Library prosa.classic.analysis.apa.interference_bound_edf
Require Import prosa.classic.util.all.
Require Import prosa.classic.model.arrival.basic.job prosa.classic.model.arrival.basic.task prosa.classic.model.priority prosa.classic.model.arrival.basic.task_arrival.
Require Import prosa.classic.model.schedule.global.workload prosa.classic.model.schedule.global.response_time
prosa.classic.model.schedule.global.schedulability.
Require Import prosa.classic.model.schedule.global.basic.schedule.
Require Import prosa.classic.model.schedule.apa.platform prosa.classic.model.schedule.apa.interference
prosa.classic.model.schedule.apa.interference_edf prosa.classic.model.schedule.apa.affinity.
Require Import prosa.classic.analysis.apa.workload_bound prosa.classic.analysis.apa.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 Affinity
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_floor d_tsk p_other) × e_other +
minn e_other ((d_tsk %% p_other) - (d_other - R_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.
Context {num_cpus: nat}.
Variable alpha: task_affinity sporadic_task num_cpus.
Let task_with_response_time := (sporadic_task × time)%type.
Variable tsk: sporadic_task.
Variable alpha': affinity num_cpus.
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 tsk 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_in alpha' := different_task_in alpha tsk alpha'.
Definition total_interference_bound_edf :=
\sum_((tsk_other, R_other) <- R_prev | other_task_in alpha' tsk_other)
interference_bound_edf (tsk_other, R_other).
End TotalInterference.
End TotalInterferenceBoundEDF.
Section ProofSpecificBound.
Import Schedule Interference Platform SporadicTaskset Affinity.
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.
Variable 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_sequential_jobs: sequential_jobs sched.
Hypothesis H_at_least_one_cpu: num_cpus > 0.
Variable alpha: task_affinity sporadic_task num_cpus.
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: apa_work_conserving job_arrival job_cost job_task arr_seq
sched alpha.
Hypothesis H_edf_weak_APA_scheduler:
respects_JLFP_policy_under_weak_APA job_arrival job_cost job_task arr_seq
sched alpha (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 alpha 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_floor D_i p_k.
Let interference_caused_by := job_interference job_arrival job_cost job_task sched alpha 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.
by apply interference_le_interference_joblist.
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 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.
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.
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.
}
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 [FSTarr [_ [ 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) (R := R_k)
(job_arrival0 := job_arrival); try (by done); apply ltnW ) ||
by apply cumulative_service_after_job_rt_zero with (job_cost := job_cost) (R := R_k)
(job_arrival := job_arrival); try (by done); apply ltnW.
Qed.
End FactsAboutFirstJob.
Section InterferenceSingleJob.
Hypothesis H_only_one_job: size sorted_jobs = 1.
Lemma interference_bound_edf_simpl_when_there's_one_job :
D_i %% p_k - (D_k - R_k) = D_i - (D_k - R_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.
Section ResponseTimeOfSingleJobBounded.
Hypothesis H_j_fst_completed_by_rt_bound :
completed job_cost sched j_fst (a_fst + R_k).
Lemma interference_bound_edf_holds_for_single_job_that_completes_on_time :
job_interference job_arrival job_cost job_task sched alpha j_i j_fst t1 t2
≤ D_i - (D_k - R_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 [FSTarr [_ [ 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).
destruct (D_k - R_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_task sched
alpha j_i j_fst (a_fst + R_k) t2).
{
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 by done.
rewrite -addnA leq_add2l.
by apply ltnW; rewrite -ltn_subRL.
}
apply leq_trans with (n := service_during sched j_fst (a_fst + R_k) t2);
first by apply job_interference_le_service.
unfold service_during; rewrite leqn0; apply/eqP.
try ( by apply cumulative_service_after_job_rt_zero with (job_cost0 := job_cost) (R := R_k)
(job_arrival0 := job_arrival); try (by done); apply leqnn ) ||
by apply cumulative_service_after_job_rt_zero with (job_cost := job_cost) (R := R_k)
(job_arrival := job_arrival); try (by done); apply leqnn.
}
{
rewrite -(leq_add2r (D_k - R_k)) addnBAC // -addnBA // subnn addn0.
assert (SUBST: D_k - R_k = \sum_(a_fst + R_k ≤ i < a_fst + D_k) 1).
{
rewrite big_const_nat iter_addn mul1n addn0.
rewrite addnC -subnBA; last by apply leq_addr.
by rewrite addnC -addnBA // subnn addn0.
}
apply leq_trans with (n := job_interference job_arrival job_cost job_task sched alpha j_i j_fst t1
(a_fst + D_k) + (D_k - R_k)).
{
rewrite leq_add2r.
destruct (t2 ≤ a_fst + R_k) eqn:LEt2.
{
apply extend_sum; first by apply leqnn.
apply leq_trans with (n := a_fst + R_k); first by done.
by rewrite leq_add2l; 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 + R_k);
[simpl | by apply AFTERt1 | by apply ltnW].
apply leq_trans with (n := job_interference job_arrival job_cost job_task sched alpha j_i j_fst t1
(a_fst + R_k) + service_during sched j_fst (a_fst + R_k) t2).
{
rewrite leq_add2l.
by apply job_interference_le_service.
}
unfold service_during.
rewrite (cumulative_service_after_job_rt_zero job_arrival job_cost _ _ _ R_k) //.
rewrite addn0; apply extend_sum; first by apply leqnn.
by rewrite leq_add2l; apply H_R_k_le_deadline.
