Library prosa.classic.analysis.global.basic.bertogna_fp_theory
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.workload prosa.classic.model.schedule.global.schedulability
prosa.classic.model.schedule.global.response_time.
Require Import prosa.classic.model.schedule.global.basic.schedule prosa.classic.model.schedule.global.basic.platform
prosa.classic.model.schedule.global.basic.constrained_deadlines prosa.classic.model.schedule.global.basic.interference.
Require Import prosa.classic.analysis.global.basic.workload_bound
prosa.classic.analysis.global.basic.interference_bound_fp.
From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq fintype bigop div path.
Module ResponseTimeAnalysisFP.
Export Job SporadicTaskset ScheduleOfSporadicTask Workload Interference
InterferenceBoundFP Platform Schedulability ResponseTime
Priority TaskArrival WorkloadBound ConstrainedDeadlines.
Section ResponseTimeBound.
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 ts: taskset_of sporadic_task.
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.
Hypothesis H_all_jobs_from_taskset:
∀ j, arrives_in arr_seq j → job_task j \in ts.
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_sequential_jobs: sequential_jobs sched.
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 higher_eq_priority: FP_policy sporadic_task.
Hypothesis H_work_conserving: work_conserving job_arrival job_cost arr_seq sched.
Hypothesis H_respects_FP_policy:
respects_FP_policy job_arrival job_cost job_task arr_seq sched higher_eq_priority.
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.
Variable tsk: sporadic_task.
Hypothesis task_in_ts: tsk \in ts.
Let is_hp_task := higher_priority_task higher_eq_priority tsk.
Let task_with_response_time := (sporadic_task × time)%type.
Variable hp_bounds: seq task_with_response_time.
Hypothesis H_response_time_of_interfering_tasks_is_known:
∀ hp_tsk R,
(hp_tsk, R) \in hp_bounds →
response_time_bounded_by hp_tsk R.
Hypothesis H_hp_bounds_has_interfering_tasks:
∀ hp_tsk,
hp_tsk \in ts →
is_hp_task hp_tsk →
∃ R, (hp_tsk, R) \in hp_bounds.
Hypothesis H_response_time_bounds_ge_cost:
∀ hp_tsk R,
(hp_tsk, R) \in hp_bounds → R ≥ task_cost hp_tsk.
Hypothesis H_interfering_tasks_miss_no_deadlines:
∀ hp_tsk R,
(hp_tsk, R) \in hp_bounds → R ≤ task_deadline hp_tsk.
Variable R: time.
Hypothesis H_response_time_recurrence_holds :
R = task_cost tsk +
div_floor
(total_interference_bound_fp task_cost task_period tsk hp_bounds R)
num_cpus.
Hypothesis H_response_time_no_larger_than_deadline:
R ≤ task_deadline tsk.
Section Lemmas.
Variable j: Job.
Hypothesis H_j_arrives: arrives_in arr_seq j.
Hypothesis H_job_of_tsk: job_task j = tsk.
Hypothesis H_j_not_completed: ~~ completed job_cost sched j (job_arrival j + R).
Hypothesis H_previous_jobs_of_tsk_completed :
∀ j0,
arrives_in arr_seq j0 →
job_task j0 = tsk →
job_arrival j0 < job_arrival j →
completed job_cost sched j0 (job_arrival j0 + R).
Let x (tsk_other: sporadic_task) :=
task_interference job_arrival job_cost job_task sched j tsk_other
(job_arrival j) (job_arrival j + R).
Let X := total_interference job_arrival job_cost sched j (job_arrival j) (job_arrival j + R).
Let workload_bound (tsk_other: sporadic_task) (R_other: time) :=
W task_cost task_period tsk_other R_other R.
Let hp_tasks := [seq tsk_other <- ts | is_hp_task tsk_other].
Section LemmasAboutHPTasks.
Variable tsk_other: sporadic_task.
Variable R_other: time.
Hypothesis H_response_time_of_tsk_other: (tsk_other, R_other) \in hp_bounds.
Lemma bertogna_fp_workload_bounds_interference :
x tsk_other ≤ workload_bound tsk_other R_other.
Proof.
unfold response_time_bounded_by, is_response_time_bound_of_task,
completed, completed_jobs_dont_execute, valid_sporadic_job in ×.
rename H_valid_job_parameters into PARAMS,
H_all_jobs_from_taskset into FROMTS,
H_valid_task_parameters into TASK_PARAMS,
H_constrained_deadlines into RESTR,
H_response_time_of_interfering_tasks_is_known into RESP,
H_interfering_tasks_miss_no_deadlines into NOMISS,
H_response_time_bounds_ge_cost into GE_COST.
unfold x, workload_bound.
destruct ([∃ t: 'I_(job_arrival j + R),
task_is_scheduled job_task sched tsk_other t]) eqn: SCHED;
last first.
{
apply negbT in SCHED; rewrite negb_exists in SCHED.
move: SCHED ⇒ /forallP SCHED.
apply leq_trans with (n := 0); last by done.
apply leq_trans with (n := \sum_(job_arrival j ≤ t < job_arrival j + R) 0);
last by rewrite big1.
apply leq_sum_nat; move ⇒ i /andP [_ LTi] _.
specialize (SCHED (Ordinal LTi)).
rewrite negb_exists in SCHED; move: SCHED ⇒ /forallP SCHED.
rewrite big1 //; intros cpu _.
specialize (SCHED cpu); apply negbTE in SCHED.
by rewrite SCHED andbF.
}
move: SCHED ⇒ /existsP [t /existsP [cpu SCHED]].
unfold task_scheduled_on in SCHED.
destruct (sched cpu t) as [j0 |] eqn:SCHED'; last by done.
assert (INts: tsk_other \in ts).
