src/HOL/UNITY/Transformers.thy
author paulson
Thu Mar 06 15:08:38 2003 +0100 (2003-03-06)
changeset 13851 f6923453953a
parent 13832 e7649436869c
child 13853 89131afa9f01
permissions -rw-r--r--
new UNITY examples theory
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(*  Title:      HOL/UNITY/Transformers
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    ID:         $Id$
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   2003  University of Cambridge
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Predicate Transformers from 
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    David Meier and Beverly Sanders,
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    Composing Leads-to Properties
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    Theoretical Computer Science 243:1-2 (2000), 339-361.
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*)
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header{*Predicate Transformers*}
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theory Transformers = Comp:
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subsection{*Defining the Predicate Transformers @{term wp},
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   @{term awp} and  @{term wens}*}
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constdefs
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  wp :: "[('a*'a) set, 'a set] => 'a set"  
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    --{*Dijkstra's weakest-precondition operator (for an individual command)*}
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    "wp act B == - (act^-1 `` (-B))"
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  awp :: "['a program, 'a set] => 'a set"  
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    --{*Dijkstra's weakest-precondition operator (for a program)*}
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    "awp F B == (\<Inter>act \<in> Acts F. wp act B)"
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  wens :: "['a program, ('a*'a) set, 'a set] => 'a set"  
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    --{*The weakest-ensures transformer*}
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    "wens F act B == gfp(\<lambda>X. (wp act B \<inter> awp F (B \<union> X)) \<union> B)"
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text{*The fundamental theorem for wp*}
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theorem wp_iff: "(A <= wp act B) = (act `` A <= B)"
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by (force simp add: wp_def) 
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lemma Compl_Domain_subset_wp: "- (Domain act) \<subseteq> wp act B"
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by (force simp add: wp_def) 
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lemma wp_empty [simp]: "wp act {} = - (Domain act)"
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by (force simp add: wp_def)
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text{*The identity relation is the skip action*}
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lemma wp_Id [simp]: "wp Id B = B"
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by (simp add: wp_def) 
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lemma wp_totalize_act:
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     "wp (totalize_act act) B = (wp act B \<inter> Domain act) \<union> (B - Domain act)"
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by (simp add: wp_def totalize_act_def, blast)
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lemma awp_Int_eq: "awp F (A\<inter>B) = awp F A \<inter> awp F B"
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by (simp add: awp_def wp_def, blast) 
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text{*The fundamental theorem for awp*}
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theorem awp_iff: "(A <= awp F B) = (F \<in> A co B)"
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by (simp add: awp_def constrains_def wp_iff INT_subset_iff) 
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theorem stable_iff_awp: "(A \<subseteq> awp F A) = (F \<in> stable A)"
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by (simp add: awp_iff stable_def) 
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lemma awp_mono: "(A \<subseteq> B) ==> awp F A \<subseteq> awp F B"
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by (simp add: awp_def wp_def, blast)
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lemma wens_unfold:
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     "wens F act B = (wp act B \<inter> awp F (B \<union> wens F act B)) \<union> B"
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apply (simp add: wens_def) 
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apply (rule gfp_unfold) 
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apply (simp add: mono_def wp_def awp_def, blast) 
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done
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lemma wens_Id [simp]: "wens F Id B = B"
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by (simp add: wens_def gfp_def wp_def awp_def, blast)
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text{*These two theorems justify the claim that @{term wens} returns the
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weakest assertion satisfying the ensures property*}
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lemma ensures_imp_wens: "F \<in> A ensures B ==> \<exists>act \<in> Acts F. A \<subseteq> wens F act B"
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apply (simp add: wens_def ensures_def transient_def, clarify) 
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apply (rule rev_bexI, assumption) 
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apply (rule gfp_upperbound)
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apply (simp add: constrains_def awp_def wp_def, blast)
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done
