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Xiaojie Guo
rtproofs
Commits
a96c1e6a
Commit
a96c1e6a
authored
Jan 05, 2016
by
Felipe Cerqueira
Browse files
Comment FP workload code
parent
b5b3c3ed
Changes
1
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workload_fp.v
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a96c1e6a
...
...
@@ 128,11 +128,9 @@ Module WorkloadBoundFP.
Variable
num_cpus
:
nat
.
Variable
rate
:
Job
>
processor
num_cpus
>
nat
.
Variable
schedule_of_platform
:
schedule
num_cpus
arr_seq
>
Prop
.
(*
Assume
any schedule of a given platform. *)
(*
Consider
any schedule of a given platform. *)
Variable
sched
:
schedule
num_cpus
arr_seq
.
Hypothesis
sched_of_platform
:
schedule_of_platform
sched
.
(* Assumption: jobs only execute if they arrived.
This is used to eliminate jobs that arrive after end of the interval t1 + delta. *)
...
...
@@ 222,7 +220,9 @@ Module WorkloadBoundFP.
(* Now, let's consider the list of interfering jobs sorted by arrival time. *)
Let
order
:
=
fun
(
x
y
:
JobIn
arr_seq
)
=>
job_arrival
x
<=
job_arrival
y
.
Let
sorted_jobs
:
=
(
sort
order
interfering_jobs
).
(* The first step consists in simplifying the sum corresponding
to the workload. *)
Section
SimplifyJobSequence
.
(* Remove the elements that we don't care about from the sum *)
...
...
@@ 237,6 +237,7 @@ Module WorkloadBoundFP.
by
rewrite
big_const_seq
iter_addn
mul0n
add0n
add0n
big_filter
.
Qed
.
(* Consider the sum over the sorted sequence of jobs. *)
Lemma
workload_bound_simpl_by_sorting_interfering_jobs
:
\
sum_
(
i
<
interfering_jobs
)
service_during
rate
sched
i
t1
t2
=
\
sum_
(
i
<
sorted_jobs
)
service_during
rate
sched
i
t1
t2
.
...
...
@@ 252,6 +253,8 @@ Module WorkloadBoundFP.
by
apply
perm_eq_mem
;
rewrite
(
perm_sort
order
).
Qed
.
(* Remember that all jobs in the sorted sequence is an
interfering job of task tsk. *)
Lemma
workload_bound_all_jobs_from_tsk
:
forall
j_i
,
j_i
\
in
sorted_jobs
>
...
...
@@ 264,7 +267,7 @@ Module WorkloadBoundFP.
by
move
:
LTi
=>
/
andP
[/
andP
[/
eqP
JOBi
SERVi
]
INi
]
;
repeat
split
.
Qed
.
(* Remember that
th
e jobs are ordered by arrival. *)
(* Remember that
consecutiv
e jobs are ordered by arrival. *)
Lemma
workload_bound_jobs_ordered_by_arrival
:
forall
i
elem
,
i
<
(
size
sorted_jobs
).
1
>
...
...
@@ 277,7 +280,9 @@ Module WorkloadBoundFP.
Qed
.
End
SimplifyJobSequence
.
(* Next, we show that if the number of jobs is no larger than n_k,
the workload bound trivially holds. *)
Section
WorkloadNotManyJobs
.
Lemma
workload_bound_holds_for_at_most_n_k_jobs
:
...
...
@@ 302,6 +307,8 @@ Module WorkloadBoundFP.
End
WorkloadNotManyJobs
.
(* Otherwise, assume that the number of jobs is larger than n_k >= 0.
First, consider the simple case with only one job. *)
Section
WorkloadSingleJob
.
(* Assume that there's at least one job in the sorted list. *)
...
...
@@ 309,7 +316,8 @@ Module WorkloadBoundFP.
Variable
elem
:
JobIn
arr_seq
.
Let
j_fst
:
=
nth
elem
sorted_jobs
0
.
(* The first job is an interfering job of task tsk. *)
Lemma
workload_bound_j_fst_is_job_of_tsk
:
job_task
j_fst
=
tsk
/\
service_during
rate
sched
j_fst
t1
t2
!=
0
/\
...
...
@@ 317,44 +325,48 @@ Module WorkloadBoundFP.
Proof
.
by
apply
workload_bound_all_jobs_from_tsk
,
mem_nth
.
Qed
.
(* The workload bound holds for the single job. *)
Lemma
workload_bound_holds_for_a_single_job
:
\
sum_
(
0
<=
i
<
1
)
service_during
rate
sched
(
nth
elem
sorted_jobs
i
)
t1
t2
<=
workload_bound
.
Proof
.
unfold
workload_bound
,
W
;
fold
n_k
.
have
INfst
:
=
workload_bound_j_fst_is_job_of_tsk
;
des
.
rewrite
big_nat_recr
//
big_geq
//
[
nth
]
lock
/=

