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Xiaojie Guo
rtproofs
Commits
f5bb80b1
Commit
f5bb80b1
authored
Jan 07, 2016
by
Felipe Cerqueira
Browse files
Fix some somments
parent
f35d7321
Changes
1
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Inline
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bertogna_fp_comp.v
View file @
f5bb80b1
...
...
@@ 34,9 +34,9 @@ Module ResponseTimeIterationFP.
(* Next we define the fixedpoint iteration for computing
Bertogna's responsetime bound for any task in ts. *)
(* First, given a sequence of pairs R_prev =
[
..., (tsk_hp, R_hp)
]
of
(* First, given a sequence of pairs R_prev =
<
..., (tsk_hp, R_hp)
>
of
responsetime bounds for the higherpriority tasks, we define an
iteration that computes the responsetime bound of the
single task t
sk:
iteration that computes the responsetime bound of the
current ta
sk:
R_tsk (0) = task_cost tsk
R_tsk (step + 1) = f (R step),
...
...
@@ 86,8 +86,8 @@ Module ResponseTimeIterationFP.
(* 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
. *)
nonetheless since the list of responsetime bounds is
computed with
a specific algorithm and there are no lemmas in the library for that
. *)
Section
SimpleLemmas
.
(* First, we show that R_list of the prefix is the prefix of R_list. *)
...
...
@@ 290,7 +290,7 @@ Module ResponseTimeIterationFP.
}
Qed
.
(* S
imple
lemma about unfold the iteration one step. *)
(* S
hort
lemma about unfold
ing
the iteration one step. *)
Lemma
per_task_rta_fold
:
forall
tsk
rt_bounds
,
task_cost
tsk
+
...
...
@@ 323,8 +323,8 @@ Module ResponseTimeIterationFP.
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.*)
(* Then, the tasks in the prefix of R_list
are
exactly
interfering tasks
under FP scheduling.*)
Lemma
R_list_unzip1
:
[
seq
tsk_hp
<
rcons
ts_hp
tsk

is_interfering_task_fp
higher_eq_priority
tsk
tsk_hp
]
=
unzip1
hp_bounds
.
...
...
@@ 407,6 +407,7 @@ Module ResponseTimeIterationFP.
End
HighPriorityTasks
.
(* In this section, we show that the fixedpoint iteration converges. *)
Section
Convergence
.
(* Consider any valid set of higherpriority tasks. *)
...
...
@@ 418,13 +419,13 @@ Module ResponseTimeIterationFP.
Variable
rt_bounds
:
seq
task_with_response_time
.
Hypothesis
H_test_succeeds
:
R_list
ts_hp
=
Some
rt_bounds
.
(* Consider any task tsk. *)
(* Consider any task tsk
to be analyzed
. *)
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. *)
(* Assume that
f (max_steps tsk) is
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. *)
...
...
@@ 489,7 +490,7 @@ Module ResponseTimeIterationFP.
by
apply
bertogna_fp_comp_f_monotonic
,
leqnSn
.
Qed
.
(* In the end, the responsetime bound must exceed the deadline. *)
(* In the end, the responsetime bound must exceed the deadline.
Contradiction!
*)
Lemma
bertogna_fp_comp_rt_exceeds_deadline
:
f
(
max_steps
tsk
)
>
task_deadline
tsk
.
Proof
.
...
...
@@ 535,9 +536,10 @@ Module ResponseTimeIterationFP.
Variable
ts
:
taskset_of
sporadic_task
.
(* Assume that higher_eq_priority is a total order.
Actually, it just needs to be total over the task set,
but to weaken the assumption, I have to reprove many lemmas
about ordering in ssreflect. This can be done later. *)
TODO: it doesn't have to be total over the entire domain, but
only within the task set.
But to weaken the hypothesis, we need to reprove some lemmas
from ssreflect. *)
Hypothesis
H_reflexive
:
reflexive
higher_eq_priority
.
Hypothesis
H_transitive
:
transitive
higher_eq_priority
.
Hypothesis
H_unique_priorities
:
antisymmetric
higher_eq_priority
.
...
...
@@ 603,15 +605,15 @@ Module ResponseTimeIterationFP.
job_misses_no_deadline
job_cost
job_deadline
rate
sched
.
(* In the following lemma, we prove that any responsetime bound contained
in R_list is safe. The proof follows by induction o
f
the task set:
in R_list is safe. The proof follows by induction o
n
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.
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
.
T
here's no clean way of breaking this
up
into small lemmas. *)
The proof is only long because of the dozens of hypothesis that we need to supply
,
so t
here's no clean way of breaking this
down
into small lemmas. *)
Lemma
R_list_has_response_time_bounds
:
forall
rt_bounds
tsk
R
,
R_list
ts
=
Some
rt_bounds
>
...
...
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