}
}
unfold job_interference.
rewrite → big_cat_nat with (n := a_fst + R_k);
[simpl| by apply AFTERt1 | by rewrite leq_add2l; apply H_R_k_le_deadline].
apply leq_trans with (n := job_interference job_arrival job_cost job_task sched alpha j_i j_fst t1
(a_fst+R_k) + service_during sched j_fst (a_fst+R_k) (a_fst+D_k) + (D_k-R_k));
first by rewrite leq_add2r leq_add2l job_interference_le_service.
unfold service_during.
rewrite (cumulative_service_after_job_rt_zero job_arrival job_cost _ _ _ R_k) // addn0.
apply leq_trans with (n := (\sum_(t1 ≤ t < a_fst + R_k) 1) +
\sum_(a_fst + R_k ≤ t < a_fst + D_k) 1).
{
apply leq_add; last by rewrite SUBST.
rewrite big_const_nat iter_addn mul1n addn0.
rewrite -{1}[a_fst + R_k](addKn t1) -addnBA //.
by apply job_interference_le_delta.
}
rewrite -big_cat_nat;
[simpl | by apply AFTERt1 | by rewrite leq_add2l; apply H_R_k_le_deadline ].
rewrite big_const_nat iter_addn mul1n addn0 leq_subLR.
by unfold D_i, D_k, t1, a_fst; rewrite -interference_bound_edf_j_fst_deadline
-interference_bound_edf_j_i_deadline.
}
Qed.
End ResponseTimeOfSingleJobBounded.
Section ResponseTimeOfSingleJobNotBounded.
Hypothesis H_j_fst_not_complete_by_rt_bound :
~~ completed job_cost sched j_fst (a_fst + R_k).
Lemma interference_bound_edf_response_time_bound_of_j_fst_after_interval :
job_arrival j_fst + R_k ≥ job_arrival 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.
Lemma interference_bound_edf_holds_for_single_job_with_big_slack :
D_i < D_k - R_k →
interference_caused_by j_fst t1 t2 = 0.
Proof.
intro LTdk.
rewrite ltn_subRL in LTdk.
rewrite -(ltn_add2l a_fst) 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.
by apply interference_bound_edf_response_time_bound_of_j_fst_after_interval.
}
apply/eqP; rewrite -[_ _ _ _ == 0]negbK; apply/negP; red; intro BUG.
try ( apply interference_under_edf_implies_shorter_deadlines with
(arr_seq0 := arr_seq) (job_deadline0 := job_deadline) in BUG; try (by done) ) ||
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.
by apply (leq_trans LTdk) in BUG; rewrite ltnn in BUG.
Qed.
Lemma interference_bound_edf_holds_for_single_job_with_small_slack :
D_i ≥ D_k - R_k →
interference_caused_by j_fst t1 t2 ≤ D_i - (D_k - R_k).
Proof.
intro LEdk.
have FST := interference_bound_edf_j_fst_is_job_of_tsk_k.
destruct FST as [FSTarr [FSTtask [LEdl _]]].
have LTr := interference_bound_edf_response_time_bound_of_j_fst_after_interval.
apply leq_subRL_impl.
apply leq_trans with (n := job_interference job_arrival job_cost job_task sched alpha j_i j_fst t1
(job_arrival j_fst + R_k) + (D_k - R_k));
first by rewrite leq_add2r; apply extend_sum; [by apply leqnn|].
apply leq_trans with (n := \sum_(t1 ≤ t < a_fst + R_k) 1 +
\sum_(a_fst + R_k ≤ t < a_fst + D_k)1).
{
apply leq_add.
{
rewrite big_const_nat iter_addn mul1n addn0.
rewrite -{1}[job_arrival j_fst + R_k](addKn t1) -addnBA;
first by apply job_interference_le_delta.
by apply leq_trans with (n := t1 + delta); first by apply leq_addr.
}
rewrite big_const_nat iter_addn mul1n addn0 addnC.
rewrite -subnBA; last by apply leq_addr.
by rewrite addnC -addnBA // subnn addn0.
}
rewrite -big_cat_nat; simpl; last 2 first.
{
apply leq_trans with (n := t1 + delta); first by apply leq_addr.
by apply interference_bound_edf_response_time_bound_of_j_fst_after_interval.
}
by rewrite 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.
try ( by apply interference_under_edf_implies_shorter_deadlines with
(arr_seq0 := arr_seq) (job_deadline0 := job_deadline) in LEdl ) ||
by apply interference_under_edf_implies_shorter_deadlines with
(arr_seq := arr_seq) (job_deadline := job_deadline) in LEdl.
Qed.
End ResponseTimeOfSingleJobNotBounded.
Lemma interference_bound_edf_interference_of_j_fst_limited_by_slack :
interference_caused_by j_fst t1 t2 ≤ D_i - (D_k - R_k).
Proof.
destruct (completed job_cost sched j_fst (a_fst + 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)) 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.
}
by rewrite ONE; apply SLACK.
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; try 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_n_k_covers_middle_jobs_plus_one :
n_k ≥ num_mid_jobs.+1.
Proof.
have DIST := interference_bound_edf_many_periods_in_between.
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 ×.
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 D_k) -addnA [D_i + _]addnC addnA in DIST.
apply leq_ltn_trans with (m := job_arrival j_i + D_i) in DIST; last first.
