Theory WFair

theory WFair
imports UNITY
(*  Title:      HOL/UNITY/WFair.thy
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1998  University of Cambridge

Conditional Fairness versions of transient, ensures, leadsTo.

From Misra, "A Logic for Concurrent Programming", 1994
*)

section‹Progress›

theory WFair imports UNITY begin

text‹The original version of this theory was based on weak fairness.  (Thus,
the entire UNITY development embodied this assumption, until February 2003.)
Weak fairness states that if a command is enabled continuously, then it is
eventually executed.  Ernie Cohen suggested that I instead adopt unconditional
fairness: every command is executed infinitely often.  

In fact, Misra's paper on "Progress" seems to be ambiguous about the correct
interpretation, and says that the two forms of fairness are equivalent.  They
differ only on their treatment of partial transitions, which under
unconditional fairness behave magically.  That is because if there are partial
transitions then there may be no fair executions, making all leads-to
properties hold vacuously.

Unconditional fairness has some great advantages.  By distinguishing partial
transitions from total ones that are the identity on part of their domain, it
is more expressive.  Also, by simplifying the definition of the transient
property, it simplifies many proofs.  A drawback is that some laws only hold
under the assumption that all transitions are total.  The best-known of these
is the impossibility law for leads-to.
›

definition

  ‹This definition specifies conditional fairness.  The rest of the theory
      is generic to all forms of fairness.  To get weak fairness, conjoin
      the inclusion below with @{term "A ⊆ Domain act"}, which specifies 
      that the action is enabled over all of @{term A}.›
  transient :: "'a set => 'a program set" where
    "transient A == {F. ∃act∈Acts F. act``A ⊆ -A}"

definition
  ensures :: "['a set, 'a set] => 'a program set"       (infixl "ensures" 60) where
    "A ensures B == (A-B co A ∪ B) ∩ transient (A-B)"


inductive_set
  leads :: "'a program => ('a set * 'a set) set"
    ‹LEADS-TO constant for the inductive definition›
  for F :: "'a program"
  where

    Basis:  "F ∈ A ensures B ==> (A,B) ∈ leads F"

  | Trans:  "[| (A,B) ∈ leads F;  (B,C) ∈ leads F |] ==> (A,C) ∈ leads F"

  | Union:  "∀A ∈ S. (A,B) ∈ leads F ==> (Union S, B) ∈ leads F"


definition leadsTo :: "['a set, 'a set] => 'a program set" (infixl "leadsTo" 60) where
     ‹visible version of the LEADS-TO relation›
    "A leadsTo B == {F. (A,B) ∈ leads F}"
  
definition wlt :: "['a program, 'a set] => 'a set" where
     ‹predicate transformer: the largest set that leads to @{term B}›
    "wlt F B == ⋃{A. F ∈ A leadsTo B}"

notation leadsTo  (infixl "⟼" 60)


subsection‹transient›

lemma stable_transient: 
    "[| F ∈ stable A; F ∈ transient A |] ==> ∃act∈Acts F. A ⊆ - (Domain act)"
apply (simp add: stable_def constrains_def transient_def, clarify)
apply (rule rev_bexI, auto)  
done

lemma stable_transient_empty: 
    "[| F ∈ stable A; F ∈ transient A; all_total F |] ==> A = {}"
apply (drule stable_transient, assumption)
apply (simp add: all_total_def)
done

lemma transient_strengthen: 
    "[| F ∈ transient A; B ⊆ A |] ==> F ∈ transient B"
apply (unfold transient_def, clarify)
apply (blast intro!: rev_bexI)
done

lemma transientI: 
    "[| act: Acts F;  act``A ⊆ -A |] ==> F ∈ transient A"
by (unfold transient_def, blast)

lemma transientE: 
    "[| F ∈ transient A;   
        !!act. [| act: Acts F;  act``A ⊆ -A |] ==> P |]  
     ==> P"
by (unfold transient_def, blast)

lemma transient_empty [simp]: "transient {} = UNIV"
by (unfold transient_def, auto)


