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Marco Maida
PROSA  Formally Proven Schedulability Analysis
Commits
f35d7321
Commit
f35d7321
authored
Jan 07, 2016
by
Felipe Cerqueira
Browse files
Comment FP computation
parent
066919f0
Changes
1
Show whitespace changes
Inline
Sidebyside
bertogna_fp_comp.v
View file @
f35d7321
...
...
@@ 6,6 +6,8 @@ Module ResponseTimeIterationFP.
Import
Schedule
ResponseTimeAnalysisFP
WorkloadBound
.
(* In this section, we define the algorithm of Bertogna and Cirinei's
responsetime analysis for FP scheduling. *)
Section
Analysis
.
Context
{
sporadic_task
:
eqType
}.
...
...
@@ 13,6 +15,8 @@ Module ResponseTimeIterationFP.
Variable
task_period
:
sporadic_task
>
nat
.
Variable
task_deadline
:
sporadic_task
>
nat
.
(* During the iterations of the algorithm, we pass around pairs
of tasks and computed responsetime bounds. *)
Let
task_with_response_time
:
=
(
sporadic_task
*
nat
)%
type
.
Context
{
Job
:
eqType
}.
...
...
@@ 20,7 +24,10 @@ Module ResponseTimeIterationFP.
Variable
job_deadline
:
Job
>
nat
.
Variable
job_task
:
Job
>
sporadic_task
.
(* Consider a platform with num_cpus processors, ...*)
Variable
num_cpus
:
nat
.
(* ..., and priorities based on an FP policy. *)
Variable
higher_eq_priority
:
fp_policy
sporadic_task
.
Hypothesis
H_valid_policy
:
valid_fp_policy
higher_eq_priority
.
...
...
@@ 28,13 +35,14 @@ Module ResponseTimeIterationFP.
Bertogna's responsetime bound for any task in ts. *)
(* First, given a sequence of pairs R_prev = [..., (tsk_hp, R_hp)] of
responsetime bounds for the higherpriority tasks, we compute
the responsetime bound of tsk using the following iteration:
responsetime bounds for the higherpriority tasks, we define an
iteration that computes the responsetime bound of the single task tsk:
R_tsk (0) = task_cost tsk
R_tsk (step + 1) = f (R step),
R_tsk < f^step (R_tsk),
where f is the responsetime recurrence,
step is the number of iterations,
and f^0 = task_cost tsk. *)
where f is the responsetime recurrence, step is the number of iterations,
and R_tsk (0) is the initial state. *)
Definition
per_task_rta
(
tsk
:
sporadic_task
)
(
R_prev
:
seq
task_with_response_time
)
(
step
:
nat
)
:
=
iter
step
...
...
@@ 46,20 +54,18 @@ Module ResponseTimeIterationFP.
(
task_cost
tsk
).
(* To ensure that the iteration converges, we will apply per_task_rta
a "sufficient" number of times: task_deadline tsk + 1.
Note that (deadline + 1) is a pessimistic bound on the number of
steps, but we don't care about precise runtime complexity here. *)
Definition
max_steps
(
tsk
:
sporadic_task
)
:
=
task_deadline
tsk
+
1
.
a "sufficient" number of times: task_deadline tsk  task_cost tsk.
This corresponds to the time complexity of the iteration. *)
Definition
max_steps
(
tsk
:
sporadic_task
)
:
=
task_deadline
tsk

