(* Title: HOL/UNITY/Comp/Priority.thy
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.*}
type_synonym state = "(vertex*vertex)set"
type_synonym command = "vertex=>(state*state)set"
consts
init :: "(vertex*vertex)set"
--{* the initial state *}
text{*Following the definitions given in section 4.4 *}
definition highest :: "[vertex, (vertex*vertex)set]=>bool"
where "highest i r <-> A i r = {}"
--{* i has highest priority in r *}
definition lowest :: "[vertex, (vertex*vertex)set]=>bool"
where "lowest i r <-> R i r = {}"
--{* i has lowest priority in r *}
definition act :: command
where "act i = {(s, s'). s'=reverse i s & highest i s}"
definition Component :: "vertex=>state program"
where "Component i = mk_total_program({init}, {act i}, UNIV)"
--{* All components start with the same initial state *}
text{*Some Abbreviations *}
definition Highest :: "vertex=>state set"
where "Highest i = {s. highest i s}"
definition Lowest :: "vertex=>state set"
where "Lowest i = {s. lowest i s}"
definition Acyclic :: "state set"
where "Acyclic = {s. acyclic s}"
definition Maximal :: "state set"
--{* Every ``above'' set has a maximal vertex*}
where "Maximal = (\<Inter>i. {s. ~highest i s-->(\<exists>j \<in> above i s. highest j s)})"
definition Maximal' :: "state set"
--{* Maximal vertex: equivalent definition*}
where "Maximal' = (\<Inter>i. Highest i Un (\<Union>j. {s. j \<in> above i s} Int Highest j))"
definition Safety :: "state set"
where "Safety = (\<Inter>i. {s. highest i s --> (\<forall>j \<in> neighbors i s. ~highest j s)})"
(* Composition of a finite set of component;
the vertex 'UNIV' is finite by assumption *)
definition system :: "state program"
where "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 \<in> 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 \<noteq> {}} 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 \<in> 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 \<in> Highest i co Highest i Un Lowest i"
by (simp add: Component_def, safety, fast)
text{* property 7: local axiom *}
lemma locality: "Component i \<in> stable {s. \<forall>j k. j\<noteq>i & k\<noteq>i--> ((j,k):s) = b j k}"
by (simp add: Component_def, safety)
subsection{*System properties*}
text{* property 8: strictly universal *}
lemma Safety: "system \<in> 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 \<in> {s. s = q} co {s. s=q} Un {s. \<exists>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 \<in> -Highest i Int {s. j\<notin>above i s} co {s. j\<notin>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 \<in> -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]:
"\<forall>i. system \<in> 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\<notin>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 \<in> 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 \<in> 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 \<in> 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\<notin>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 \<in> Highest i leadsTo (\<Inter>k. {s. i\<notin>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 \<in> Highest j Int {s. j \<in> above i s} leadsTo {s. j\<notin>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 \<in> (\<Union>j. {s. above i s = x} Int {s. j \<in> 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\<noteq>{} ==>
Acyclic Int {s. above i s = x} <=
(\<Union>j. {s. above i s = x} Int {s. j \<in> 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\<noteq>{}==> system \<in> 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 \<in> 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