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lemma wens_ensures: "act \<in> Acts F ==> F \<in> (wens F act B) ensures B"
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by (simp add: wens_def gfp_def constrains_def awp_def wp_def
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              ensures_def transient_def, blast) 
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text{*These two results constitute assertion (4.13) of the thesis*}
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lemma wens_mono: "(A \<subseteq> B) ==> wens F act A \<subseteq> wens F act B"
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apply (simp add: wens_def wp_def awp_def) 
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apply (rule gfp_mono, blast) 
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done
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lemma wens_weakening: "B \<subseteq> wens F act B"
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by (simp add: wens_def gfp_def, blast)
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text{*Assertion (6), or 4.16 in the thesis*}
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lemma subset_wens: "A-B \<subseteq> wp act B \<inter> awp F (B \<union> A) ==> A \<subseteq> wens F act B" 
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apply (simp add: wens_def wp_def awp_def) 
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apply (rule gfp_upperbound, blast) 
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done
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text{*Assertion 4.17 in the thesis*}
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lemma Diff_wens_constrains: "F \<in> (wens F act A - A) co wens F act A" 
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by (simp add: wens_def gfp_def wp_def awp_def constrains_def, blast)
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text{*Assertion (7): 4.18 in the thesis.  NOTE that many of these results
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hold for an arbitrary action.  We often do not require @{term "act \<in> Acts F"}*}
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lemma stable_wens: "F \<in> stable A ==> F \<in> stable (wens F act A)"
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apply (simp add: stable_def)
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apply (drule constrains_Un [OF Diff_wens_constrains [of F act A]]) 
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apply (simp add: Un_Int_distrib2 Compl_partition2) 
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apply (erule constrains_weaken, blast) 
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apply (simp add: Un_subset_iff wens_weakening) 
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done
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text{*Assertion 4.20 in the thesis.*}
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lemma wens_Int_eq_lemma:
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      "[|T-B \<subseteq> awp F T; act \<in> Acts F|]
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       ==> T \<inter> wens F act B \<subseteq> wens F act (T\<inter>B)"
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apply (rule subset_wens) 
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apply (rule_tac P="\<lambda>x. ?f x \<subseteq> ?b" in ssubst [OF wens_unfold])
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apply (simp add: wp_def awp_def, blast)
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done
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text{*Assertion (8): 4.21 in the thesis. Here we indeed require
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      @{term "act \<in> Acts F"}*}
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lemma wens_Int_eq:
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      "[|T-B \<subseteq> awp F T; act \<in> Acts F|]
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       ==> T \<inter> wens F act B = T \<inter> wens F act (T\<inter>B)"
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apply (rule equalityI)
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 apply (simp_all add: Int_lower1 Int_subset_iff) 
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 apply (rule wens_Int_eq_lemma, assumption+) 
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apply (rule subset_trans [OF _ wens_mono [of "T\<inter>B" B]], auto) 
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done
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subsection{*Defining the Weakest Ensures Set*}
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consts
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  wens_set :: "['a program, 'a set] => 'a set set"
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inductive "wens_set F B"
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 intros 
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  Basis: "B \<in> wens_set F B"
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  Wens:  "[|X \<in> wens_set F B; act \<in> Acts F|] ==> wens F act X \<in> wens_set F B"
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  Union: "W \<noteq> {} ==> \<forall>U \<in> W. U \<in> wens_set F B ==> \<Union>W \<in> wens_set F B"
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lemma wens_set_imp_co: "A \<in> wens_set F B ==> F \<in> (A-B) co A"
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apply (erule wens_set.induct) 
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  apply (simp add: constrains_def)
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 apply (drule_tac act1=act and A1=X 
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        in constrains_Un [OF Diff_wens_constrains]) 
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 apply (erule constrains_weaken, blast) 
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 apply (simp add: Un_subset_iff wens_weakening) 
