete_Partial_Order.thy: complete partial orders over arbitrary chains, with fixpoint theorem
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237 
qed 
c58951943cba
Complete_Partial_Order.thy: complete partial orders over arbitrary chains, with fixpoint theorem
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238 

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239 
lemma admissible_disj: 
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240 
fixes P Q :: "'a \<Rightarrow> bool" 
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Complete_Partial_Order.thy: complete partial orders over arbitrary chains, with fixpoint theorem
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241 
assumes P: "admissible (\<lambda>x. P x)" 
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242 
assumes Q: "admissible (\<lambda>x. Q x)" 
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243 
shows "admissible (\<lambda>x. P x \<or> Q x)" 
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Complete_Partial_Order.thy: complete partial orders over arbitrary chains, with fixpoint theorem
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244 
proof (rule admissibleI) 
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245 
fix A :: "'a set" assume A: "chain (op \<le>) A" 
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246 
assume "\<forall>x\<in>A. P x \<or> Q x" 
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247 
hence "(\<forall>x\<in>A. \<exists>y\<in>A. x \<le> y \<and> P y) \<or> (\<forall>x\<in>A. \<exists>y\<in>A. x \<le> y \<and> Q y)" 
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248 
using chainD[OF A] by blast 
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249 
hence "Sup A = Sup {x \<in> A. P x} \<or> Sup A = Sup {x \<in> A. Q x}" 
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250 
using admissible_disj_lemma [OF A] by fast 
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251 
thus "P (Sup A) \<or> Q (Sup A)" 
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252 
apply (rule disjE, simp_all) 
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253 
apply (rule disjI1, rule admissibleD [OF P chain_compr [OF A]], simp) 
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254 
apply (rule disjI2, rule admissibleD [OF Q chain_compr [OF A]], simp) 
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255 
done 
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Complete_Partial_Order.thy: complete partial orders over arbitrary chains, with fixpoint theorem
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256 
qed 
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Complete_Partial_Order.thy: complete partial orders over arbitrary chains, with fixpoint theorem
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257 

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258 
end 
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259 

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260 
instance complete_lattice \<subseteq> ccpo 
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261 
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 admissible 
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Complete_Partial_Order.thy: complete partial orders over arbitrary chains, with fixpoint theorem
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Complete_Partial_Order.thy: complete partial orders over arbitrary chains, with fixpoint theorem
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end 