text‹This equation recovers the notion of weak fairness.  A totalized
      program satisfies a transient assertion just if the original program
      contains a suitable action that is also enabled.›
lemma totalize_transient_iff:
   "(totalize F ∈ transient A) = (∃act∈Acts F. A ⊆ Domain act & act``A ⊆ -A)"
apply (simp add: totalize_def totalize_act_def transient_def 
                 Un_Image, safe)
apply (blast intro!: rev_bexI)+
done

lemma totalize_transientI: 
    "[| act: Acts F;  A ⊆ Domain act;  act``A ⊆ -A |] 
     ==> totalize F ∈ transient A"
by (simp add: totalize_transient_iff, blast)

subsection‹ensures›

lemma ensuresI: 
    "[| F ∈ (A-B) co (A ∪ B); F ∈ transient (A-B) |] ==> F ∈ A ensures B"
by (unfold ensures_def, blast)

lemma ensuresD: 
    "F ∈ A ensures B ==> F ∈ (A-B) co (A ∪ B) & F ∈ transient (A-B)"
by (unfold ensures_def, blast)

lemma ensures_weaken_R: 
    "[| F ∈ A ensures A'; A'<=B' |] ==> F ∈ A ensures B'"
apply (unfold ensures_def)
apply (blast intro: constrains_weaken transient_strengthen)
done

text‹The L-version (precondition strengthening) fails, but we have this›
lemma stable_ensures_Int: 
    "[| F ∈ stable C;  F ∈ A ensures B |]    
    ==> F ∈ (C ∩ A) ensures (C ∩ B)"
apply (unfold ensures_def)
apply (auto simp add: ensures_def Int_Un_distrib [symmetric] Diff_Int_distrib [symmetric])
prefer 2 apply (blast intro: transient_strengthen)
apply (blast intro: stable_constrains_Int constrains_weaken)
done

lemma stable_transient_ensures:
     "[| F ∈ stable A;  F ∈ transient C;  A ⊆ B ∪ C |] ==> F ∈ A ensures B"
apply (simp add: ensures_def stable_def)
apply (blast intro: constrains_weaken transient_strengthen)
done

lemma ensures_eq: "(A ensures B) = (A unless B) ∩ transient (A-B)"
by (simp (no_asm) add: ensures_def unless_def)


subsection‹leadsTo›

lemma leadsTo_Basis [intro]: "F ∈ A ensures B ==> F ∈ A leadsTo B"
apply (unfold leadsTo_def)
apply (blast intro: leads.Basis)
done

lemma leadsTo_Trans: 
     "[| F ∈ A leadsTo B;  F ∈ B leadsTo C |] ==> F ∈ A leadsTo C"
apply (unfold leadsTo_def)
apply (blast intro: leads.Trans)
done

lemma leadsTo_Basis':
     "[| F ∈ A co A ∪ B; F ∈ transient A |] ==> F ∈ A leadsTo B"
apply (drule_tac B = "A-B" in constrains_weaken_L)
apply (drule_tac [2] B = "A-B" in transient_strengthen)
apply (rule_tac [3] ensuresI [THEN leadsTo_Basis])
apply (blast+)
done

lemma transient_imp_leadsTo: "F ∈ transient A ==> F ∈ A leadsTo (-A)"
by (simp (no_asm_simp) add: leadsTo_Basis ensuresI Compl_partition)

text‹Useful with cancellation, disjunction›
lemma leadsTo_Un_duplicate: "F ∈ A leadsTo (A' ∪ A') ==> F ∈ A leadsTo A'"
by (simp add: Un_ac)

lemma leadsTo_Un_duplicate2:
     "F ∈ A leadsTo (A' ∪ C ∪ C) ==> F ∈ A leadsTo (A' ∪ C)"
by (simp add: Un_ac)

text‹The Union introduction rule as we should have liked to state it›
lemma leadsTo_Union: 
    "(!!A. A ∈ S ==> F ∈ A leadsTo B) ==> F ∈ (⋃S) leadsTo B"
apply (unfold leadsTo_def)
apply (blast intro: leads.Union)
done

lemma leadsTo_Union_Int: 
 "(!!A. A ∈ S ==> F ∈ (A ∩ C) leadsTo B) ==> F ∈ (⋃S ∩ C) leadsTo B"
apply (unfold leadsTo_def)
apply (simp only: Int_Union_Union)
apply (blast intro: leads.Union)
done

