src/HOL/UNITY/Transformers.thy
author haftmann
Wed Feb 17 21:51:56 2016 +0100 (2016-02-17)
changeset 62343 24106dc44def
parent 59807 22bc39064290
child 63146 f1ecba0272f9
permissions -rw-r--r--
prefer abbreviations for compound operators INFIMUM and SUPREMUM
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(*  Title:      HOL/UNITY/Transformers.thy
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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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    David Meier,
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    Progress Properties in Program Refinement and Parallel Composition
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    Swiss Federal Institute of Technology Zurich (1997)
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*)
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section{*Predicate Transformers*}
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theory Transformers imports Comp begin
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subsection{*Defining the Predicate Transformers @{term wp},
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   @{term awp} and  @{term wens}*}
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definition wp :: "[('a*'a) set, 'a set] => 'a set" where  
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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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definition awp :: "['a program, 'a set] => 'a set" where
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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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definition wens :: "['a program, ('a*'a) set, 'a set] => 'a set" where
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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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text{*This lemma is a good deal more intuitive than the definition!*}
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lemma in_wp_iff: "(a \<in> wp act B) = (\<forall>x. (a,x) \<in> act --> x \<in> B)"
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by (simp add: wp_def, blast)
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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_subset: "(awp F A \<subseteq> A)"
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by (force simp add: awp_def wp_def)
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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_constrains: "(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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lemma awp_iff_stable: "(A \<subseteq> awp F A) = (F \<in> stable A)"
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by (simp add: awp_iff_constrains stable_def) 
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lemma stable_imp_awp_ident: "F \<in> stable A ==> awp F A = A"
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apply (rule equalityI [OF awp_subset]) 
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apply (simp add: awp_iff_stable) 
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done
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lemma wp_mono: "(A \<subseteq> B) ==> wp act A \<subseteq> wp act B"
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by (simp add: wp_def, blast)
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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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  --{*Proved instantly, yet remarkably fragile. If @{text Un_subset_iff}
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      is declared as an iff-rule, then it's almost impossible to prove. 
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      One proof is via @{text meson} after expanding all definitions, but it's
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      slow!*}
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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: 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" for f 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) 
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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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inductive_set
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  wens_set :: "['a program, 'a set] => 'a set set"
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  for F :: "'a program" and B :: "'a set"
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where
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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: 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) 
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 apply (clarify, 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: "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 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) 
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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, 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_Join:
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  assumes leadsTo: "F \<in> A leadsTo B"
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      and 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 "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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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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done
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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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text{*Next, we express the @{term "wens_set"} for single-assignment programs*}
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definition wens_single_finite :: "[('a*'a) set, 'a set, nat] => 'a set" where  
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    "wens_single_finite act B k == \<Union>i \<in> atMost k. (wp act ^^ i) B"
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definition wens_single :: "[('a*'a) set, 'a set] => 'a set" where
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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) 
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apply (simp add: wens_single_def wp_UN_eq, clarify) 
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apply (rule_tac a="Suc xa" in UN_I, auto) 
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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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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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   377
           wens F act (wens_single_finite act B k)"
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   378
by (simp add: wens_single_finite_Suc_eq_wens) 
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   380
lemma wens_single_finite_Un_eq:
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      "single_valued act
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       ==> wens_single_finite act B k \<union> wp act (wens_single_finite act B k)
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   383
           \<in> range (wens_single_finite act B)"
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   384
by (simp add: wens_single_finite_Suc [symmetric]) 
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   385
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   386
lemma wens_single_eq_Union:
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      "wens_single act B = \<Union>range (wens_single_finite act B)" 
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   388
by (simp add: wens_single_finite_def wens_single_def, blast) 
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   389
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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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   392
apply (auto simp add: wens_single_finite_def) 
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   393
apply (blast intro: le_trans) 
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   394
done
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   395
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   396
lemma wens_single_finite_mono:
