Library prosa.classic.analysis.global.parallel.interference_bound_edf
Require Import prosa.classic.util.all.
Require Import prosa.classic.model.arrival.basic.task prosa.classic.model.arrival.basic.job 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.basic.schedule prosa.classic.model.schedule.global.basic.platform
prosa.classic.model.schedule.global.basic.interference prosa.classic.model.schedule.global.basic.interference_edf.
Require Import prosa.classic.analysis.global.parallel.workload_bound
prosa.classic.analysis.global.parallel.interference_bound.
From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq fintype bigop div path.
Module InterferenceBoundEDF.
Import Job SporadicTaskset Schedule ScheduleOfSporadicTask Schedulability
WorkloadBound ResponseTime Priority
TaskArrival Interference InterferenceEDF.
Export InterferenceBoundGeneric.
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 tsk: sporadic_task.
Variable delta: time.
Variable tsk_other: sporadic_task.
Variable R_other: time.
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
(div_ceil (d_tsk + R_other - d_other + 1) p_other) × e_other.
End SpecificBoundDef.
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 tsk: sporadic_task.
Let task_with_response_time := (sporadic_task × time)%type.
Variable R_prev: seq task_with_response_time.
Variable delta: time.
Section RecallInterferenceBounds.
Variable tsk_R: task_with_response_time.
Let tsk_other := fst tsk_R.
Let R_other := snd tsk_R.
Let basic_interference_bound := interference_bound_generic task_cost task_period delta tsk_R.
Let edf_specific_bound := edf_specific_interference_bound task_cost task_period task_deadline tsk tsk_other R_other.
Definition interference_bound_edf :=
minn basic_interference_bound edf_specific_bound.
End RecallInterferenceBounds.
Section TotalInterference.
Let other_task := different_task tsk.
Definition total_interference_bound_edf :=
\sum_((tsk_other, R_other) <- R_prev | other_task tsk_other)
interference_bound_edf (tsk_other, R_other).
End TotalInterference.
End TotalInterferenceBoundEDF.
Section ProofSpecificBound.
Import Schedule 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.
Context {Job: eqType}.
Variable job_arrival: Job → time.
Variable job_cost: Job → time.
Variable job_deadline: Job → time.
Variable job_task: Job → sporadic_task.
Context {arr_seq: arrival_sequence Job}.
Hypothesis H_sporadic_tasks:
sporadic_task_model task_period job_arrival job_task arr_seq.
Hypothesis H_valid_job_parameters:
∀ j,
arrives_in arr_seq j →
valid_sporadic_job task_cost task_deadline job_cost job_deadline job_task j.
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.
Hypothesis H_jobs_must_arrive_to_execute:
jobs_must_arrive_to_execute job_arrival sched.
Hypothesis H_completed_jobs_dont_execute:
completed_jobs_dont_execute job_cost sched.
Hypothesis H_at_least_one_cpu: num_cpus > 0.
Variable ts: taskset_of sporadic_task.
Hypothesis all_jobs_from_taskset:
∀ 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:
∀ 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.
Hypothesis H_work_conserving: work_conserving job_arrival job_cost arr_seq sched.
Hypothesis H_edf_scheduler:
respects_JLFP_policy job_arrival job_cost arr_seq sched (EDF job_arrival job_deadline).
Variable tsk_i: sporadic_task.
Hypothesis H_tsk_i_in_task_set: tsk_i \in ts.
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.
Variable tsk_k: sporadic_task.
Hypothesis H_tsk_k_in_task_set: tsk_k \in ts.
Variable R_k: time.
Hypothesis H_R_k_le_deadline: R_k ≤ task_deadline tsk_k.
Variable delta: time.
Hypothesis H_delta_le_deadline: delta ≤ task_deadline tsk_i.
Hypothesis H_all_previous_jobs_completed_on_time :
∀ j_k,
arrives_in arr_seq j_k →
job_task j_k = tsk_k →
job_arrival j_k + R_k < job_arrival j_i + delta →
completed job_cost sched j_k (job_arrival j_k + R_k).
Section MainProof.
Let x :=
task_interference job_arrival job_cost job_task sched j_i
tsk_k (job_arrival j_i) (job_arrival j_i + delta).
Let interference_bound :=
edf_specific_interference_bound task_cost task_period task_deadline tsk_i tsk_k R_k.
Let t1 := job_arrival j_i.
Let t2 := job_arrival j_i + delta.
Let D_i := task_deadline tsk_i.
Let D_k := task_deadline tsk_k.
Let p_k := task_period tsk_k.
Let n_k := div_ceil (D_i + R_k - D_k + 1) p_k.
Let interference_caused_by := job_interference job_arrival job_cost sched j_i.
Let interfering_jobs :=
filter (fun j' ⇒
(job_task j' == tsk_k) && (interference_caused_by j' t1 t2 != 0))
(jobs_scheduled_between sched t1 t2).
Let earlier_arrival := fun x y ⇒ job_arrival x ≤ job_arrival y.
Let sorted_jobs := sort earlier_arrival interfering_jobs.
Section SimplifyJobSequence.
