replaced apply-style proof for instance Multiset :: plus_ac0 by recommended Isar proof style
(* Title: HOL/UNITY/Priority
ID: $Id$
Author: Sidi O Ehmety, Cambridge University Computer Laboratory
Copyright 2001 University of Cambridge
*)
header{*The priority system*}
theory Priority = PriorityAux:
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
(* the initial state *)
init :: "(vertex*vertex)set"
constdefs
(* from the definitions given in section 4.4 *)
(* i has highest priority in r *)
highest :: "[vertex, (vertex*vertex)set]=>bool"
"highest i r == A i r = {}"
(* i has lowest priority in r *)
lowest :: "[vertex, (vertex*vertex)set]=>bool"
"lowest i r == R i r = {}"
act :: command
"act i == {(s, s'). s'=reverse i s & highest i s}"
(* All components start with the same initial state *)
Component :: "vertex=>state program"
"Component i == mk_total_program({init}, {act i}, UNIV)"
(* Abbreviations *)
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}"
(* Every above set has a maximal vertex: two equivalent defs. *)
Maximal :: "state set"
"Maximal == \<Inter>i. {s. ~highest i s-->(\<exists>j \<in> above i s. highest j s)}"
Maximal' :: "state set"
"Maximal' == \<Inter>i. Highest i Un (\<Union>j. {s. j \<in> above i s} Int Highest j)"
Safety :: "state set"
"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 *)
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*}
(* neighbors is stable *)
lemma Component_neighbors_stable: "Component i \<in> stable {s. neighbors k s = n}"
by (simp add: Component_def, constrains, auto)
(* 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, constrains)
(* 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
(* 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
(* 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, constrains, fast)
(* 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, constrains)
subsection{*System properties*}
(* property 8: strictly universal *)
lemma Safety: "system \<in> stable Safety"
apply (unfold Safety_def)
apply (rule stable_INT)
apply (simp add: system_def, constrains, fast)
done
(* 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, constrains, blast)
(* 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,
constrains)
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
(* 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,
constrains, 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
(* 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)
(* propert 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
(* 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)
(* 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])
(*** The main result: above set decreases ***)
(* 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