task_cost
tsk
.
(* Next we compute the responsetime bounds for the entire task set.
Since highpriority tasks may not be schedulable, we allow the
computation to fail.
Thus, given the responsetime bound of previous tasks, we either
(a) append the responsetime bound (tsk, R) of the current task
(a) append the
computed
responsetime bound (tsk, R) of the current task
to the list of pairs, or,
(b) return None if the responsetime analysis failed. *)
Definition
R_list_helper
:
=
fun
hp_pairs
tsk
=>
Definition
R_list_helper
hp_pairs
tsk
:
=
if
hp_pairs
is
Some
rt_bounds
then
let
R
:
=
per_task_rta
tsk
rt_bounds
(
max_steps
tsk
)
in
if
R
<=
task_deadline
tsk
then
...
...
@@ 67,9 +73,10 @@ Module ResponseTimeIterationFP.
else
None
else
None
.
(* To return the complete list of responsetime bounds for any task set,
we just apply foldl (reduce) using the function above. *)
Definition
R_list
(
ts
:
taskset_of
sporadic_task
)
:
option
(
seq
task_with_response_time
)
:
=
(* The responsetime analysis for a given task set is defined
as a leftfold (reduce) based on the function above.
This either returns a list of task and responsetime bounds, or None. *)
Definition
R_list
(
ts
:
taskset_of
sporadic_task
)
:
=
foldl
R_list_helper
(
Some
[
::
])
ts
.
(* The schedulability test simply checks if we got a list of
...
...
@@ 77,16 +84,13 @@ Module ResponseTimeIterationFP.
Definition
fp_schedulable
(
ts
:
taskset_of
sporadic_task
)
:
=
R_list
ts
!=
None
.
Section
AuxiliaryLemmas
.
(* In this section, we prove several helper lemmas about the
list of responsetime bounds, such as:
(1) Equality among tasks in R_list and in the task set.
(2) If (tsk, R) \in R_list, then R <= task_deadline tsk.
(3) If (tsk, R) \in R_list, then R >= task_cost tsk.
(4) If per_task_rta returns a bound <= deadline, then the
iteration reached a fixedpoint. *)
(* In the following section, we prove several helper lemmas about the
list of responsetime bounds. The results seem trivial, but must be proven
nonetheless since the list of responsetime bounds is a result of an
iterative procedure. *)
Section
SimpleLemmas
.
(* First, we show that R_list of the prefix is the prefix of R_list. *)
Lemma
R_list_rcons_prefix
:
forall
ts'
hp_bounds
tsk1
tsk2
R
,
R_list
(
rcons
ts'
tsk1
)
=
Some
(
rcons
hp_bounds
(
tsk2
,
R
))
>
...
...
@@ 105,6 +109,7 @@ Module ResponseTimeIterationFP.
by
f_equal
.
Qed
.
(* R_list returns the same tasks as the original task set. *)
Lemma
R_list_rcons_task
:
forall
ts'
hp_bounds
tsk1
tsk2
R
,
R_list
(
rcons
ts'
tsk1
)
=
Some
(
rcons
hp_bounds
(
tsk2
,
R
))
>
...
...
@@ 123,6 +128,8 @@ Module ResponseTimeIterationFP.
by
inversion
EQ
.
Qed
.
(* The responsetime bounds computed using R_list are based on the pertask
fixedpoint iteration. *)
Lemma
R_list_rcons_response_time
:
forall
ts'
hp_bounds
tsk
R
,
R_list
(
rcons
ts'
tsk
)
=
Some
(
rcons
hp_bounds
(
tsk
,
R
))
>
...
...
@@ 139,6 +146,8 @@ Module ResponseTimeIterationFP.
by
inversion
EQ2
;
rewrite
EQ1
.
Qed
.
(* If the analysis suceeds, the computed responsetime bounds are no larger
than the deadline. *)
Lemma
R_list_le_deadline
:
forall
ts'
rt_bounds
tsk
R
,
R_list
ts'
=
Some
rt_bounds
>
...
...
@@ 177,6 +186,8 @@ Module ResponseTimeIterationFP.
}
Qed
.
(* If the analysis succeeds, the computed responsetime bounds are no smaller
than the task cost. *)
Lemma
R_list_ge_cost
:
forall
ts'
rt_bounds
tsk
R
,
R_list
ts'
=
Some
rt_bounds
>
...
...
@@ 216,6 +227,7 @@ Module ResponseTimeIterationFP.
}
Qed
.
(* R_list contains a responsetime bound for every tasks in the original task set. *)
Lemma
R_list_non_empty
:
forall
ts'
rt_bounds
tsk
,
R_list
ts'
=
Some
rt_bounds
>
...
...
@@ 278,88 +290,7 @@ Module ResponseTimeIterationFP.
}
Qed
.
(* To prove convergence of R, we first show convergence of rt_rec. *)
Lemma
per_task_rta_converges
:
forall
ts'
tsk
rt_bounds
,
valid_sporadic_taskset
task_cost
task_period
task_deadline
ts'
>
R_list
ts'
=
Some
rt_bounds
>
per_task_rta
tsk
rt_bounds
(
max_steps
tsk
)
<=
task_deadline
tsk
>
per_task_rta
tsk
rt_bounds
(
max_steps
tsk
)
=
per_task_rta
tsk
rt_bounds
(
max_steps
tsk
).+
1
.
Proof
.
unfold
valid_sporadic_taskset
,
is_valid_sporadic_task
in
*.
(* To simplify, let's call the function f.*)
intros
ts'
tsk
rt_bounds
VALID
SOME
LE
;
set
(
f
:
=
per_task_rta
tsk
rt_bounds
)
;
fold
f
in
LE
.
(* First prove that f is monotonic.*)
assert
(
MON
:
forall
x1
x2
,
x1
<=
x2
>
f
x1
<=
f
x2
).
{
intros
x1
x2
LEx
;
unfold
f
,
per_task_rta
.
apply
fun_mon_iter_mon
;
[
by
ins

