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Require Import prosa.classic.util.all.
Require Import prosa.classic.model.arrival.basic.task prosa.classic.model.priority prosa.classic.model.arrival.basic.task_arrival.
Require Import prosa.classic.model.schedule.global.response_time prosa.classic.model.schedule.global.workload
prosa.classic.model.schedule.global.schedulability.
Require Import prosa.classic.model.schedule.global.jitter.job prosa.classic.model.schedule.global.jitter.schedule
prosa.classic.model.schedule.global.jitter.platform prosa.classic.model.schedule.global.jitter.interference
prosa.classic.model.schedule.global.jitter.interference_edf.
Require Import prosa.classic.analysis.global.jitter.workload_bound prosa.classic.analysis.global.jitter.interference_bound.
From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq fintype bigop div path.
Module InterferenceBoundEDFJitter.
Import JobWithJitter SporadicTaskset ScheduleWithJitter ScheduleOfSporadicTask Schedulability
ResponseTime WorkloadBoundJitter Priority TaskArrival Interference InterferenceEDF.
Export InterferenceBoundJitter.
(* In this section, we define Bertogna and Cirinei's EDF-specific
interference bound. *)
Section SpecificBoundDef.
Context {sporadic_task: eqType}.
Variable task_cost: sporadic_task -> time.
Variable task_period: sporadic_task -> time.
Variable task_deadline: sporadic_task -> time.
Variable task_jitter: sporadic_task -> time.
(* Let tsk be the task to be analyzed. *)
Variable tsk: sporadic_task.
(* Consider the interference incurred by tsk in a window of length delta... *)
Variable delta: time.
(* due to a different task tsk_other, with response-time bound R_other. *)
Variable tsk_other: sporadic_task.
Variable R_other: time.
(* Bertogna and Cirinei define the following bound for task interference
under EDF scheduling. *)
Definition edf_specific_interference_bound :=
let d_tsk := task_deadline tsk in
let e_other := task_cost tsk_other in
let p_other := task_period tsk_other in
let d_other := task_deadline tsk_other in
let j_other := task_jitter tsk_other in
(div_floor d_tsk p_other) * e_other +
minn e_other ((d_tsk %% p_other) - (d_other - R_other - j_other)).
End SpecificBoundDef.
(* Next, we define the total interference bound for EDF, which combines the generic
and the EDF-specific bounds. *)
Section TotalInterferenceBoundEDF.
Context {sporadic_task: eqType}.
Variable task_cost: sporadic_task -> time.
Variable task_period: sporadic_task -> time.
Variable task_deadline: sporadic_task -> time.
Variable task_jitter: sporadic_task -> time.
(* Let tsk be the task to be analyzed. *)
Variable tsk: sporadic_task.
Let task_with_response_time := (sporadic_task * time)%type.
(* Assume a known response-time bound for each interfering task ... *)
Variable R_prev: seq task_with_response_time.
(* ... and an interval length delta. *)
Variable delta: time.
Variable tsk_R: task_with_response_time.
Let tsk_other := fst tsk_R.
Let R_other := snd tsk_R.
(* By combining Bertogna's interference bound for a work-conserving
scheduler ... *)
Let basic_interference_bound := interference_bound_generic task_cost task_period task_jitter tsk delta tsk_R.
(* ... with and EDF-specific interference bound, ... *)
Let edf_specific_bound := edf_specific_interference_bound task_cost task_period task_deadline task_jitter tsk tsk_other R_other.
(* Bertogna and Cirinei define the following interference bound
under EDF scheduling. *)
Definition interference_bound_edf :=
minn basic_interference_bound edf_specific_bound.
(* Next we define the computation of the total interference for APA scheduling. *)
Section TotalInterference.
(* Recall the definition of a different task (with respect to tsk). *)
Let other_task := different_task tsk.
(* The total interference incurred by tsk is bounded by the sum
of individual task interferences of the other tasks. *)
Definition total_interference_bound_edf :=
\sum_((tsk_other, R_other) <- R_prev | other_task tsk_other)
interference_bound_edf (tsk_other, R_other).
End TotalInterferenceBoundEDF.
(* In this section, we show that the EDF-specific interference bound is safe. *)
Section ProofSpecificBound.
Import ScheduleWithJitter Interference Platform SporadicTaskset.
Context {sporadic_task: eqType}.
Variable task_cost: sporadic_task -> time.
Variable task_period: sporadic_task -> time.
Variable task_deadline: sporadic_task -> time.
Variable task_jitter: sporadic_task -> time.
Context {Job: eqType}.
Variable job_arrival: Job -> time.
Variable job_cost: Job -> time.
Variable job_deadline: Job -> time.
Variable job_task: Job -> sporadic_task.
Variable job_jitter: Job -> time.
(* Assume any job arrival sequence... *)
Context {arr_seq: arrival_sequence Job}.
(* ... in which jobs arrive sporadically and have valid parameters. *)
Hypothesis H_sporadic_tasks:
sporadic_task_model task_period job_arrival job_task arr_seq.
Hypothesis H_valid_job_parameters:
forall j,
arrives_in arr_seq j ->
valid_sporadic_job_with_jitter task_cost task_deadline task_jitter job_cost job_deadline job_task job_jitter j.
(* Consider any schedule such that...*)
Variable num_cpus: nat.
Variable sched: schedule Job num_cpus.
Hypothesis H_jobs_come_from_arrival_sequence:
jobs_come_from_arrival_sequence sched arr_seq.
(* ...jobs do not execute before jitter nor longer than their execution costs. *)
Hypothesis H_jobs_execute_after_jitter:
jobs_execute_after_jitter job_arrival job_jitter sched.
Hypothesis H_completed_jobs_dont_execute:
completed_jobs_dont_execute job_cost sched.