lemma leadsTo_UN: 
    "(!!i. i ∈ I ==> F ∈ (A i) leadsTo B) ==> F ∈ (⋃i ∈ I. A i) leadsTo B"
apply (blast intro: leadsTo_Union)
done

text‹Binary union introduction rule›
lemma leadsTo_Un:
     "[| F ∈ A leadsTo C; F ∈ B leadsTo C |] ==> F ∈ (A ∪ B) leadsTo C"
  using leadsTo_Union [of "{A, B}" F C] by auto

lemma single_leadsTo_I: 
     "(!!x. x ∈ A ==> F ∈ {x} leadsTo B) ==> F ∈ A leadsTo B"
by (subst UN_singleton [symmetric], rule leadsTo_UN, blast)


text‹The INDUCTION rule as we should have liked to state it›
lemma leadsTo_induct: 
  "[| F ∈ za leadsTo zb;   
      !!A B. F ∈ A ensures B ==> P A B;  
      !!A B C. [| F ∈ A leadsTo B; P A B; F ∈ B leadsTo C; P B C |]  
               ==> P A C;  
      !!B S. ∀A ∈ S. F ∈ A leadsTo B & P A B ==> P (⋃S) B  
   |] ==> P za zb"
apply (unfold leadsTo_def)
apply (drule CollectD, erule leads.induct)
apply (blast+)
done


lemma subset_imp_ensures: "A ⊆ B ==> F ∈ A ensures B"
by (unfold ensures_def constrains_def transient_def, blast)

lemmas subset_imp_leadsTo = subset_imp_ensures [THEN leadsTo_Basis]

lemmas leadsTo_refl = subset_refl [THEN subset_imp_leadsTo]

lemmas empty_leadsTo = empty_subsetI [THEN subset_imp_leadsTo, simp]

lemmas leadsTo_UNIV = subset_UNIV [THEN subset_imp_leadsTo, simp]



(** Variant induction rule: on the preconditions for B **)

text‹Lemma is the weak version: can't see how to do it in one step›
lemma leadsTo_induct_pre_lemma: 
  "[| F ∈ za leadsTo zb;   
      P zb;  
      !!A B. [| F ∈ A ensures B;  P B |] ==> P A;  
      !!S. ∀A ∈ S. P A ==> P (⋃S)  
   |] ==> P za"
txt‹by induction on this formula›
apply (subgoal_tac "P zb --> P za")
txt‹now solve first subgoal: this formula is sufficient›
apply (blast intro: leadsTo_refl)
apply (erule leadsTo_induct)
apply (blast+)
done

lemma leadsTo_induct_pre: 
  "[| F ∈ za leadsTo zb;   
      P zb;  
      !!A B. [| F ∈ A ensures B;  F ∈ B leadsTo zb;  P B |] ==> P A;  
      !!S. ∀A ∈ S. F ∈ A leadsTo zb & P A ==> P (⋃S)  
   |] ==> P za"
apply (subgoal_tac "F ∈ za leadsTo zb & P za")
apply (erule conjunct2)
apply (erule leadsTo_induct_pre_lemma)
prefer 3 apply (blast intro: leadsTo_Union)
prefer 2 apply (blast intro: leadsTo_Trans)
apply (blast intro: leadsTo_refl)
done


lemma leadsTo_weaken_R: "[| F ∈ A leadsTo A'; A'<=B' |] ==> F ∈ A leadsTo B'"
by (blast intro: subset_imp_leadsTo leadsTo_Trans)

lemma leadsTo_weaken_L:
     "[| F ∈ A leadsTo A'; B ⊆ A |] ==> F ∈ B leadsTo A'"
by (blast intro: leadsTo_Trans subset_imp_leadsTo)

text‹Distributes over binary unions›
lemma leadsTo_Un_distrib:
     "F ∈ (A ∪ B) leadsTo C  =  (F ∈ A leadsTo C & F ∈ B leadsTo C)"
by (blast intro: leadsTo_Un leadsTo_weaken_L)