{
rewrite leq_add2r; apply (leq_trans AFTERt1).
by rewrite leq_add2l; apply H_R_k_le_deadline.
}
have LST := interference_bound_edf_j_lst_is_job_of_tsk_k.
destruct LST as [LSTarr [_ [ 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 DIST.
Qed.
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.
Lemma interference_bound_edf_n_k_equals_num_mid_jobs_plus_one :
n_k = num_mid_jobs.+1.
Proof.
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.
by move: NK; rewrite leq_eqVlt orbC; move ⇒ /orP NK; des;
[by rewrite SIZE ltnS leqNgt NK in NUM | by rewrite NK].
Qed.
Section InterferenceOfFirstJob.
Lemma interference_bound_edf_remainder_ge_slack :
D_k - R_k ≤ D_i %% p_k.
Proof.
have AFTERt1 :=
interference_bound_edf_j_fst_completion_implies_rt_bound_inside_interval
interference_bound_edf_j_fst_completed_on_time.
have NK := interference_bound_edf_n_k_equals_num_mid_jobs_plus_one.
have DIST := interference_bound_edf_many_periods_in_between.
rewrite -NK in DIST.
rewrite -subndiv_eq_mod leq_subLR.
fold (div_floor D_i p_k) n_k.
rewrite addnBA; last by apply leq_trunc_div.
apply leq_trans with (n := R_k + D_i - (a_lst - a_fst)); last by apply leq_sub2l.
rewrite subnBA; last by apply interference_bound_edf_j_fst_before_j_lst.
rewrite -(leq_add2r a_lst) addnBAC; last first.
{
apply leq_trans with (n := t2);
[by apply ltnW, interference_bound_edf_last_job_arrives_before_end_of_interval|].
rewrite addnC addnA.
apply leq_trans with (n := t1 + D_i).
unfold t2; rewrite leq_add2l; apply H_delta_le_deadline.
by rewrite leq_add2r; apply AFTERt1.
}
rewrite -addnBA // subnn addn0 [D_k + _]addnC.
apply leq_trans with (n := t1 + D_i);
last by rewrite -addnA [D_i + _]addnC addnA leq_add2r addnC AFTERt1.
have LST := interference_bound_edf_j_lst_is_job_of_tsk_k.
destruct LST as [LSTarr [_ [ LSTserv _]]].
unfold D_i, D_k, a_lst, t1; rewrite -interference_bound_edf_j_lst_deadline
-interference_bound_edf_j_i_deadline.
try ( by apply interference_under_edf_implies_shorter_deadlines with
(arr_seq0 := arr_seq) (job_deadline0 := job_deadline) in LSTserv ) ||
by apply interference_under_edf_implies_shorter_deadlines with
(arr_seq := arr_seq) (job_deadline := job_deadline) in LSTserv.
Qed.
Lemma interference_bound_edf_simpl_by_moving_to_left_side :
interference_caused_by j_fst t1 t2 + (D_k - R_k) + D_i %/ p_k × p_k ≤ D_i →
interference_caused_by j_fst t1 t2 ≤ D_i %% p_k - (D_k - R_k).
Proof.
intro LE.
apply leq_subRL_impl.
by rewrite -subndiv_eq_mod; apply leq_subRL_impl.
Qed.
Lemma interference_bound_edf_interference_of_j_fst_bounded_by_response_time :
interference_caused_by j_fst t1 t2 ≤ \sum_(t1 ≤ t < a_fst + R_k) 1.
Proof.
assert (AFTERt1: t1 ≤ a_fst + R_k).
{
apply interference_bound_edf_j_fst_completion_implies_rt_bound_inside_interval.
by apply interference_bound_edf_j_fst_completed_on_time.
}
destruct (leqP t2 (a_fst + R_k)) as [LEt2 | GTt2].
{
apply leq_trans with (n := job_interference job_arrival job_cost job_task sched alpha j_i j_fst t1
(a_fst + R_k));
first by apply extend_sum; rewrite ?leqnn.
simpl_sum_const; rewrite -{1}[_ + R_k](addKn t1) -addnBA //.
by apply job_interference_le_delta.
}
{
unfold interference_caused_by, job_interference.
rewrite → big_cat_nat with (n := a_fst + R_k);
[simpl | by apply AFTERt1 | by apply ltnW].
rewrite -[\sum_(_ ≤ _ < _) 1]addn0; apply leq_add.
{
simpl_sum_const; rewrite -{1}[_ + R_k](addKn t1) -addnBA //.
by apply job_interference_le_delta.
}
apply leq_trans with (n := service_during sched j_fst (a_fst + R_k) t2);
first by apply job_interference_le_service.
rewrite leqn0; apply/eqP.
(try ( apply cumulative_service_after_job_rt_zero with (job_cost0 := job_cost) (R := R_k)
(job_arrival0 := job_arrival) ) ||
apply cumulative_service_after_job_rt_zero with (job_cost := job_cost) (R := R_k)
(job_arrival := job_arrival)); [ by done | | by apply leqnn].
by apply interference_bound_edf_j_fst_completed_on_time.
}
Qed.