lock
add0n
.
destruct
n_k
;
last
first
.
{
rewrite
[
service_during
_
_
_
_
_
]
add0n
;
rewrite
leq_add
//.
rewrite
[
service_during
_
_
_
_
_
]
add0n
[
_
*
task_cost
tsk
]
mulSnr
.
apply
leq_add
;
first
by
done
.
by
eapply
cumulative_service_le_task_cost
;
[
by
apply
INfst

by
apply
H_jobs_have_valid_parameters
].
}
{
rewrite
2
!
mul0n
addn0
subn0
leq_min
;
apply
/
andP
;
split
.
Proof
.
unfold
workload_bound
,
W
;
fold
n_k
.
have
INfst
:
=
workload_bound_j_fst_is_job_of_tsk
;
des
.
rewrite
big_nat_recr
//
big_geq
//
[
nth
]
lock
/=

lock
add0n
.
destruct
n_k
;
last
first
.
{
rewrite
[
service_during
_
_
_
_
_
]
add0n
;
rewrite
leq_add
//.
rewrite
[
service_during
_
_
_
_
_
]
add0n
[
_
*
task_cost
tsk
]
mulSnr
.
apply
leq_add
;
first
by
done
.
by
eapply
cumulative_service_le_task_cost
;
[
by
apply
INfst

by
apply
H_jobs_have_valid_parameters
].
[
by
apply
INfst

by
apply
H_jobs_have_valid_parameters
].
}
{
rewrite

addnBA
//
[
service_during
_
_
_
_
_
]
addn0
.
apply
leq_add
;
last
by
done
.
apply
leq_trans
with
(
n
:
=
\
sum_
(
t1
<=
t
<
t2
)
1
).
by
apply
leq_sum
;
ins
;
apply
service_at_le_max_rate
.
by
unfold
t2
;
rewrite
big_const_nat
iter_addn
mul1n
addn0
addnC

addnBA
//
subnn
addn0
.
rewrite
2
!
mul0n
addn0
subn0
leq_min
;
apply
/
andP
;
split
.
{
by
eapply
cumulative_service_le_task_cost
;
[
by
apply
INfst

by
apply
H_jobs_have_valid_parameters
].
}
{
rewrite

addnBA
//
[
service_during
_
_
_
_
_
]
addn0
.
apply
leq_add
;
last
by
done
.
apply
leq_trans
with
(
n
:
=
\
sum_
(
t1
<=
t
<
t2
)
1
).
by
apply
leq_sum
;
ins
;
apply
service_at_le_max_rate
.
by
unfold
t2
;
rewrite
big_const_nat
iter_addn
mul1n
addn0
addnC

