Theory Priority

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theory Priority
imports PriorityAux
begin

(*  Title:      HOL/UNITY/Priority
    ID:         $Id: Priority.thy,v 1.8 2005/06/17 14:13:10 haftmann Exp $
    Author:     Sidi O Ehmety, Cambridge University Computer Laboratory
    Copyright   2001  University of Cambridge
*)

header{*The priority system*}

theory Priority imports PriorityAux begin

text{*From Charpentier and Chandy,
Examples of Program Composition Illustrating the Use of Universal Properties
   In J. Rolim (editor), Parallel and Distributed Processing,
   Spriner LNCS 1586 (1999), pages 1215-1227.*}

types state = "(vertex*vertex)set"
types command = "vertex=>(state*state)set"
  
consts
  init :: "(vertex*vertex)set"  
  --{* the initial state *}

text{*Following the definitions given in section 4.4 *}

constdefs
  highest :: "[vertex, (vertex*vertex)set]=>bool"
  "highest i r == A i r = {}"
    --{* i has highest priority in r *}
  
  lowest :: "[vertex, (vertex*vertex)set]=>bool"
  "lowest i r == R i r = {}"
    --{* i has lowest priority in r *}

  act :: command
  "act i == {(s, s'). s'=reverse i s & highest i s}"

  Component :: "vertex=>state program"
  "Component i == mk_total_program({init}, {act i}, UNIV)"
    --{* All components start with the same initial state *}


text{*Some Abbreviations *}
constdefs
  Highest :: "vertex=>state set"
  "Highest i == {s. highest i s}"

  Lowest :: "vertex=>state set"
  "Lowest i == {s. lowest i s}"

  Acyclic :: "state set"
  "Acyclic == {s. acyclic s}"


  Maximal :: "state set"
    --{* Every ``above'' set has a maximal vertex*}
  "Maximal == \<Inter>i. {s. ~highest i s-->(∃j ∈ above i  s. highest j s)}"

  Maximal' :: "state set"
    --{* Maximal vertex: equivalent definition*}
  "Maximal' == \<Inter>i. Highest i Un (\<Union>j. {s. j ∈ above i s} Int Highest j)"

  
  Safety :: "state set"
  "Safety == \<Inter>i. {s. highest i s --> (∀j ∈ neighbors i s. ~highest j s)}"


  (* Composition of a finite set of component;
     the vertex 'UNIV' is finite by assumption *)
  
  system :: "state program"
  "system == JN i. Component i"


declare highest_def [simp] lowest_def [simp]
declare Highest_def [THEN def_set_simp, simp] 
    and Lowest_def  [THEN def_set_simp, simp]

declare Component_def [THEN def_prg_Init, simp]
declare act_def [THEN def_act_simp, simp]



subsection{*Component correctness proofs*}

text{* neighbors is stable  *}
lemma Component_neighbors_stable: "Component i ∈ stable {s. neighbors k s = n}"
by (simp add: Component_def, safety, auto)

text{* property 4 *}
lemma Component_waits_priority: "Component i: {s. ((i,j):s) = b} Int (- Highest i) co {s. ((i,j):s)=b}"
by (simp add: Component_def, safety)

text{* property 5: charpentier and Chandy mistakenly express it as
 'transient Highest i'. Consider the case where i has neighbors *}
lemma Component_yields_priority: 
 "Component i: {s. neighbors i s ≠ {}} Int Highest i  
               ensures - Highest i"
apply (simp add: Component_def)
apply (ensures_tac "act i", blast+) 
done

text{* or better *}
lemma Component_yields_priority': "Component i ∈ Highest i ensures Lowest i"
apply (simp add: Component_def)
apply (ensures_tac "act i", blast+) 
done

text{* property 6: Component doesn't introduce cycle *}
lemma Component_well_behaves: "Component i ∈ Highest i co Highest i Un Lowest i"
by (simp add: Component_def, safety, fast)

text{* property 7: local axiom *}
lemma locality: "Component i ∈ stable {s. ∀j k. j≠i & k≠i--> ((j,k):s) = b j k}"
by (simp add: Component_def, safety)


subsection{*System  properties*}
text{* property 8: strictly universal *}

lemma Safety: "system ∈ stable Safety"
apply (unfold Safety_def)
apply (rule stable_INT)
apply (simp add: system_def, safety, fast)
done

text{* property 13: universal *}
lemma p13: "system ∈ {s. s = q} co {s. s=q} Un {s. ∃i. derive i q s}"
by (simp add: system_def Component_def mk_total_program_def totalize_JN, safety, blast)

text{* property 14: the 'above set' of a Component that hasn't got 
      priority doesn't increase *}
lemma above_not_increase: 
     "system ∈ -Highest i Int {s. j∉above i s} co {s. j∉above i s}"
apply (insert reach_lemma [of concl: j])
apply (simp add: system_def Component_def mk_total_program_def totalize_JN, 
       safety)
apply (simp add: trancl_converse, blast) 
done