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apply (rule constrains_weaken) 
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apply (rule_tac I=W and A="\<lambda>v. v-B" and A'="\<lambda>v. v" in constrains_UN, blast+)
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done
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lemma wens_set_imp_leadsTo: "A \<in> wens_set F B ==> F \<in> A leadsTo B"
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apply (erule wens_set.induct)
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  apply (rule leadsTo_refl)  
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 apply (blast intro: wens_ensures leadsTo_Trans) 
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apply (blast intro: leadsTo_Union) 
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done
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lemma leadsTo_imp_wens_set: "F \<in> A leadsTo B ==> \<exists>C \<in> wens_set F B. A \<subseteq> C"
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apply (erule leadsTo_induct_pre)
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  apply (blast dest!: ensures_imp_wens intro: wens_set.Basis wens_set.Wens, clarify) 
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 apply (drule ensures_weaken_R, assumption)  
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 apply (blast dest!: ensures_imp_wens intro: wens_set.Wens)
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apply (case_tac "S={}") 
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 apply (simp, blast intro: wens_set.Basis)
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apply (clarsimp dest!: bchoice simp: ball_conj_distrib Bex_def) 
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apply (rule_tac x = "\<Union>{Z. \<exists>U\<in>S. Z = f U}" in exI)
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apply (blast intro: wens_set.Union) 
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done
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text{*Assertion (9): 4.27 in the thesis.*}
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lemma leadsTo_iff_wens_set: "(F \<in> A leadsTo B) = (\<exists>C \<in> wens_set F B. A \<subseteq> C)"
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by (blast intro: leadsTo_imp_wens_set leadsTo_weaken_L wens_set_imp_leadsTo) 
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text{*This is the result that requires the definition of @{term wens_set} to
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  require @{term W} to be non-empty in the Unio case, for otherwise we should
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  always have @{term "{} \<in> wens_set F B"}.*}
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lemma wens_set_imp_subset: "A \<in> wens_set F B ==> B \<subseteq> A"
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apply (erule wens_set.induct) 
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  apply (blast intro: wens_weakening [THEN subsetD])+
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done
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subsection{*Properties Involving Program Union*}
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text{*Assertion (4.30) of thesis, reoriented*}
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lemma awp_Join_eq: "awp (F\<squnion>G) B = awp F B \<inter> awp G B"
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by (simp add: awp_def wp_def, blast)
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lemma wens_subset: 
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     "wens F act B - B \<subseteq> wp act B \<inter> awp F (B \<union> wens F act B)"
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by (subst wens_unfold, fast) 
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text{*Assertion (4.31)*}
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lemma subset_wens_Join:
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      "[|A = T \<inter> wens F act B;  T-B \<subseteq> awp F T; A-B \<subseteq> awp G (A \<union> B)|] 
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       ==> A \<subseteq> wens (F\<squnion>G) act B"
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apply (subgoal_tac "(T \<inter> wens F act B) - B \<subseteq> 
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                    wp act B \<inter> awp F (B \<union> wens F act B) \<inter> awp F T") 
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 apply (rule subset_wens) 
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 apply (simp add: awp_Join_eq awp_Int_eq Int_subset_iff Un_commute)
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 apply (simp add: awp_def wp_def, blast) 
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apply (insert wens_subset [of F act B], blast) 
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done
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text{*Assertion (4.32)*}
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lemma wens_Join_subset: "wens (F\<squnion>G) act B \<subseteq> wens F act B"
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apply (simp add: wens_def) 
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apply (rule gfp_mono)
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apply (auto simp add: awp_Join_eq)  
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done
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text{*Lemma, because the inductive step is just too messy.*}
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lemma wens_Union_inductive_step:
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  assumes awpF: "T-B \<subseteq> awp F T"
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      and awpG: "!!X. X \<in> wens_set F B ==> (T\<inter>X) - B \<subseteq> awp G (T\<inter>X)"
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  shows "[|X \<in> wens_set F B; act \<in> Acts F; Y \<subseteq> X; T\<inter>X = T\<inter>Y|]
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         ==> wens (F\<squnion>G) act Y \<subseteq> wens F act X \<and>