{
move: SCHED ⇒ /eqP <-. apply FROMTS, (H_jobs_come_from_arrival_sequence j0 t).
by apply/existsP; ∃ cpu; apply/eqP.
}
apply leq_trans with (n := workload job_task sched tsk_other
(job_arrival j) (job_arrival j + R));
first by apply task_interference_le_workload.
by ( try ( apply workload_bounded_by_W with (task_deadline0 := task_deadline) (arr_seq0 := arr_seq)
(job_arrival0 := job_arrival) (job_cost0 := job_cost) (job_deadline0 := job_deadline) ) ||
apply workload_bounded_by_W with (task_deadline := task_deadline) (arr_seq := arr_seq)
(job_arrival := job_arrival) (job_cost := job_cost) (job_deadline := job_deadline));
try (by ins); last 2 first;
[ by ins; apply GE_COST
| by ins; apply NOMISS
| by ins; apply TASK_PARAMS
| by ins; apply RESTR
| by ins; apply RESP with (hp_tsk := tsk_other)].
Qed.
End LemmasAboutHPTasks.
Section DerivingContradiction.
Lemma bertogna_fp_too_much_interference : X ≥ R - task_cost tsk + 1.
Proof.
rename H_completed_jobs_dont_execute into COMP,
H_valid_job_parameters into PARAMS,
H_response_time_recurrence_holds into REC,
H_job_of_tsk into JOBtsk, H_j_not_completed into NOTCOMP.
unfold completed, valid_sporadic_job in ×.
unfold X, total_interference; rewrite addn1.
rewrite -(ltn_add2r (task_cost tsk)).
rewrite addnBAC; last by rewrite [R](REC) // leq_addr.
rewrite -addnBA // subnn addn0.
move: (NOTCOMP) ⇒ /negP NOTCOMP'.
rewrite -ltnNge in NOTCOMP.
apply leq_ltn_trans with (n := (\sum_(job_arrival j ≤ t < job_arrival j + R)
backlogged job_arrival job_cost sched j t) +
service sched j (job_arrival j + R)); last first.
{
rewrite -addn1 -addnA leq_add2l addn1.
apply leq_trans with (n := job_cost j); first by done.
by specialize (PARAMS j H_j_arrives); des; rewrite -JOBtsk.
}
unfold service; rewrite service_before_arrival_eq_service_during //.
rewrite -big_split /=.
apply leq_trans with (n := \sum_(job_arrival j ≤ i < job_arrival j + R) 1);
first by rewrite big_const_nat iter_addn mul1n addn0 addKn.
rewrite big_nat_cond [\sum_(_ ≤ _ < _ | true) _]big_nat_cond.
apply leq_sum; move ⇒ i /andP [/andP [GEi LTi] _].
destruct (backlogged job_arrival job_cost sched j i) eqn:BACK;
first by rewrite -addn1 addnC; apply leq_add.
apply negbT in BACK.
rewrite add0n lt0n -not_scheduled_no_service negbK.
rewrite /backlogged negb_and negbK in BACK.
move: BACK ⇒ /orP [/negP NOTPENDING | SCHED]; last by done.
exfalso; apply NOTPENDING; unfold pending; apply/andP; split; first by done.
apply/negP; red; intro BUG; apply NOTCOMP'.
by apply completion_monotonic with (t := i); try (by done); apply ltnW.
Qed.
Lemma bertogna_fp_interference_by_different_tasks :
∀ t j_other,
job_arrival j ≤ t < job_arrival j + R →
arrives_in arr_seq j_other →
backlogged job_arrival job_cost sched j t →
scheduled sched j_other t →
job_task j_other != tsk.
Proof.
rename H_all_jobs_from_taskset into FROMTS,
H_valid_task_parameters into PARAMS,
H_job_of_tsk into JOBtsk, H_sporadic_tasks into SPO,
H_work_conserving into WORK,
H_constrained_deadlines into CONSTR,
H_previous_jobs_of_tsk_completed into PREV,
H_response_time_no_larger_than_deadline into NOMISS.
move ⇒ t j_other /andP [LEt GEt] ARRother BACK SCHED.
apply/eqP; red; intro SAMEtsk.
move: SCHED ⇒ /existsP [cpu SCHED].
assert (SCHED': scheduled sched j_other t).
by apply/existsP; ∃ cpu.
clear SCHED; rename SCHED' into SCHED.
move: (SCHED) ⇒ PENDING.
try ( apply scheduled_implies_pending with (job_arrival0 := job_arrival)
(job_cost0 := job_cost) in PENDING; try (by done) ) ||
apply scheduled_implies_pending with (job_arrival := job_arrival)
(job_cost := job_cost) in PENDING; try (by done).
destruct (ltnP (job_arrival j_other) (job_arrival j)) as [BEFOREother | BEFOREj].
{
move: (BEFOREother) ⇒ LT; rewrite -(ltn_add2r R) in LT.
specialize (PREV j_other ARRother SAMEtsk BEFOREother).
move: PENDING ⇒ /andP [_ /negP NOTCOMP]; apply NOTCOMP.
try ( apply completion_monotonic with (t0 := job_arrival j_other + R); try (by done) ) ||
apply completion_monotonic with (t := job_arrival j_other + R); try (by done).
apply leq_trans with (n := job_arrival j); last by done.
apply leq_trans with (n := job_arrival j_other + task_deadline tsk);
first by rewrite leq_add2l; apply NOMISS.
apply leq_trans with (n := job_arrival j_other + task_period tsk);
first by rewrite leq_add2l; apply CONSTR; rewrite -JOBtsk FROMTS.
rewrite -SAMEtsk; apply SPO; try (by done); [ | by rewrite JOBtsk | by apply ltnW].
by intro EQ; subst j_other; rewrite ltnn in BEFOREother.