by
ins
;
apply
leq_addr
].
clear
LEx
x1
x2
;
intros
x1
x2
LEx
.
unfold
div_floor
,
total_interference_bound_fp
.
rewrite
big_seq_cond
[
\
sum_
(
i
<
_

let
'
(
tsk_other
,
_
)
:
=
i
in
_
&&
(
tsk_other
!=
tsk
))
_
]
big_seq_cond
.
rewrite
leq_add2l
leq_div2r
//
leq_sum
//.
intros
i
;
destruct
(
i
\
in
rt_bounds
)
eqn
:
HP
;
last
by
rewrite
andFb
.
destruct
i
as
[
i
R
]
;
intros
_
.
have
GE_COST
:
=
(
R_list_ge_cost
ts'
rt_bounds
i
R
).
have
INts
:
=
(
R_list_non_empty
ts'
rt_bounds
i
SOME
).
destruct
INts
as
[
_
EX
]
;
exploit
EX
;
[
by
exists
R

intro
IN
].
unfold
interference_bound
;
simpl
.
rewrite
leq_min
;
apply
/
andP
;
split
.
{
apply
leq_trans
with
(
n
:
=
W
task_cost
task_period
i
R
x1
)
;
first
by
apply
geq_minl
.
specialize
(
VALID
i
IN
)
;
des
.
by
apply
W_monotonic
;
try
(
by
ins
)
;
[
by
apply
GE_COST

by
apply
leqnn
].
}
{
apply
leq_trans
with
(
n
:
=
x1

task_cost
tsk
+
1
)
;
first
by
apply
geq_minr
.
by
rewrite
leq_add2r
leq_sub2r
//.
}
}
(* Either f converges by the deadline or not. *)
unfold
max_steps
in
*
;
rewrite
>
addn1
in
*.
destruct
([
exists
k
in
'
I_
(
task_deadline
tsk
).+
1
,
f
k
==
f
k
.+
1
])
eqn
:
EX
.
{
move
:
EX
=>
/
exists_inP
EX
;
destruct
EX
as
[
k
_
ITERk
].
move
:
ITERk
=>
/
eqP
ITERk
.
by
apply
iter_fix
with
(
k
:
=
k
)
;
[
by
ins