Lemma interference_bound_edf_bounding_interference_with_interval_lengths :
interference_caused_by j_fst t1 t2 + (D_k - R_k) + D_i %/ p_k × p_k ≤
\sum_(t1 ≤ t < a_fst + R_k) 1
+ \sum_(a_fst + R_k ≤ t < a_fst + D_k) 1
+ \sum_(a_fst + D_k ≤ t < a_lst + D_k) 1.
Proof.
apply leq_trans with (n := \sum_(t1 ≤ t < a_fst + R_k) 1 + (D_k - R_k) +
D_i %/ p_k × p_k).
{
rewrite 2!leq_add2r.
apply interference_bound_edf_interference_of_j_fst_bounded_by_response_time.
}
apply leq_trans with (n := \sum_(t1 ≤ t < a_fst + R_k) 1 + (D_k - R_k) +
(a_lst - a_fst)).
{
rewrite leq_add2l; fold (div_floor D_i p_k) n_k.
rewrite interference_bound_edf_n_k_equals_num_mid_jobs_plus_one.
by apply interference_bound_edf_many_periods_in_between.
}
apply leq_trans with (n := \sum_(t1 ≤ t < a_fst + R_k) 1 +
\sum_(a_fst + R_k ≤ t < a_fst + D_k) 1 + \sum_(a_fst + D_k ≤ t < a_lst + D_k) 1).
{
by rewrite -2!addnA leq_add2l; apply leq_add;
rewrite big_const_nat iter_addn mul1n addn0;
rewrite ?subnDl ?subnDr leqnn.
}
by apply leqnn.
Qed.
Lemma interference_bound_edf_simpl_by_concatenation_of_intervals :
\sum_(t1 ≤ t < a_fst + R_k) 1
+ \sum_(a_fst + R_k ≤ t < a_fst + D_k) 1
+ \sum_(a_fst + D_k ≤ t < a_lst + D_k) 1 = (a_lst + D_k) - t1.
Proof.
assert (AFTERt1: t1 ≤ a_fst + R_k).
{
apply interference_bound_edf_j_fst_completion_implies_rt_bound_inside_interval.
by apply interference_bound_edf_j_fst_completed_on_time.
}
rewrite -big_cat_nat;
[simpl | by apply AFTERt1 | by rewrite leq_add2l; apply H_R_k_le_deadline].
rewrite -big_cat_nat; simpl; last 2 first.
{
apply leq_trans with (n := a_fst + R_k); first by apply AFTERt1.
by rewrite leq_add2l; apply H_R_k_le_deadline.
}
{
rewrite leq_add2r; unfold a_fst, a_lst, j_fst, j_lst.
rewrite -[num_mid_jobs.+1]add0n; apply prev_le_next;
last by rewrite add0n H_at_least_two_jobs ltnSn.
by ins; apply interference_bound_edf_jobs_ordered_by_arrival.
}
by rewrite big_const_nat iter_addn mul1n addn0.
Qed.
Lemma interference_bound_edf_interference_of_j_fst_limited_by_remainder_and_slack :
interference_caused_by j_fst t1 t2 ≤ D_i %% p_k - (D_k - R_k).
Proof.
apply interference_bound_edf_simpl_by_moving_to_left_side.
apply (leq_trans interference_bound_edf_bounding_interference_with_interval_lengths).
rewrite interference_bound_edf_simpl_by_concatenation_of_intervals leq_subLR.
have LST := interference_bound_edf_j_lst_is_job_of_tsk_k.
destruct LST as [LSTarr [_ [ LSTserv _]]].
unfold D_i, D_k, a_lst, t1; rewrite -interference_bound_edf_j_lst_deadline
-interference_bound_edf_j_i_deadline.
try ( by apply interference_under_edf_implies_shorter_deadlines
with (arr_seq0 := arr_seq) (job_deadline0 := job_deadline) in LSTserv ) ||
by apply interference_under_edf_implies_shorter_deadlines
with (arr_seq := arr_seq) (job_deadline := job_deadline) in LSTserv.
Qed.
End InterferenceOfFirstJob.
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.
rewrite [nth]lock big_nat_recl // big_nat_recr // /= -lock.
rewrite addnA addnC addnA.
have NK := interference_bound_edf_n_k_equals_num_mid_jobs_plus_one.
unfold interference_bound, edf_specific_interference_bound.
fold D_i D_k p_k n_k.
rewrite addnC addnA; apply leq_add;
first by rewrite addnC interference_bound_edf_holds_for_middle_and_last_jobs.
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 H_at_least_two_jobs.
}
by apply interference_bound_edf_interference_of_j_fst_limited_by_remainder_and_slack.
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.
{
rewrite leq_min; apply/andP; split.
apply leq_trans with (n := (minn (W task_cost task_period (fst (tsk_other, R))
(snd (tsk_other, R)) delta) (delta - task_cost tsk + 1)));
first by apply geq_minl.
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 := minn (W task_cost task_period (fst (tsk_other, R)) (snd (tsk_other, R)) delta) (delta - task_cost tsk + 1));
first by apply geq_minl.
apply leq_trans with (n := delta - task_cost tsk + 1);
first by apply geq_minr.
by rewrite leq_add2r leq_sub2r.
}
{
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_add2l leq_min; apply/andP; split; first by apply geq_minl.
apply leq_trans with (n := task_deadline tsk %% task_period tsk_other -
(task_deadline tsk_other - R));
[by apply geq_minr | by rewrite 2?leq_sub2l 2?leq_sub2r // leq_sub2l].
}
Qed.
End MonotonicitySpecificBound.