}
{
move: PENDING ⇒ /andP [ARRIVED _].
exploit (SPO j j_other); try (by done); [ | by rewrite SAMEtsk | ]; last first.
{
apply/negP; rewrite -ltnNge.
apply leq_ltn_trans with (n := t); first by done.
apply leq_trans with (n := job_arrival j + R); first by done.
by rewrite leq_add2l; apply leq_trans with (n := task_deadline tsk);
[by apply NOMISS | by rewrite JOBtsk CONSTR // -JOBtsk FROMTS].
}
by red; intros EQtsk; subst j_other; rewrite /backlogged SCHED andbF in BACK.
}
Qed.
Let other_scheduled_task (t: time) (tsk_other: sporadic_task) :=
task_is_scheduled job_task sched tsk_other t &&
is_hp_task tsk_other.
Lemma bertogna_fp_all_cpus_are_busy:
∀ t,
job_arrival j ≤ t < job_arrival j + R →
backlogged job_arrival job_cost sched j t →
count (other_scheduled_task t) ts = num_cpus.
Proof.
rename H_valid_task_parameters into PARAMS,
H_all_jobs_from_taskset into FROMTS,
H_job_of_tsk into JOBtsk,
H_sporadic_tasks into SPO,
H_valid_job_parameters into JOBPARAMS,
H_constrained_deadlines into RESTR,
H_hp_bounds_has_interfering_tasks into HAS,
H_interfering_tasks_miss_no_deadlines into NOMISS,
H_response_time_of_interfering_tasks_is_known into PREV.
unfold sporadic_task_model, is_response_time_bound_of_task in ×.
move ⇒ t /andP [LEt LTt] BACK.
(try ( apply platform_fp_cpus_busy_with_interfering_tasks with (task_cost0 := task_cost)
(task_period0 := task_period) (task_deadline0 := task_deadline) (job_task0 := job_task)
(arr_seq0 := arr_seq) (ts0 := ts) (tsk0 := tsk) (higher_eq_priority0 := higher_eq_priority)
in BACK ) ||
apply platform_fp_cpus_busy_with_interfering_tasks with (task_cost := task_cost)
(task_period := task_period) (task_deadline := task_deadline) (job_task := job_task)
(arr_seq := arr_seq) (ts := ts) (tsk := tsk) (higher_eq_priority := higher_eq_priority)
in BACK); try (by done); first by apply PARAMS.
{
apply leq_trans with (n := job_arrival j + R); first by done.
rewrite leq_add2l.
by apply leq_trans with (n := task_deadline tsk); last by apply RESTR.
}
{
intros j_other tsk_other ARRother JOBother INTERF.
feed (HAS tsk_other); first by rewrite -JOBother FROMTS.
move: (HAS INTERF) ⇒ [R' IN].
( try ( apply completion_monotonic with (t0 := job_arrival j_other + R'); try (by done) ) ||
apply completion_monotonic with (t := job_arrival j_other + R'); try (by done));
last by apply PREV with (hp_tsk := tsk_other).
{
rewrite leq_add2l.
apply leq_trans with (n := task_deadline tsk_other); first by apply NOMISS.
by apply RESTR; rewrite -JOBother FROMTS.
}
}
{
ins; ( try ( apply completion_monotonic with (t0 := job_arrival j0 + R) ) ||
apply completion_monotonic with (t := job_arrival j0 + R)); try (by done);
last by apply H_previous_jobs_of_tsk_completed.
rewrite leq_add2l.
by apply leq_trans with (n := task_deadline tsk); last by apply RESTR.
}
Qed.
Lemma bertogna_fp_interference_on_all_cpus :
\sum_(tsk_k <- hp_tasks) x tsk_k = X × num_cpus.
Proof.
have DIFFTASK := bertogna_fp_interference_by_different_tasks.
rename H_work_conserving into WORK, H_respects_FP_policy into FP,
H_jobs_come_from_arrival_sequence into SEQ,
H_all_jobs_from_taskset into FROMTS, H_job_of_tsk into JOBtsk.
unfold sporadic_task_model in ×.
unfold x, X, total_interference, task_interference.
rewrite -big_mkcond -exchange_big big_distrl /= mul1n.
rewrite [\sum_(_ ≤ _ < _ | backlogged _ _ _ _ _) _]big_mkcond.
apply eq_big_nat; move ⇒ t /andP [GEt LTt].
destruct (backlogged job_arrival job_cost sched j t) eqn:BACK;
last by rewrite big1 //; ins; rewrite big1.
rewrite big_mkcond /=.
rewrite exchange_big /=.
apply eq_trans with (y := \sum_(cpu < num_cpus) 1); last by simpl_sum_const.
apply eq_bigr; intros cpu _.
move: (WORK j t H_j_arrives BACK cpu) ⇒ [j_other /eqP SCHED]; unfold scheduled_on in ×.
rewrite (bigD1_seq (job_task j_other)) /=; last by rewrite filter_uniq; destruct ts.
{
rewrite (eq_bigr (fun i ⇒ 0));
last by intros i DIFF; rewrite /task_scheduled_on SCHED;apply/eqP;rewrite eqb0 eq_sym.
rewrite big_const_seq iter_addn mul0n 2!addn0; apply/eqP; rewrite eqb1.
by unfold task_scheduled_on; rewrite SCHED.
}
have ARRother: arrives_in arr_seq j_other.
by apply (SEQ j_other t); apply/existsP; ∃ cpu; apply/eqP.
rewrite mem_filter; apply/andP; split; last by apply FROMTS.
apply/andP; split.
{
rewrite -JOBtsk; apply FP with (t := t); try by done.
by apply/existsP; ∃ cpu; apply/eqP.
}
apply DIFFTASK with (t := t); try (by done); first by auto.
by apply/existsP; ∃ cpu; apply/eqP.