(* Also assume that jobs are sequential and that there exists
at least one processor. *)
Hypothesis H_sequential_jobs: sequential_jobs sched.
Hypothesis H_at_least_one_cpu: num_cpus > 0.
(* Assume that we have a task set where all tasks have valid
parameters and constrained deadlines. *)
Variable ts: taskset_of sporadic_task.
Hypothesis all_jobs_from_taskset:
forall j,
arrives_in arr_seq j -> job_task j \in ts.
Hypothesis H_valid_task_parameters:
valid_sporadic_taskset task_cost task_period task_deadline ts.
Hypothesis H_constrained_deadlines:
forall tsk, tsk \in ts -> task_deadline tsk <= task_period tsk.
Let no_deadline_is_missed_by_tsk (tsk: sporadic_task) :=
task_misses_no_deadline job_arrival job_cost job_deadline job_task arr_seq sched tsk.
Let response_time_bounded_by (tsk: sporadic_task) :=
is_response_time_bound_of_task job_arrival job_cost job_task arr_seq sched tsk.
(* Assume that we have a work-conserving EDF scheduler. *)
Hypothesis H_work_conserving: work_conserving job_arrival job_cost job_jitter arr_seq sched.
Hypothesis H_edf_policy: respects_JLFP_policy job_arrival job_cost job_jitter arr_seq
sched (EDF job_arrival job_deadline).
(* Let tsk_i be the task to be analyzed, ...*)
Variable tsk_i: sporadic_task.
Hypothesis H_tsk_i_in_task_set: tsk_i \in ts.
(* and j_i one of its jobs. *)
Variable j_i: Job.
Hypothesis H_j_i_arrives: arrives_in arr_seq j_i.
Hypothesis H_job_of_tsk_i: job_task j_i = tsk_i.
(* Let tsk_k denote any interfering task, ... *)
Variable tsk_k: sporadic_task.
Hypothesis H_tsk_k_in_task_set: tsk_k \in ts.
(* ...and R_k its response-time bound. *)
Variable R_k: time.
Hypothesis H_R_k_le_deadline: task_jitter tsk_k + R_k <= task_deadline tsk_k.
(* Consider a time window of length delta <= D_i, starting with j_i's arrival time. *)
Variable delta: time.
Hypothesis H_delta_le_deadline: delta <= task_deadline tsk_i.
(* Assume that the jobs of tsk_k satisfy the response-time bound before the end of the interval *)
Hypothesis H_all_previous_jobs_completed_on_time :
forall j_k,
arrives_in arr_seq j_k ->
job_task j_k = tsk_k ->
job_arrival j_k + task_jitter tsk_k + R_k < job_arrival j_i + task_jitter tsk_i + delta ->
completed job_cost sched j_k (job_arrival j_k + task_jitter tsk_k + R_k).
(* In this section, we prove that Bertogna and Cirinei's EDF interference bound
indeed bounds the interference caused by task tsk_k in the interval [t1, t1 + delta). *)
Section MainProof.
(* Let t1 be the first point in time where j can actually be scheduled. *)
Let t1 := job_arrival j_i + job_jitter j_i.
Let t2 := t1 + delta.
(* Let's call x the task interference incurred by job j due to tsk_k. *)
Let x :=
task_interference job_arrival job_cost job_task job_jitter sched j_i tsk_k t1 t2.
(* Also, recall the EDF-specific interference bound for EDF. *)
Let interference_bound :=
edf_specific_interference_bound task_cost task_period task_deadline task_jitter tsk_i tsk_k R_k.
(* Let's simplify the names a bit. *)
Let a_i := job_arrival j_i.
Let D_i := task_deadline tsk_i.
Let D_k := task_deadline tsk_k.
Let p_k := task_period tsk_k.
Let J_i := task_jitter tsk_i.
Let J_k := task_jitter tsk_k.
Let n_k := div_floor D_i p_k.
(* Let's give a simpler name to job interference. *)
Let interference_caused_by := job_interference job_arrival job_cost job_jitter sched j_i.
(* Identify the subset of jobs that actually cause interference *)
Let interfering_jobs :=
filter (fun j' =>
(job_task j' == tsk_k) && (interference_caused_by j' t1 t2 != 0))
(jobs_scheduled_between sched t1 t2).
(* Now, consider the list of interfering jobs sorted by arrival time. *)
Let earlier_arrival := fun x y => job_arrival x <= job_arrival y.
Let sorted_jobs := (sort earlier_arrival interfering_jobs).
(* Now we proceed with the proof.
The first step consists in simplifying the sum corresponding to the workload. *)
Section SimplifyJobSequence.
(* Use the alternative definition of task interference, based on
individual job interference. *)
Lemma interference_bound_edf_use_another_definition :
x <= \sum_(j <- jobs_scheduled_between sched t1 t2 | job_task j == tsk_k)
interference_caused_by j t1 t2.
Proof.
apply interference_le_interference_joblist.
(* Remove the elements that we don't care about from the sum *)
Lemma interference_bound_edf_simpl_by_filtering_interfering_jobs :
\sum_(j <- jobs_scheduled_between sched t1 t2 | job_task j == tsk_k)
interference_caused_by j t1 t2 =
\sum_(j <- interfering_jobs) interference_caused_by j t1 t2.
Proof.
unfold interfering_jobs; rewrite big_filter.
rewrite big_mkcond; rewrite [\sum_(_ <- _ | _) _]big_mkcond /=.
apply eq_bigr; intros i _; clear -i.
destruct (job_task i == tsk_k); rewrite ?andTb ?andFb; last by done.
destruct (interference_caused_by i t1 t2 != 0) eqn:DIFF; first by done.
by apply negbT in DIFF; rewrite negbK in DIFF; apply/eqP.
Qed.