End InterferenceBoundEDF.
Require Import prosa.classic.model.arrival.basic.job prosa.classic.model.arrival.basic.task prosa.classic.model.priority prosa.classic.model.arrival.basic.task_arrival.
Require Import prosa.classic.model.schedule.global.workload prosa.classic.model.schedule.global.response_time
prosa.classic.model.schedule.global.schedulability.
Require Import prosa.classic.model.schedule.global.basic.schedule.
Require Import prosa.classic.model.schedule.apa.platform prosa.classic.model.schedule.apa.interference
prosa.classic.model.schedule.apa.interference_edf prosa.classic.model.schedule.apa.affinity.
Require Import prosa.classic.analysis.apa.workload_bound prosa.classic.analysis.apa.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 Affinity
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_floor d_tsk p_other) × e_other +
minn e_other ((d_tsk %% p_other) - (d_other - R_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.
Context {num_cpus: nat}.
Variable alpha: task_affinity sporadic_task num_cpus.
Let task_with_response_time := (sporadic_task × time)%type.
Variable tsk: sporadic_task.
Variable alpha': affinity num_cpus.
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 tsk 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_in alpha' := different_task_in alpha tsk alpha'.
Definition total_interference_bound_edf :=
\sum_((tsk_other, R_other) <- R_prev | other_task_in alpha' tsk_other)
interference_bound_edf (tsk_other, R_other).
End TotalInterference.
End TotalInterferenceBoundEDF.
Section ProofSpecificBound.
Import Schedule Interference Platform SporadicTaskset Affinity.
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.
Variable 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_sequential_jobs: sequential_jobs sched.
Hypothesis H_at_least_one_cpu: num_cpus > 0.
Variable alpha: task_affinity sporadic_task num_cpus.
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: apa_work_conserving job_arrival job_cost job_task arr_seq
sched alpha.
Hypothesis H_edf_weak_APA_scheduler:
respects_JLFP_policy_under_weak_APA job_arrival job_cost job_task arr_seq
sched alpha (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 alpha 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_floor D_i p_k.
Let interference_caused_by := job_interference job_arrival job_cost job_task sched alpha 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.
by apply interference_le_interference_joblist.
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 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.
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.
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.
}
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 [FSTarr [_ [ 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) (R := R_k)
(job_arrival0 := job_arrival); try (by done); apply ltnW ) ||
by apply cumulative_service_after_job_rt_zero with (job_cost := job_cost) (R := R_k)
(job_arrival := job_arrival); try (by done); apply ltnW.
Qed.
End FactsAboutFirstJob.
Section InterferenceSingleJob.
Hypothesis H_only_one_job: size sorted_jobs = 1.
Lemma interference_bound_edf_simpl_when_there's_one_job :
D_i %% p_k - (D_k - R_k) = D_i - (D_k - R_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.
Section ResponseTimeOfSingleJobBounded.
Hypothesis H_j_fst_completed_by_rt_bound :
completed job_cost sched j_fst (a_fst + R_k).
Lemma interference_bound_edf_holds_for_single_job_that_completes_on_time :
job_interference job_arrival job_cost job_task sched alpha j_i j_fst t1 t2
≤ D_i - (D_k - R_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 [FSTarr [_ [ 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).
destruct (D_k - R_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_task sched
alpha j_i j_fst (a_fst + R_k) t2).
{
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 by done.
rewrite -addnA leq_add2l.
by apply ltnW; rewrite -ltn_subRL.
}
apply leq_trans with (n := service_during sched j_fst (a_fst + R_k) t2);
first by apply job_interference_le_service.
unfold service_during; rewrite leqn0; apply/eqP.
try ( by apply cumulative_service_after_job_rt_zero with (job_cost0 := job_cost) (R := R_k)
(job_arrival0 := job_arrival); try (by done); apply leqnn ) ||
by apply cumulative_service_after_job_rt_zero with (job_cost := job_cost) (R := R_k)
(job_arrival := job_arrival); try (by done); apply leqnn.
}
{
rewrite -(leq_add2r (D_k - R_k)) addnBAC // -addnBA // subnn addn0.
assert (SUBST: D_k - R_k = \sum_(a_fst + R_k ≤ i < a_fst + D_k) 1).
{
rewrite big_const_nat iter_addn mul1n addn0.
rewrite addnC -subnBA; last by apply leq_addr.
by rewrite addnC -addnBA // subnn addn0.
}
apply leq_trans with (n := job_interference job_arrival job_cost job_task sched alpha j_i j_fst t1
(a_fst + D_k) + (D_k - R_k)).
{
rewrite leq_add2r.
destruct (t2 ≤ a_fst + R_k) eqn:LEt2.
{
apply extend_sum; first by apply leqnn.
apply leq_trans with (n := a_fst + R_k); first by done.
by rewrite leq_add2l; 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 + R_k);
[simpl | by apply AFTERt1 | by apply ltnW].
apply leq_trans with (n := job_interference job_arrival job_cost job_task sched alpha j_i j_fst t1
(a_fst + R_k) + service_during sched j_fst (a_fst + R_k) t2).
{
rewrite leq_add2l.
by apply job_interference_le_service.
}
unfold service_during.
rewrite (cumulative_service_after_job_rt_zero job_arrival job_cost _ _ _ R_k) //.
rewrite addn0; apply extend_sum; first by apply leqnn.
by rewrite leq_add2l; apply H_R_k_le_deadline.