by
apply
ltnW
,
ltn_ord
].
}
apply
negbT
in
EX
;
rewrite
negb_exists_in
in
EX
.
move
:
EX
=>
/
forall_inP
EX
.
assert
(
GROWS
:
forall
k
:
'
I_
(
task_deadline
tsk
).+
1
,
f
k
<
f
k
.+
1
).
{
intros
k
;
rewrite
ltn_neqAle
;
apply
/
andP
;
split
;
first
by
apply
EX
.
apply
MON
,
leqnSn
.
}
(* If it doesn't converge, then it becomes larger than the deadline.
But initialy we assumed otherwise. Contradiction! *)
assert
(
BY1
:
f
(
task_deadline
tsk
).+
1
>
task_deadline
tsk
).
{
clear
MON
LE
EX
.
induction
(
task_deadline
tsk
).+
1
;
first
by
ins
.
apply
leq_ltn_trans
with
(
n
:
=
f
n
)
;
last
by
apply
(
GROWS
(
Ordinal
(
ltnSn
n
))).
apply
IHn
;
intros
k
.
by
apply
(
GROWS
(
widen_ord
(
leqnSn
n
)
k
)).
}
by
apply
leq_ltn_trans
with
(
m
:
=
f
(
task_deadline
tsk
).+
1
)
in
BY1
;
[
by
rewrite
ltnn
in
BY1

by
ins
].
Qed
.
(* Simple lemma about unfold the iteration one step. *)
Lemma
per_task_rta_fold
:
forall
tsk
rt_bounds
,
task_cost
tsk
+
...
...
@@ 367,28 +298,50 @@ Module ResponseTimeIterationFP.
(
per_task_rta
tsk
rt_bounds
(
max_steps
tsk
))
higher_eq_priority
)
num_cpus
=
per_task_rta
tsk
rt_bounds
(
max_steps
tsk
).+
1
.
Proof
.
by
ins
.
by
done
.
Qed
.
End
SimpleLemmas
.
(* In this section, we prove that if the task set is sorted by priority,
the tasks in R_list are interfering tasks. *)
Section
HighPriorityTasks
.
(* Consider a list of previous tasks and a task tsk to be analyzed. *)
Variable
ts_hp
:
taskset_of
sporadic_task
.
Variable
tsk
:
sporadic_task
.
(* Assume that the task set doesn't contain duplicates and is sorted by priority, ... *)
Hypothesis
H_task_set_is_a_set
:
uniq
(
rcons
ts_hp
tsk
).
Hypothesis
H_task_set_is_sorted
:
sorted
higher_eq_priority
(
rcons
ts_hp
tsk
).
(* ...the priority order is transitive, ...*)
Hypothesis
H_priority_transitive
:
transitive
higher_eq_priority
.
(* ... and that the responsetime analysis succeeds. *)
Variable
hp_bounds
:
seq
task_with_response_time
.
Variable
R
:
time
.
Hypothesis
H_analysis_succeeds
:
R_list
(
rcons
ts_hp
tsk
)
=
Some
(
rcons
hp_bounds
(
tsk
,
R
)).
(* Then, the list of tasks in the prefix of R_list is exactly the set of
interfering tasks under FP scheduling.*)
Lemma
R_list_unzip1
:
forall
ts'
tsk
hp_bounds
R
,
transitive
higher_eq_priority
>
uniq
(
rcons
ts'
tsk
)
>
sorted
higher_eq_priority
(
rcons
ts'
tsk
)
>
R_list
(
rcons
ts'
tsk
)
=
Some
(
rcons
hp_bounds
(
tsk
,
R
))
>
[
seq
tsk_hp
<
rcons
ts'
tsk

is_interfering_task_fp
higher_eq_priority
tsk
tsk_hp
]
=
[
seq
tsk_hp
<
rcons
ts_hp
tsk