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             T \<inter> wens F act X = T \<inter> wens (F\<squnion>G) act Y"
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apply (subgoal_tac "wens (F\<squnion>G) act Y \<subseteq> wens F act X")  
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 prefer 2
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 apply (blast dest: wens_mono intro: wens_Join_subset [THEN subsetD], simp)
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apply (rule equalityI) 
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 prefer 2 apply blast
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apply (simp add: Int_lower1 Int_subset_iff) 
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apply (frule wens_set_imp_subset) 
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apply (subgoal_tac "T-X \<subseteq> awp F T")  
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 prefer 2 apply (blast intro: awpF [THEN subsetD]) 
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apply (rule_tac B = "wens (F\<squnion>G) act (T\<inter>X)" in subset_trans) 
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 prefer 2 apply (blast intro!: wens_mono)
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apply (subst wens_Int_eq, assumption+) 
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apply (rule subset_wens_Join [of _ T], simp) 
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 apply blast
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apply (subgoal_tac "T \<inter> wens F act (T\<inter>X) \<union> T\<inter>X = T \<inter> wens F act X")
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 prefer 2
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 apply (subst wens_Int_eq [symmetric], assumption+) 
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 apply (blast intro: wens_weakening [THEN subsetD], simp) 
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apply (blast intro: awpG [THEN subsetD] wens_set.Wens)
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done
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theorem wens_Union:
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  assumes awpF: "T-B \<subseteq> awp F T"
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      and awpG: "!!X. X \<in> wens_set F B ==> (T\<inter>X) - B \<subseteq> awp G (T\<inter>X)"
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      and major: "X \<in> wens_set F B"
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  shows "\<exists>Y \<in> wens_set (F\<squnion>G) B. Y \<subseteq> X & T\<inter>X = T\<inter>Y"
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apply (rule wens_set.induct [OF major])
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  txt{*Basis: trivial*}
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  apply (blast intro: wens_set.Basis)
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 txt{*Inductive step*}
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 apply clarify 
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 apply (rule_tac x = "wens (F\<squnion>G) act Y" in rev_bexI)
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  apply (force intro: wens_set.Wens)
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 apply (simp add: wens_Union_inductive_step [OF awpF awpG]) 
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txt{*Union: by Axiom of Choice*}
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apply (simp add: ball_conj_distrib Bex_def) 
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apply (clarify dest!: bchoice) 
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apply (rule_tac x = "\<Union>{Z. \<exists>U\<in>W. Z = f U}" in exI)
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apply (blast intro: wens_set.Union) 
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done
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theorem leadsTo_Union:
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  assumes awpF: "T-B \<subseteq> awp F T"
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      and awpG: "!!X. X \<in> wens_set F B ==> (T\<inter>X) - B \<subseteq> awp G (T\<inter>X)"
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      and leadsTo: "F \<in> A leadsTo B"
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  shows "F\<squnion>G \<in> T\<inter>A leadsTo B"
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apply (rule leadsTo [THEN leadsTo_imp_wens_set, THEN bexE]) 
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apply (rule wens_Union [THEN bexE]) 
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   apply (rule awpF) 
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  apply (erule awpG, assumption)
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apply (blast intro: wens_set_imp_leadsTo [THEN leadsTo_weaken_L])  
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done
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subsection {*The Set @{term "wens_set F B"} for a Single-Assignment Program*}
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text{*Thesis Section 4.3.3*}
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text{*We start by proving laws about single-assignment programs*}
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lemma awp_single_eq [simp]:
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     "awp (mk_program (init, {act}, allowed)) B = B \<inter> wp act B"
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by (force simp add: awp_def wp_def) 
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lemma wp_Un_subset: "wp act A \<union> wp act B \<subseteq> wp act (A \<union> B)"
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by (force simp add: wp_def)
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lemma wp_Un_eq: "single_valued act ==> wp act (A \<union> B) = wp act A \<union> wp act B"
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apply (rule equalityI)