Qed.
Let num_tasks_exceeding delta := count (fun i ⇒ x i ≥ delta) (hp_tasks).
Lemma bertogna_fp_interference_in_non_full_processors :
∀ delta,
0 < num_tasks_exceeding delta < num_cpus →
\sum_(i <- hp_tasks | x i < delta) x i ≥ delta × (num_cpus - num_tasks_exceeding delta).
Proof.
have INV := bertogna_fp_all_cpus_are_busy.
rename H_all_jobs_from_taskset into FROMTS, H_jobs_come_from_arrival_sequence into FROMSEQ,
H_valid_task_parameters into PARAMS, H_job_of_tsk into JOBtsk,
H_sporadic_tasks into SPO, H_previous_jobs_of_tsk_completed into BEFOREok,
H_response_time_no_larger_than_deadline into NOMISS,
H_constrained_deadlines into CONSTR, H_sequential_jobs into SEQ,
H_respects_FP_policy into FP, H_hp_bounds_has_interfering_tasks into HASHP,
H_interfering_tasks_miss_no_deadlines into NOMISSHP.
unfold sporadic_task_model in ×.
move ⇒ delta /andP [HAS LT].
rewrite -has_count in HAS.
set some_interference_A := fun t ⇒
has (fun tsk_k ⇒ backlogged job_arrival job_cost sched j t &&
(x tsk_k ≥ delta) &&
task_is_scheduled job_task sched tsk_k t) hp_tasks.
set total_interference_B := fun t ⇒
backlogged job_arrival job_cost sched j t ×
count (fun tsk_k ⇒ (x tsk_k < delta) &&
task_is_scheduled job_task sched tsk_k t) hp_tasks.
apply leq_trans with ((\sum_(job_arrival j ≤ t < job_arrival j + R)
some_interference_A t) × (num_cpus - num_tasks_exceeding delta)).
{
rewrite leq_mul2r; apply/orP; right.
move: HAS ⇒ /hasP HAS; destruct HAS as [tsk_a INa LEa].
apply leq_trans with (n := x tsk_a); first by apply LEa.
unfold x, task_interference, some_interference_A.
apply leq_sum_nat; move ⇒ t /andP [GEt LTt] _.
destruct (backlogged job_arrival job_cost sched j t) eqn:BACK;
last by rewrite (eq_bigr (fun x ⇒ 0)); [by simpl_sum_const | by ins].
destruct ([∃ cpu, task_scheduled_on job_task sched tsk_a cpu t]) eqn:SCHED;
last first.
{
apply negbT in SCHED; rewrite negb_exists in SCHED; move: SCHED ⇒ /forallP ALL.
rewrite (eq_bigr (fun x ⇒ 0)); first by simpl_sum_const.
by intros cpu _; specialize (ALL cpu); apply negbTE in ALL; rewrite ALL.
}
move: SCHED ⇒ /existsP [cpu SCHED].
apply leq_trans with (n := 1); last first.
{
rewrite lt0b; apply/hasP; ∃ tsk_a; first by done.
by rewrite LEa 2!andTb; apply/existsP; ∃ cpu.
}
rewrite (bigD1 cpu) /= // SCHED.
rewrite (eq_bigr (fun x ⇒ 0)); first by simpl_sum_const; rewrite leq_b1.
intros cpu' DIFF.
apply/eqP; rewrite eqb0; apply/negP.
intros SCHED'.
move: DIFF ⇒ /negP DIFF; apply DIFF; apply/eqP.
unfold task_scheduled_on in ×.
destruct (sched cpu t) as [j1|] eqn:SCHED1; last by done.
destruct (sched cpu' t) as [j2|] eqn:SCHED2; last by done.
move: SCHED SCHED' ⇒ /eqP JOB /eqP JOB'.
subst tsk_a; symmetry in JOB'.
have ARR1: arrives_in arr_seq j1.
by apply (FROMSEQ j1 t); apply/existsP; ∃ cpu; apply/eqP.
have ARR2: arrives_in arr_seq j2.
by apply (FROMSEQ j2 t); apply/existsP; ∃ cpu'; apply/eqP.
assert (PENDING1: pending job_arrival job_cost sched j1 t).
{
apply scheduled_implies_pending; try by done.
by apply/existsP; ∃ cpu; apply/eqP.
}
assert (PENDING2: pending job_arrival job_cost sched j2 t).
{
apply scheduled_implies_pending; try by done.
by apply/existsP; ∃ cpu'; apply/eqP.
}
assert (BUG: j1 = j2).
{
destruct (job_task j1 == tsk) eqn:SAMEtsk.
{
move: SAMEtsk ⇒ /eqP SAMEtsk.
move: (PENDING1) ⇒ SAMEjob.
( try ( apply platform_fp_no_multiple_jobs_of_tsk with (task_cost0 := task_cost)
(arr_seq0 := arr_seq) (task_period0 := task_period) (task_deadline0 := task_deadline)
(job_task0 := job_task) (tsk0 := tsk) (j0 := j) in SAMEjob ) ||
apply platform_fp_no_multiple_jobs_of_tsk with (task_cost := task_cost)
(arr_seq := arr_seq) (task_period := task_period) (task_deadline := task_deadline)
(job_task := job_task) (tsk := tsk) (j := j) in SAMEjob); try (by done);
[ | by apply PARAMS | |]; last 2 first.
{
apply (leq_trans LTt); rewrite leq_add2l.
by apply leq_trans with (n := task_deadline tsk); last by apply CONSTR.
}
{
intros j0 ARR0 JOB0 LT0.