(* Then, we consider the sum over the sorted sequence of jobs. *)
Lemma interference_bound_edf_simpl_by_sorting_interfering_jobs :
\sum_(j <- interfering_jobs) interference_caused_by j t1 t2 =
\sum_(j <- sorted_jobs) interference_caused_by j t1 t2.
Proof.
by rewrite (perm_big sorted_jobs) /=; last by rewrite -(perm_sort earlier_arrival).
(* Note that both sequences have the same set of elements. *)
Lemma interference_bound_edf_job_in_same_sequence :
forall j,
(j \in interfering_jobs) = (j \in sorted_jobs).
Proof.
by apply perm_mem; rewrite -(perm_sort earlier_arrival).
(* Also recall that all jobs in the sorted sequence is an interfering job of tsk_k, ... *)
Lemma interference_bound_edf_all_jobs_from_tsk_k :
forall j,
j \in sorted_jobs ->
job_task j = tsk_k /\
interference_caused_by j t1 t2 != 0 /\
j \in jobs_scheduled_between sched t1 t2.
Proof.
intros j LT.
rewrite -interference_bound_edf_job_in_same_sequence mem_filter in LT.
move: LT => /andP [/andP [/eqP JOBi SERVi] INi]; repeat split; try (by done).
unfold jobs_scheduled_between in *; rewrite mem_undup in INi.
apply mem_bigcat_nat_exists in INi; des.
rewrite mem_scheduled_jobs_eq_scheduled in INi.
by apply (H_jobs_come_from_arrival_sequence j i).
(* ...and consecutive jobs are ordered by arrival. *)
Lemma interference_bound_edf_jobs_ordered_by_arrival :
forall i elem,
i < (size sorted_jobs).-1 ->
earlier_arrival (nth elem sorted_jobs i) (nth elem sorted_jobs i.+1).
Proof.
intros i elem LT.
assert (SORT: sorted earlier_arrival sorted_jobs).
by apply sort_sorted; unfold total, earlier_arrival; ins; apply leq_total.
by destruct sorted_jobs; simpl in *; [by rewrite ltn0 in LT | by apply/pathP].
Qed.
(* Also, for any job of task tsk_k, the interference is bounded by the task cost. *)
Lemma interference_bound_edf_interference_le_task_cost :
interference_caused_by j t1 t2 <= task_cost tsk_k.
Proof.
rename H_valid_job_parameters into PARAMS.
unfold valid_sporadic_job_with_jitter, valid_sporadic_job in *.
intros j IN.
feed (interference_bound_edf_all_jobs_from_tsk_k j);
first by rewrite -interference_bound_edf_job_in_same_sequence.
move => [ARRj [TSKj [_ SCHEDj]]].
specialize (PARAMS j ARRj); des.
apply leq_trans with (n := service_during sched j t1 t2);
first by apply job_interference_le_service.
by apply cumulative_service_le_task_cost with (job_task := job_task)
(task_deadline := task_deadline) (job_cost := job_cost)
(job_deadline := job_deadline); try (by done).
End SimplifyJobSequence.
(* Next, we show that if the number of jobs is no larger than n_k,
the workload bound trivially holds. *)
Section InterferenceFewJobs.
Hypothesis H_few_jobs: size sorted_jobs <= n_k.
Lemma interference_bound_edf_holds_for_at_most_n_k_jobs :
\sum_(j <- sorted_jobs) interference_caused_by j t1 t2 <=
interference_bound.
Proof.
rewrite -[\sum_(_ <- _ | _) _]addn0 leq_add //.
apply leq_trans with (n := \sum_(x <- sorted_jobs) task_cost tsk_k);
last by rewrite big_const_seq iter_addn addn0 mulnC leq_mul2r; apply/orP; right.
{
rewrite [\sum_(_ <- _) interference_caused_by _ _ _]big_seq_cond.
rewrite [\sum_(_ <- _) task_cost _]big_seq_cond.
apply leq_sum; intros i; move/andP => [INi _].
rewrite -interference_bound_edf_job_in_same_sequence in INi.
by apply interference_bound_edf_interference_le_task_cost.
End InterferenceFewJobs.
(* Otherwise, assume that the number of jobs is larger than n_k >= 0. *)
Section InterferenceManyJobs.
Hypothesis H_many_jobs: n_k < size sorted_jobs.
(* This trivially implies that there's at least one job. *)
Lemma interference_bound_edf_at_least_one_job: size sorted_jobs > 0.
Proof.
by apply leq_ltn_trans with (n := n_k).
Qed.
(* Let j_fst be the first job, and a_fst its arrival time. *)
Let j_fst := nth elem sorted_jobs 0.
Let a_fst := job_arrival j_fst.
(* In this section, we prove some basic lemmas about j_fst. *)
Section FactsAboutFirstJob.
(* The first job is an interfering job of task tsk_k. *)
Lemma interference_bound_edf_j_fst_is_job_of_tsk_k :
job_task j_fst = tsk_k /\
interference_caused_by j_fst t1 t2 != 0 /\
j_fst \in jobs_scheduled_between sched t1 t2.
Proof.
by apply interference_bound_edf_all_jobs_from_tsk_k, mem_nth,
interference_bound_edf_at_least_one_job.
Qed.
(* The deadline of j_fst is the deadline of tsk_k. *)
Lemma interference_bound_edf_j_fst_deadline :
job_deadline j_fst = task_deadline tsk_k.
Proof.
unfold valid_sporadic_job_with_jitter, valid_sporadic_job in *.
rename H_valid_job_parameters into PARAMS.
have FST := interference_bound_edf_j_fst_is_job_of_tsk_k.
destruct FST as [FSTarr [FSTtask _]].
by specialize (PARAMS j_fst FSTarr); des; rewrite PARAMS2 FSTtask.
Qed.