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.
unfold x, task_interference, interference_caused_by, job_interference.
rewrite [\sum_(_ <- _ sched _ _ | _) _]exchange_big /=.
rewrite big_nat_cond [\sum_(_ ≤ _ < _ | true) _]big_nat_cond.
apply leq_sum. move ⇒ t /andP [LEt _].
rewrite exchange_big /=.
apply leq_sum; intros cpu _.
destruct (backlogged job_arrival job_cost sched j_i t) eqn:BACK;
last by rewrite andFb (eq_bigr (fun x ⇒ 0));
first by rewrite big_const_seq iter_addn mul0n addn0.
rewrite andTb.
destruct (task_scheduled_on job_task sched tsk_k cpu t) eqn:SCHED;
last by done.
unfold task_scheduled_on in ×.
destruct (sched cpu t) eqn:SOME; last by done.
rewrite big_mkcond /= (bigD1_seq s) /=; last by apply undup_uniq.
{
rewrite SCHED -addn1 addnC; apply leq_add; last by done.
apply eq_leq; symmetry; apply/eqP; rewrite eqb1.
by unfold scheduled_on; apply/eqP.
}
{
unfold jobs_scheduled_between.
rewrite mem_undup; apply mem_bigcat_nat with (j := t);
first by done.
apply mem_bigcat_ord with (j := cpu); first by apply ltn_ord.
by unfold make_sequence; rewrite SOME mem_seq1 eq_refl.
}
Qed.
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.
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).
Qed.
Lemma interference_bound_edf_job_in_same_sequence :
∀ j,
(j \in interfering_jobs) = (j \in sorted_jobs).
Proof.
by apply perm_mem; rewrite -(perm_sort earlier_arrival).
Qed.
Lemma interference_bound_edf_all_jobs_from_tsk_k :
∀ j,
j \in sorted_jobs →
arrives_in arr_seq j ∧
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).
Qed.
Lemma interference_bound_edf_jobs_ordered_by_arrival :
∀ 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.
Lemma interference_bound_edf_interference_le_task_cost :
∀ j,
j \in interfering_jobs →
interference_caused_by j t1 t2 ≤ task_cost tsk_k.
Proof.
rename H_valid_job_parameters into PARAMS.
intros j INj.
feed (interference_bound_edf_all_jobs_from_tsk_k j);
first by rewrite -interference_bound_edf_job_in_same_sequence.
move ⇒ [ARRj [TSKj _]].
specialize (PARAMS j ARRj); des.
apply leq_trans with (n := service_during sched j t1 t2);
first by apply job_interference_le_service.
try ( by apply cumulative_service_le_task_cost with (job_task0 := job_task)
(task_deadline0 := task_deadline) (job_cost0 := job_cost) (job_deadline0 := job_deadline) ) ||
by apply cumulative_service_le_task_cost with (job_task := job_task)
(task_deadline := task_deadline) (job_cost := job_cost) (job_deadline := job_deadline).
Qed.
End SimplifyJobSequence.
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.
unfold interference_bound, edf_specific_interference_bound; fold D_i p_k n_k.
apply leq_trans with (n := \sum_(x <- sorted_jobs) task_cost tsk_k);
last first.
{
rewrite big_const_seq iter_addn addn0 count_predT mulnC.
by rewrite leq_mul2r; apply/orP; right.
}
{
rewrite big_seq_cond [\sum_(_ <- _ | true)_]big_seq_cond.
apply leq_sum; move ⇒ j /andP [IN _].
apply interference_bound_edf_interference_le_task_cost.
by rewrite interference_bound_edf_job_in_same_sequence.
}
Qed.
End InterferenceFewJobs.
Section InterferenceManyJobs.
Hypothesis H_many_jobs: n_k < size sorted_jobs.
Lemma interference_bound_edf_at_least_one_job: size sorted_jobs > 0.
Proof.
by apply leq_ltn_trans with (n := n_k).
Qed.
Variable elem: Job.
Let j_fst := nth elem sorted_jobs 0.
Let a_fst := job_arrival j_fst.
Section FactsAboutFirstJob.
Lemma interference_bound_edf_j_fst_is_job_of_tsk_k :
arrives_in arr_seq j_fst ∧
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.
Lemma interference_bound_edf_j_fst_deadline :
job_deadline j_fst = task_deadline tsk_k.
Proof.
unfold 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 PARAMS1 FSTtask.
Qed.
Lemma interference_bound_edf_j_i_deadline :
job_deadline j_i = task_deadline tsk_i.
Proof.
unfold 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 PARAMS1 JOBtsk.
Qed.
Lemma interference_bound_edf_j_fst_completion_implies_rt_bound_inside_interval :
completed job_cost sched j_fst (a_fst + R_k) →
t1 ≤ a_fst + 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.
rewrite leqn0; apply/eqP.
try ( by apply cumulative_service_after_job_rt_zero with (job_cost0 := job_cost)
(job_arrival0 := job_arrival) (R := R_k);
try (by done); apply ltnW ) ||
by apply cumulative_service_after_job_rt_zero with (job_cost := job_cost)
(job_arrival := job_arrival) (R := R_k);
try (by done); apply ltnW.
Qed.
End FactsAboutFirstJob.
Section InterferenceSingleJob.
Hypothesis H_only_one_job: size sorted_jobs = 1.
Lemma interference_bound_edf_holds_for_a_single_job :
interference_caused_by j_fst t1 t2 ≤ interference_bound.
Proof.
unfold valid_sporadic_taskset, is_valid_sporadic_task in ×.
rename H_many_jobs into NUM,
H_valid_task_parameters into PARAMS,
H_only_one_job into SIZE.
apply leq_trans with (n := task_cost tsk_k).