( try ( apply completion_monotonic with (t0 := job_arrival j0 + R) ) ||
apply completion_monotonic with (t := job_arrival j0 + R)); try (by done);
last by apply BEFOREok.
rewrite leq_add2l.
by apply leq_trans with (n := task_deadline tsk); last by apply CONSTR.
}
move: BACK ⇒ /andP [_ /negP NOTSCHED]; exfalso; apply NOTSCHED.
by rewrite -SAMEjob; apply/existsP; ∃ cpu; apply/eqP.
}
{
assert (INTERF: is_hp_task (job_task j1)).
{
apply/andP; split; last by rewrite SAMEtsk.
rewrite -JOBtsk; apply FP with (t := t); try (by done).
by apply/existsP; ∃ cpu; apply/eqP.
}
( try ( apply platform_fp_no_multiple_jobs_of_interfering_tasks with
(job_arrival0 := job_arrival) (arr_seq0 := arr_seq) (task_period0 := task_period)
(tsk0 := tsk) (higher_eq_priority0 := higher_eq_priority)
(job_cost0 := job_cost) (job_task0 := job_task) (sched0 := sched) (t0 := t) ) ||
apply platform_fp_no_multiple_jobs_of_interfering_tasks with
(job_arrival := job_arrival) (arr_seq := arr_seq) (task_period := task_period)
(tsk := tsk) (higher_eq_priority := higher_eq_priority)
(job_cost := job_cost) (job_task := job_task) (sched := sched) (t := t));
rewrite ?JOBtsk ?SAMEtsk //.
{
intros j0 tsk0 ARR0 JOB0 INTERF0.
feed (HASHP tsk0); first by rewrite -JOB0 FROMTS.
move: (HASHP INTERF0) ⇒ [R0 IN0].
( try ( apply completion_monotonic with (t0 := job_arrival j0 + R0); try (by done) ) ||
apply completion_monotonic with (t := job_arrival j0 + R0); try (by done));
last by eapply H_response_time_of_interfering_tasks_is_known; first by apply IN0.
rewrite leq_add2l.
by apply leq_trans with (n := task_deadline tsk0);
[by apply NOMISSHP | by apply CONSTR; rewrite -JOB0 FROMTS].
}
}
}
by subst j2; apply SEQ with (j := j1) (t := t).
}
apply leq_trans with (\sum_(job_arrival j ≤ t < job_arrival j + R)
total_interference_B t).
{
rewrite big_distrl /=.
apply leq_sum_nat; move ⇒ t LEt _.
unfold some_interference_A, total_interference_B.
destruct (backlogged job_arrival job_cost sched j t) eqn:BACK;
[rewrite mul1n /= | by rewrite has_pred0 //].
destruct (has (fun tsk_k : sporadic_task ⇒ (delta ≤ x tsk_k) &&
task_is_scheduled job_task sched tsk_k t) hp_tasks) eqn:HAS';
last by done.
rewrite mul1n; move: HAS ⇒ /hasP [tsk_k INk LEk].
unfold num_tasks_exceeding.
apply leq_trans with (n := num_cpus -
count (fun i ⇒ (x i ≥ delta) &&
task_is_scheduled job_task sched i t) hp_tasks).
{
apply leq_sub2l.
rewrite -2!sum1_count big_mkcond /=.
rewrite [\sum_(_ <- _ | _ ≤ _)_]big_mkcond /=.
apply leq_sum; intros i _.
by destruct (task_is_scheduled job_task sched i t);
[by rewrite andbT | by rewrite andbF].
}
rewrite -count_filter -[count _ hp_tasks]count_filter.
eapply leq_trans with (n := count (predC (fun tsk ⇒ delta ≤ x tsk)) _);
last by apply eq_leq, eq_in_count; red; ins; rewrite ltnNge.
rewrite leq_subLR count_predC size_filter.
apply leq_trans with (n := count (other_scheduled_task t) ts);
[by rewrite INV | by rewrite count_filter].
}
{
unfold x at 2, total_interference_B.
rewrite exchange_big /=; apply leq_sum; intros t _.
destruct (backlogged job_arrival job_cost sched j t) eqn:BACK; last by ins.
rewrite mul1n -sum1_count.
rewrite big_mkcond [\sum_(i <- hp_tasks | _ < _) _]big_mkcond /=.
apply leq_sum_seq; move ⇒ tsk_k IN _.
destruct (x tsk_k < delta); [rewrite andTb | by rewrite andFb].
destruct (task_is_scheduled job_task sched tsk_k t) eqn:SCHED; last by done.
move: SCHED ⇒ /existsP [cpu SCHED].
by rewrite (bigD1 cpu) /= // SCHED.
}
Qed.
Lemma bertogna_fp_minimum_exceeds_interference :
∀ delta,
\sum_(tsk_k <- hp_tasks) x tsk_k ≥ delta × num_cpus →
\sum_(tsk_k <- hp_tasks) minn (x tsk_k) delta ≥
delta × num_cpus.
Proof.
intros delta SUMLESS.
set more_interf := fun tsk_k ⇒ x tsk_k ≥ delta.
rewrite [\sum_(_ <- _) minn _ _](bigID more_interf) /=.
unfold more_interf, minn.
rewrite [\sum_(_ <- _ | delta ≤ _)_](eq_bigr (fun i ⇒ delta));
last by intros i COND; rewrite leqNgt in COND; destruct (delta > x i).
rewrite [\sum_(_ <- _ | ~~_)_](eq_big (fun i ⇒ x i < delta)
(fun i ⇒ x i));
[| by red; ins; rewrite ltnNge
| by intros i COND; rewrite -ltnNge in COND; rewrite COND].
destruct (~~ has (fun i ⇒ delta ≤ x i) hp_tasks) eqn:HASa.