(* The deadline of j_i is the deadline of tsk_i. *)
Lemma interference_bound_edf_j_i_deadline :
job_deadline j_i = task_deadline tsk_i.
Proof.
unfold valid_sporadic_job_with_jitter, valid_sporadic_job in *.
rename H_valid_job_parameters into PARAMS,
H_job_of_tsk_i into JOBtsk.
by specialize (PARAMS j_i H_j_i_arrives); des; rewrite PARAMS2 JOBtsk.
Qed.
(* If j_fst completes by its response-time bound, then t1 <= a_fst + R_k,
where t1 is the beginning of the time window (arrival of j_i). *)
Lemma interference_bound_edf_j_fst_completion_implies_rt_bound_inside_interval :
completed job_cost sched j_fst (a_fst + J_k + R_k) ->
t1 <= a_fst + J_k + R_k.
Proof.
intros RBOUND.
rewrite leqNgt; apply/negP; unfold not; intro BUG.
have FST := interference_bound_edf_j_fst_is_job_of_tsk_k.
destruct FST as [_ [_ [FSTserv _]]].
move: FSTserv => /negP FSTserv; apply FSTserv.
rewrite -leqn0; apply leq_trans with (n := service_during sched j_fst t1 t2);
first by apply job_interference_le_service.
by apply cumulative_service_after_job_rt_zero with
(job_arrival := job_arrival) (job_cost := job_cost) (R := J_k + R_k);
try (by done); rewrite addnA; last by apply ltnW.
(* Now, let's prove the interference bound for the particular case of a single job.
This case must be solved separately because the single job can simultaneously
be carry-in and carry-out job, so its response time is not necessarily
bounded by R_k (from the hypothesis H_all_previous_jobs_completed_on_time). *)
Section InterferenceSingleJob.
(* Assume that there's at least one job in the sorted list. *)
Hypothesis H_only_one_job: size sorted_jobs = 1.
(* Since there's only one job, we simplify the terms in the interference bound. *)
Lemma interference_bound_edf_simpl_when_there's_one_job :
D_i %% p_k - (D_k - R_k - J_k) = D_i - (D_k - R_k - J_k).
Proof.
rename H_many_jobs into NUM,
H_valid_task_parameters into TASK_PARAMS,
H_tsk_k_in_task_set into INk.
unfold valid_sporadic_taskset, is_valid_sporadic_task,
interference_bound, edf_specific_interference_bound in *.
rewrite H_only_one_job in NUM.
rewrite ltnS leqn0 in NUM; move: NUM => /eqP EQnk.
move: EQnk => /eqP EQnk; unfold n_k, div_floor in EQnk.
rewrite -leqn0 leqNgt divn_gt0 in EQnk;
last by specialize (TASK_PARAMS tsk_k INk); des.
by rewrite -ltnNge in EQnk; rewrite modn_small //.
Qed.
(* Next, we show that if j_fst completes by its response-time bound R_k,
then then interference bound holds. *)
Section ResponseTimeOfSingleJobBounded.
completed job_cost sched j_fst (a_fst + J_k + R_k).
Lemma interference_bound_edf_holds_for_single_job_that_completes_on_time :
job_interference job_arrival job_cost job_jitter sched j_i j_fst t1 t2
<= D_i - (D_k - R_k - J_k).
Proof.
rename H_j_fst_completed_by_rt_bound into RBOUND.
have AFTERt1 :=
interference_bound_edf_j_fst_completion_implies_rt_bound_inside_interval RBOUND.
have FST := interference_bound_edf_j_fst_is_job_of_tsk_k.
destruct FST as [ARRfst [_ [LEdl _]]].
apply interference_under_edf_implies_shorter_deadlines with
(arr_seq := arr_seq) (job_deadline := job_deadline) in LEdl; try (by done).
destruct (D_k - R_k - J_k <= D_i) eqn:LEdk; last first.
apply negbT in LEdk; rewrite -ltnNge in LEdk.
apply leq_trans with (n := 0); last by done.
apply leq_trans with (n := job_interference job_arrival job_cost
job_jitter sched j_i j_fst (a_fst + J_k + R_k) t2); last first.
{
apply leq_trans with (n := service_during sched j_fst (a_fst + J_k + R_k) t2);
first by apply job_interference_le_service.
unfold service_during; rewrite leqn0; apply/eqP.
by apply cumulative_service_after_job_rt_zero with
(job_arrival := job_arrival) (job_cost := job_cost) (R := J_k + R_k);
try (by done); rewrite addnA // leqnn.
}
{
apply extend_sum; last by apply leqnn.
rewrite -(leq_add2r D_i).
rewrite interference_bound_edf_j_fst_deadline
interference_bound_edf_j_i_deadline in LEdl.
apply leq_trans with (n := a_fst + D_k); last first.
{
apply leq_trans with (n := job_arrival j_i + D_i); first by done.
by rewrite leq_add2r leq_addr.
}
rewrite -2!addnA leq_add2l.
rewrite addnA.
by apply ltnW; rewrite -ltn_subRL addnC subnDA.
apply leq_trans with (n := job_interference job_arrival job_cost job_jitter sched j_i j_fst a_i t2);
first by apply extend_sum; [by apply leq_addr | by apply leqnn].
rewrite -(leq_add2r (D_k - R_k - J_k)) addnBAC // -addnBA // subnn addn0.
assert (SUBST: D_k - R_k - J_k = \sum_(a_fst + J_k + R_k <= i < a_fst + D_k) 1).
{
rewrite big_const_nat iter_addn mul1n addn0.
rewrite [_ + D_k]addnC.
rewrite -subnBA; last by rewrite -addnA leq_addr.
rewrite [a_fst + _]addnC -addnA [a_fst + R_k]addnC addnA.
rewrite -addnBA // subnn addn0.
by rewrite [_ + R_k]addnC subnDA.
