author | wenzelm |
Tue, 03 Jul 2007 22:27:27 +0200 | |
changeset 23560 | e43686246de4 |
parent 19741 | f65265d71426 |
child 26359 | 6d437bde2f1d |
permissions | -rw-r--r-- |
3433
2de17c994071
added deadlock freedom, polished definitions and proofs
mueller
parents:
diff
changeset
|
1 |
(* Title: HOLCF/IOA/meta_theory/Deadlock.thy |
2de17c994071
added deadlock freedom, polished definitions and proofs
mueller
parents:
diff
changeset
|
2 |
ID: $Id$ |
12218 | 3 |
Author: Olaf Müller |
17233 | 4 |
*) |
3433
2de17c994071
added deadlock freedom, polished definitions and proofs
mueller
parents:
diff
changeset
|
5 |
|
17233 | 6 |
header {* Deadlock freedom of I/O Automata *} |
3433
2de17c994071
added deadlock freedom, polished definitions and proofs
mueller
parents:
diff
changeset
|
7 |
|
17233 | 8 |
theory Deadlock |
9 |
imports RefCorrectness CompoScheds |
|
10 |
begin |
|
11 |
||
19741 | 12 |
text {* input actions may always be added to a schedule *} |
13 |
||
14 |
lemma scheds_input_enabled: |
|
15 |
"[| Filter (%x. x:act A)$sch : schedules A; a:inp A; input_enabled A; Finite sch|] |
|
16 |
==> Filter (%x. x:act A)$sch @@ a>>nil : schedules A" |
|
17 |
apply (simp add: schedules_def has_schedule_def) |
|
18 |
apply (tactic "safe_tac set_cs") |
|
19 |
apply (frule inp_is_act) |
|
20 |
apply (simp add: executions_def) |
|
21 |
apply (tactic {* pair_tac "ex" 1 *}) |
|
22 |
apply (rename_tac s ex) |
|
23 |
apply (subgoal_tac "Finite ex") |
|
24 |
prefer 2 |
|
25 |
apply (simp add: filter_act_def) |
|
26 |
defer |
|
27 |
apply (rule_tac [2] Map2Finite [THEN iffD1]) |
|
28 |
apply (rule_tac [2] t = "Map fst$ex" in subst) |
|
29 |
prefer 2 apply (assumption) |
|
30 |
apply (erule_tac [2] FiniteFilter) |
|
31 |
(* subgoal 1 *) |
|
32 |
apply (frule exists_laststate) |
|
33 |
apply (erule allE) |
|
34 |
apply (erule exE) |
|
35 |
(* using input-enabledness *) |
|
36 |
apply (simp add: input_enabled_def) |
|
37 |
apply (erule conjE)+ |
|
38 |
apply (erule_tac x = "a" in allE) |
|
39 |
apply simp |
|
40 |
apply (erule_tac x = "u" in allE) |
|
41 |
apply (erule exE) |
|
42 |
(* instantiate execution *) |
|
43 |
apply (rule_tac x = " (s,ex @@ (a,s2) >>nil) " in exI) |
|
44 |
apply (simp add: filter_act_def MapConc) |
|
45 |
apply (erule_tac t = "u" in lemma_2_1) |
|
46 |
apply simp |
|
47 |
apply (rule sym) |
|
48 |
apply assumption |
|
49 |
done |
|
50 |
||
51 |
text {* |
|
52 |
Deadlock freedom: component B cannot block an out or int action |
|
53 |
of component A in every schedule. |
|
54 |
Needs compositionality on schedule level, input-enabledness, compatibility |
|
55 |
and distributivity of is_exec_frag over @@ |
|
56 |
*} |
|
57 |
||
58 |
declare split_if [split del] |
|
59 |
lemma IOA_deadlock_free: "[| a : local A; Finite sch; sch : schedules (A||B); |
|
60 |
Filter (%x. x:act A)$(sch @@ a>>nil) : schedules A; compatible A B; input_enabled B |] |
|
61 |
==> (sch @@ a>>nil) : schedules (A||B)" |
|
62 |
apply (simp add: compositionality_sch locals_def) |
|
63 |
apply (rule conjI) |
|
64 |
(* a : act (A||B) *) |
|
65 |
prefer 2 |
|
66 |
apply (simp add: actions_of_par) |
|
67 |
apply (blast dest: int_is_act out_is_act) |
|
68 |
||
69 |
(* Filter B (sch@@[a]) : schedules B *) |
|
70 |
||
71 |
apply (case_tac "a:int A") |
|
72 |
apply (drule intA_is_not_actB) |
|
73 |
apply (assumption) (* --> a~:act B *) |
|
74 |
apply simp |
|
75 |
||
76 |
(* case a~:int A , i.e. a:out A *) |
|
77 |
apply (case_tac "a~:act B") |
|
78 |
apply simp |
|
79 |
(* case a:act B *) |
|
80 |
apply simp |
|
81 |
apply (subgoal_tac "a:out A") |
|
82 |
prefer 2 apply (blast) |
|
83 |
apply (drule outAactB_is_inpB) |
|
84 |
apply assumption |
|
85 |
apply assumption |
|
86 |
apply (rule scheds_input_enabled) |
|
87 |
apply simp |
|
88 |
apply assumption+ |
|
89 |
done |
|
90 |
||
91 |
declare split_if [split] |
|
92 |
||
17233 | 93 |
end |