src/HOLCF/Pcpo.thy
author huffman
Fri, 01 Jul 2005 02:30:59 +0200
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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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theory Pcpo
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imports Porder
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begin
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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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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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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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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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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)"
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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:
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  "chain Y \<Longrightarrow> max_in_chain i Y = ((\<Squnion>i. Y i) = ((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 "\<sqsubseteq>"} relation between two chains is preserved by their lubs *}
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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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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 *}
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lemma lub_mono2:
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  "\<lbrakk>\<exists>j::nat. \<forall>i>j. X i = Y i; chain (X::nat=>'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 (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 simp
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apply (erule is_ub_thelub)
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apply (rule_tac y = "X (Suc j)" in trans_less)
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apply (erule chain_mono)
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apply (erule not_less_eq [THEN iffD1])
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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:
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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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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 (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 (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. ALL 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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text {* useful lemmas about @{term \<bottom>} *}
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lemma eq_UU_iff: "(x = \<bottom>) = (x \<sqsubseteq> \<bottom>)"
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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 \<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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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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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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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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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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lemma chain_mono2: 
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 "\<lbrakk>\<exists>j::nat. 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 {* Chain-finite 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: "\<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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lemma flat_imp_chfin: 
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     "\<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
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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 (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) \<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)
9e125b675253 converted to new-style theory
huffman
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   322
done
15576
efb95d0d01f7 converted to new-style theories, and combined numbered files
huffman
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diff changeset
   323
16626
d28314d2dce3 cleaned up
huffman
parents: 16203
diff changeset
   324
end