{
apply interference_bound_edf_interference_le_task_cost.
rewrite interference_bound_edf_job_in_same_sequence.
by apply mem_nth; rewrite SIZE.
}
{
unfold interference_bound, edf_specific_interference_bound.
rewrite -{1}[task_cost tsk_k]mul1n.
rewrite leq_mul2r; apply/orP; right.
exploit (PARAMS tsk_i); [by done | intro PARAMSi]; des.
exploit (PARAMS tsk_k); [by done | intro PARAMSk]; des.
apply ceil_neq0; last by done.
rewrite -subnBA; last by done.
by rewrite addn1 ltnS.
}
Qed.
End InterferenceSingleJob.
Section InterferenceTwoOrMoreJobs.
Variable num_mid_jobs: nat.
Hypothesis H_at_least_two_jobs : size sorted_jobs = num_mid_jobs.+2.
Let j_lst := nth elem sorted_jobs num_mid_jobs.+1.
Let a_lst := job_arrival j_lst.
Section FactsAboutFirstAndLastJobs.
Lemma interference_bound_edf_j_lst_is_job_of_tsk_k :
arrives_in arr_seq j_lst ∧
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.
Lemma interference_bound_edf_j_lst_deadline :
job_deadline j_lst = task_deadline tsk_k.
Proof.
unfold 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 PARAMS1 LSTtask.
Qed.
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.
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.
exploit interference_bound_edf_all_jobs_from_tsk_k.
{
apply mem_nth; instantiate (1 := num_mid_jobs.+1).
by rewrite -(ltn_add2r 1) addn1 H_at_least_two_jobs addn1.
}
instantiate (1 := elem); move ⇒ [LSTarr [LSTtsk [/eqP LSTserv LSTin]]].
apply LSTserv; apply/eqP; rewrite -leqn0.
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.
try ( by apply cumulative_service_before_job_arrival_zero with (job_arrival0 := job_arrival) ) ||
by apply cumulative_service_before_job_arrival_zero with (job_arrival := job_arrival).
Qed.
Lemma interference_bound_edf_j_fst_completed_on_time :
completed job_cost sched j_fst (a_fst + R_k).
Proof.
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.
{
rewrite ltnNge; apply/negP; red; intro BUG; apply SNDserv.
apply/eqP; rewrite -leqn0; apply leq_trans with (n := service_during
sched j_snd t1 t2);
first by apply job_interference_le_service.
rewrite leqn0; apply/eqP.
try ( by apply cumulative_service_before_job_arrival_zero with (job_arrival0 := job_arrival) ) ||
by apply cumulative_service_before_job_arrival_zero with (job_arrival := job_arrival).
}
apply leq_trans with (n := a_fst + p_k).
{
by rewrite leq_add2l; apply leq_trans with (n := D_k);
[by apply H_R_k_le_deadline | by apply H_constrained_deadlines].
}
unfold p_k; rewrite -FST0.
apply H_sporadic_tasks; try (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.
Lemma interference_bound_edf_many_periods_in_between :
a_lst - a_fst ≥ num_mid_jobs.+1 × p_k.
Proof.
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.
set cur := nth elem sorted_jobs i.
set next := nth elem sorted_jobs i.+1.
assert (ARRle: job_arrival cur ≤ job_arrival next).
by unfold cur, next; apply interference_bound_edf_jobs_ordered_by_arrival.
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]]].
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.
by rewrite CURtsk.
}
by rewrite leq_subRL_impl // addnC /p_k -CURtsk.
Qed.
Lemma interference_bound_edf_slack_le_delta:
D_k - R_k ≤ D_i.
Proof.
have AFTERt1 :=
interference_bound_edf_j_fst_completion_implies_rt_bound_inside_interval
interference_bound_edf_j_fst_completed_on_time.
rewrite leq_subLR -(leq_add2r a_fst).
rewrite -addnA [R_k + _]addnC -addnA.
apply leq_trans with (n := D_i + t1);
last by rewrite leq_add2l.
have FST := interference_bound_edf_j_fst_is_job_of_tsk_k.
destruct FST as [ARRfst [_ [ LEdl _]]].
try ( apply interference_under_edf_implies_shorter_deadlines with
(arr_seq0 := arr_seq) (job_deadline0 := job_deadline) in LEdl; try (by done) ) ||
apply interference_under_edf_implies_shorter_deadlines with
(arr_seq := arr_seq) (job_deadline := job_deadline) in LEdl; try (by done).
rewrite addnC [D_i + _]addnC.
unfold D_k, D_i.
by rewrite -interference_bound_edf_j_fst_deadline
-interference_bound_edf_j_i_deadline.
Qed.
Lemma interference_bound_edf_n_k_covers_all_jobs :
n_k ≥ num_mid_jobs.+2.
Proof.
have AFTERt1 :=
interference_bound_edf_j_fst_completion_implies_rt_bound_inside_interval
interference_bound_edf_j_fst_completed_on_time.
have SLACK := interference_bound_edf_slack_le_delta.
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; rewrite ltnS; intro LTnk.
assert (BUG: a_lst - a_fst > D_i + R_k - D_k).
{
apply leq_trans with (n := num_mid_jobs.+1 × p_k); last by done.
apply leq_trans with (n := n_k × p_k);
last by rewrite leq_mul2r; apply/orP; right.
unfold n_k, div_ceil.
feed (TASK_PARAMS tsk_k); [by done | des].
destruct (p_k %| D_i + R_k - D_k + 1) eqn:DIV.