{
rewrite [\sum_(_ <- _ | _ ≤ _) _]big_hasC; last by apply HASa.
rewrite big_seq_cond; move: HASa ⇒ /hasPn HASa.
rewrite add0n (eq_bigl (fun i ⇒ (i \in hp_tasks) && true));
last by red; intros tsk_k; destruct (tsk_k \in hp_tasks) eqn:INk;
[by rewrite andTb ltnNge; apply HASa | by rewrite andFb].
by rewrite -big_seq_cond.
} apply negbFE in HASa.
destruct (num_tasks_exceeding delta ≥ num_cpus) eqn:CARD.
{
apply leq_trans with (delta × num_tasks_exceeding delta);
first by rewrite leq_mul2l; apply/orP; right.
unfold num_tasks_exceeding; rewrite -sum1_count big_distrr /=.
rewrite -[\sum_(_ <- _ | _) _]addn0.
by apply leq_add; [by apply leq_sum; ins; rewrite muln1|by ins].
} apply negbT in CARD; rewrite -ltnNge in CARD.
rewrite big_const_seq iter_addn addn0; fold num_tasks_exceeding.
apply leq_trans with (n := delta × num_tasks_exceeding delta +
delta × (num_cpus - num_tasks_exceeding delta));
first by rewrite -mulnDr subnKC //; apply ltnW.
rewrite leq_add2l; apply bertogna_fp_interference_in_non_full_processors.
by apply/andP; split; first by rewrite -has_count.
Qed.
Lemma bertogna_fp_sum_exceeds_total_interference:
\sum_((tsk_k, R_k) <- hp_bounds)
minn (x tsk_k) (R - task_cost tsk + 1) >
total_interference_bound_fp task_cost task_period tsk hp_bounds R.
Proof.
have EXCEEDS := bertogna_fp_minimum_exceeds_interference.
have ALLBUSY := bertogna_fp_interference_on_all_cpus.
have TOOMUCH := bertogna_fp_too_much_interference.
rename H_hp_bounds_has_interfering_tasks into HAS,
H_response_time_recurrence_holds into REC.
apply leq_trans with (n := \sum_(tsk_k <- hp_tasks) minn (x tsk_k) (R - task_cost tsk + 1));
last first.
{
rewrite (eq_bigr (fun i ⇒ minn (x (fst i)) (R - task_cost tsk + 1)));
last by ins; destruct i.
have MAP := @big_map _ 0 addn _ _ (fun x ⇒ fst x) hp_bounds (fun x ⇒ true) (fun y ⇒ minn (x y) (R - task_cost tsk + 1)).
rewrite -MAP.
apply leq_sum_sub_uniq; first by apply filter_uniq; destruct ts.
red; move ⇒ tsk0 IN0.
rewrite mem_filter in IN0; move: IN0 ⇒ /andP [INTERF0 IN0].
apply/mapP.
feed (HAS tsk0); first by done.
move: (HAS INTERF0) ⇒ [R0 IN].
by ∃ (tsk0, R0).
}
apply ltn_div_trunc with (d := num_cpus);
first by apply H_at_least_one_cpu.
rewrite -(ltn_add2l (task_cost tsk)) -REC.
rewrite -addn1 -leq_subLR.
rewrite -[R + 1 - _]addnBAC; last by rewrite REC; apply leq_addr.
rewrite leq_divRL; last by apply H_at_least_one_cpu.
apply EXCEEDS.
apply leq_trans with (n := X × num_cpus); last by rewrite ALLBUSY.
by rewrite leq_mul2r; apply/orP; right; apply TOOMUCH.
Qed.
Lemma bertogna_fp_exists_task_that_exceeds_bound :
∃ tsk_k R_k,
(tsk_k, R_k) \in hp_bounds ∧
(minn (x tsk_k) (R - task_cost tsk + 1) >
minn (workload_bound tsk_k R_k) (R - task_cost tsk + 1)).
Proof.
have SUM := bertogna_fp_sum_exceeds_total_interference.
rename H_hp_bounds_has_interfering_tasks into HASHP.
assert (HAS: has (fun tup : task_with_response_time ⇒
let (tsk_k, R_k) := tup in
(minn (x tsk_k) (R - task_cost tsk + 1) >
minn (workload_bound tsk_k R_k)(R - task_cost tsk + 1)))
hp_bounds).
{
apply/negP; unfold not; intro NOTHAS.
move: NOTHAS ⇒ /negP /hasPn ALL.
rewrite -[_ < _]negbK in SUM.
move: SUM ⇒ /negP SUM; apply SUM; rewrite -leqNgt.
rewrite (eq_bigr (fun i ⇒ minn (x (fst i)) (R - task_cost tsk + 1)));
last by ins; destruct i.
unfold total_interference_bound_fp.
rewrite big_seq_cond.
rewrite [\sum_(_ <- _ | true)_]big_seq_cond.
apply leq_sum.
intros p; rewrite andbT; intros IN.
by specialize (ALL p IN); destruct p; rewrite leqNgt.
}
move: HAS ⇒ /hasP HAS; destruct HAS as [[tsk_k R_k] HPk MINk]; ∃ tsk_k, R_k.
by repeat split.
Qed.
End DerivingContradiction.
End Lemmas.
Theorem bertogna_cirinei_response_time_bound_fp :
response_time_bounded_by tsk R.
Proof.
have WORKLOAD := bertogna_fp_workload_bounds_interference.
have EX := bertogna_fp_exists_task_that_exceeds_bound.
rename H_response_time_bounds_ge_cost into GE_COST,
H_interfering_tasks_miss_no_deadlines into NOMISS,
H_response_time_recurrence_holds into REC,
H_response_time_of_interfering_tasks_is_known into RESP,
H_hp_bounds_has_interfering_tasks into HAS,
H_response_time_no_larger_than_deadline into LEdl.
intros j ARRj JOBtsk.
remember (job_arrival j + R) as ctime.
generalize dependent j.
induction ctime as [ctime IH] using strong_ind.
intros j ARRj JOBtsk EQc; subst ctime.
assert (BEFOREok: ∀ j0,
arrives_in arr_seq j0 →
job_task j0 = tsk →
job_arrival j0 < job_arrival j →
service sched j0 (job_arrival j0 + R) ≥ job_cost j0).
