author  paulson 
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parent 22577  1a08fce38565 
child 25131  2c8caac48ade 
permissions  rwrr 
2640  1 
(* Title: HOLCF/Pcpo.thy 
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ID: $Id$ 

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Author: Franz Regensburger 

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*) 

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header {* Classes cpo and pcpo *} 
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15577  8 
theory Pcpo 
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imports Porder 

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begin 

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Franz Regensburger's HigherOrder Logic of Computable Functions embedding LCF
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subsection {* Complete partial orders *} 
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text {* The class cpo of chain complete partial orders *} 
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axclass cpo < po 
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 {* class axiom: *} 
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cpo: "chain S \<Longrightarrow> \<exists>x. range S << x" 
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text {* in cpo's everthing equal to THE lub has lub properties for every chain *} 
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16626  22 
lemma thelubE: "\<lbrakk>chain S; (\<Squnion>i. S i) = (l::'a::cpo)\<rbrakk> \<Longrightarrow> range S << l" 
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by (blast dest: cpo intro: lubI) 
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text {* Properties of the lub *} 
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16626  27 
lemma is_ub_thelub: "chain (S::nat \<Rightarrow> 'a::cpo) \<Longrightarrow> S x \<sqsubseteq> (\<Squnion>i. S i)" 
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by (blast dest: cpo intro: lubI [THEN is_ub_lub]) 
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16626  30 
lemma is_lub_thelub: 
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"\<lbrakk>chain (S::nat \<Rightarrow> 'a::cpo); range S < x\<rbrakk> \<Longrightarrow> (\<Squnion>i. S i) \<sqsubseteq> x" 

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by (blast dest: cpo intro: lubI [THEN is_lub_lub]) 
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16626  34 
lemma lub_range_mono: 
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"\<lbrakk>range X \<subseteq> range Y; chain Y; chain (X::nat \<Rightarrow> 'a::cpo)\<rbrakk> 

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\<Longrightarrow> (\<Squnion>i. X i) \<sqsubseteq> (\<Squnion>i. Y i)" 

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apply (erule is_lub_thelub) 
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apply (rule ub_rangeI) 

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apply (subgoal_tac "\<exists>j. X i = Y j") 
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apply clarsimp 
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apply (erule is_ub_thelub) 

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apply auto 

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done 

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lemma lub_range_shift: 
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"chain (Y::nat \<Rightarrow> 'a::cpo) \<Longrightarrow> (\<Squnion>i. Y (i + j)) = (\<Squnion>i. Y i)" 

15563  47 
apply (rule antisym_less) 
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apply (rule lub_range_mono) 

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apply fast 

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apply assumption 

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apply (erule chain_shift) 

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apply (rule is_lub_thelub) 

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apply assumption 

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apply (rule ub_rangeI) 

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apply (rule_tac y="Y (i + j)" in trans_less) 
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apply (erule chain_mono3) 
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apply (rule le_add1) 

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apply (rule is_ub_thelub) 
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apply (erule chain_shift) 

15563  60 
done 
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16626  62 
lemma maxinch_is_thelub: 
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"chain Y \<Longrightarrow> max_in_chain i Y = ((\<Squnion>i. Y i) = ((Y i)::'a::cpo))" 

15563  64 
apply (rule iffI) 
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apply (fast intro!: thelubI lub_finch1) 

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apply (unfold max_in_chain_def) 

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apply (safe intro!: antisym_less) 

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apply (fast elim!: chain_mono3) 

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apply (drule sym) 

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apply (force elim!: is_ub_thelub) 

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done 

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text {* the @{text "\<sqsubseteq>"} relation between two chains is preserved by their lubs *} 
15563  74 

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lemma lub_mono: 
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"\<lbrakk>chain (X::nat \<Rightarrow> 'a::cpo); chain Y; \<forall>k. X k \<sqsubseteq> Y k\<rbrakk> 

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\<Longrightarrow> (\<Squnion>i. X i) \<sqsubseteq> (\<Squnion>i. Y i)" 

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apply (erule is_lub_thelub) 
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apply (rule ub_rangeI) 

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apply (rule trans_less) 

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apply (erule spec) 