- by rewrite dvdn_eq in DIV; move: DIV ⇒ /eqP DIV; rewrite DIV addn1.
- by rewrite -addn1; apply ltnW, ltn_ceil.
}
rewrite leq_subLR in SLACK.
rewrite -(leq_add2r a_fst) addnBAC in BUG;
last by apply interference_bound_edf_j_fst_before_j_lst.
rewrite -[a_lst + _ - _]subnBA // subnn subn0 in BUG.
rewrite addnC addnS in BUG.
rewrite addnBA // in BUG; last by rewrite addnC.
rewrite -(ltn_add2r D_k) in BUG.
rewrite addnBAC in BUG; last first.
{
rewrite [D_i + R_k]addnC.
by apply leq_trans with (n := R_k + D_i);
last by apply leq_addl.
}
rewrite -addnBA // subnn addn0 in BUG.
rewrite [D_i + _]addnC addnA in BUG.
apply leq_ltn_trans with (m := t1 + D_i) in BUG;
last by rewrite leq_add2r.
have LST := interference_bound_edf_j_lst_is_job_of_tsk_k.
destruct LST as [ARRlst [_ [ LEdl _]]].
try ( apply interference_under_edf_implies_shorter_deadlines with
(arr_seq0 := arr_seq) (job_deadline0 := job_deadline) in LEdl; try (by done) ) ||
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 rewrite ltnNge LEdl in BUG.
Qed.
Lemma interference_bound_edf_holds_for_multiple_jobs :
\sum_(0 ≤ i < num_mid_jobs.+2)
interference_caused_by (nth elem sorted_jobs i) t1 t2
≤ interference_bound.
Proof.
apply leq_trans with (n := num_mid_jobs.+2 × task_cost tsk_k); last first.
{
rewrite leq_mul2r; apply/orP; right.
by apply interference_bound_edf_n_k_covers_all_jobs.
}
{
apply leq_trans with (n := \sum_(0 ≤ i < num_mid_jobs.+2) task_cost tsk_k);
last by rewrite big_const_nat iter_addn addn0 mulnC subn0.
rewrite big_nat_cond [\sum_(0 ≤ i < _ | true) _]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.
by apply mem_nth; rewrite H_at_least_two_jobs.
}
Qed.
End InterferenceTwoOrMoreJobs.
End InterferenceManyJobs.
Theorem interference_bound_edf_bounds_interference :
x ≤ interference_bound.
Proof.
apply (leq_trans interference_bound_edf_use_another_definition).
rewrite interference_bound_edf_simpl_by_filtering_interfering_jobs.
rewrite interference_bound_edf_simpl_by_sorting_interfering_jobs.
destruct (size sorted_jobs ≤ n_k) eqn:NUM;
first by apply interference_bound_edf_holds_for_at_most_n_k_jobs.
apply negbT in NUM; rewrite -ltnNge in NUM.
assert (EX: ∃ elem: Job, True).
destruct sorted_jobs as [| j]; [by rewrite ltn0 in NUM | by ∃ j].
destruct EX as [elem _].
rewrite (big_nth elem).
destruct (size sorted_jobs) as [| n] eqn:SIZE;
first by rewrite big_geq.
destruct n as [| num_mid_jobs].
{
rewrite big_nat_recr // big_geq //.
rewrite [nth]lock /= -lock add0n.
by apply interference_bound_edf_holds_for_a_single_job; rewrite SIZE.
}
{
by apply interference_bound_edf_holds_for_multiple_jobs; first by rewrite SIZE.
}
Qed.
End MainProof.
End ProofSpecificBound.
Section MonotonicitySpecificBound.
Context {sporadic_task: eqType}.
Variable task_cost: sporadic_task → time.
Variable task_period: sporadic_task → time.
Variable task_deadline: sporadic_task → time.
Variable tsk tsk_other: sporadic_task.
Hypothesis H_period_positive: task_period tsk_other > 0.
Variable delta delta' R R': time.
Hypothesis H_delta_monotonic: delta ≤ delta'.
Hypothesis H_response_time_monotonic: R ≤ R'.
Hypothesis H_cost_le_rt_bound: task_cost tsk_other ≤ R.
Lemma interference_bound_edf_monotonic :
interference_bound_edf task_cost task_period task_deadline tsk delta (tsk_other, R) ≤
interference_bound_edf task_cost task_period task_deadline tsk delta' (tsk_other, R').
Proof.
rename H_response_time_monotonic into LEr, H_delta_monotonic into LEx,
H_cost_le_rt_bound into LEcost, H_period_positive into GEperiod.
unfold interference_bound_edf, interference_bound_generic.
rewrite leq_min; apply/andP; split.
{
apply leq_trans with (n := W task_cost task_period (fst (tsk_other, R))
(snd (tsk_other, R)) delta);
[by apply geq_minl | by apply W_monotonic].
}
{
apply leq_trans with (n := edf_specific_interference_bound task_cost task_period
task_deadline tsk tsk_other R);
first by apply geq_minr.
unfold edf_specific_interference_bound; simpl.
rewrite leq_mul2r; apply/orP; right.
apply leq_divceil2r; first by done.
by rewrite leq_add2r leq_sub2r // leq_add2l.
}
Qed.