is_interfering_task_fp
higher_eq_priority
tsk
tsk_hp
]
=
unzip1
hp_bounds
.
Proof
.
intros
ts
tsk
hp_bounds
R
TRANS
.
revert
tsk
hp_bounds
R
.
induction
ts
as
[
ts'
tsk_lst
]
using
last_ind
.
{
intros
tsk
hp_bounds
R
_
_
SOME
;
simpl
in
*.
rename
H_priority_transitive
into
TRANS
,
H_task_set_is_a_set
into
UNIQ
,
H_analysis_succeeds
into
SOME
,
H_task_set_is_sorted
into
SORT
.
revert
tsk
hp_bounds
R
UNIQ
SORT
SOME
.
induction
ts_hp
as
[
ts_hp'
tsk_lst
]
using
last_ind
.
{
intros
tsk
hp_bounds
R
UNIQ
SORT
SOME
;
simpl
in
*.
unfold
is_interfering_task_fp
.
rewrite
eq_refl
andbF
.
destruct
hp_bounds
;
first
by
ins
.
unfold
R_list
in
SOME
;
inversion
SOME
;
desf
.
by
destruct
hp_bounds
.
by
destruct
l
.
}
{
intros
tsk
hp_bounds
R
UNIQ
SORTED
SOME
.
...
...
@@ 401,11 +354,11 @@ Module ResponseTimeIterationFP.
inversion
SOME
;
desf
.
by
destruct
l
.
}
generalize
SOME
;
apply
R_list_rcons_prefix
,
R_list_rcons_task
in
SOME
;
subst
tsk_lst'
;
intro
SOME
.
specialize
(
IHts
tsk_lst
hp_bounds
R_lst
)
.
rewrite
filter_rcons
in
IHts
.
unfold
is_interfering_task_fp
in
IHt
s
.
rewrite
eq_refl
andbF
in
IHt
s
.
generalize
SOME
;
apply
R_list_rcons_prefix
,
R_list_rcons_task
in
SOME
;
subst
tsk_lst'
;
intro
SOME
.
specialize
(
IHt
tsk_lst
hp_bounds
R_lst
)
.
rewrite
filter_rcons
in
IHt
.
unfold
is_interfering_task_fp
in
IHt
;
rewrite
eq_refl
andbF
in
IHt
.
assert
(
NOTHP
:
is_interfering_task_fp
higher_eq_priority
tsk
tsk
=
false
).
{
by
unfold
is_interfering_task_fp
;
rewrite
eq_refl
andbF
.
...
...
@@ 424,42 +377,159 @@ Module ResponseTimeIterationFP.
}
}
rewrite
filter_rcons
HP
;
clear
HP
.
unfold
unzip1
;
rewrite
map_rcons
/=
;
f_equal
.
assert
(
SHIFT
:
[
seq
tsk_hp
<
ts'

is_interfering_task_fp
higher_eq_priority
tsk
tsk_hp
]
=
[
seq
tsk_hp
<
ts'

is_interfering_task_fp
higher_eq_priority
tsk_lst
tsk_hp
]).
assert
(
SHIFT
:
[
seq
tsk_hp
<
ts_hp'

is_interfering_task_fp
higher_eq_priority
tsk
tsk_hp
]
=
[
seq
tsk_hp
<
ts_hp'