lemma above_not_increase':
     "system ∈ -Highest i Int {s. above i s = x} co {s. above i s <= x}"
apply (insert above_not_increase [of i])
apply (simp add: trancl_converse constrains_def, blast)
done



text{* p15: universal property: all Components well behave  *}
lemma system_well_behaves [rule_format]:
     "∀i. system ∈ Highest i co Highest i Un Lowest i"
apply clarify
apply (simp add: system_def Component_def mk_total_program_def totalize_JN, 
       safety, auto)
done


lemma Acyclic_eq: "Acyclic = (\<Inter>i. {s. i∉above i s})"
by (auto simp add: Acyclic_def acyclic_def trancl_converse)


lemmas system_co =
      constrains_Un [OF above_not_increase [rule_format] system_well_behaves] 

lemma Acyclic_stable: "system ∈ stable Acyclic"
apply (simp add: stable_def Acyclic_eq) 
apply (auto intro!: constrains_INT system_co [THEN constrains_weaken] 
            simp add: image0_r_iff_image0_trancl trancl_converse)
done


lemma Acyclic_subset_Maximal: "Acyclic <= Maximal"
apply (unfold Acyclic_def Maximal_def, clarify)
apply (drule above_lemma_b, auto)
done

text{* property 17: original one is an invariant *}
lemma Acyclic_Maximal_stable: "system ∈ stable (Acyclic Int Maximal)"
by (simp add: Acyclic_subset_Maximal [THEN Int_absorb2] Acyclic_stable)


text{* property 5: existential property *}

lemma Highest_leadsTo_Lowest: "system ∈ Highest i leadsTo Lowest i"
apply (simp add: system_def Component_def mk_total_program_def totalize_JN)
apply (ensures_tac "act i", auto)
done

text{* a lowest i can never be in any abover set *} 
lemma Lowest_above_subset: "Lowest i <= (\<Inter>k. {s. i∉above k s})"
by (auto simp add: image0_r_iff_image0_trancl trancl_converse)

text{* property 18: a simpler proof than the original, one which uses psp *}
lemma Highest_escapes_above: "system ∈ Highest i leadsTo (\<Inter>k. {s. i∉above k s})"
apply (rule leadsTo_weaken_R)
apply (rule_tac [2] Lowest_above_subset)
apply (rule Highest_leadsTo_Lowest)
done

lemma Highest_escapes_above':
     "system ∈ Highest j Int {s. j ∈ above i s} leadsTo {s. j∉above i s}"
by (blast intro: leadsTo_weaken [OF Highest_escapes_above Int_lower1 INT_lower])

subsection{*The main result: above set decreases*}

text{* The original proof of the following formula was wrong *}

lemma Highest_iff_above0: "Highest i = {s. above i s ={}}"
by (auto simp add: image0_trancl_iff_image0_r)

lemmas above_decreases_lemma = 
     psp [THEN leadsTo_weaken, OF Highest_escapes_above' above_not_increase'] 


lemma above_decreases: 
     "system ∈ (\<Union>j. {s. above i s = x} Int {s. j ∈ above i s} Int Highest j)  
               leadsTo {s. above i s < x}"
apply (rule leadsTo_UN)
apply (rule single_leadsTo_I, clarify)
apply (rule_tac x2 = "above i x" in above_decreases_lemma)
apply (simp_all (no_asm_use) add: Highest_iff_above0)
apply blast+
done

(** Just a massage of conditions to have the desired form ***)
lemma Maximal_eq_Maximal': "Maximal = Maximal'"
by (unfold Maximal_def Maximal'_def Highest_def, blast)

lemma Acyclic_subset:
   "x≠{} ==>  
    Acyclic Int {s. above i s = x} <=  
    (\<Union>j. {s. above i s = x} Int {s. j ∈ above i s} Int Highest j)"
apply (rule_tac B = "Maximal' Int {s. above i s = x}" in subset_trans)
apply (simp (no_asm) add: Maximal_eq_Maximal' [symmetric])
apply (blast intro: Acyclic_subset_Maximal [THEN subsetD])
apply (simp (no_asm) del: above_def add: Maximal'_def Highest_iff_above0)
apply blast
done

lemmas above_decreases' = leadsTo_weaken_L [OF above_decreases Acyclic_subset]
lemmas above_decreases_psp = psp_stable [OF above_decreases' Acyclic_stable]

lemma above_decreases_psp': 
"x≠{}==> system ∈ Acyclic Int {s. above i s = x} leadsTo 
                   Acyclic Int {s. above i s < x}"
by (erule above_decreases_psp [THEN leadsTo_weaken], blast, auto)


lemmas finite_psubset_induct = wf_finite_psubset [THEN leadsTo_wf_induct]