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 apply (force simp add: wp_def single_valued_def) 
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apply (rule wp_Un_subset) 
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   299
done
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lemma wp_UN_subset: "(\<Union>i\<in>I. wp act (A i)) \<subseteq> wp act (\<Union>i\<in>I. A i)"
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by (force simp add: wp_def)
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lemma wp_UN_eq:
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     "[|single_valued act; I\<noteq>{}|]
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      ==> wp act (\<Union>i\<in>I. A i) = (\<Union>i\<in>I. wp act (A i))"
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apply (rule equalityI)
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 prefer 2 apply (rule wp_UN_subset) 
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 apply (simp add: wp_def Image_INT_eq) 
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   310
done
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   311
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lemma wens_single_eq:
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     "wens (mk_program (init, {act}, allowed)) act B = B \<union> wp act B"
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by (simp add: wens_def gfp_def wp_def, blast)
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paulson@13832
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text{*Next, we express the @{term "wens_set"} for single-assignment programs*}
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constdefs
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  wens_single_finite :: "[('a*'a) set, 'a set, nat] => 'a set"  
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    "wens_single_finite act B k == \<Union>i \<in> atMost k. ((wp act)^i) B"
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  wens_single :: "[('a*'a) set, 'a set] => 'a set"  
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    "wens_single act B == \<Union>i. ((wp act)^i) B"
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lemma wens_single_Un_eq:
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      "single_valued act
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       ==> wens_single act B \<union> wp act (wens_single act B) = wens_single act B"
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apply (rule equalityI)
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 apply (simp_all add: Un_upper1 Un_subset_iff) 
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apply (simp add: wens_single_def wp_UN_eq, clarify) 
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apply (rule_tac a="Suc(i)" in UN_I, auto) 
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   333
done
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lemma atMost_nat_nonempty: "atMost (k::nat) \<noteq> {}"
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by force
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lemma wens_single_finite_0 [simp]: "wens_single_finite act B 0 = B"
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by (simp add: wens_single_finite_def)
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lemma wens_single_finite_Suc:
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      "single_valued act
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       ==> wens_single_finite act B (Suc k) =
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           wens_single_finite act B k \<union> wp act (wens_single_finite act B k)"
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apply (simp add: wens_single_finite_def image_def 
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                 wp_UN_eq [OF _ atMost_nat_nonempty]) 
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apply (force elim!: le_SucE)
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done
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lemma wens_single_finite_Suc_eq_wens:
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     "single_valued act
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       ==> wens_single_finite act B (Suc k) =
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           wens (mk_program (init, {act}, allowed)) act 
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                (wens_single_finite act B k)"
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by (simp add: wens_single_finite_Suc wens_single_eq) 
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   357
lemma def_wens_single_finite_Suc_eq_wens:
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     "[|F = mk_program (init, {act}, allowed); single_valued act|]
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       ==> wens_single_finite act B (Suc k) =
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   360
           wens F act (wens_single_finite act B k)"
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   361
by (simp add: wens_single_finite_Suc_eq_wens) 
paulson@13851
   362
paulson@13832
   363
lemma wens_single_finite_Un_eq:
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      "single_valued act
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   365
       ==> wens_single_finite act B k \<union> wp act (wens_single_finite act B k)
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   366
           \<in> range (wens_single_finite act B)"
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by (simp add: wens_single_finite_Suc [symmetric]) 
paulson@13832
   368
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   369
lemma wens_single_eq_Union:
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      "wens_single act B = \<Union>range (wens_single_finite act B)" 
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   371
by (simp add: wens_single_finite_def wens_single_def, blast) 
paulson@13832
   372
paulson@13832