is_interfering_task_fp
higher_eq_priority
tsk_lst
tsk_hp
]).
{
apply
eq_in_filter
;
red
.
unfold
is_interfering_task_fp
;
intros
x
INx
.
rewrite
2
!
rcons_uniq
mem_rcons
in_cons
negb_or
in
UNIQ
.
move
:
UNIQ
=>
/
andP
[/
andP
[
NEQ
NOTIN
]
/
andP
[
NOTIN'
UNIQ
]].
destruct
(
x
==
tsk
)
eqn
:
EQtsk
.
{
move
:
EQtsk
=>
/
eqP
EQtsk
;
subst
.
by
rewrite
INx
in
NOTIN
.
}
destruct
(
x
==
tsk_lst
)
eqn
:
EQlst
.
{
move
:
EQlst
=>
/
eqP
EQlst
;
subst
.
by
rewrite
INx
in
NOTIN'
.
}
destruct
(
x
==
tsk
)
eqn
:
EQtsk
;
first
by
move
:
EQtsk
=>
/
eqP
EQtsk
;
subst
;
rewrite
INx
in
NOTIN
.
destruct
(
x
==
tsk_lst
)
eqn
:
EQlst
;
first
by
move
:
EQlst
=>
/
eqP
EQlst
;
subst
;
rewrite
INx
in
NOTIN'
.
rewrite
2
!
andbT
.
generalize
SORTED
;
intro
SORTED'
.
have
bla
:
=
order_sorted_rcons
.
apply
order_sorted_rcons
with
(
x0
:
=
x
)
in
SORTED
;
try
(
by
ins
)
;
last
by
rewrite
mem_rcons
in_cons
;
apply
/
orP
;
right
.
rewrite
SORTED
.
apply
sorted_rcons_prefix
in
SORTED'
.
by
apply
order_sorted_rcons
with
(
x0
:
=
x
)
in
SORTED'
.
}
rewrite
SHIFT
;
clear
SHIFT
.
apply
IHt
s
.
apply
IHt
.
by
rewrite
rcons_uniq
in
UNIQ
;
move
:
UNIQ
=>
/
andP
[
_
UNIQ
].
by
apply
sorted_rcons_prefix
in
SORTED
.
by
apply
R_list_rcons_prefix
in
SOME
.
}
Qed
.
End
AuxiliaryLemmas
.
End
HighPriorityTasks
.
Section
Convergence
.
(* Consider any valid set of higherpriority tasks. *)
Variable
ts_hp
:
taskset_of
sporadic_task
.
Hypothesis
valid_task_parameters
:
valid_sporadic_taskset
task_cost
task_period
task_deadline
ts_hp
.
(* Assume that the responsetime analysis succeeds. *)
Variable
rt_bounds
:
seq
task_with_response_time
.
Hypothesis
H_test_succeeds
:
R_list
ts_hp
=
Some
rt_bounds
.
(* Consider any task tsk. *)
Variable
tsk
:
sporadic_task
.
(* To simplify, let f denote the fixedpoint iteration. *)
Let
f
:
=
per_task_rta
tsk
rt_bounds
.
(* Assume that the iteration reaches a value no larger than the deadline. *)
Hypothesis
H_no_larger_than_deadline
:
f
(
max_steps
tsk
)
<=
task_deadline
tsk
.
(* First, we show that f is monotonically increasing. *)
Lemma
bertogna_fp_comp_f_monotonic
:
forall
x1
x2
,
x1
<=
x2
>
f
x1
<=
f
x2
.
Proof
.
unfold
valid_sporadic_taskset
,
is_valid_sporadic_task
in
*.
rename
H_test_succeeds
into
SOME
,
valid_task_parameters
into
VALID
.
intros
x1
x2
LEx
;
unfold
f
,
per_task_rta
.
apply
fun_mon_iter_mon
;
[
by
ins

by
ins
;
apply
leq_addr
].
clear
LEx
x1
x2
;
intros
x1
x2
LEx
.
unfold
div_floor
,
total_interference_bound_fp
.
rewrite
big_seq_cond
[
\
sum_
(
i
<
_

let
'
(
tsk_other
,
_
)
:
=
i
in
_
&&
(
tsk_other
!=
tsk
))
_
]
big_seq_cond
.
rewrite
leq_add2l
leq_div2r
//
leq_sum
//.
intros
i
;
destruct
(
i
\
in
rt_bounds
)
eqn
:
HP
;
last
by
rewrite
andFb
.
destruct
i
as
[
i
R
]
;
intros
_
.
have
GE_COST
:
=
(
R_list_ge_cost
ts_hp
rt_bounds
i
R
).
have
INts
:
=
(
R_list_non_empty
ts_hp
rt_bounds
i
SOME
).
destruct
INts
as
[
_
EX
]
;
exploit
EX
;
[
by
exists
R

intro
IN
].
unfold
interference_bound
;
simpl
.
rewrite
leq_min
;
apply
/
andP
;
split
.
{
apply
leq_trans
with
(
n
:
=
W
task_cost
task_period
i
R
x1
)
;
first
by
apply
geq_minl
.
specialize
(
VALID
i
IN
)
;
des
.
by
apply
W_monotonic
;
try
(
by
ins
)
;
[
by
apply
GE_COST

