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Sophie Quinton
rtproofs
Commits
d9829b6d
Commit
d9829b6d
authored
Sep 20, 2018
by
Sergey Bozhko
Committed by
Sergey Bozhko
Apr 05, 2019
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Add notion of Request Bound Function
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model/schedule/uni/limited/rbf.v
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d9829b6d
Require
Import
rt
.
util
.
all
.
Require
Import
rt
.
model
.
time
rt
.
model
.
arrival
.
basic
.
job
rt
.
model
.
arrival
.
basic
.
task_arrival
rt
.
model
.
priority
rt
.
model
.
arrival
.
basic
.
arrival_sequence
.
Require
Import
rt
.
model
.
schedule
.
uni
.
schedule
.
Require
Import
rt
.
model
.
arrival
.
curves
.
bounds
.
Require
Import
rt
.
analysis
.
uni
.
arrival_curves
.
workload_bound
.
From
mathcomp
Require
Import
ssreflect
ssrbool
eqtype
ssrnat
seq
path
fintype
bigop
.
Module
RBF
.
Import
Job
Time
ArrivalSequence
ArrivalCurves
TaskArrival
Priority
MaxArrivalsWorkloadBound
.
(* In this section, we prove some properties of Request Bound Functions (RBF). *)
Section
RBFProperties
.
Context
{
Task
:
eqType
}.
Variable
task_cost
:
Task
>
time
.
Context
{
Job
:
eqType
}.
Variable
job_arrival
:
Job
>
time
.
Variable
job_task
:
Job
>
Task
.
(* Consider any arrival sequence. *)
Variable
arr_seq
:
arrival_sequence
Job
.
Hypothesis
H_arrival_times_are_consistent
:
arrival_times_are_consistent
job_arrival
arr_seq
.
(* Consider an FP policy that indicates a higherorequal priority relation,
and assume that the relation is reflexive and transitive. *)
Variable
higher_eq_priority
:
FP_policy
Task
.
Hypothesis
H_priority_is_reflexive
:
FP_is_reflexive
higher_eq_priority
.
Hypothesis
H_priority_is_transitive
:
FP_is_transitive
higher_eq_priority
.
(* Let tsk be any task. *)
Variable
tsk
:
Task
.
(* Let max_arrivals be a proper arrival curve for task tsk, i.e.,
[max_arrival tsk] is (1) an arrival bound of tsk, and (2) it is
a monotonic function that equals 0 for the empty interval delta = 0. *)
Variable
max_arrivals
:
Task
>
time
>
nat
.
Hypothesis
H_proper_arrival_curve
:
proper_arrival_curve
job_task
arr_seq
max_arrivals
tsk
.
(* Let's define some local names for clarity. *)
Let
task_rbf
:
=
task_request_bound_function
task_cost
max_arrivals
tsk
.
(* We prove that [task_rbf 0] is equal to 0. *)
Lemma
task_rbf_0_zero
:
task_rbf
0
=
0
.
Proof
.
rewrite
/
task_rbf
/
task_request_bound_function
.
apply
/
eqP
;
rewrite
muln_eq0
;
apply
/
orP
;
right
;
apply
/
eqP
.
by
move
:
H_proper_arrival_curve
=>
[
_
[
T
_
]]
;
apply
T
.
Qed
.
(* We prove that task_rbf is monotone. *)
Lemma
task_rbf_monotone
:
monotone
task_rbf
leq
.
Proof
.
rewrite
/
monotone
;
intros
.
rewrite
/
task_rbf
/
task_request_bound_function
leq_mul2l
.
apply
/
orP
;
right
.
by
move
:
H_proper_arrival_curve
=>
[
_
T
]
;
apply
T
.
Qed
.
(* Consider any job j of tsk. *)
Variable
j
:
Job
.
Hypothesis
H_j_arrives
:
arrives_in
arr_seq
j
.
Hypothesis
H_job_of_tsk
:
job_task
j
=
tsk
.
(* Then we prove that task_rbf 1 is greater than or equal to task cost. *)
Lemma
task_rbf_1_ge_task_cost
:
task_rbf
1
>=
task_cost
tsk
.
Proof
.
have
ALT
:
forall
n
,
n
=
0
\/
n
>
0
.
{
by
clear
;
intros
n
;
destruct
n
;
[
left

right
].
}
specialize
(
ALT
(
task_cost
tsk
))
;
destruct
ALT
as
[
Z

POS
]
;
first
by
rewrite
Z
.
rewrite
leqNgt
;
apply
/
negP
;
intros
CONTR
.
move
:
H_proper_arrival_curve
=>
[
ARRB
_
].
specialize
(
ARRB
(
job_arrival
j
)
(
job_arrival
j
+
1
)).
feed
ARRB
;
first
by
rewrite
leq_addr
.
rewrite
addKn
in
ARRB
.
move
:
CONTR
;
rewrite
/
task_rbf
/
task_request_bound_function
;
move
=>
CONTR
.
move
:
CONTR
;
rewrite
{
2
}[
task_cost
tsk
]
muln1
ltn_mul2l
;
move
=>
/
andP
[
_
CONTR
].
move
:
CONTR
;
rewrite

addn1
{
3
}[
1
]
add0n
leq_add2r
leqn0
;
move
=>
/
eqP
CONTR
.
move
:
ARRB
;
rewrite
CONTR
leqn0
eqn0Ngt
;
move
=>
/
negP
T
;
apply
:
T
.
rewrite
/
num_arrivals_of_task

has_predT
.
rewrite
/
arrivals_of_task_between
/
is_job_of_task
.
apply
/
hasP
;
exists
j
;
last
by
done
.
rewrite
/
jobs_arrived_between
addn1
big_nat_recl
;
last
by
done
.
rewrite
big_geq
?cats0
;
last
by
done
.
rewrite
mem_filter
.
apply
/
andP
;
split
.

by
apply
/
eqP
.

move
:
H_j_arrives
=>
[
t
ARR
].
move
:
(
ARR
)
=>
CONS
.
apply
H_arrival_times_are_consistent
in
CONS
.
by
rewrite
CONS
.
Qed
.
End
RBFProperties
.
End
RBF
.
\ No newline at end of file
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