End MonotonicitySpecificBound.
End InterferenceBoundEDF.
Require Import prosa.classic.model.arrival.basic.task prosa.classic.model.arrival.basic.job 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.basic.schedule prosa.classic.model.schedule.global.basic.platform
prosa.classic.model.schedule.global.basic.interference prosa.classic.model.schedule.global.basic.interference_edf.
Require Import prosa.classic.analysis.global.parallel.workload_bound
prosa.classic.analysis.global.parallel.interference_bound.
From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq fintype bigop div path.
Module InterferenceBoundEDF.
Import Job SporadicTaskset Schedule ScheduleOfSporadicTask Schedulability
WorkloadBound ResponseTime Priority
TaskArrival Interference InterferenceEDF.
Export InterferenceBoundGeneric.
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 tsk: sporadic_task.
Variable delta: time.
Variable tsk_other: sporadic_task.
Variable R_other: time.
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
(div_ceil (d_tsk + R_other - d_other + 1) p_other) × e_other.
End SpecificBoundDef.
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 tsk: sporadic_task.
Let task_with_response_time := (sporadic_task × time)%type.
Variable R_prev: seq task_with_response_time.
Variable delta: time.
Section RecallInterferenceBounds.
Variable tsk_R: task_with_response_time.
Let tsk_other := fst tsk_R.
Let R_other := snd tsk_R.
Let basic_interference_bound := interference_bound_generic task_cost task_period delta tsk_R.
Let edf_specific_bound := edf_specific_interference_bound task_cost task_period task_deadline tsk tsk_other R_other.
Definition interference_bound_edf :=
minn basic_interference_bound edf_specific_bound.
End RecallInterferenceBounds.
Section TotalInterference.
Let other_task := different_task tsk.
Definition total_interference_bound_edf :=
\sum_((tsk_other, R_other) <- R_prev | other_task tsk_other)
interference_bound_edf (tsk_other, R_other).
End TotalInterference.
End TotalInterferenceBoundEDF.
Section ProofSpecificBound.
Import Schedule 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.
Context {Job: eqType}.
Variable job_arrival: Job → time.
Variable job_cost: Job → time.
Variable job_deadline: Job → time.
Variable job_task: Job → sporadic_task.
Context {arr_seq: arrival_sequence Job}.
Hypothesis H_sporadic_tasks:
sporadic_task_model task_period job_arrival job_task arr_seq.
Hypothesis H_valid_job_parameters:
∀ j,
arrives_in arr_seq j →
valid_sporadic_job task_cost task_deadline job_cost job_deadline job_task j.
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.
Hypothesis H_jobs_must_arrive_to_execute:
jobs_must_arrive_to_execute job_arrival sched.
Hypothesis H_completed_jobs_dont_execute:
completed_jobs_dont_execute job_cost sched.
Hypothesis H_at_least_one_cpu: num_cpus > 0.
Variable ts: taskset_of sporadic_task.
Hypothesis all_jobs_from_taskset:
∀ 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:
∀ 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.
Hypothesis H_work_conserving: work_conserving job_arrival job_cost arr_seq sched.
Hypothesis H_edf_scheduler:
respects_JLFP_policy job_arrival job_cost arr_seq sched (EDF job_arrival job_deadline).
Variable tsk_i: sporadic_task.
Hypothesis H_tsk_i_in_task_set: tsk_i \in ts.
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.
Variable tsk_k: sporadic_task.
Hypothesis H_tsk_k_in_task_set: tsk_k \in ts.
Variable R_k: time.
Hypothesis H_R_k_le_deadline: R_k ≤ task_deadline tsk_k.
Variable delta: time.
Hypothesis H_delta_le_deadline: delta ≤ task_deadline tsk_i.
Hypothesis H_all_previous_jobs_completed_on_time :
∀ j_k,
arrives_in arr_seq j_k →
job_task j_k = tsk_k →
job_arrival j_k + R_k < job_arrival j_i + delta →
completed job_cost sched j_k (job_arrival j_k + R_k).
Section MainProof.
Let x :=
task_interference job_arrival job_cost job_task sched j_i
tsk_k (job_arrival j_i) (job_arrival j_i + delta).
Let interference_bound :=
edf_specific_interference_bound task_cost task_period task_deadline tsk_i tsk_k R_k.
Let t1 := job_arrival j_i.
Let t2 := job_arrival j_i + delta.
Let D_i := task_deadline tsk_i.
Let D_k := task_deadline tsk_k.
Let p_k := task_period tsk_k.
Let n_k := div_ceil (D_i + R_k - D_k + 1) p_k.
Let interference_caused_by := job_interference job_arrival job_cost sched j_i.
Let interfering_jobs :=
filter (fun j' ⇒
(job_task j' == tsk_k) && (interference_caused_by j' t1 t2 != 0))
(jobs_scheduled_between sched t1 t2).
Let earlier_arrival := fun x y ⇒ job_arrival x ≤ job_arrival y.
Let sorted_jobs := sort earlier_arrival interfering_jobs.
Section SimplifyJobSequence.