}
apply leq_trans with (n := job_interference job_arrival job_cost job_jitter sched j_i
j_fst a_i (a_fst + D_k) + (D_k - R_k - J_k)).
rewrite leq_add2r.
destruct (t2 <= a_fst + J_k + R_k) eqn:LEt2.
apply extend_sum; first by apply leqnn.
apply leq_trans with (n := a_fst + J_k + R_k); first by done.
rewrite -addnA leq_add2l.
by apply H_R_k_le_deadline.
unfold job_interference.
apply negbT in LEt2; rewrite -ltnNge in LEt2.
rewrite -> big_cat_nat with (n := a_fst + J_k + R_k);
[simpl | | by apply ltnW]; last first.
by apply leq_trans with (n := t1); first by apply leq_addr.
apply leq_trans with (n := job_interference job_arrival job_cost job_jitter sched j_i j_fst a_i
(a_fst + J_k + R_k) + service_during sched j_fst (a_fst + J_k + R_k) t2).
rewrite leq_add2l.
by apply job_interference_le_service.
rewrite -> cumulative_service_after_job_rt_zero with (job_cost := job_cost)
(job_arrival := job_arrival) (R := J_k + R_k);
rewrite ?addnA //.
rewrite addn0; apply extend_sum; first by done.
rewrite -addnA leq_add2l.
by apply H_R_k_le_deadline.
unfold job_interference.
rewrite -> big_cat_nat with (n := a_fst + J_k + R_k);
[simpl | | ]; last first.
{
rewrite -addnA leq_add2l.
by apply H_R_k_le_deadline.
}
{
by apply leq_trans with (n := t1); first by apply leq_addr.
}
apply leq_trans with (n := job_interference job_arrival job_cost job_jitter sched j_i j_fst a_i
(a_fst + J_k + R_k) + service_during sched j_fst (a_fst + J_k + R_k) (a_fst + D_k) + (D_k - R_k - J_k)).
{
rewrite leq_add2r leq_add2l.
by apply job_interference_le_service.
}
unfold service_during.
rewrite -> cumulative_service_after_job_rt_zero with (job_arrival := job_arrival)
(job_cost := job_cost) (R:=J_k + R_k);
rewrite addn0.
apply leq_trans with (n := (\sum_(a_i <= t < a_fst + J_k + R_k) 1) +
\sum_(a_fst + J_k + R_k <= t < a_fst + D_k) 1).
{
apply leq_add; last by rewrite SUBST.
simpl_sum_const; rewrite -{1}[_ + R_k](addKn a_i) -addnBA //;
last by apply leq_trans with (n := t1); first by apply leq_addr.
by apply job_interference_le_delta.
rewrite -big_cat_nat; simpl; last by rewrite -addnA leq_add2l H_R_k_le_deadline.
simpl_sum_const; rewrite leq_subLR; unfold D_i, D_k, t1, a_fst.
by rewrite -interference_bound_edf_j_fst_deadline
-interference_bound_edf_j_i_deadline.
}
by apply leq_trans with (n := t1); first by apply leq_addr.
Qed.
End ResponseTimeOfSingleJobBounded.
(* Else, if j_fst did not complete by its response-time bound, then
we need a separate proof. *)
Section ResponseTimeOfSingleJobNotBounded.
Hypothesis H_j_fst_not_complete_by_rt_bound :
~~ completed job_cost sched j_fst (a_fst + J_k + R_k).
(* This trivially implies that a_fst + R_k lies after the end of the interval,
otherwise j_fst would have completed by its response-time bound. *)
Lemma interference_bound_edf_response_time_bound_of_j_fst_after_interval :
job_arrival j_fst + J_k + R_k >= job_arrival j_i + J_i + delta.
Proof.
have FST := interference_bound_edf_j_fst_is_job_of_tsk_k.
destruct FST as [FSTarr [FSTtask _]].
rewrite leqNgt; apply/negP; intro LT.
move: H_j_fst_not_complete_by_rt_bound => /negP BUG; apply BUG.
by apply H_all_previous_jobs_completed_on_time.
Qed.
(* If the slack is too big (D_i < D_k - R_k - J_k), j_fst causes no interference. *)
Lemma interference_bound_edf_holds_for_single_job_with_big_slack :
D_i < D_k - R_k - J_k ->
interference_caused_by j_fst t1 t2 = 0.
Proof.
unfold valid_sporadic_job_with_jitter, valid_sporadic_job in *.
have PARAMS := H_valid_job_parameters j_i H_j_i_arrives; des.
have FST := interference_bound_edf_j_fst_is_job_of_tsk_k.
destruct FST as [FSTarr [FSTtask _]].
rewrite 2!ltn_subRL addnA [R_k + _]addnC in LTdk.
rewrite -(ltn_add2l a_fst) 2!addnA in LTdk.
apply leq_ltn_trans with (m := t1 + D_i) in LTdk; last first.
{
rewrite leq_add2r.
apply leq_trans with (n := t1 + delta); first by apply leq_addr.
apply leq_trans with (n := job_arrival j_i + J_i + delta).
{
unfold J_i;rewrite leq_add2r leq_add2l.
rewrite -H_job_of_tsk_i; apply PARAMS0.
}
by apply interference_bound_edf_response_time_bound_of_j_fst_after_interval.
apply/eqP; rewrite -[_ _ _ _ == 0]negbK; apply/negP; red; intro BUG.
apply interference_under_edf_implies_shorter_deadlines with
(arr_seq := arr_seq) (job_deadline := job_deadline) in BUG; try (by done).
rewrite interference_bound_edf_j_fst_deadline
interference_bound_edf_j_i_deadline in BUG.
apply leq_ltn_trans with (m := job_arrival j_i + D_i) in LTdk;
last by rewrite leq_add2r leq_addr.
by apply (leq_trans LTdk) in BUG; rewrite ltnn in BUG.