}
}
unfold job_interference.
rewrite → big_cat_nat with (n := a_fst + R_k);
[simpl| by apply AFTERt1 | by rewrite leq_add2l; apply H_R_k_le_deadline].
apply leq_trans with (n := job_interference job_arrival job_cost job_task sched alpha j_i j_fst t1
(a_fst+R_k) + service_during sched j_fst (a_fst+R_k) (a_fst+D_k) + (D_k-R_k));
first by rewrite leq_add2r leq_add2l job_interference_le_service.
unfold service_during.
rewrite (cumulative_service_after_job_rt_zero job_arrival job_cost _ _ _ R_k) // addn0.
apply leq_trans with (n := (\sum_(t1 ≤ t < a_fst + R_k) 1) +
\sum_(a_fst + R_k ≤ t < a_fst + D_k) 1).
{
apply leq_add; last by rewrite SUBST.
rewrite big_const_nat iter_addn mul1n addn0.
rewrite -{1}[a_fst + R_k](addKn t1) -addnBA //.
by apply job_interference_le_delta.
}
rewrite -big_cat_nat;
[simpl | by apply AFTERt1 | by rewrite leq_add2l; apply H_R_k_le_deadline ].
rewrite big_const_nat iter_addn mul1n addn0 leq_subLR.
by unfold D_i, D_k, t1, a_fst; rewrite -interference_bound_edf_j_fst_deadline
-interference_bound_edf_j_i_deadline.
}
Qed.
End ResponseTimeOfSingleJobBounded.
Section ResponseTimeOfSingleJobNotBounded.
Hypothesis H_j_fst_not_complete_by_rt_bound :
~~ completed job_cost sched j_fst (a_fst + R_k).
Lemma interference_bound_edf_response_time_bound_of_j_fst_after_interval :
job_arrival j_fst + R_k ≥ job_arrival 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.
Lemma interference_bound_edf_holds_for_single_job_with_big_slack :
D_i < D_k - R_k →
interference_caused_by j_fst t1 t2 = 0.
Proof.
intro LTdk.
rewrite ltn_subRL in LTdk.
rewrite -(ltn_add2l a_fst) 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.
by apply interference_bound_edf_response_time_bound_of_j_fst_after_interval.
}
apply/eqP; rewrite -[_ _ _ _ == 0]negbK; apply/negP; red; intro BUG.
try ( apply interference_under_edf_implies_shorter_deadlines with
(arr_seq0 := arr_seq) (job_deadline0 := job_deadline) in BUG; try (by done) ) ||
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.
by apply (leq_trans LTdk) in BUG; rewrite ltnn in BUG.
Qed.
Lemma interference_bound_edf_holds_for_single_job_with_small_slack :
D_i ≥ D_k - R_k →
interference_caused_by j_fst t1 t2 ≤ D_i - (D_k - R_k).
Proof.
intro LEdk.
have FST := interference_bound_edf_j_fst_is_job_of_tsk_k.
destruct FST as [FSTarr [FSTtask [LEdl _]]].
have LTr := interference_bound_edf_response_time_bound_of_j_fst_after_interval.
apply leq_subRL_impl.
apply leq_trans with (n := job_interference job_arrival job_cost job_task sched alpha j_i j_fst t1
(job_arrival j_fst + R_k) + (D_k - R_k));
first by rewrite leq_add2r; apply extend_sum; [by apply leqnn|].
apply leq_trans with (n := \sum_(t1 ≤ t < a_fst + R_k) 1 +
\sum_(a_fst + R_k ≤ t < a_fst + D_k)1).
{
apply leq_add.
{
rewrite big_const_nat iter_addn mul1n addn0.
rewrite -{1}[job_arrival j_fst + R_k](addKn t1) -addnBA;
first by apply job_interference_le_delta.
by apply leq_trans with (n := t1 + delta); first by apply leq_addr.
}
rewrite big_const_nat iter_addn mul1n addn0 addnC.
rewrite -subnBA; last by apply leq_addr.
by rewrite addnC -addnBA // subnn addn0.
}
rewrite -big_cat_nat; simpl; last 2 first.
{
apply leq_trans with (n := t1 + delta); first by apply leq_addr.
by apply interference_bound_edf_response_time_bound_of_j_fst_after_interval.
}
by rewrite 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.
try ( by apply interference_under_edf_implies_shorter_deadlines with
(arr_seq0 := arr_seq) (job_deadline0 := job_deadline) in LEdl ) ||
by apply interference_under_edf_implies_shorter_deadlines with
(arr_seq := arr_seq) (job_deadline := job_deadline) in LEdl.
Qed.
End ResponseTimeOfSingleJobNotBounded.
Lemma interference_bound_edf_interference_of_j_fst_limited_by_slack :
interference_caused_by j_fst t1 t2 ≤ D_i - (D_k - R_k).
Proof.
destruct (completed job_cost sched j_fst (a_fst + 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)) 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.
}
by rewrite ONE; apply SLACK.
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; try 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_n_k_covers_middle_jobs_plus_one :
n_k ≥ num_mid_jobs.+1.
Proof.
have DIST := interference_bound_edf_many_periods_in_between.
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 ×.
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 D_k) -addnA [D_i + _]addnC addnA in DIST.
apply leq_ltn_trans with (m := job_arrival j_i + D_i) in DIST; last first.