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apply (erule is_ub_thelub) 

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done 

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text {* the = relation between two chains is preserved by their lubs *} 
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16626  87 
lemma lub_equal: 
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"\<lbrakk>chain (X::nat \<Rightarrow> 'a::cpo); chain Y; \<forall>k. X k = Y k\<rbrakk> 

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\<Longrightarrow> (\<Squnion>i. X i) = (\<Squnion>i. Y i)" 

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by (simp only: expand_fun_eq [symmetric]) 
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text {* more results about mono and = of lubs of chains *} 
3326  93 

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lemma lub_mono2: 
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"\<lbrakk>\<exists>j. \<forall>i>j. X i = Y i; chain (X::nat \<Rightarrow> 'a::cpo); chain Y\<rbrakk> 
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\<Longrightarrow> (\<Squnion>i. X i) \<sqsubseteq> (\<Squnion>i. Y i)" 
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apply (erule exE) 
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apply (subgoal_tac "(\<Squnion>i. X (i + Suc j)) \<sqsubseteq> (\<Squnion>i. Y (i + Suc j))") 
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apply (thin_tac "\<forall>i>j. X i = Y i") 

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apply (simp only: lub_range_shift) 

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apply simp 
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done 
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lemma lub_equal2: 
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"\<lbrakk>\<exists>j. \<forall>i>j. X i = Y i; chain (X::nat \<Rightarrow> 'a::cpo); chain Y\<rbrakk> 

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\<Longrightarrow> (\<Squnion>i. X i) = (\<Squnion>i. Y i)" 

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by (blast intro: antisym_less lub_mono2 sym) 
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16626  109 
lemma lub_mono3: 
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"\<lbrakk>chain (Y::nat \<Rightarrow> 'a::cpo); chain X; \<forall>i. \<exists>j. Y i \<sqsubseteq> X j\<rbrakk> 

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\<Longrightarrow> (\<Squnion>i. Y i) \<sqsubseteq> (\<Squnion>i. X i)" 

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apply (erule is_lub_thelub) 
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apply (rule ub_rangeI) 
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apply (erule allE) 

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apply (erule exE) 

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apply (erule trans_less) 
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apply (erule is_ub_thelub) 

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done 
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lemma ch2ch_lub: 
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fixes Y :: "nat \<Rightarrow> nat \<Rightarrow> 'a::cpo" 

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assumes 1: "\<And>j. chain (\<lambda>i. Y i j)" 

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assumes 2: "\<And>i. chain (\<lambda>j. Y i j)" 

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shows "chain (\<lambda>i. \<Squnion>j. Y i j)" 

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apply (rule chainI) 

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apply (rule lub_mono [rule_format, OF 2 2]) 

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apply (rule chainE [OF 1]) 

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done 

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lemma diag_lub: 
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fixes Y :: "nat \<Rightarrow> nat \<Rightarrow> 'a::cpo" 

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assumes 1: "\<And>j. chain (\<lambda>i. Y i j)" 

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assumes 2: "\<And>i. chain (\<lambda>j. Y i j)" 

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shows "(\<Squnion>i. \<Squnion>j. Y i j) = (\<Squnion>i. Y i i)" 

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proof (rule antisym_less) 

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have 3: "chain (\<lambda>i. Y i i)" 

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apply (rule chainI) 

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apply (rule trans_less) 

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apply (rule chainE [OF 1]) 

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apply (rule chainE [OF 2]) 

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done 

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have 4: "chain (\<lambda>i. \<Squnion>j. Y i j)" 

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by (rule ch2ch_lub [OF 1 2]) 
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show "(\<Squnion>i. \<Squnion>j. Y i j) \<sqsubseteq> (\<Squnion>i. Y i i)" 
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apply (rule is_lub_thelub [OF 4]) 

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apply (rule ub_rangeI) 

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apply (rule lub_mono3 [rule_format, OF 2 3]) 
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apply (rule exI) 
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apply (rule trans_less) 

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apply (rule chain_mono3 [OF 1 le_maxI1]) 

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apply (rule chain_mono3 [OF 2 le_maxI2]) 

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done 

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show "(\<Squnion>i. Y i i) \<sqsubseteq> (\<Squnion>i. \<Squnion>j. Y i j)" 