{
by ins; apply IH; try (by done); rewrite ltn_add2r.
} clear IH.
unfold response_time_bounded_by, is_response_time_bound_of_task,
completed, completed_jobs_dont_execute, valid_sporadic_job in ×.
destruct (completed job_cost sched j (job_arrival j + R)) eqn:NOTCOMP;
first by done.
apply negbT in NOTCOMP; exfalso.
specialize (EX j ARRj JOBtsk NOTCOMP BEFOREok).
destruct EX as [tsk_k [R_k [HPk LTmin]]].
unfold minn at 1 in LTmin.
specialize (WORKLOAD j tsk_k R_k HPk).
destruct (W task_cost task_period tsk_k R_k R < R - task_cost tsk + 1); rewrite leq_min in LTmin;
last by move: LTmin ⇒ /andP [_ BUG]; rewrite ltnn in BUG.
move: LTmin ⇒ /andP [BUG _]; des.
apply leq_trans with (p := W task_cost task_period tsk_k R_k R) in BUG; last by done.
by rewrite ltnn in BUG.
Qed.
End ResponseTimeBound.
End ResponseTimeAnalysisFP.
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.workload prosa.classic.model.schedule.global.schedulability
prosa.classic.model.schedule.global.response_time.
Require Import prosa.classic.model.schedule.global.basic.schedule prosa.classic.model.schedule.global.basic.platform
prosa.classic.model.schedule.global.basic.constrained_deadlines prosa.classic.model.schedule.global.basic.interference.
Require Import prosa.classic.analysis.global.basic.workload_bound
prosa.classic.analysis.global.basic.interference_bound_fp.
From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq fintype bigop div path.
Module ResponseTimeAnalysisFP.
Export Job SporadicTaskset ScheduleOfSporadicTask Workload Interference
InterferenceBoundFP Platform Schedulability ResponseTime
Priority TaskArrival WorkloadBound ConstrainedDeadlines.
Section ResponseTimeBound.
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 ts: taskset_of sporadic_task.
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.
Hypothesis H_all_jobs_from_taskset:
∀ j, arrives_in arr_seq j → job_task j \in ts.
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_sequential_jobs: sequential_jobs sched.
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 higher_eq_priority: FP_policy sporadic_task.
Hypothesis H_work_conserving: work_conserving job_arrival job_cost arr_seq sched.
Hypothesis H_respects_FP_policy:
respects_FP_policy job_arrival job_cost job_task arr_seq sched higher_eq_priority.
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.
Variable tsk: sporadic_task.
Hypothesis task_in_ts: tsk \in ts.
Let is_hp_task := higher_priority_task higher_eq_priority tsk.
Let task_with_response_time := (sporadic_task × time)%type.
Variable hp_bounds: seq task_with_response_time.
Hypothesis H_response_time_of_interfering_tasks_is_known:
∀ hp_tsk R,
(hp_tsk, R) \in hp_bounds →
response_time_bounded_by hp_tsk R.
Hypothesis H_hp_bounds_has_interfering_tasks:
∀ hp_tsk,
hp_tsk \in ts →
is_hp_task hp_tsk →
∃ R, (hp_tsk, R) \in hp_bounds.
Hypothesis H_response_time_bounds_ge_cost:
∀ hp_tsk R,
(hp_tsk, R) \in hp_bounds → R ≥ task_cost hp_tsk.
Hypothesis H_interfering_tasks_miss_no_deadlines:
∀ hp_tsk R,
(hp_tsk, R) \in hp_bounds → R ≤ task_deadline hp_tsk.
Variable R: time.
Hypothesis H_response_time_recurrence_holds :
R = task_cost tsk +
div_floor
(total_interference_bound_fp task_cost task_period tsk hp_bounds R)
num_cpus.
Hypothesis H_response_time_no_larger_than_deadline:
R ≤ task_deadline tsk.
Section Lemmas.
Variable j: Job.
Hypothesis H_j_arrives: arrives_in arr_seq j.
Hypothesis H_job_of_tsk: job_task j = tsk.
Hypothesis H_j_not_completed: ~~ completed job_cost sched j (job_arrival j + R).
Hypothesis H_previous_jobs_of_tsk_completed :
∀ j0,
arrives_in arr_seq j0 →
job_task j0 = tsk →
job_arrival j0 < job_arrival j →
completed job_cost sched j0 (job_arrival j0 + R).
Let x (tsk_other: sporadic_task) :=
task_interference job_arrival job_cost job_task sched j tsk_other
(job_arrival j) (job_arrival j + R).
Let X := total_interference job_arrival job_cost sched j (job_arrival j) (job_arrival j + R).
Let workload_bound (tsk_other: sporadic_task) (R_other: time) :=
W task_cost task_period tsk_other R_other R.
Let hp_tasks := [seq tsk_other <- ts | is_hp_task tsk_other].
Section LemmasAboutHPTasks.
Variable tsk_other: sporadic_task.
Variable R_other: time.
Hypothesis H_response_time_of_tsk_other: (tsk_other, R_other) \in hp_bounds.
Lemma bertogna_fp_workload_bounds_interference :
x tsk_other ≤ workload_bound tsk_other R_other.
