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Marco Maida
PROSA  Formally Proven Schedulability Analysis
Commits
d574638e
Commit
d574638e
authored
Jan 19, 2016
by
Felipe Cerqueira
Browse files
Add lemma about pending job + comments
parent
8e5fabfe
Changes
1
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schedule.v
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d574638e
...
...
@@ 117,8 +117,8 @@ Module Schedule.
End
ValidSchedules
.
(* In this section, we prove some basic lemmas about
service
. *)
Section
Service
Lemmas
.
(* In this section, we prove some basic lemmas about
a job
. *)
Section
Job
Lemmas
.
(* Consider an arrival sequence, ...*)
Context
{
Job
:
eqType
}.
...
...
@@ 253,7 +253,7 @@ Module Schedule.
(* Assume that completed jobs do not execute. *)
Hypothesis
H_completed_jobs
:
completed_jobs_dont_execute
job_cost
sched
.
(* Then, after job j completes, it remains completed. *)
Lemma
completion_monotonic
:
forall
t
t'
,
...
...
@@ 333,7 +333,41 @@ Module Schedule.
End
Arrival
.
End
ServiceLemmas
.
Section
Pending
.
(* Assume that jobs must arrive to execute. *)
Hypothesis
H_jobs_must_arrive
:
jobs_must_arrive_to_execute
sched
.
(* Assume that completed jobs do not execute. *)
Hypothesis
H_completed_jobs
:
completed_jobs_dont_execute
job_cost
sched
.
(* Then, if job j is scheduled, it must be pending. *)
Lemma
scheduled_implies_pending
:
forall
t
,
scheduled
sched
j
t
>
pending
job_cost
sched
j
t
.
Proof
.
rename
H_jobs_must_arrive
into
ARRIVE
,
H_completed_jobs
into
COMP
.
unfold
jobs_must_arrive_to_execute
,
completed_jobs_dont_execute
in
*.
intros
t
SCHED
.
unfold
pending
;
apply
/
andP
;
split
;
first
by
apply
ARRIVE
.
apply
/
negP
;
unfold
not
;
intro
COMPLETED
.
have
BUG
:
=
COMP
j
t
.+
1
.
rewrite
leqNgt
in
BUG
;
move
:
BUG
=>
/
negP
BUG
;
apply
BUG
.
unfold
service
;
rewrite

addn1
big_nat_recr
//
/=.
apply
leq_add
;
first
by
move
:
COMPLETED
=>
/
eqP
COMPLETED
;
rewrite

COMPLETED
.
rewrite
lt0n
;
apply
/
eqP
;
red
;
move
=>
/
eqP
NOSERV
.
rewrite

not_scheduled_no_service
in
NOSERV
.
by
rewrite
SCHED
in
NOSERV
.
Qed
.
End
Pending
.
End
JobLemmas
.
(* In this section, we prove some lemmas about the list of jobs
scheduled at time t. *)
...
...
@@ 347,6 +381,7 @@ Module Schedule.
Context
{
num_cpus
:
nat
}.
Variable
sched
:
schedule
num_cpus
arr_seq
.
(* A job is in the list of scheduled jobs iff it is scheduled. *)
Lemma
mem_scheduled_jobs_eq_scheduled
:
forall
j
t
,
j
\
in
jobs_scheduled_at
sched
t
=
scheduled
sched
j
t
.
...
...
@@ 369,9 +404,11 @@ Module Schedule.
Section
Uniqueness
.
(* Suppose there's no job parallelism. *)
Hypothesis
H_no_parallelism
:
jobs_dont_execute_in_parallel
sched
.
(* Then, the list of jobs scheduled at any time t has no duplicates. *)
Lemma
scheduled_jobs_uniq
:
forall
t
,
uniq
(
jobs_scheduled_at
sched
t
).
...
...
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