lemma leadsTo_UN_distrib:
     "F ∈ (⋃i ∈ I. A i) leadsTo B  =  (∀i ∈ I. F ∈ (A i) leadsTo B)"
by (blast intro: leadsTo_UN leadsTo_weaken_L)

lemma leadsTo_Union_distrib:
     "F ∈ (⋃S) leadsTo B  =  (∀A ∈ S. F ∈ A leadsTo B)"
by (blast intro: leadsTo_Union leadsTo_weaken_L)


lemma leadsTo_weaken:
     "[| F ∈ A leadsTo A'; B ⊆ A; A'<=B' |] ==> F ∈ B leadsTo B'"
by (blast intro: leadsTo_weaken_R leadsTo_weaken_L leadsTo_Trans)


text‹Set difference: maybe combine with ‹leadsTo_weaken_L›??›
lemma leadsTo_Diff:
     "[| F ∈ (A-B) leadsTo C; F ∈ B leadsTo C |]   ==> F ∈ A leadsTo C"
by (blast intro: leadsTo_Un leadsTo_weaken)

lemma leadsTo_UN_UN:
   "(!! i. i ∈ I ==> F ∈ (A i) leadsTo (A' i))  
    ==> F ∈ (⋃i ∈ I. A i) leadsTo (⋃i ∈ I. A' i)"
apply (blast intro: leadsTo_Union leadsTo_weaken_R)
done

text‹Binary union version›
lemma leadsTo_Un_Un:
     "[| F ∈ A leadsTo A'; F ∈ B leadsTo B' |]  
      ==> F ∈ (A ∪ B) leadsTo (A' ∪ B')"
by (blast intro: leadsTo_Un leadsTo_weaken_R)


(** The cancellation law **)

lemma leadsTo_cancel2:
     "[| F ∈ A leadsTo (A' ∪ B); F ∈ B leadsTo B' |]  
      ==> F ∈ A leadsTo (A' ∪ B')"
by (blast intro: leadsTo_Un_Un subset_imp_leadsTo leadsTo_Trans)

lemma leadsTo_cancel_Diff2:
     "[| F ∈ A leadsTo (A' ∪ B); F ∈ (B-A') leadsTo B' |]  
      ==> F ∈ A leadsTo (A' ∪ B')"
apply (rule leadsTo_cancel2)
prefer 2 apply assumption
apply (simp_all (no_asm_simp))
done

lemma leadsTo_cancel1:
     "[| F ∈ A leadsTo (B ∪ A'); F ∈ B leadsTo B' |]  
    ==> F ∈ A leadsTo (B' ∪ A')"
apply (simp add: Un_commute)
apply (blast intro!: leadsTo_cancel2)
done

lemma leadsTo_cancel_Diff1:
     "[| F ∈ A leadsTo (B ∪ A'); F ∈ (B-A') leadsTo B' |]  
    ==> F ∈ A leadsTo (B' ∪ A')"
apply (rule leadsTo_cancel1)
prefer 2 apply assumption
apply (simp_all (no_asm_simp))
done


text‹The impossibility law›
lemma leadsTo_empty: "[|F ∈ A leadsTo {}; all_total F|] ==> A={}"
apply (erule leadsTo_induct_pre)
apply (simp_all add: ensures_def constrains_def transient_def all_total_def, clarify)
apply (drule bspec, assumption)+
apply blast
done

subsection‹PSP: Progress-Safety-Progress›

text‹Special case of PSP: Misra's "stable conjunction"›
lemma psp_stable: 
   "[| F ∈ A leadsTo A'; F ∈ stable B |]  
    ==> F ∈ (A ∩ B) leadsTo (A' ∩ B)"
apply (unfold stable_def)
apply (erule leadsTo_induct)
prefer 3 apply (blast intro: leadsTo_Union_Int)
prefer 2 apply (blast intro: leadsTo_Trans)
apply (rule leadsTo_Basis)
apply (simp add: ensures_def Diff_Int_distrib2 [symmetric] Int_Un_distrib2 [symmetric])
apply (blast intro: transient_strengthen constrains_Int)
done

lemma psp_stable2: 
   "[| F ∈ A leadsTo A'; F ∈ stable B |] ==> F ∈ (B ∩ A) leadsTo (B ∩ A')"
by (simp add: psp_stable Int_ac)