Qed.
(* Else, if the slack is small, j_fst causes interference for no longer than
D_i - (D_k - R_k - J_k). *)
Lemma interference_bound_edf_holds_for_single_job_with_small_slack :
D_i >= D_k - R_k - J_k ->
interference_caused_by j_fst t1 t2 <= D_i - (D_k - R_k - J_k).
unfold valid_sporadic_job_with_jitter, valid_sporadic_job in *.
have PARAMS := H_valid_job_parameters j_i H_j_i_arrives; des.
intro LEdk.
have FST := interference_bound_edf_j_fst_is_job_of_tsk_k.
destruct FST as [FSTtask [_ [LEdl _]]].
have LTr := interference_bound_edf_response_time_bound_of_j_fst_after_interval.
apply leq_trans with (n := job_interference job_arrival job_cost job_jitter sched j_i
j_fst t1 (job_arrival j_fst + J_k + R_k) + (D_k - R_k - J_k)).
{
rewrite leq_add2r; apply extend_sum; first by apply leqnn.
apply leq_trans with (n := job_arrival j_i + J_i + delta);
last by done.
rewrite leq_add2r leq_add2l.
by unfold J_i; rewrite -H_job_of_tsk_i; apply PARAMS0.
}
apply leq_trans with (n := \sum_(t1 <= t < a_fst + J_k + R_k) 1 +
\sum_(a_fst + J_k + R_k <= t < a_fst + D_k)1).
apply leq_add; unfold job_interference.
{
simpl_sum_const.
rewrite -{1}[job_arrival j_fst + J_k + R_k](addKn t1) -addnBA;
first by apply job_interference_le_delta.
apply leq_trans with (n := a_i + J_i + delta); last by done.
apply leq_trans with (n := a_i + J_i); last by apply leq_addr.
by rewrite leq_add2l /J_i -H_job_of_tsk_i; apply PARAMS0.
}
simpl_sum_const; rewrite addnC.
rewrite -subnBA; last by rewrite -addnA leq_addr.
rewrite [a_fst + _]addnC -addnA [a_fst + _]addnC addnA.
rewrite -addnBA // subnn addn0.
by rewrite addnC subnDA.
}
rewrite -big_cat_nat; simpl; last 2 first.
{
apply leq_trans with (n := t1 + delta); first by apply leq_addr.
apply leq_trans with (n := job_arrival j_i + J_i + delta).
{
rewrite leq_add2r leq_add2l.
unfold J_i; rewrite -H_job_of_tsk_i; apply PARAMS0.
}
by apply interference_bound_edf_response_time_bound_of_j_fst_after_interval.
}
{
by rewrite -addnA leq_add2l; apply H_R_k_le_deadline.
}
rewrite big_const_nat iter_addn mul1n addn0 leq_subLR.
unfold D_i, D_k, t1, a_fst.
rewrite -interference_bound_edf_j_fst_deadline
-interference_bound_edf_j_i_deadline.
apply leq_trans with (n := a_i + job_deadline j_i);
last by rewrite leq_add2r leq_addr.
by apply interference_under_edf_implies_shorter_deadlines with (arr_seq := arr_seq)
(job_arrival := job_arrival) (job_deadline := job_deadline) in LEdl.
Qed.
End ResponseTimeOfSingleJobNotBounded.
(* By combining the results above, we prove that the interference caused by the single job
is bounded by D_i - (D_k - R_k), ... *)
Lemma interference_bound_edf_interference_of_j_fst_limited_by_slack :
interference_caused_by j_fst t1 t2 <= D_i - (D_k - R_k - J_k).
destruct (completed job_cost sched j_fst (a_fst + J_k + R_k)) eqn:COMP;
first by apply interference_bound_edf_holds_for_single_job_that_completes_on_time.
apply negbT in COMP.
destruct (ltnP D_i (D_k - R_k - J_k)) as [LEdk | LTdk].
by rewrite interference_bound_edf_holds_for_single_job_with_big_slack.
by apply interference_bound_edf_holds_for_single_job_with_small_slack.
Qed.
Lemma interference_bound_edf_holds_for_a_single_job :
interference_caused_by j_fst t1 t2 <= interference_bound.
Proof.
have ONE := interference_bound_edf_simpl_when_there's_one_job.
have SLACK := interference_bound_edf_interference_of_j_fst_limited_by_slack.
rename H_many_jobs into NUM, H_only_one_job into SIZE.
unfold interference_caused_by, interference_bound, edf_specific_interference_bound.
fold D_i D_k p_k n_k.
rewrite SIZE ltnS leqn0 in NUM; move: NUM => /eqP EQnk.
rewrite EQnk mul0n add0n.
rewrite leq_min; apply/andP; split.
apply interference_bound_edf_interference_le_task_cost.
rewrite interference_bound_edf_job_in_same_sequence.
by apply mem_nth; rewrite SIZE.
(* Next, consider the other case where there are at least two jobs:
the first job j_fst, and the last job j_lst. *)
Section InterferenceTwoOrMoreJobs.
(* Assume there are at least two jobs. *)
Variable num_mid_jobs: nat.
Hypothesis H_at_least_two_jobs : size sorted_jobs = num_mid_jobs.+2.
(* Let j_lst be the last job of the sequence and a_lst its arrival time. *)
Let j_lst := nth elem sorted_jobs num_mid_jobs.+1.
Let a_lst := job_arrival j_lst.
(* In this section, we prove some basic lemmas about the first and last jobs. *)
Section FactsAboutFirstAndLastJobs.
(* The last job is an interfering job of task tsk_k. *)
Lemma interference_bound_edf_j_lst_is_job_of_tsk_k :
job_task j_lst = tsk_k /\
interference_caused_by j_lst t1 t2 != 0 /\
j_lst \in jobs_scheduled_between sched t1 t2.