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     "m \<le> n ==> wens_single_finite act B m \<subseteq> wens_single_finite act B n"
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   398
by (force simp add:  wens_single_finite_eq_Union [of act B n]) 
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   399
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   400
lemma wens_single_finite_subset_wens_single:
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   401
      "wens_single_finite act B k \<subseteq> wens_single act B"
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   402
by (simp add: wens_single_eq_Union, blast)
paulson@13832
   403
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   404
lemma subset_wens_single_finite:
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   405
      "[|W \<subseteq> wens_single_finite act B ` (atMost k); single_valued act; W\<noteq>{}|]
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   406
       ==> \<exists>m. \<Union>W = wens_single_finite act B m"
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   407
apply (induct k)
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   408
 apply (rule_tac x=0 in exI, simp, blast)
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   409
apply (auto simp add: atMost_Suc)
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   410
apply (case_tac "wens_single_finite act B (Suc k) \<in> W")
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   411
 prefer 2 apply blast
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   412
apply (drule_tac x="Suc k" in spec)
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   413
apply (erule notE, rule equalityI)
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   414
 prefer 2 apply blast
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   415
apply (subst wens_single_finite_eq_Union)
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   416
apply (simp add: atMost_Suc, blast)
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   417
done
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   418
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   419
text{*lemma for Union case*}
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   420
lemma Union_eq_wens_single:
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      "\<lbrakk>\<forall>k. \<not> W \<subseteq> wens_single_finite act B ` {..k};
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   422
        W \<subseteq> insert (wens_single act B)
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   423
            (range (wens_single_finite act B))\<rbrakk>
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   424
       \<Longrightarrow> \<Union>W = wens_single act B"
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   425
apply (cases "wens_single act B \<in> W")
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   426
 apply (blast dest: wens_single_finite_subset_wens_single [THEN subsetD]) 
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   427
apply (simp add: wens_single_eq_Union) 
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   428
apply (rule equalityI, blast) 
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   429
apply (simp add: UN_subset_iff, clarify)
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   430
apply (subgoal_tac "\<exists>y\<in>W. \<exists>n. y = wens_single_finite act B n & i\<le>n")  
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   431
 apply (blast intro: wens_single_finite_mono [THEN subsetD]) 
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   432
apply (drule_tac x=i in spec)
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   433
apply (force simp add: atMost_def)
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   434
done
paulson@13832
   435
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   436
lemma wens_set_subset_single:
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   437
      "single_valued act
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   438
       ==> wens_set (mk_program (init, {act}, allowed)) B \<subseteq> 
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   439
           insert (wens_single act B) (range (wens_single_finite act B))"
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   440
apply (rule subsetI)  
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   441
apply (erule wens_set.induct)
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   442
  txt{*Basis*} 
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   443
  apply (fastforce simp add: wens_single_finite_def)
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   444
 txt{*Wens inductive step*}
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   445
 apply (case_tac "acta = Id", simp)
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   446
 apply (simp add: wens_single_eq)
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   447
 apply (elim disjE)
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   448
 apply (simp add: wens_single_Un_eq)
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   449
 apply (force simp add: wens_single_finite_Un_eq)
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   450
txt{*Union inductive step*}
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   451
apply (case_tac "\<exists>k. W \<subseteq> wens_single_finite act B ` (atMost k)")
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   452
 apply (blast dest!: subset_wens_single_finite, simp) 
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   453
apply (rule disjI1 [OF Union_eq_wens_single], blast+)
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   454
done
paulson@13832
   455
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   456
lemma wens_single_finite_in_wens_set:
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   457
      "single_valued act \<Longrightarrow>
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   458
         wens_single_finite act B k
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   459
         \<in> wens_set (mk_program (init, {act}, allowed)) B"
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   460
apply (induct_tac k) 
paulson@13832
   461
 apply (simp add: wens_single_finite_def wens_set.Basis)
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   462
apply (simp add: wens_set.Wens
paulson@13832
   463
                 wens_single_finite_Suc_eq_wens [of act B _ init allowed]) 
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   464
done
paulson@13832
   465
paulson@13832
   466
lemma single_subset_wens_set:
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   467
      "single_valued act
paulson@13832
   468
       ==> insert (wens_single act B) (range (wens_single_finite act B)) \<subseteq> 
paulson@13832
   469
           wens_set (mk_program (init, {act}, allowed)) B"
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   470
apply (simp add: image_def wens_single_eq_Union) 
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   471
apply (blast intro: wens_set.Union wens_single_finite_in_wens_set)
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   472
done
paulson@13832
   473
paulson@13832
   474
text{*Theorem (4.29)*}
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   475
theorem wens_set_single_eq:
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   476
     "[|F = mk_program (init, {act}, allowed); single_valued act|]
paulson@13851
   477
      ==> wens_set F B =
paulson@13851
   478
          insert (wens_single act B) (range (wens_single_finite act B))"
paulson@13832
   479
apply (rule equalityI)
paulson@13851
   480
 apply (simp add: wens_set_subset_single) 
paulson@13851
   481
apply (erule ssubst, erule single_subset_wens_set) 
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   482
done
paulson@13832
   483
paulson@13853
   484
text{*Generalizing Misra's Fixed Point Union Theorem (4.41)*}
paulson@13853
   485
paulson@13866
   486
lemma fp_leadsTo_Join:
paulson@13853
   487
    "[|T-B \<subseteq> awp F T; T-B \<subseteq> FP G; F \<in> A leadsTo B|] ==> F\<squnion>G \<in> T\<inter>A leadsTo B"
paulson@13866
   488
apply (rule leadsTo_Join, assumption, blast)
paulson@13866
   489
apply (simp add: FP_def awp_iff_constrains stable_def constrains_def, blast) 
paulson@13853
   490
done
paulson@13853
   491
paulson@13821
   492
end