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.
unfold x, task_interference, interference_caused_by, job_interference.
rewrite [\sum_(_ <- _ sched _ _ | _) _]exchange_big /=.
rewrite big_nat_cond [\sum_(_ ≤ _ < _ | true) _]big_nat_cond.
apply leq_sum. move ⇒ t /andP [LEt _].
rewrite exchange_big /=.
apply leq_sum; intros cpu _.
destruct (backlogged job_arrival job_cost sched j_i t) eqn:BACK;
last by rewrite andFb (eq_bigr (fun x ⇒ 0));
first by rewrite big_const_seq iter_addn mul0n addn0.
rewrite andTb.
destruct (task_scheduled_on job_task sched tsk_k cpu t) eqn:SCHED;
last by done.
unfold task_scheduled_on in ×.
destruct (sched cpu t) eqn:SOME; last by done.
rewrite big_mkcond /= (bigD1_seq s) /=; last by apply undup_uniq.
{
rewrite SCHED -addn1 addnC; apply leq_add; last by done.
apply eq_leq; symmetry; apply/eqP; rewrite eqb1.
by unfold scheduled_on; apply/eqP.
}
{
unfold jobs_scheduled_between.
rewrite mem_undup; apply mem_bigcat_nat with (j := t);
first by done.
apply mem_bigcat_ord with (j := cpu); first by apply ltn_ord.
by unfold make_sequence; rewrite SOME mem_seq1 eq_refl.
}
Qed.
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.
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).
Qed.
Lemma interference_bound_edf_job_in_same_sequence :
∀ j,
(j \in interfering_jobs) = (j \in sorted_jobs).
Proof.
by apply perm_mem; rewrite -(perm_sort earlier_arrival).
Qed.
Lemma interference_bound_edf_all_jobs_from_tsk_k :
∀ j,
j \in sorted_jobs →
arrives_in arr_seq j ∧
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).
Qed.
Lemma interference_bound_edf_jobs_ordered_by_arrival :
∀ 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.
Lemma interference_bound_edf_interference_le_task_cost :
∀ j,
j \in interfering_jobs →
interference_caused_by j t1 t2 ≤ task_cost tsk_k.
Proof.
rename H_valid_job_parameters into PARAMS.
intros j INj.
feed (interference_bound_edf_all_jobs_from_tsk_k j);
first by rewrite -interference_bound_edf_job_in_same_sequence.
move ⇒ [ARRj [TSKj _]].
specialize (PARAMS j ARRj); des.
apply leq_trans with (n := service_during sched j t1 t2);
first by apply job_interference_le_service.
try ( by apply cumulative_service_le_task_cost with (job_task0 := job_task)
(task_deadline0 := task_deadline) (job_cost0 := job_cost) (job_deadline0 := job_deadline) ) ||
by apply cumulative_service_le_task_cost with (job_task := job_task)
(task_deadline := task_deadline) (job_cost := job_cost) (job_deadline := job_deadline).
Qed.
End SimplifyJobSequence.
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.
unfold interference_bound, edf_specific_interference_bound; fold D_i p_k n_k.
apply leq_trans with (n := \sum_(x <- sorted_jobs) task_cost tsk_k);
last first.
{
rewrite big_const_seq iter_addn addn0 count_predT mulnC.
by rewrite leq_mul2r; apply/orP; right.
}
{
rewrite big_seq_cond [\sum_(_ <- _ | true)_]big_seq_cond.
apply leq_sum; move ⇒ j /andP [IN _].
apply interference_bound_edf_interference_le_task_cost.
by rewrite interference_bound_edf_job_in_same_sequence.
}
Qed.
End InterferenceFewJobs.
Section InterferenceManyJobs.
Hypothesis H_many_jobs: n_k < size sorted_jobs.
Lemma interference_bound_edf_at_least_one_job: size sorted_jobs > 0.
Proof.
by apply leq_ltn_trans with (n := n_k).
Qed.
Variable elem: Job.
Let j_fst := nth elem sorted_jobs 0.
Let a_fst := job_arrival j_fst.
Section FactsAboutFirstJob.
Lemma interference_bound_edf_j_fst_is_job_of_tsk_k :
arrives_in arr_seq j_fst ∧
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.
Lemma interference_bound_edf_j_fst_deadline :
job_deadline j_fst = task_deadline tsk_k.
Proof.
unfold 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 PARAMS1 FSTtask.
Qed.
Lemma interference_bound_edf_j_i_deadline :
job_deadline j_i = task_deadline tsk_i.
Proof.
unfold 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 PARAMS1 JOBtsk.
Qed.
Lemma interference_bound_edf_j_fst_completion_implies_rt_bound_inside_interval :
completed job_cost sched j_fst (a_fst + R_k) →
t1 ≤ a_fst + 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.
rewrite leqn0; apply/eqP.
try ( by apply cumulative_service_after_job_rt_zero with (job_cost0 := job_cost)
(job_arrival0 := job_arrival) (R := R_k);
try (by done); apply ltnW ) ||
by apply cumulative_service_after_job_rt_zero with (job_cost := job_cost)
(job_arrival := job_arrival) (R := R_k);
try (by done); apply ltnW.
Qed.
End FactsAboutFirstJob.
Section InterferenceSingleJob.
Hypothesis H_only_one_job: size sorted_jobs = 1.
Lemma interference_bound_edf_holds_for_a_single_job :
interference_caused_by j_fst t1 t2 ≤ interference_bound.
Proof.
unfold valid_sporadic_taskset, is_valid_sporadic_task in ×.
rename H_many_jobs into NUM,
H_valid_task_parameters into PARAMS,
H_only_one_job into SIZE.
apply leq_trans with (n := task_cost tsk_k).