Proof.
apply interference_bound_edf_all_jobs_from_tsk_k, mem_nth.
by rewrite H_at_least_two_jobs.
Qed.
(* The deadline of j_lst is the deadline of tsk_k. *)
Lemma interference_bound_edf_j_lst_deadline :
job_deadline j_lst = task_deadline tsk_k.
Proof.
unfold valid_sporadic_job_with_jitter, valid_sporadic_job in *.
rename H_valid_job_parameters into PARAMS.
have LST := interference_bound_edf_j_lst_is_job_of_tsk_k.
destruct LST as [LSTarr [LSTtask _]].
by specialize (PARAMS j_lst LSTarr); des; rewrite PARAMS2 LSTtask.
Qed.
(* The first job arrives before the last job. *)
Lemma interference_bound_edf_j_fst_before_j_lst :
job_arrival j_fst <= job_arrival j_lst.
Proof.
rename H_at_least_two_jobs into SIZE.
unfold j_fst, j_lst; rewrite -[num_mid_jobs.+1]add0n.
apply prev_le_next; last by rewrite SIZE leqnn.
by intros i LT; apply interference_bound_edf_jobs_ordered_by_arrival.
Qed.
(* The last job arrives before the end of the interval. *)
Lemma interference_bound_edf_last_job_arrives_before_end_of_interval :
job_arrival j_lst < t2.
Proof.
rewrite leqNgt; apply/negP; unfold not; intro LT2.
feed (interference_bound_edf_all_jobs_from_tsk_k j_lst).
by apply mem_nth; rewrite -(ltn_add2r 1) addn1 H_at_least_two_jobs addn1.
move => [LSTarr [LSTtsk [/eqP LSTserv LSTin]]].
apply leq_trans with (n := service_during sched j_lst t1 t2);
first by apply job_interference_le_service.
rewrite leqn0; apply/eqP; unfold service_during.
rewrite (cumulative_service_before_job_arrival_zero job_arrival) //.
by apply arrival_before_jitter with (job_jitter := job_jitter).
Qed.
(* Since there are multiple jobs, j_fst is far enough from the end of
(by the assumption H_all_previous_jobs_completed_on_time). *)
Lemma interference_bound_edf_j_fst_completed_on_time :
completed job_cost sched j_fst (a_fst + J_k + R_k).
have PARAMS := H_valid_job_parameters j_i H_j_i_arrives.
unfold valid_sporadic_job_with_jitter, valid_sporadic_job in *; des.
have FST := interference_bound_edf_j_fst_is_job_of_tsk_k; des.
set j_snd := nth elem sorted_jobs 1.
exploit interference_bound_edf_all_jobs_from_tsk_k.
{
by apply mem_nth; instantiate (1 := 1); rewrite H_at_least_two_jobs.
}
instantiate (1 := elem); move => [SNDarr [SNDtsk [/eqP SNDserv _]]].
apply H_all_previous_jobs_completed_on_time; try (by done).
apply leq_ltn_trans with (n := job_arrival j_snd); last first.
{
apply leq_trans with (n := t2); last first.
{
unfold t2, t1; rewrite leq_add2r leq_add2l.
by rewrite -H_job_of_tsk_i; apply PARAMS0.
}
rewrite ltnNge; apply/negP; red; intro BUG; apply SNDserv.
apply/eqP; rewrite -leqn0; apply leq_trans with (n := service_during
first by apply job_interference_le_service.
apply cumulative_service_before_job_arrival_zero with (job_arrival := job_arrival);
by apply arrival_before_jitter with (job_jitter := job_jitter).
}
apply leq_trans with (n := a_fst + p_k).
{
apply leq_trans with (n := job_arrival j_fst + D_k);
first by rewrite -addnA leq_add2l.
by rewrite leq_add2l; apply H_constrained_deadlines.
}
(* Since jobs are sporadic, we know that the first job arrives
at least p_k units before the second. *)
unfold p_k; rewrite -FST0.
apply H_sporadic_tasks; [| by done | by done | by rewrite SNDtsk | ]; last first.
{
apply interference_bound_edf_jobs_ordered_by_arrival.
by rewrite H_at_least_two_jobs.
}
red; move => /eqP BUG.
by rewrite nth_uniq in BUG; rewrite ?SIZE //;
[ by apply interference_bound_edf_at_least_one_job
| by rewrite H_at_least_two_jobs
| by rewrite sort_uniq; apply filter_uniq, undup_uniq].
Qed.
End FactsAboutFirstAndLastJobs.
(* Next, we prove that the distance between the first and last jobs is at least
num_mid_jobs + 1 periods. *)
Lemma interference_bound_edf_many_periods_in_between :
a_lst - a_fst >= num_mid_jobs.+1 * p_k.
unfold a_fst, a_lst, j_fst, j_lst.
assert (EQnk: num_mid_jobs.+1=(size sorted_jobs).-1).
by rewrite H_at_least_two_jobs.
rewrite EQnk telescoping_sum;
last by ins; apply interference_bound_edf_jobs_ordered_by_arrival.
rewrite -[_ * _ tsk_k]addn0 mulnC -iter_addn -{1}[_.-1]subn0 -big_const_nat.
rewrite big_nat_cond [\sum_(0 <= i < _)(_-_)]big_nat_cond.
apply leq_sum; intros i; rewrite andbT; move => /andP LT; des.
(* To simplify, call the jobs 'cur' and 'next' *)
set cur := nth elem sorted_jobs i.
set next := nth elem sorted_jobs i.+1.
(* Show that cur arrives earlier than next *)
assert (ARRle: job_arrival cur <= job_arrival next).
by unfold cur, next; apply interference_bound_edf_jobs_ordered_by_arrival.