by
apply
leqnn
].
}
{
apply
leq_trans
with
(
n
:
=
x1

task_cost
tsk
+
1
)
;
first
by
apply
geq_minr
.
by
rewrite
leq_add2r
leq_sub2r
//.
}
Qed
.
(* If the iteration converged at an earlier step, then it remains stable. *)
Lemma
bertogna_fp_comp_f_converges_early
:
(
exists
k
,
k
<
max_steps
tsk
/\
f
k
=
f
k
.+
1
)
>
f
(
max_steps
tsk
)
=
f
(
max_steps
tsk
).+
1
.
Proof
.
by
intros
EX
;
des
;
apply
iter_fix
with
(
k
:
=
k
)
;
[
by
done

by
apply
ltnW
].
Qed
.
(* Else, we derive a contradiction. *)
Section
DerivingContradiction
.
Section
Proof
.
(* Assume instead that the iteration continued to diverge. *)
Hypothesis
H_keeps_diverging
:
forall
k
,
k
<
max_steps
tsk
>
f
k
!=
f
k
.+
1
.
(* By monotonicity, it follows that the value always increases. *)
Lemma
bertogna_fp_comp_f_increases
:
forall
k
,
k
<
max_steps
tsk
>
f
k
<
f
k
.+
1
.
Proof
.
intros
k
LT
.
rewrite
ltn_neqAle
;
apply
/
andP
;
split
.
by
apply
H_keeps_diverging
.
by
apply
bertogna_fp_comp_f_monotonic
,
leqnSn
.
Qed
.
(* In the end, the responsetime bound must exceed the deadline. *)
Lemma
bertogna_fp_comp_rt_exceeds_deadline
:
f
(
max_steps
tsk
)
>
task_deadline
tsk
.
Proof
.
unfold
max_steps
.
induction
(
task_deadline
tsk
).
{
simpl
.
admit
.
}
admit
.
Qed
.
End
DerivingContradiction
.
(* Using the lemmas above, we prove the convergence of the iteration after max_steps. *)
Lemma
per_task_rta_converges
:
f
(
max_steps
tsk
)
=
f
(
max_steps
tsk
).+
1
.
Proof
.
unfold
valid_sporadic_taskset
,
is_valid_sporadic_task
in
*.
rename
H_no_larger_than_deadline
into
LE
.
(* Either f converges by the deadline or not. *)
destruct
([
exists
k
in
'
I_
(
max_steps
tsk
),
f
k
==
f
k
.+
1
])
eqn
:
EX
.
{
move
:
EX
=>
/
exists_inP
EX
;
destruct
EX
as
[
k
_
ITERk
].
apply
bertogna_fp_comp_f_converges_early
.
by
exists
k
;
split
;
[
by
apply
ltn_ord