   373
lemma wens_single_finite_eq_Union:
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     "wens_single_finite act B n = (\<Union>k\<in>atMost n. wens_single_finite act B k)"
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   375
apply (auto simp add: wens_single_finite_def) 
paulson@13832
   376
apply (blast intro: le_trans) 
paulson@13832
   377
done
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   378
paulson@13832
   379
lemma wens_single_finite_mono:
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   380
     "m \<le> n ==> wens_single_finite act B m \<subseteq> wens_single_finite act B n"
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   381
by (force simp add:  wens_single_finite_eq_Union [of act B n]) 
paulson@13832
   382
paulson@13832
   383
lemma wens_single_finite_subset_wens_single:
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   384
      "wens_single_finite act B k \<subseteq> wens_single act B"
paulson@13832
   385
by (simp add: wens_single_eq_Union, blast) 
paulson@13832
   386
paulson@13832
   387
lemma subset_wens_single_finite:
paulson@13832
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      "[|W \<subseteq> wens_single_finite act B ` (atMost k); single_valued act; W\<noteq>{}|]
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   389
       ==> \<exists>m. \<Union>W = wens_single_finite act B m"
paulson@13851
   390
apply (induct k)
paulson@13851
   391
 apply (rule_tac x=0 in exI, simp) 
paulson@13851
   392
 apply blast 
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   393
apply (auto simp add: atMost_Suc) 
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   394
apply (case_tac "wens_single_finite act B (Suc n) \<in> W") 
paulson@13832
   395
 prefer 2 apply blast 
paulson@13832
   396
apply (drule_tac x="Suc n" in spec)
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   397
apply (erule notE, rule equalityI)
paulson@13832
   398
 prefer 2 apply blast 
paulson@13832
   399
apply (subst wens_single_finite_eq_Union) 
paulson@13832
   400
apply (simp add: atMost_Suc, blast) 
paulson@13832
   401
done
paulson@13832
   402
paulson@13832
   403
text{*lemma for Union case*}
paulson@13832
   404
lemma Union_eq_wens_single:
paulson@13832
   405
      "\<lbrakk>\<forall>k. \<not> W \<subseteq> wens_single_finite act B ` {..k};
paulson@13832
   406
        W \<subseteq> insert (wens_single act B)
paulson@13832
   407
            (range (wens_single_finite act B))\<rbrakk>
paulson@13832
   408
       \<Longrightarrow> \<Union>W = wens_single act B"
paulson@13832
   409
apply (case_tac "wens_single act B \<in> W")
paulson@13832
   410
 apply (blast dest: wens_single_finite_subset_wens_single [THEN subsetD]) 
paulson@13832
   411
apply (simp add: wens_single_eq_Union) 
paulson@13851
   412
apply (rule equalityI, blast) 
paulson@13832
   413
apply (simp add: UN_subset_iff, clarify)
paulson@13832
   414
apply (subgoal_tac "\<exists>y\<in>W. \<exists>n. y = wens_single_finite act B n & i\<le>n")  
paulson@13851
   415
 apply (blast intro: wens_single_finite_mono [THEN subsetD]) 
paulson@13832
   416
apply (drule_tac x=i in spec)
paulson@13832
   417
apply (force simp add: atMost_def)
paulson@13832
   418
done
paulson@13832
   419
paulson@13832
   420
lemma wens_set_subset_single:
paulson@13832
   421
      "single_valued act
paulson@13832
   422
       ==> wens_set (mk_program (init, {act}, allowed)) B \<subseteq> 
paulson@13832
   423
           insert (wens_single act B) (range (wens_single_finite act B))"
paulson@13832
   424
apply (rule subsetI)  
paulson@13832
   425
apply (erule wens_set.induct)
paulson@13832
   426
  txt{*Basis*} 
paulson@13832
   427
  apply (force simp add: wens_single_finite_def)
paulson@13832
   428
 txt{*Wens inductive step*}
paulson@13832
   429
 apply (case_tac "acta = Id", simp)   
paulson@13832
   430
 apply (simp add: wens_single_eq)
paulson@13832
   431
 apply (elim disjE)   
paulson@13832
   432
 apply (simp add: wens_single_Un_eq)
paulson@13832
   433
 apply (force simp add: wens_single_finite_Un_eq)
paulson@13832
   434
txt{*Union inductive step*}
paulson@13832
   435
apply (case_tac "\<exists>k. W \<subseteq> wens_single_finite act B ` (atMost k)")
paulson@13832
   436
 apply (blast dest!: subset_wens_single_finite, simp) 
paulson@13832
   437
apply (rule disjI1 [OF Union_eq_wens_single], blast+)
paulson@13832
   438
done
paulson@13832
   439
paulson@13832
   440
lemma wens_single_finite_in_wens_set:
paulson@13832
   441
      "single_valued act \<Longrightarrow>
paulson@13832
   442
         wens_single_finite act B k
paulson@13832
   443
         \<in> wens_set (mk_program (init, {act}, allowed)) B"
paulson@13832
   444
apply (induct_tac k) 
paulson@13832
   445
 apply (simp add: wens_single_finite_def wens_set.Basis)
paulson@13832
   446
apply (simp add: wens_set.Wens
paulson@13832
   447
                 wens_single_finite_Suc_eq_wens [of act B _ init allowed]) 
paulson@13832
   448
done
paulson@13832
   449
paulson@13832
   450
lemma single_subset_wens_set:
paulson@13832
   451
      "single_valued act
paulson@13832
   452
       ==> insert (wens_single act B) (range (wens_single_finite act B)) \<subseteq> 
paulson@13832
   453
           wens_set (mk_program (init, {act}, allowed)) B"
paulson@13832
   454
apply (simp add: wens_single_eq_Union UN_eq) 
paulson@13832
   455
apply (blast intro: wens_set.Union wens_single_finite_in_wens_set)
paulson@13832
   456
done
paulson@13832
   457
paulson@13832
   458
text{*Theorem (4.29)*}
paulson@13832
   459
theorem wens_set_single_eq:
paulson@13851
   460
     "[|F = mk_program (init, {act}, allowed); single_valued act|]
paulson@13851
   461
      ==> wens_set F B =
paulson@13851
   462
          insert (wens_single act B) (range (wens_single_finite act B))"
paulson@13832
   463
apply (rule equalityI)
paulson@13851
   464
 apply (simp add: wens_set_subset_single) 
paulson@13851
   465
apply (erule ssubst, erule single_subset_wens_set) 
paulson@13832
   466
done
paulson@13832
   467
paulson@13821
   468
end