{
apply interference_bound_edf_interference_le_task_cost.
rewrite interference_bound_edf_job_in_same_sequence.
by apply mem_nth; rewrite SIZE.
}
{
unfold interference_bound, edf_specific_interference_bound.
rewrite -{1}[task_cost tsk_k]mul1n.
rewrite leq_mul2r; apply/orP; right.
exploit (PARAMS tsk_i); [by done | intro PARAMSi]; des.
exploit (PARAMS tsk_k); [by done | intro PARAMSk]; des.
apply ceil_neq0; last by done.
rewrite -subnBA; last by done.
by rewrite addn1 ltnS.
}
Qed.
End InterferenceSingleJob.
Section InterferenceTwoOrMoreJobs.
Variable num_mid_jobs: nat.
Hypothesis H_at_least_two_jobs : size sorted_jobs = num_mid_jobs.+2.
Let j_lst := nth elem sorted_jobs num_mid_jobs.+1.
Let a_lst := job_arrival j_lst.
Section FactsAboutFirstAndLastJobs.
Lemma interference_bound_edf_j_lst_is_job_of_tsk_k :
arrives_in arr_seq j_lst ∧
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.
Lemma interference_bound_edf_j_lst_deadline :
job_deadline j_lst = task_deadline tsk_k.
Proof.
unfold 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 PARAMS1 LSTtask.
Qed.
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.
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.
exploit interference_bound_edf_all_jobs_from_tsk_k.
{
apply mem_nth; instantiate (1 := num_mid_jobs.+1).
by rewrite -(ltn_add2r 1) addn1 H_at_least_two_jobs addn1.
}
instantiate (1 := elem); move ⇒ [LSTarr [LSTtsk [/eqP LSTserv LSTin]]].
apply LSTserv; apply/eqP; rewrite -leqn0.
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.
try ( by apply cumulative_service_before_job_arrival_zero with (job_arrival0 := job_arrival) ) ||
by apply cumulative_service_before_job_arrival_zero with (job_arrival := job_arrival).
Qed.
Lemma interference_bound_edf_j_fst_completed_on_time :
completed job_cost sched j_fst (a_fst + R_k).
Proof.
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.
{
rewrite ltnNge; apply/negP; red; intro BUG; apply SNDserv.
apply/eqP; rewrite -leqn0; apply leq_trans with (n := service_during
sched j_snd t1 t2);
first by apply job_interference_le_service.
rewrite leqn0; apply/eqP.
try ( by apply cumulative_service_before_job_arrival_zero with (job_arrival0 := job_arrival) ) ||
by apply cumulative_service_before_job_arrival_zero with (job_arrival := job_arrival).
}
apply leq_trans with (n := a_fst + p_k).
{
by rewrite leq_add2l; apply leq_trans with (n := D_k);
[by apply H_R_k_le_deadline | by apply H_constrained_deadlines].
}
unfold p_k; rewrite -FST0.
apply H_sporadic_tasks; try (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.
Lemma interference_bound_edf_many_periods_in_between :
a_lst - a_fst ≥ num_mid_jobs.+1 × p_k.
Proof.
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.
set cur := nth elem sorted_jobs i.
set next := nth elem sorted_jobs i.+1.
assert (ARRle: job_arrival cur ≤ job_arrival next).
by unfold cur, next; apply interference_bound_edf_jobs_ordered_by_arrival.
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]]].
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.
by rewrite CURtsk.
}
by rewrite leq_subRL_impl // addnC /p_k -CURtsk.
Qed.
Lemma interference_bound_edf_slack_le_delta:
D_k - R_k ≤ D_i.
Proof.
have AFTERt1 :=
interference_bound_edf_j_fst_completion_implies_rt_bound_inside_interval
interference_bound_edf_j_fst_completed_on_time.
rewrite leq_subLR -(leq_add2r a_fst).
rewrite -addnA [R_k + _]addnC -addnA.
apply leq_trans with (n := D_i + t1);
last by rewrite leq_add2l.
have FST := interference_bound_edf_j_fst_is_job_of_tsk_k.
destruct FST as [ARRfst [_ [ LEdl _]]].
try ( apply interference_under_edf_implies_shorter_deadlines with
(arr_seq0 := arr_seq) (job_deadline0 := job_deadline) in LEdl; try (by done) ) ||
apply interference_under_edf_implies_shorter_deadlines with
(arr_seq := arr_seq) (job_deadline := job_deadline) in LEdl; try (by done).
rewrite addnC [D_i + _]addnC.
unfold D_k, D_i.
by rewrite -interference_bound_edf_j_fst_deadline
-interference_bound_edf_j_i_deadline.
Qed.
Lemma interference_bound_edf_n_k_covers_all_jobs :
n_k ≥ num_mid_jobs.+2.
Proof.
have AFTERt1 :=
interference_bound_edf_j_fst_completion_implies_rt_bound_inside_interval
interference_bound_edf_j_fst_completed_on_time.
have SLACK := interference_bound_edf_slack_le_delta.
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; rewrite ltnS; intro LTnk.
assert (BUG: a_lst - a_fst > D_i + R_k - D_k).
{
apply leq_trans with (n := num_mid_jobs.+1 × p_k); last by done.
apply leq_trans with (n := n_k × p_k);
last by rewrite leq_mul2r; apply/orP; right.
unfold n_k, div_ceil.
feed (TASK_PARAMS tsk_k); [by done | des].
destruct (p_k %| D_i + R_k - D_k + 1) eqn:DIV.