by
apply
/
eqP
].
}
(* If not, then we reach a contradiction *)
apply
negbT
in
EX
;
rewrite
negb_exists_in
in
EX
.
move
:
EX
=>
/
forall_inP
EX
.
rewrite
leqNgt
in
LE
;
move
:
LE
=>
/
negP
LE
.
exfalso
;
apply
LE
.
by
apply
bertogna_fp_comp_rt_exceeds_deadline
.
Qed
.
End
Convergence
.
Section
MainProof
.
(* Consider a task set ts. *)
Variable
ts
:
taskset_of
sporadic_task
.
...
...
@@ 532,10 +602,16 @@ Module ResponseTimeIterationFP.
Definition
no_deadline_missed_by_job
:
=
job_misses_no_deadline
job_cost
job_deadline
rate
sched
.
Section
HelperLemma
.
(* In the following lemma, we prove that any responsetime bound contained
in R_list is safe. The proof follows by induction of the task set:
Induction hypothesis: all higherpriority tasks have safe responsetime bounds.
Inductive step: We prove that the responsetime bound of the current task is safe.
(* The following lemma states that the responsetime bounds
computed using R_list are valid. *)
Note that the inductive step is a direct application of the main Theorem from
bertogna_fp_theory.v.
The proof is only long because of the dozens of hypothesis that we need to supply.
There's no clean way of breaking this up into small lemmas. *)
Lemma
R_list_has_response_time_bounds
:
forall
rt_bounds
tsk
R
,
R_list
ts
=
Some
rt_bounds
>
...
...
@@ 544,18 +620,12 @@ Module ResponseTimeIterationFP.
job_task
j
=
tsk
>
completed
job_cost
rate
sched
j
(
job_arrival
j
+
R
).
Proof
.
rename
H_valid_job_parameters
into
JOBPARAMS
,
H_valid_task_parameters
into
TASKPARAMS
,
H_restricted_deadlines
into
RESTR
,
H_completed_jobs_dont_execute
into
COMP
,
H_jobs_must_arrive_to_execute
into
MUSTARRIVE
,
H_global_scheduling_invariant
into
INVARIANT
,
H_sorted_ts
into
SORT
,
H_transitive
into
TRANS
,
H_unique_priorities
into
UNIQ
,
H_total
into
TOTAL
,
H_all_jobs_from_taskset
into
ALLJOBS
,
H_ts_is_a_set
into
SET
.
rename
H_valid_job_parameters
into
JOBPARAMS
,
H_valid_task_parameters
into
TASKPARAMS
,
H_restricted_deadlines
into
RESTR
,
H_completed_jobs_dont_execute
into
COMP
,
H_jobs_must_arrive_to_execute
into
MUSTARRIVE
,
H_sorted_ts
into
SORT
,
H_global_scheduling_invariant
into
INVARIANT
,
H_transitive
into
TRANS
,
H_unique_priorities
into
UNIQ
,
H_total
into
TOTAL
,
H_all_jobs_from_taskset
into
ALLJOBS
,
H_ts_is_a_set
into
SET
.
clear
ALLJOBS
.
unfold
fp_schedulable
,
R_list
in
*.
...
...
@@ 634,12 +704,7 @@ Module ResponseTimeIterationFP.
by
ins
;
apply
R_list_le_deadline
with
(
ts'
:
=
ts'
)
(
rt_bounds
:
=
hp_bounds
).
{
rewrite
[
R_lst
](
R_list_rcons_response_time
ts'
hp_bounds
tsk_lst
)
;
last
by
ins
.
rewrite
per_task_rta_fold
.
apply
per_task_rta_converges
with
(
ts'
:
=
ts'
)
;
try
(
by
ins
).
{
red
;
ins
;
apply
TASKPARAMS
.
by
rewrite
mem_rcons
in_cons
;
apply
/
orP
;
right
.
}
rewrite
per_task_rta_fold
;
apply
per_task_rta_converges
.
apply
R_list_le_deadline
with
(
ts'
:
=
rcons
ts'
tsk_lst
)
(
rt_bounds
:
=
rcons
hp_bounds
(
tsk_lst
,
R_lst
))
;
first
by
apply
SOME'
.
...
...
@@ 651,12 +716,10 @@ Module ResponseTimeIterationFP.
}
Qed
.
End
HelperLemma
.
(* If the schedulability test suceeds, ...*)
(* Finally, if the schedulability test suceeds, ...*)
Hypothesis
H_test_succeeds
:
fp_schedulable
ts
.
(*...,
then
no task misses its deadline,... *)
(*..., no task misses its deadline,... *)
Theorem
taskset_schedulable_by_fp_rta
:
forall
tsk
,
tsk
\
in
ts
>
no_deadline_missed_by_task
tsk
.
Proof
.
...
...
@@ 724,8 +787,8 @@ Module ResponseTimeIterationFP.
Qed
.
(* For completeness, since all jobs of the arrival sequence
are spawned by the task set, we conclude that no job
misses
its deadline. *)
are spawned by the task set, we
also
conclude that no job
in
the schedule misses
its deadline. *)
Theorem
jobs_schedulable_by_fp_rta
:
forall
(
j
:
JobIn
arr_seq
),
no_deadline_missed_by_job
j
.
Proof
.
...
...
@@ 736,7 +799,7 @@ Module ResponseTimeIterationFP.
by
apply
H_all_jobs_from_taskset
.
Qed
.
End
Proof
.
End
Main
Proof
.
End
Analysis
.
...
...
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