lemma Progress: "system ∈ Acyclic leadsTo Highest i"
apply (rule_tac f = "%s. above i s" in finite_psubset_induct)
apply (simp del: above_def
            add: Highest_iff_above0 vimage_def finite_psubset_def, clarify)
apply (case_tac "m={}")
apply (rule Int_lower2 [THEN [2] leadsTo_weaken_L])
apply (force simp add: leadsTo_refl)
apply (rule_tac A' = "Acyclic Int {x. above i x < m}" in leadsTo_weaken_R)
apply (blast intro: above_decreases_psp')+
done


text{*We have proved all (relevant) theorems given in the paper.  We didn't
assume any thing about the relation @{term r}.  It is not necessary that
@{term r} be a priority relation as assumed in the original proof.  It
suffices that we start from a state which is finite and acyclic.*}


end

Component correctness proofs

lemma Component_neighbors_stable:

  Component i ∈ stable {s. neighbors k s = n}

lemma Component_waits_priority:

  Component i ∈ {s. ((i, j) ∈ s) = b} ∩ - Highest i co {s. ((i, j) ∈ s) = b}

lemma Component_yields_priority:

  Component i ∈ {s. neighbors i s  {}} ∩ Highest i ensures - Highest i

lemma Component_yields_priority':

  Component i ∈ Highest i ensures Lowest i

lemma Component_well_behaves:

  Component i ∈ Highest i co Highest i ∪ Lowest i

lemma locality:

  Component i ∈ stable {s. ∀j k. j  ik  i --> ((j, k) ∈ s) = b j k}

System properties

lemma Safety:

  system ∈ stable Safety

lemma p13:

  system ∈ {s. s = q} co {s. s = q} ∪ {s. ∃i. derive i q s}

lemma above_not_increase:

  system ∈ - Highest i ∩ {s. j  above i s} co {s. j  above i s}

lemma above_not_increase':

  system ∈ - Highest i ∩ {s. above i s = x} co {s. above i s  x}

lemma system_well_behaves:

  system ∈ Highest i co Highest i ∪ Lowest i

lemma Acyclic_eq:

  Acyclic = (INT i. {s. i  above i s})

lemma system_co:

  system
  ∈ - Highest i2 ∩ {s. j2  above i2 s} ∪ Highest i1 co
    {s. j2  above i2 s} ∪ (Highest i1 ∪ Lowest i1)

lemma Acyclic_stable:

  system ∈ stable Acyclic

lemma Acyclic_subset_Maximal:

  Acyclic  Maximal

lemma Acyclic_Maximal_stable:

  system ∈ stable (Acyclic ∩ Maximal)

lemma Highest_leadsTo_Lowest:

  system ∈ Highest i leadsTo Lowest i

lemma Lowest_above_subset:

  Lowest i  (INT k. {s. i  above k s})

lemma Highest_escapes_above:

  system ∈ Highest i leadsTo (INT k. {s. i  above k s})

lemma Highest_escapes_above':

  system ∈ Highest j ∩ {s. j ∈ above i s} leadsTo {s. j  above i s}

The main result: above set decreases

lemma Highest_iff_above0:

  Highest i = {s. above i s = {}}

lemma above_decreases_lemma:

  [| B  Highest j3 ∩ {s. j3 ∈ above i3 s} ∩ {s. above i2 s  x2};
     {s. j3  above i3 s} ∩ (- Highest i2 ∩ {s. above i2 s = x2}) ∪
     ({s. above i2 s  x2} - - Highest i2 ∩ {s. above i2 s = x2})
      B' |]
  ==> system ∈ B leadsTo B'

lemma above_decreases:

  system
  ∈ (UN j. {s. above i s = x} ∩ {s. j ∈ above i s} ∩ Highest j) leadsTo
    {s. above i s  x}

lemma Maximal_eq_Maximal':

  Maximal = Maximal'

lemma Acyclic_subset:

  x  {}
  ==> Acyclic ∩ {s. above i s = x}
       (UN j. {s. above i s = x} ∩ {s. j ∈ above i s} ∩ Highest j)

lemma above_decreases':

  x1  {} ==> system ∈ Acyclic ∩ {s. above i1 s = x1} leadsTo {s. above i1 s  x1}

lemma above_decreases_psp:

  x2  {}
  ==> system
      ∈ Acyclic ∩ {s. above i2 s = x2} ∩ Acyclic leadsTo
        {s. above i2 s  x2} ∩ Acyclic

lemma above_decreases_psp':

  x  {}
  ==> system ∈ Acyclic ∩ {s. above i s = x} leadsTo Acyclic ∩ {s. above i s  x}

lemma finite_psubset_induct:

  m. FAf -` {m} leadsTo Af -` finite_psubset^-1 `` {m} ∪ B
  ==> FA leadsTo B

lemma Progress:

  system ∈ Acyclic leadsTo Highest i