{
rewrite leq_add2r; apply (leq_trans AFTERt1).
by rewrite leq_add2l; apply H_R_k_le_deadline.
}
have LST := interference_bound_edf_j_lst_is_job_of_tsk_k.
destruct LST as [LSTarr [_ [ 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 DIST.
Qed.
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.
Lemma interference_bound_edf_n_k_equals_num_mid_jobs_plus_one :
n_k = num_mid_jobs.+1.
Proof.
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.
by move: NK; rewrite leq_eqVlt orbC; move ⇒ /orP NK; des;
[by rewrite SIZE ltnS leqNgt NK in NUM | by rewrite NK].
Qed.
Section InterferenceOfFirstJob.
Lemma interference_bound_edf_remainder_ge_slack :
D_k - R_k ≤ D_i %% p_k.
Proof.
have AFTERt1 :=
interference_bound_edf_j_fst_completion_implies_rt_bound_inside_interval
interference_bound_edf_j_fst_completed_on_time.
have NK := interference_bound_edf_n_k_equals_num_mid_jobs_plus_one.
have DIST := interference_bound_edf_many_periods_in_between.
rewrite -NK in DIST.
rewrite -subndiv_eq_mod leq_subLR.
fold (div_floor D_i p_k) n_k.
rewrite addnBA; last by apply leq_trunc_div.
apply leq_trans with (n := R_k + D_i - (a_lst - a_fst)); last by apply leq_sub2l.
rewrite subnBA; last by apply interference_bound_edf_j_fst_before_j_lst.
rewrite -(leq_add2r a_lst) addnBAC; last first.
{
apply leq_trans with (n := t2);
[by apply ltnW, interference_bound_edf_last_job_arrives_before_end_of_interval|].
rewrite addnC addnA.
apply leq_trans with (n := t1 + D_i).
unfold t2; rewrite leq_add2l; apply H_delta_le_deadline.
by rewrite leq_add2r; apply AFTERt1.
}
rewrite -addnBA // subnn addn0 [D_k + _]addnC.
apply leq_trans with (n := t1 + D_i);
last by rewrite -addnA [D_i + _]addnC addnA leq_add2r addnC AFTERt1.
have LST := interference_bound_edf_j_lst_is_job_of_tsk_k.
destruct LST as [LSTarr [_ [ LSTserv _]]].
unfold D_i, D_k, a_lst, t1; rewrite -interference_bound_edf_j_lst_deadline
-interference_bound_edf_j_i_deadline.
try ( by apply interference_under_edf_implies_shorter_deadlines with
(arr_seq0 := arr_seq) (job_deadline0 := job_deadline) in LSTserv ) ||
by apply interference_under_edf_implies_shorter_deadlines with
(arr_seq := arr_seq) (job_deadline := job_deadline) in LSTserv.
Qed.
Lemma interference_bound_edf_simpl_by_moving_to_left_side :
interference_caused_by j_fst t1 t2 + (D_k - R_k) + D_i %/ p_k × p_k ≤ D_i →
interference_caused_by j_fst t1 t2 ≤ D_i %% p_k - (D_k - R_k).
Proof.
intro LE.
apply leq_subRL_impl.
by rewrite -subndiv_eq_mod; apply leq_subRL_impl.
Qed.
Lemma interference_bound_edf_interference_of_j_fst_bounded_by_response_time :
interference_caused_by j_fst t1 t2 ≤ \sum_(t1 ≤ t < a_fst + R_k) 1.
Proof.
assert (AFTERt1: t1 ≤ a_fst + R_k).
{
apply interference_bound_edf_j_fst_completion_implies_rt_bound_inside_interval.
by apply interference_bound_edf_j_fst_completed_on_time.
}
destruct (leqP t2 (a_fst + R_k)) as [LEt2 | GTt2].
{
apply leq_trans with (n := job_interference job_arrival job_cost job_task sched alpha j_i j_fst t1
(a_fst + R_k));
first by apply extend_sum; rewrite ?leqnn.
simpl_sum_const; rewrite -{1}[_ + R_k](addKn t1) -addnBA //.
by apply job_interference_le_delta.
}
{
unfold interference_caused_by, job_interference.
rewrite → big_cat_nat with (n := a_fst + R_k);
[simpl | by apply AFTERt1 | by apply ltnW].
rewrite -[\sum_(_ ≤ _ < _) 1]addn0; apply leq_add.
{
simpl_sum_const; rewrite -{1}[_ + R_k](addKn t1) -addnBA //.
by apply job_interference_le_delta.
}
apply leq_trans with (n := service_during sched j_fst (a_fst + R_k) t2);
first by apply job_interference_le_service.
rewrite leqn0; apply/eqP.
(try ( apply cumulative_service_after_job_rt_zero with (job_cost0 := job_cost) (R := R_k)
(job_arrival0 := job_arrival) ) ||
apply cumulative_service_after_job_rt_zero with (job_cost := job_cost) (R := R_k)
(job_arrival := job_arrival)); [ by done | | by apply leqnn].
by apply interference_bound_edf_j_fst_completed_on_time.
}
Qed.
Lemma interference_bound_edf_bounding_interference_with_interval_lengths :
interference_caused_by j_fst t1 t2 + (D_k - R_k) + D_i %/ p_k × p_k ≤
\sum_(t1 ≤ t < a_fst + R_k) 1
+ \sum_(a_fst + R_k ≤ t < a_fst + D_k) 1
+ \sum_(a_fst + D_k ≤ t < a_lst + D_k) 1.