lemma psp_ensures: 
   "[| F ∈ A ensures A'; F ∈ B co B' |]  
    ==> F ∈ (A ∩ B') ensures ((A' ∩ B) ∪ (B' - B))"
apply (unfold ensures_def constrains_def, clarify) (*speeds up the proof*)
apply (blast intro: transient_strengthen)
done

lemma psp:
     "[| F ∈ A leadsTo A'; F ∈ B co B' |]  
      ==> F ∈ (A ∩ B') leadsTo ((A' ∩ B) ∪ (B' - B))"
apply (erule leadsTo_induct)
  prefer 3 apply (blast intro: leadsTo_Union_Int)
 txt‹Basis case›
 apply (blast intro: psp_ensures)
txt‹Transitivity case has a delicate argument involving "cancellation"›
apply (rule leadsTo_Un_duplicate2)
apply (erule leadsTo_cancel_Diff1)
apply (simp add: Int_Diff Diff_triv)
apply (blast intro: leadsTo_weaken_L dest: constrains_imp_subset)
done

lemma psp2:
     "[| F ∈ A leadsTo A'; F ∈ B co B' |]  
    ==> F ∈ (B' ∩ A) leadsTo ((B ∩ A') ∪ (B' - B))"
by (simp (no_asm_simp) add: psp Int_ac)

lemma psp_unless: 
   "[| F ∈ A leadsTo A';  F ∈ B unless B' |]  
    ==> F ∈ (A ∩ B) leadsTo ((A' ∩ B) ∪ B')"

apply (unfold unless_def)
apply (drule psp, assumption)
apply (blast intro: leadsTo_weaken)
done


subsection‹Proving the induction rules›

(** The most general rule: r is any wf relation; f is any variant function **)

lemma leadsTo_wf_induct_lemma:
     "[| wf r;      
         ∀m. F ∈ (A ∩ f-`{m}) leadsTo                      
                    ((A ∩ f-`(r^-1 `` {m})) ∪ B) |]  
      ==> F ∈ (A ∩ f-`{m}) leadsTo B"
apply (erule_tac a = m in wf_induct)
apply (subgoal_tac "F ∈ (A ∩ (f -` (r^-1 `` {x}))) leadsTo B")
 apply (blast intro: leadsTo_cancel1 leadsTo_Un_duplicate)
apply (subst vimage_eq_UN)
apply (simp only: UN_simps [symmetric])
apply (blast intro: leadsTo_UN)
done


(** Meta or object quantifier ? **)
lemma leadsTo_wf_induct:
     "[| wf r;      
         ∀m. F ∈ (A ∩ f-`{m}) leadsTo                      
                    ((A ∩ f-`(r^-1 `` {m})) ∪ B) |]  
      ==> F ∈ A leadsTo B"
apply (rule_tac t = A in subst)
 defer 1
 apply (rule leadsTo_UN)
 apply (erule leadsTo_wf_induct_lemma)
 apply assumption
apply fast (*Blast_tac: Function unknown's argument not a parameter*)
done


lemma bounded_induct:
     "[| wf r;      
         ∀m ∈ I. F ∈ (A ∩ f-`{m}) leadsTo                    
                      ((A ∩ f-`(r^-1 `` {m})) ∪ B) |]  
      ==> F ∈ A leadsTo ((A - (f-`I)) ∪ B)"
apply (erule leadsTo_wf_induct, safe)
apply (case_tac "m ∈ I")
apply (blast intro: leadsTo_weaken)
apply (blast intro: subset_imp_leadsTo)
done