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apply (rule lub_mono [rule_format, OF 3 4]) 
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apply (rule is_ub_thelub [OF 2]) 
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done 

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qed 

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lemma ex_lub: 

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fixes Y :: "nat \<Rightarrow> nat \<Rightarrow> 'a::cpo" 

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assumes 1: "\<And>j. chain (\<lambda>i. Y i j)" 

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assumes 2: "\<And>i. chain (\<lambda>j. Y i j)" 

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shows "(\<Squnion>i. \<Squnion>j. Y i j) = (\<Squnion>j. \<Squnion>i. Y i j)" 

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by (simp add: diag_lub 1 2) 

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subsection {* Pointed cpos *} 
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text {* The class pcpo of pointed cpos *} 
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axclass pcpo < cpo 
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least: "\<exists>x. \<forall>y. x \<sqsubseteq> y" 
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constdefs 
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UU :: "'a::pcpo" 

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"UU \<equiv> THE x. \<forall>y. x \<sqsubseteq> y" 
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syntax (xsymbols) 
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UU :: "'a::pcpo" ("\<bottom>") 
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text {* derive the old rule minimal *} 
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lemma UU_least: "\<forall>z. \<bottom> \<sqsubseteq> z" 
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apply (unfold UU_def) 
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apply (rule theI') 
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apply (rule ex_ex1I) 
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apply (rule least) 
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apply (blast intro: antisym_less) 
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done 
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lemma minimal [iff]: "\<bottom> \<sqsubseteq> x" 
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by (rule UU_least [THEN spec]) 

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lemma UU_reorient: "(\<bottom> = x) = (x = \<bottom>)" 
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by auto 

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ML_setup {* 

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local 

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val meta_UU_reorient = thm "UU_reorient" RS eq_reflection; 

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fun reorient_proc sg _ (_ $ t $ u) = 

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case u of 
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Const("Pcpo.UU",_) => NONE 
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 Const("HOL.zero", _) => NONE 
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 Const("HOL.one", _) => NONE 
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 Const("Numeral.number_of", _) $ _ => NONE 
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 _ => SOME meta_UU_reorient; 
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in 
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val UU_reorient_simproc = 

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Simplifier.simproc @{theory} "UU_reorient_simproc" ["UU=x"] reorient_proc 
16739  210 
end; 
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Addsimprocs [UU_reorient_simproc]; 

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*} 

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16626  215 
text {* useful lemmas about @{term \<bottom>} *} 
15563  216 

19440  217 
lemma less_UU_iff [simp]: "(x \<sqsubseteq> \<bottom>) = (x = \<bottom>)" 
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by (simp add: po_eq_conv) 

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16626  220 
lemma eq_UU_iff: "(x = \<bottom>) = (x \<sqsubseteq> \<bottom>)" 
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by simp 
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16626  223 
lemma UU_I: "x \<sqsubseteq> \<bottom> \<Longrightarrow> x = \<bottom>" 
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by (subst eq_UU_iff) 
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lemma not_less2not_eq: "\<not> (x::'a::po) \<sqsubseteq> y \<Longrightarrow> x \<noteq> y" 
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by auto 
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16626  229 
lemma chain_UU_I: "\<lbrakk>chain Y; (\<Squnion>i. Y i) = \<bottom>\<rbrakk> \<Longrightarrow> \<forall>i. Y i = \<bottom>" 
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apply (rule allI) 
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apply (rule UU_I) 
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apply (erule subst) 
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apply (erule is_ub_thelub) 

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done 

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16626  236 
lemma chain_UU_I_inverse: "\<forall>i::nat. Y i = \<bottom> \<Longrightarrow> (\<Squnion>i. Y i) = \<bottom>" 
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apply (rule lub_chain_maxelem) 
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apply (erule spec) 