Proof.
apply leq_trans with (n := \sum_(t1 ≤ t < a_fst + R_k) 1 + (D_k - R_k) +
D_i %/ p_k × p_k).
{
rewrite 2!leq_add2r.
apply interference_bound_edf_interference_of_j_fst_bounded_by_response_time.
}
apply leq_trans with (n := \sum_(t1 ≤ t < a_fst + R_k) 1 + (D_k - R_k) +
(a_lst - a_fst)).
{
rewrite leq_add2l; fold (div_floor D_i p_k) n_k.
rewrite interference_bound_edf_n_k_equals_num_mid_jobs_plus_one.
by apply interference_bound_edf_many_periods_in_between.
}
apply leq_trans with (n := \sum_(t1 ≤ t < a_fst + R_k) 1 +
\sum_(a_fst + R_k ≤ t < a_fst + D_k) 1 + \sum_(a_fst + D_k ≤ t < a_lst + D_k) 1).
{
by rewrite -2!addnA leq_add2l; apply leq_add;
rewrite big_const_nat iter_addn mul1n addn0;
rewrite ?subnDl ?subnDr leqnn.
}
by apply leqnn.
Qed.
Lemma interference_bound_edf_simpl_by_concatenation_of_intervals :
\sum_(t1 ≤ t < a_fst + R_k) 1
+ \sum_(a_fst + R_k ≤ t < a_fst + D_k) 1
+ \sum_(a_fst + D_k ≤ t < a_lst + D_k) 1 = (a_lst + D_k) - t1.
Proof.
assert (AFTERt1: t1 ≤ a_fst + R_k).
{
apply interference_bound_edf_j_fst_completion_implies_rt_bound_inside_interval.
by apply interference_bound_edf_j_fst_completed_on_time.
}
rewrite -big_cat_nat;
[simpl | by apply AFTERt1 | by rewrite leq_add2l; apply H_R_k_le_deadline].
rewrite -big_cat_nat; simpl; last 2 first.
{
apply leq_trans with (n := a_fst + R_k); first by apply AFTERt1.
by rewrite leq_add2l; apply H_R_k_le_deadline.
}
{
rewrite leq_add2r; unfold a_fst, a_lst, j_fst, j_lst.
rewrite -[num_mid_jobs.+1]add0n; apply prev_le_next;
last by rewrite add0n H_at_least_two_jobs ltnSn.
by ins; apply interference_bound_edf_jobs_ordered_by_arrival.
}
by rewrite big_const_nat iter_addn mul1n addn0.
Qed.
Lemma interference_bound_edf_interference_of_j_fst_limited_by_remainder_and_slack :
interference_caused_by j_fst t1 t2 ≤ D_i %% p_k - (D_k - R_k).
Proof.
apply interference_bound_edf_simpl_by_moving_to_left_side.
apply (leq_trans interference_bound_edf_bounding_interference_with_interval_lengths).
rewrite interference_bound_edf_simpl_by_concatenation_of_intervals leq_subLR.
have LST := interference_bound_edf_j_lst_is_job_of_tsk_k.
destruct LST as [LSTarr [_ [ LSTserv _]]].
unfold D_i, D_k, a_lst, t1; rewrite -interference_bound_edf_j_lst_deadline
-interference_bound_edf_j_i_deadline.
try ( by apply interference_under_edf_implies_shorter_deadlines
with (arr_seq0 := arr_seq) (job_deadline0 := job_deadline) in LSTserv ) ||
by apply interference_under_edf_implies_shorter_deadlines
with (arr_seq := arr_seq) (job_deadline := job_deadline) in LSTserv.
Qed.
End InterferenceOfFirstJob.
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.
rewrite [nth]lock big_nat_recl // big_nat_recr // /= -lock.
rewrite addnA addnC addnA.
have NK := interference_bound_edf_n_k_equals_num_mid_jobs_plus_one.
unfold interference_bound, edf_specific_interference_bound.
fold D_i D_k p_k n_k.
rewrite addnC addnA; apply leq_add;
first by rewrite addnC interference_bound_edf_holds_for_middle_and_last_jobs.
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 H_at_least_two_jobs.
}
by apply interference_bound_edf_interference_of_j_fst_limited_by_remainder_and_slack.
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.
{
rewrite leq_min; apply/andP; split.
apply leq_trans with (n := (minn (W task_cost task_period (fst (tsk_other, R))
(snd (tsk_other, R)) delta) (delta - task_cost tsk + 1)));
first by apply geq_minl.
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 := minn (W task_cost task_period (fst (tsk_other, R)) (snd (tsk_other, R)) delta) (delta - task_cost tsk + 1));
first by apply geq_minl.
apply leq_trans with (n := delta - task_cost tsk + 1);
first by apply geq_minr.
by rewrite leq_add2r leq_sub2r.
}
{
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_add2l leq_min; apply/andP; split; first by apply geq_minl.
apply leq_trans with (n := task_deadline tsk %% task_period tsk_other -
(task_deadline tsk_other - R));
[by apply geq_minr | by rewrite 2?leq_sub2l 2?leq_sub2r // leq_sub2l].
}
Qed.
End MonotonicitySpecificBound.
End InterferenceBoundEDF.