(* Show that both cur and next are in the arrival sequence *)
assert (INnth: cur \in interfering_jobs /\ next \in interfering_jobs).
{
rewrite 2!interference_bound_edf_job_in_same_sequence; split.
by apply mem_nth, (ltn_trans LT0); destruct sorted_jobs; ins.
by apply mem_nth; destruct sorted_jobs; ins.
}
rewrite 2?mem_filter in INnth; des.
feed (interference_bound_edf_all_jobs_from_tsk_k cur).
by apply mem_nth, (ltn_trans LT0); destruct sorted_jobs.
intros [CURarr [CURtsk [_ CURin]]].
feed (interference_bound_edf_all_jobs_from_tsk_k next).
by apply mem_nth; destruct sorted_jobs.
intros [NEXTarr [NEXTtsk [_ NEXTin]]].
(* Use the sporadic task model to conclude that cur and next are separated
by at least (task_period tsk) units. Of course this only holds if cur != next.
Since we don't know much about the list (except that it's sorted), we must
also prove that it doesn't contain duplicates. *)
assert (CUR_LE_NEXT: job_arrival cur + task_period (job_task cur) <= job_arrival next).
apply H_sporadic_tasks; try (by done).
unfold cur, next, not; intro EQ; move: EQ => /eqP EQ.
rewrite nth_uniq in EQ; first by move: EQ => /eqP EQ; ssrlia.
by apply ltn_trans with (n := (size sorted_jobs).-1); destruct sorted_jobs; ins.
by destruct sorted_jobs; ins.
by rewrite sort_uniq -/interfering_jobs filter_uniq // undup_uniq.
(* Using the lemma above, we prove that the ratio n_k is at least the number of
middle jobs + 1, ... *)
Lemma interference_bound_edf_n_k_covers_middle_jobs_plus_one :
n_k >= num_mid_jobs.+1.
Proof.
have AFTERt1 :=
interference_bound_edf_j_fst_completion_implies_rt_bound_inside_interval
interference_bound_edf_j_fst_completed_on_time.
rename H_valid_task_parameters into TASK_PARAMS,
H_tsk_k_in_task_set into INk.
unfold valid_sporadic_taskset, is_valid_sporadic_task,
interference_bound, edf_specific_interference_bound in *.
have DIST := interference_bound_edf_many_periods_in_between.
rewrite leqNgt; apply/negP; unfold not; intro LTnk; unfold n_k in LTnk.
rewrite ltn_divLR in LTnk; last by specialize (TASK_PARAMS tsk_k INk); des.
apply (leq_trans LTnk) in DIST; rewrite ltn_subRL in DIST.
rewrite -(ltn_add2r (J_k + R_k)) -addnA [D_i + _]addnC addnA in DIST.
unfold t1 in *.
apply leq_ltn_trans with (m := job_arrival j_i + D_i) in DIST; last first.
{
rewrite addnA leq_add2r.
by apply leq_trans with (n := job_arrival j_i + job_jitter j_i);
first by apply leq_addr.
apply leq_trans with (p := a_lst + D_k) in DIST;
last by rewrite leq_add2l.
have LST := interference_bound_edf_j_lst_is_job_of_tsk_k.
destruct LST as [LSTarr [_ [LEdl _]]].
apply interference_under_edf_implies_shorter_deadlines with
(arr_seq := arr_seq) (job_deadline := job_deadline) in LEdl; try (by done).
unfold D_i, D_k in DIST; rewrite interference_bound_edf_j_lst_deadline
interference_bound_edf_j_i_deadline in LEdl.
by apply (leq_trans DIST) in LEdl; rewrite ltnn in LEdl.
(* ... which allows bounding the interference of the middle and last jobs
using n_k multiplied by the cost. *)
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Lemma interference_bound_edf_holds_for_middle_and_last_jobs :
interference_caused_by j_lst t1 t2 +
\sum_(0 <= i < num_mid_jobs)
interference_caused_by (nth elem sorted_jobs i.+1) t1 t2
<= n_k * task_cost tsk_k.
Proof.
apply leq_trans with (n := num_mid_jobs.+1 * task_cost tsk_k); last first.
{
rewrite leq_mul2r; apply/orP; right.
by apply interference_bound_edf_n_k_covers_middle_jobs_plus_one.
}
rewrite mulSn; apply leq_add.
{
apply interference_bound_edf_interference_le_task_cost.
rewrite interference_bound_edf_job_in_same_sequence.
by apply mem_nth; rewrite H_at_least_two_jobs.
}
{
apply leq_trans with (n := \sum_(0 <= i < num_mid_jobs) task_cost tsk_k);
last by rewrite big_const_nat iter_addn addn0 mulnC subn0.
rewrite big_nat_cond [\sum_(0 <= i < num_mid_jobs) task_cost _]big_nat_cond.
apply leq_sum; intros i; rewrite andbT; move => /andP LT; des.
apply interference_bound_edf_interference_le_task_cost.
rewrite interference_bound_edf_job_in_same_sequence.
apply mem_nth; rewrite H_at_least_two_jobs.
by rewrite ltnS; apply leq_trans with (n := num_mid_jobs).
}
Qed.
(* Now, since n_k < sorted_jobs = num_mid_jobs + 2, it follows that
n_k = num_mid_jobs + 1. *)
Lemma interference_bound_edf_n_k_equals_num_mid_jobs_plus_one :
n_k = num_mid_jobs.+1.
have NK := interference_bound_edf_n_k_covers_middle_jobs_plus_one.
rename H_many_jobs into NUM, H_at_least_two_jobs into SIZE.
move: NK; rewrite leq_eqVlt orbC; move => /orP NK; des;
first by rewrite SIZE ltnS leqNgt NK in NUM.