- by rewrite dvdn_eq in DIV; move: DIV ⇒ /eqP DIV; rewrite DIV addn1.
- by rewrite -addn1; apply ltnW, ltn_ceil.
}
rewrite leq_subLR in SLACK.
rewrite -(leq_add2r a_fst) addnBAC in BUG;
last by apply interference_bound_edf_j_fst_before_j_lst.
rewrite -[a_lst + _ - _]subnBA // subnn subn0 in BUG.
rewrite addnC addnS in BUG.
rewrite addnBA // in BUG; last by rewrite addnC.
rewrite -(ltn_add2r D_k) in BUG.
rewrite addnBAC in BUG; last first.
{
rewrite [D_i + R_k]addnC.
by apply leq_trans with (n := R_k + D_i);
last by apply leq_addl.
}
rewrite -addnBA // subnn addn0 in BUG.
rewrite [D_i + _]addnC addnA in BUG.
apply leq_ltn_trans with (m := t1 + D_i) in BUG;
last by rewrite leq_add2r.
have LST := interference_bound_edf_j_lst_is_job_of_tsk_k.
destruct LST as [ARRlst [_ [ LEdl _]]].
try ( apply interference_under_edf_implies_shorter_deadlines with
(arr_seq0 := arr_seq) (job_deadline0 := job_deadline) in LEdl; try (by done) ) ||
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 rewrite ltnNge LEdl in BUG.
Qed.
Lemma interference_bound_edf_holds_for_multiple_jobs :
\sum_(0 ≤ i < num_mid_jobs.+2)
interference_caused_by (nth elem sorted_jobs i) t1 t2
≤ interference_bound.
Proof.
apply leq_trans with (n := num_mid_jobs.+2 × task_cost tsk_k); last first.
{
rewrite leq_mul2r; apply/orP; right.
by apply interference_bound_edf_n_k_covers_all_jobs.
}
{
apply leq_trans with (n := \sum_(0 ≤ i < num_mid_jobs.+2) task_cost tsk_k);
last by rewrite big_const_nat iter_addn addn0 mulnC subn0.
rewrite big_nat_cond [\sum_(0 ≤ i < _ | true) _]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.
by apply mem_nth; rewrite H_at_least_two_jobs.
}
Qed.
End InterferenceTwoOrMoreJobs.
End InterferenceManyJobs.
Theorem interference_bound_edf_bounds_interference :
x ≤ interference_bound.
Proof.
apply (leq_trans interference_bound_edf_use_another_definition).
rewrite interference_bound_edf_simpl_by_filtering_interfering_jobs.
rewrite interference_bound_edf_simpl_by_sorting_interfering_jobs.
destruct (size sorted_jobs ≤ n_k) eqn:NUM;
first by apply interference_bound_edf_holds_for_at_most_n_k_jobs.
apply negbT in NUM; rewrite -ltnNge in NUM.
assert (EX: ∃ elem: Job, True).
destruct sorted_jobs as [| j]; [by rewrite ltn0 in NUM | by ∃ j].
destruct EX as [elem _].
rewrite (big_nth elem).
destruct (size sorted_jobs) as [| n] eqn:SIZE;
first by rewrite big_geq.
destruct n as [| num_mid_jobs].
{
rewrite big_nat_recr // big_geq //.
rewrite [nth]lock /= -lock add0n.
by apply interference_bound_edf_holds_for_a_single_job; rewrite SIZE.
}
{
by apply interference_bound_edf_holds_for_multiple_jobs; first by rewrite SIZE.
}
Qed.
End MainProof.
End ProofSpecificBound.
Section MonotonicitySpecificBound.
Context {sporadic_task: eqType}.
Variable task_cost: sporadic_task → time.
Variable task_period: sporadic_task → time.
Variable task_deadline: sporadic_task → time.
Variable tsk tsk_other: sporadic_task.
Hypothesis H_period_positive: task_period tsk_other > 0.
Variable delta delta' R R': time.
Hypothesis H_delta_monotonic: delta ≤ delta'.
Hypothesis H_response_time_monotonic: R ≤ R'.
Hypothesis H_cost_le_rt_bound: task_cost tsk_other ≤ R.
Lemma interference_bound_edf_monotonic :
interference_bound_edf task_cost task_period task_deadline tsk delta (tsk_other, R) ≤
interference_bound_edf task_cost task_period task_deadline tsk delta' (tsk_other, R').
Proof.
rename H_response_time_monotonic into LEr, H_delta_monotonic into LEx,
H_cost_le_rt_bound into LEcost, H_period_positive into GEperiod.
unfold interference_bound_edf, interference_bound_generic.
rewrite leq_min; apply/andP; split.
{
apply leq_trans with (n := W task_cost task_period (fst (tsk_other, R))
(snd (tsk_other, R)) delta);
[by apply geq_minl | by apply W_monotonic].
}
{
apply leq_trans with (n := edf_specific_interference_bound task_cost task_period
task_deadline tsk tsk_other R);
first by apply geq_minr.
unfold edf_specific_interference_bound; simpl.
rewrite leq_mul2r; apply/orP; right.
apply leq_divceil2r; first by done.
by rewrite leq_add2r leq_sub2r // leq_add2l.
}
Qed.
End MonotonicitySpecificBound.
End InterferenceBoundEDF.