(*Alternative proof is via the lemma F ∈ (A ∩ f-`(lessThan m)) leadsTo B*)
lemma lessThan_induct: 
     "[| !!m::nat. F ∈ (A ∩ f-`{m}) leadsTo ((A ∩ f-`{..<m}) ∪ B) |]  
      ==> F ∈ A leadsTo B"
apply (rule wf_less_than [THEN leadsTo_wf_induct])
apply (simp (no_asm_simp))
apply blast
done

lemma lessThan_bounded_induct:
     "!!l::nat. [| ∀m ∈ greaterThan l.     
            F ∈ (A ∩ f-`{m}) leadsTo ((A ∩ f-`(lessThan m)) ∪ B) |]  
      ==> F ∈ A leadsTo ((A ∩ (f-`(atMost l))) ∪ B)"
apply (simp only: Diff_eq [symmetric] vimage_Compl Compl_greaterThan [symmetric])
apply (rule wf_less_than [THEN bounded_induct])
apply (simp (no_asm_simp))
done

lemma greaterThan_bounded_induct:
     "(!!l::nat. ∀m ∈ lessThan l.     
                 F ∈ (A ∩ f-`{m}) leadsTo ((A ∩ f-`(greaterThan m)) ∪ B))
      ==> F ∈ A leadsTo ((A ∩ (f-`(atLeast l))) ∪ B)"
apply (rule_tac f = f and f1 = "%k. l - k" 
       in wf_less_than [THEN wf_inv_image, THEN leadsTo_wf_induct])
apply (simp (no_asm) add:Image_singleton)
apply clarify
apply (case_tac "m<l")
 apply (blast intro: leadsTo_weaken_R diff_less_mono2)
apply (blast intro: not_le_imp_less subset_imp_leadsTo)
done


subsection‹wlt›

text‹Misra's property W3›
lemma wlt_leadsTo: "F ∈ (wlt F B) leadsTo B"
apply (unfold wlt_def)
apply (blast intro!: leadsTo_Union)
done

lemma leadsTo_subset: "F ∈ A leadsTo B ==> A ⊆ wlt F B"
apply (unfold wlt_def)
apply (blast intro!: leadsTo_Union)
done

text‹Misra's property W2›
lemma leadsTo_eq_subset_wlt: "F ∈ A leadsTo B = (A ⊆ wlt F B)"
by (blast intro!: leadsTo_subset wlt_leadsTo [THEN leadsTo_weaken_L])

text‹Misra's property W4›
lemma wlt_increasing: "B ⊆ wlt F B"
apply (simp (no_asm_simp) add: leadsTo_eq_subset_wlt [symmetric] subset_imp_leadsTo)
done


text‹Used in the Trans case below›
lemma lemma1: 
   "[| B ⊆ A2;   
       F ∈ (A1 - B) co (A1 ∪ B);  
       F ∈ (A2 - C) co (A2 ∪ C) |]  
    ==> F ∈ (A1 ∪ A2 - C) co (A1 ∪ A2 ∪ C)"
by (unfold constrains_def, clarify,  blast)

text‹Lemma (1,2,3) of Misra's draft book, Chapter 4, "Progress"›
lemma leadsTo_123:
     "F ∈ A leadsTo A'  
      ==> ∃B. A ⊆ B & F ∈ B leadsTo A' & F ∈ (B-A') co (B ∪ A')"
apply (erule leadsTo_induct)
  txt‹Basis›
  apply (blast dest: ensuresD)
 txt‹Trans›
 apply clarify
 apply (rule_tac x = "Ba ∪ Bb" in exI)
 apply (blast intro: lemma1 leadsTo_Un_Un leadsTo_cancel1 leadsTo_Un_duplicate)
txt‹Union›
apply (clarify dest!: ball_conj_distrib [THEN iffD1] bchoice)
apply (rule_tac x = "⋃A ∈ S. f A" in exI)
apply (auto intro: leadsTo_UN)
(*Blast_tac says PROOF FAILED*)
apply (rule_tac I1=S and A1="%i. f i - B" and A'1="%i. f i ∪ B" 
       in constrains_UN [THEN constrains_weaken], auto) 
done


text‹Misra's property W5›
lemma wlt_constrains_wlt: "F ∈ (wlt F B - B) co (wlt F B)"
proof -
  from wlt_leadsTo [of F B, THEN leadsTo_123]
  show ?thesis
  proof (elim exE conjE)
(* assumes have to be in exactly the form as in the goal displayed at
   this point.  Isar doesn't give you any automation. *)
    fix C
    assume wlt: "wlt F B ⊆ C"
       and lt:  "F ∈ C leadsTo B"
       and co:  "F ∈ C - B co C ∪ B"
    have eq: "C = wlt F B"
    proof -
      from lt and wlt show ?thesis 
           by (blast dest: leadsTo_eq_subset_wlt [THEN iffD1])
    qed
    from co show ?thesis by (simp add: eq wlt_increasing Un_absorb2)
  qed
qed