addnBA
//
subnn
addn0
.
}
}
}
Qed
.
Qed
.
End
WorkloadSingleJob
.
(* Next, consider the last case where there are at least two jobs:
the first job j_fst, and the last job j_lst. *)
Section
WorkloadTwoOrMoreJobs
.
(* There are at least two jobs. *)
Variable
num_mid_jobs
:
nat
.
Hypothesis
H_at_least_two_jobs
:
size
sorted_jobs
=
num_mid_jobs
.+
2
.
...
...
@@ 362,6 +374,7 @@ Module WorkloadBoundFP.
Let
j_fst
:
=
nth
elem
sorted_jobs
0
.
Let
j_lst
:
=
nth
elem
sorted_jobs
num_mid_jobs
.+
1
.
(* The last job is an interfering job of task tsk. *)
Lemma
workload_bound_j_lst_is_job_of_tsk
:
job_task
j_lst
=
tsk
/\
service_during
rate
sched
j_lst
t1
t2
!=
0
/\
...
...
@@ 371,6 +384,7 @@ Module WorkloadBoundFP.
by
rewrite
H_at_least_two_jobs
.
Qed
.
(* The response time of the first job must fall inside the interval. *)
Lemma
workload_bound_response_time_of_first_job_inside_interval
:
t1
<=
job_arrival
j_fst
+
R_tsk
.
Proof
.
...
...
@@ 387,7 +401,8 @@ Module WorkloadBoundFP.
apply
H_response_time_bound
;
first
by
done
.
by
apply
leq_trans
with
(
n
:
=
t1
)
;
last
by
apply
leq_addr
.
Qed
.
(* The arrival of the last job must also fall inside the interval. *)
Lemma
workload_bound_last_job_arrives_before_end_of_interval
:
job_arrival
j_lst
<
t2
.
Proof
.
...
...
@@ 455,6 +470,7 @@ Module WorkloadBoundFP.
}
Qed
.
(* Simplify the expression from the previous lemma. *)
Lemma
workload_bound_simpl_expression_with_first_and_last
:
job_arrival
j_fst
+
R_tsk

t1
+
(
t2

job_arrival
j_lst
)
=
delta
+
R_tsk

(
job_arrival
j_lst

job_arrival
j_fst
).
...
...
@@ 471,6 +487,7 @@ Module WorkloadBoundFP.
by
ins
;
apply
workload_bound_jobs_ordered_by_arrival
.
Qed
.
(* Bound the service of the middle jobs. *)
Lemma
workload_bound_service_of_middle_jobs
:
\
sum_
(
0
<=
i
<
num_mid_jobs
)
service_during
rate
sched
(
nth
elem
sorted_jobs
i
.+
1
)
t1
t2
<=
...
...
@@ 493,8 +510,7 @@ Module WorkloadBoundFP.
by
ins
;
des
.
Qed
.
(* Conclude that the distance between first and last is at least n + 1 periods,
where n is the number of middle jobs. *)
(* Conclude that the distance between first and last is at least num_mid_jobs + 1 periods. *)
Lemma
workload_bound_many_periods_in_between
:
job_arrival
j_lst

job_arrival
j_fst
>=
num_mid_jobs
.+
1
*
(
task_period
tsk
).
Proof
.
...
...
@@ 540,6 +556,9 @@ Module WorkloadBoundFP.
by
rewrite
subh3
//
addnC
;
move
:
INnth
=>
/
eqP
INnth
;
rewrite

INnth
.
Qed
.
(* Now, we prove an auxiliary lemma for the next result.
The statement is not meaninful, since it's part of a proof
by contradiction. *)
Lemma
workload_bound_helper_lemma
:
job_arrival
j_fst
+
task_period
tsk
+
delta
<=
job_arrival
j_lst
>
t1
<=
job_arrival
j_fst
+
task_deadline
tsk
.
...
...
@@ 625,6 +644,45 @@ Module WorkloadBoundFP.
by
apply
workload_bound_helper_lemma
.
Qed
.
(* If n_k = num_mid_jobs, then the workload bound holds. *)
Lemma
workload_bound_n_k_equals_num_mid_jobs
:
num_mid_jobs
=
n_k
>
service_during
rate
sched
j_lst
t1
t2
+
service_during
rate
sched
j_fst
t1
t2
+
\
sum_
(
0
<=
i
<
num_mid_jobs
)
service_during
rate
sched
(
nth
elem
sorted_jobs
i
.+
1
)
t1
t2
<=
workload_bound
.
Proof
.
rename
H_valid_task_parameters
into
PARAMS
.
unfold
is_valid_sporadic_task
in
*
;
des
.
unfold
workload_bound
,
W
;
fold
n_k
.
move
=>
NK
;
rewrite

