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(* Title: HOLCF/Pcpo.thy 
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ID: $Id$ 

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Author: Franz Regensburger 

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Introduction of the classes cpo and pcpo. 
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*) 
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header {* Classes cpo and pcpo *} 
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15577  10 
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 ==> ? 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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lemma thelubE: "[ chain(S); lub(range(S)) = (l::'a::cpo) ] ==> 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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lemma is_ub_thelub: "chain (S::nat => 'a::cpo) ==> S(x) << lub(range(S))" 

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by (blast dest: cpo intro: lubI [THEN is_ub_lub]) 
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lemma is_lub_thelub: "[ chain (S::nat => 'a::cpo); range(S) < x ] ==> lub(range S) << x" 

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by (blast dest: cpo intro: lubI [THEN is_lub_lub]) 
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lemma lub_range_mono: "[ range X <= range Y; chain Y; chain (X::nat=>'a::cpo) ] ==> lub(range X) << lub(range Y)" 

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apply (erule is_lub_thelub) 

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apply (rule ub_rangeI) 

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apply (subgoal_tac "? 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: "chain (Y::nat=>'a::cpo) ==> lub(range (%i. Y(i + j))) = lub(range Y)" 

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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 trans_less) 

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apply (rule_tac [2] is_ub_thelub) 

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apply (erule_tac [2] chain_shift) 

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apply (erule chain_mono3) 

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apply (rule le_add1) 

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done 

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lemma maxinch_is_thelub: "chain Y ==> max_in_chain i Y = (lub(range(Y)) = ((Y i)::'a::cpo))" 

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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 "<<"} relation between two chains is preserved by their lubs *} 
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lemma lub_mono: "[chain(C1::(nat=>'a::cpo));chain(C2); ALL k. C1(k) << C2(k)] 

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==> lub(range(C1)) << lub(range(C2))" 

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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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lemma lub_equal: "[ chain(C1::(nat=>'a::cpo));chain(C2);ALL k. C1(k)=C2(k)] 

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==> lub(range(C1))=lub(range(C2))" 

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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 *} 
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lemma lub_mono2: "[EX j. ALL i. j<i > X(i::nat)=Y(i);chain(X::nat=>'a::cpo);chain(Y)] 
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==> lub(range(X))<<lub(range(Y))" 
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apply (erule exE) 
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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 (case_tac "j<i") 

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apply (rule_tac s = "Y (i) " and t = "X (i) " in subst) 

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apply (rule sym) 

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apply fast 

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apply (rule is_ub_thelub) 

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apply assumption 

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apply (rule_tac y = "X (Suc (j))" in trans_less) 

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apply (rule chain_mono) 

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apply assumption 

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apply (rule not_less_eq [THEN subst]) 

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apply assumption 

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apply (rule_tac s = "Y (Suc (j))" and t = "X (Suc (j))" in subst) 

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apply (simp) 
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apply (erule is_ub_thelub) 
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done 

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lemma lub_equal2: "[EX j. ALL i. j<i > X(i)=Y(i); chain(X::nat=>'a::cpo); chain(Y)] 

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==> lub(range(X))=lub(range(Y))" 

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by (blast intro: antisym_less lub_mono2 sym) 
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lemma lub_mono3: "[chain(Y::nat=>'a::cpo);chain(X); 

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ALL i. EX j. Y(i)<< X(j)]==> lub(range(Y))<<lub(range(X))" 

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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 (erule allE) 

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apply (erule exE) 

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apply (rule trans_less) 

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apply (rule_tac [2] is_ub_thelub) 

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prefer 2 apply (assumption) 

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apply assumption 

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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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apply (rule chainI) 

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

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apply (rule chainE [OF 1]) 

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done 

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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 [OF 2 3, rule_format]) 

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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 [OF 3 4, rule_format]) 

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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: "? x.!y. x<<y" 
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consts 
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UU :: "'a::pcpo" 
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syntax (xsymbols) 
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UU :: "'a::pcpo" ("\<bottom>") 
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defs 
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UU_def: "UU == THE x. ALL y. x<<y" 
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text {* derive the old rule minimal *} 
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lemma UU_least: "ALL z. UU << 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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lemmas minimal = UU_least [THEN spec, standard] 
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declare minimal [iff] 
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text {* useful lemmas about @{term UU} *} 
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lemma eq_UU_iff: "(x=UU)=(x<<UU)" 

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apply (rule iffI) 

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apply (erule ssubst) 

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apply (rule refl_less) 

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apply (rule antisym_less) 

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apply assumption 

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apply (rule minimal) 

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done 

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lemma UU_I: "x << UU ==> x = UU" 

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by (subst eq_UU_iff) 
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lemma not_less2not_eq: "~(x::'a::po)<<y ==> ~x=y" 

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by auto 
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lemma chain_UU_I: "[chain(Y);lub(range(Y))=UU] ==> ALL i. Y(i)=UU" 

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apply (rule allI) 

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apply (rule antisym_less) 

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apply (rule_tac [2] minimal) 

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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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lemma chain_UU_I_inverse: "ALL i. Y(i::nat)=UU ==> lub(range(Y::(nat=>'a::pcpo)))=UU" 

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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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lemma chain_UU_I_inverse2: "~lub(range(Y::(nat=>'a::pcpo)))=UU ==> EX i.~ Y(i)=UU" 

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by (blast intro: chain_UU_I_inverse) 
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lemma notUU_I: "[ x<<y; ~x=UU ] ==> ~y=UU" 

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by (blast intro: UU_I) 
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lemma chain_mono2: 

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"[EX j. ~Y(j)=UU;chain(Y::nat=>'a::pcpo)] ==> EX j. ALL i. j<i>~Y(i)=UU" 

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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 
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chfin: "!Y. chain Y>(? n. max_in_chain n Y)" 
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axclass flat < pcpo 
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ax_flat: "! x y. x << y > (x = UU)  (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=>'a::chfin) ==> EX 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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lemma flat_imp_chfin: 
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"ALL Y::nat=>'a::flat. chain Y > (EX n. max_in_chain n Y)" 

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apply (unfold max_in_chain_def) 

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apply clarify 

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apply (case_tac "ALL i. Y (i) =UU") 

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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 (erule le_imp_less_or_eq [THEN disjE]) 
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apply safe 

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apply (blast dest: chain_mono 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 ">"} @{text adm_flat} not needed *} 
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lemma flat_eq: "(a::'a::flat) ~= UU ==> a << b = (a = b)" 

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by (safe dest!: ax_flat [rule_format]) 
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lemma chfin2finch: "chain (Y::nat=>'a::chfin) ==> 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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"[chain Y; ALL i. P (Y i); 

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(!!Y. [ chain Y; ALL i. P (Y i); ~ finite_chain Y ] ==> P (lub(range Y))) 

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] ==> P (lub (range Y))" 

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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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"[chain Y; ALL i. P (Y i); 

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(!!Y. [ chain Y; ALL i. P (Y i); 

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ALL i. EX j. i < j & Y i ~= Y j & Y i << Y j] 

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==> P (lub (range Y))) ] ==> P (lub (range Y))" 

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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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Franz Regensburger's HigherOrder Logic of Computable Functions embedding LCF
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end 