subsection‹Completion: Binary and General Finite versions›

lemma completion_lemma :
     "[| W = wlt F (B' ∪ C);      
       F ∈ A leadsTo (A' ∪ C);  F ∈ A' co (A' ∪ C);    
       F ∈ B leadsTo (B' ∪ C);  F ∈ B' co (B' ∪ C) |]  
    ==> F ∈ (A ∩ B) leadsTo ((A' ∩ B') ∪ C)"
apply (subgoal_tac "F ∈ (W-C) co (W ∪ B' ∪ C) ")
 prefer 2
 apply (blast intro: wlt_constrains_wlt [THEN [2] constrains_Un, 
                                         THEN constrains_weaken])
apply (subgoal_tac "F ∈ (W-C) co W")
 prefer 2
 apply (simp add: wlt_increasing Un_assoc Un_absorb2)
apply (subgoal_tac "F ∈ (A ∩ W - C) leadsTo (A' ∩ W ∪ C) ")
 prefer 2 apply (blast intro: wlt_leadsTo psp [THEN leadsTo_weaken])
(** LEVEL 6 **)
apply (subgoal_tac "F ∈ (A' ∩ W ∪ C) leadsTo (A' ∩ B' ∪ C) ")
 prefer 2
 apply (rule leadsTo_Un_duplicate2)
 apply (blast intro: leadsTo_Un_Un wlt_leadsTo
                         [THEN psp2, THEN leadsTo_weaken] leadsTo_refl)
apply (drule leadsTo_Diff)
apply (blast intro: subset_imp_leadsTo)
apply (subgoal_tac "A ∩ B ⊆ A ∩ W")
 prefer 2
 apply (blast dest!: leadsTo_subset intro!: subset_refl [THEN Int_mono])
apply (blast intro: leadsTo_Trans subset_imp_leadsTo)
done

lemmas completion = completion_lemma [OF refl]

lemma finite_completion_lemma:
     "finite I ==> (∀i ∈ I. F ∈ (A i) leadsTo (A' i ∪ C)) -->   
                   (∀i ∈ I. F ∈ (A' i) co (A' i ∪ C)) -->  
                   F ∈ (⋂i ∈ I. A i) leadsTo ((⋂i ∈ I. A' i) ∪ C)"
apply (erule finite_induct, auto)
apply (rule completion)
   prefer 4
   apply (simp only: INT_simps [symmetric])
   apply (rule constrains_INT, auto)
done

lemma finite_completion: 
     "[| finite I;   
         !!i. i ∈ I ==> F ∈ (A i) leadsTo (A' i ∪ C);  
         !!i. i ∈ I ==> F ∈ (A' i) co (A' i ∪ C) |]    
      ==> F ∈ (⋂i ∈ I. A i) leadsTo ((⋂i ∈ I. A' i) ∪ C)"
by (blast intro: finite_completion_lemma [THEN mp, THEN mp])

lemma stable_completion: 
     "[| F ∈ A leadsTo A';  F ∈ stable A';    
         F ∈ B leadsTo B';  F ∈ stable B' |]  
    ==> F ∈ (A ∩ B) leadsTo (A' ∩ B')"
apply (unfold stable_def)
apply (rule_tac C1 = "{}" in completion [THEN leadsTo_weaken_R])
apply (force+)
done

lemma finite_stable_completion: 
     "[| finite I;   
         !!i. i ∈ I ==> F ∈ (A i) leadsTo (A' i);  
         !!i. i ∈ I ==> F ∈ stable (A' i) |]    
      ==> F ∈ (⋂i ∈ I. A i) leadsTo (⋂i ∈ I. A' i)"
apply (unfold stable_def)
apply (rule_tac C1 = "{}" in finite_completion [THEN leadsTo_weaken_R])
apply (simp_all (no_asm_simp))
apply blast+
done

end