NK
.
apply
leq_add
;
last
by
apply
workload_bound_service_of_middle_jobs
.
apply
leq_trans
with
(
delta
+
R_tsk

(
job_arrival
j_lst

job_arrival
j_fst
)).
{
rewrite
addnC

workload_bound_simpl_expression_with_first_and_last
.
by
apply
workload_bound_service_of_first_and_last_jobs
.
}
rewrite
leq_min
;
apply
/
andP
;
split
.
{
rewrite
leq_subLR
[
_
+
task_cost
_
]
addnC

leq_subLR
.
apply
leq_trans
with
(
num_mid_jobs
.+
1
*
task_period
tsk
)
;
last
by
apply
workload_bound_many_periods_in_between
.
rewrite
NK
ltnW
//

ltn_divLR
;
last
by
apply
PARAMS0
.
by
unfold
n_k
,
max_jobs
,
div_floor
.
}
{
rewrite

subnDA
;
apply
leq_sub2l
.
apply
leq_trans
with
(
n
:
=
num_mid_jobs
.+
1
*
task_period
tsk
)
;
last
by
apply
workload_bound_many_periods_in_between
.
rewrite

addn1
addnC
mulnDl
mul1n
.
by
rewrite
leq_add2l
;
last
by
apply
PARAMS3
.
}
Qed
.
(* If n_k = num_mid_jobs + 1, then the workload bound holds. *)
Lemma
workload_bound_n_k_equals_num_mid_jobs_plus_1
:
num_mid_jobs
.+
1
=
n_k
>
service_during
rate
sched
j_lst
t1
t2
+
...
...
@@ 663,46 +721,10 @@ Module WorkloadBoundFP.
}
}
Qed
.
Lemma
workload_bound_n_k_equals_num_mid_jobs
:
num_mid_jobs
=
n_k
>
service_during
rate
sched
j_lst
t1
t2
+
service_during
rate
sched
j_fst
t1
t2
+
\
sum_
(
0
<=
i
<
num_mid_jobs
)
service_during
rate
sched
(
nth
elem
sorted_jobs
i
.+
1
)
t1
t2
<=
workload_bound
.
Proof
.
rename
H_valid_task_parameters
into
PARAMS
.
unfold
is_valid_sporadic_task
in
*
;
des
.
unfold
workload_bound
,
W
;
fold
n_k
.
move
=>
NK
;
rewrite

NK
.
apply
leq_add
;
last
by
apply
workload_bound_service_of_middle_jobs
.
apply
leq_trans
with
(
delta
+
R_tsk

(
job_arrival
j_lst

job_arrival
j_fst
)).
{
rewrite
addnC

workload_bound_simpl_expression_with_first_and_last
.
by
apply
workload_bound_service_of_first_and_last_jobs
.
}
rewrite
leq_min
;
apply
/
andP
;
split
.
{
rewrite
leq_subLR
[
_
+
task_cost
_
]
addnC

leq_subLR
.
apply
leq_trans
with
(
num_mid_jobs
.+
1
*
task_period
tsk
)
;
last
by
apply
workload_bound_many_periods_in_between
.
rewrite
NK
ltnW
//

ltn_divLR
;
last
by
apply
PARAMS0
.
by
unfold
n_k
,
max_jobs
,
div_floor
.
}
{
rewrite

subnDA
;
apply
leq_sub2l
.
apply
leq_trans
with
(
n
:
=
num_mid_jobs
.+
1
*
task_period
tsk
)
;
last
by
apply
workload_bound_many_periods_in_between
.
rewrite

addn1
addnC
mulnDl
mul1n
.
by
rewrite
leq_add2l
;
last
by
apply
PARAMS3
.
}
Qed
.
End
WorkloadTwoOrMoreJobs
.
(* Using the lemmas above, we prove the main theorem about the workload bound. *)
Theorem
workload_bounded_by_W
:
workload_of
tsk
t1
(
t1
+
delta
)
<=
workload_bound
.
Proof
.
...
...
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