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apply simp 
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done 
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16626  242 
lemma chain_UU_I_inverse2: "(\<Squnion>i. Y i) \<noteq> \<bottom> \<Longrightarrow> \<exists>i::nat. Y i \<noteq> \<bottom>" 
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by (blast intro: chain_UU_I_inverse) 
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16626  245 
lemma notUU_I: "\<lbrakk>x \<sqsubseteq> y; x \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> y \<noteq> \<bottom>" 
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by (blast intro: UU_I) 
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16627  248 
lemma chain_mono2: "\<lbrakk>\<exists>j. Y j \<noteq> \<bottom>; chain Y\<rbrakk> \<Longrightarrow> \<exists>j. \<forall>i>j. Y i \<noteq> \<bottom>" 
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by (blast dest: notUU_I chain_mono) 
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subsection {* Chainfinite and flat cpos *} 
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text {* further useful classes for HOLCF domains *} 
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axclass chfin < po 
16626  256 
chfin: "\<forall>Y. chain Y \<longrightarrow> (\<exists>n. max_in_chain n Y)" 
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axclass flat < pcpo 
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ax_flat: "\<forall>x y. x \<sqsubseteq> y \<longrightarrow> (x = \<bottom>) \<or> (x = y)" 
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text {* some properties for chfin and flat *} 
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text {* chfin types are cpo *} 
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lemma chfin_imp_cpo: 
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"chain (S::nat \<Rightarrow> 'a::chfin) \<Longrightarrow> \<exists>x. range S << x" 
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apply (frule chfin [rule_format]) 
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apply (blast intro: lub_finch1) 
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done 
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instance chfin < cpo 
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by intro_classes (rule chfin_imp_cpo) 
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text {* flat types are chfin *} 
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15563  276 
lemma flat_imp_chfin: 
16626  277 
"\<forall>Y::nat \<Rightarrow> 'a::flat. chain Y \<longrightarrow> (\<exists>n. max_in_chain n Y)" 
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apply (unfold max_in_chain_def) 
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apply clarify 

16626  280 
apply (case_tac "\<forall>i. Y i = \<bottom>") 
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apply simp 
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apply simp 
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apply (erule exE) 

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apply (rule_tac x="i" in exI) 
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apply clarify 
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apply (blast dest: chain_mono3 ax_flat [rule_format]) 
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done 
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instance flat < chfin 
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by intro_classes (rule flat_imp_chfin) 
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text {* flat subclass of chfin; @{text adm_flat} not needed *} 
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lemma flat_eq: "(a::'a::flat) \<noteq> \<bottom> \<Longrightarrow> a \<sqsubseteq> b = (a = b)" 
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by (safe dest!: ax_flat [rule_format]) 
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lemma chfin2finch: "chain (Y::nat \<Rightarrow> 'a::chfin) \<Longrightarrow> finite_chain Y" 
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by (simp add: chfin finite_chain_def) 
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text {* lemmata for improved admissibility introdution rule *} 
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lemma infinite_chain_adm_lemma: 

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"\<lbrakk>chain Y; \<forall>i. P (Y i); 
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\<And>Y. \<lbrakk>chain Y; \<forall>i. P (Y i); \<not> finite_chain Y\<rbrakk> \<Longrightarrow> P (\<Squnion>i. Y i)\<rbrakk> 

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\<Longrightarrow> P (\<Squnion>i. Y i)" 

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apply (case_tac "finite_chain Y") 
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prefer 2 apply fast 

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apply (unfold finite_chain_def) 

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apply safe 

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apply (erule lub_finch1 [THEN thelubI, THEN ssubst]) 

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apply assumption 

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apply (erule spec) 

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done 

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lemma increasing_chain_adm_lemma: 

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"\<lbrakk>chain Y; \<forall>i. P (Y i); \<And>Y. \<lbrakk>chain Y; \<forall>i. P (Y i); 
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\<forall>i. \<exists>j>i. Y i \<noteq> Y j \<and> Y i \<sqsubseteq> Y j\<rbrakk> \<Longrightarrow> P (\<Squnion>i. Y i)\<rbrakk> 

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\<Longrightarrow> P (\<Squnion>i. Y i)" 

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apply (erule infinite_chain_adm_lemma) 
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apply assumption 

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apply (erule thin_rl) 

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apply (unfold finite_chain_def) 

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apply (unfold max_in_chain_def) 

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apply (fast dest: le_imp_less_or_eq elim: chain_mono) 

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done 

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end 