Proof.
unfold response_time_bounded_by, is_response_time_bound_of_task,
completed, completed_jobs_dont_execute, valid_sporadic_job in ×.
rename H_valid_job_parameters into PARAMS,
H_all_jobs_from_taskset into FROMTS,
H_valid_task_parameters into TASK_PARAMS,
H_constrained_deadlines into RESTR,
H_response_time_of_interfering_tasks_is_known into RESP,
H_interfering_tasks_miss_no_deadlines into NOMISS,
H_response_time_bounds_ge_cost into GE_COST.
unfold x, workload_bound.
destruct ([∃ t: 'I_(job_arrival j + R),
task_is_scheduled job_task sched tsk_other t]) eqn: SCHED;
last first.
{
apply negbT in SCHED; rewrite negb_exists in SCHED.
move: SCHED ⇒ /forallP SCHED.
apply leq_trans with (n := 0); last by done.
apply leq_trans with (n := \sum_(job_arrival j ≤ t < job_arrival j + R) 0);
last by rewrite big1.
apply leq_sum_nat; move ⇒ i /andP [_ LTi] _.
specialize (SCHED (Ordinal LTi)).
rewrite negb_exists in SCHED; move: SCHED ⇒ /forallP SCHED.
rewrite big1 //; intros cpu _.
specialize (SCHED cpu); apply negbTE in SCHED.
by rewrite SCHED andbF.
}
move: SCHED ⇒ /existsP [t /existsP [cpu SCHED]].
unfold task_scheduled_on in SCHED.
destruct (sched cpu t) as [j0 |] eqn:SCHED'; last by done.
assert (INts: tsk_other \in ts).
{
move: SCHED ⇒ /eqP <-. apply FROMTS, (H_jobs_come_from_arrival_sequence j0 t).
by apply/existsP; ∃ cpu; apply/eqP.
}
apply leq_trans with (n := workload job_task sched tsk_other
(job_arrival j) (job_arrival j + R));
first by apply task_interference_le_workload.
by ( try ( apply workload_bounded_by_W with (task_deadline0 := task_deadline) (arr_seq0 := arr_seq)
(job_arrival0 := job_arrival) (job_cost0 := job_cost) (job_deadline0 := job_deadline) ) ||
apply workload_bounded_by_W with (task_deadline := task_deadline) (arr_seq := arr_seq)
(job_arrival := job_arrival) (job_cost := job_cost) (job_deadline := job_deadline));
try (by ins); last 2 first;
[ by ins; apply GE_COST
| by ins; apply NOMISS
| by ins; apply TASK_PARAMS
| by ins; apply RESTR
| by ins; apply RESP with (hp_tsk := tsk_other)].
Qed.
End LemmasAboutHPTasks.
Section DerivingContradiction.
Lemma bertogna_fp_too_much_interference : X ≥ R - task_cost tsk + 1.
Proof.
rename H_completed_jobs_dont_execute into COMP,
H_valid_job_parameters into PARAMS,
H_response_time_recurrence_holds into REC,
H_job_of_tsk into JOBtsk, H_j_not_completed into NOTCOMP.
unfold completed, valid_sporadic_job in ×.
unfold X, total_interference; rewrite addn1.
rewrite -(ltn_add2r (task_cost tsk)).
rewrite addnBAC; last by rewrite [R](REC) // leq_addr.
rewrite -addnBA // subnn addn0.
move: (NOTCOMP) ⇒ /negP NOTCOMP'.
rewrite -ltnNge in NOTCOMP.
apply leq_ltn_trans with (n := (\sum_(job_arrival j ≤ t < job_arrival j + R)
backlogged job_arrival job_cost sched j t) +
service sched j (job_arrival j + R)); last first.
{
rewrite -addn1 -addnA leq_add2l addn1.
apply leq_trans with (n := job_cost j); first by done.
by specialize (PARAMS j H_j_arrives); des; rewrite -JOBtsk.
}
unfold service; rewrite service_before_arrival_eq_service_during //.
rewrite -big_split /=.
apply leq_trans with (n := \sum_(job_arrival j ≤ i < job_arrival j + R) 1);
first by rewrite big_const_nat iter_addn mul1n addn0 addKn.
rewrite big_nat_cond [\sum_(_ ≤ _ < _ | true) _]big_nat_cond.
apply leq_sum; move ⇒ i /andP [/andP [GEi LTi] _].
destruct (backlogged job_arrival job_cost sched j i) eqn:BACK;
first by rewrite -addn1 addnC; apply leq_add.
apply negbT in BACK.
rewrite add0n lt0n -not_scheduled_no_service negbK.
rewrite /backlogged negb_and negbK in BACK.
move: BACK ⇒ /orP [/negP NOTPENDING | SCHED]; last by done.
exfalso; apply NOTPENDING; unfold pending; apply/andP; split; first by done.
apply/negP; red; intro BUG; apply NOTCOMP'.
by apply completion_monotonic with (t := i); try (by done); apply ltnW.
Qed.
Lemma bertogna_fp_interference_by_different_tasks :
∀ t j_other,
job_arrival j ≤ t < job_arrival j + R →
arrives_in arr_seq j_other →
backlogged job_arrival job_cost sched j t →
scheduled sched j_other t →
job_task j_other != tsk.
Proof.
rename H_all_jobs_from_taskset into FROMTS,
H_valid_task_parameters into PARAMS,
H_job_of_tsk into JOBtsk, H_sporadic_tasks into SPO,
H_work_conserving into WORK,
H_constrained_deadlines into CONSTR,
H_previous_jobs_of_tsk_completed into PREV,
H_response_time_no_larger_than_deadline into NOMISS.
move ⇒ t j_other /andP [LEt GEt] ARRother BACK SCHED.
apply/eqP; red; intro SAMEtsk.
move: SCHED ⇒ /existsP [cpu SCHED].
assert (SCHED': scheduled sched j_other t).
by apply/existsP; ∃ cpu.
clear SCHED; rename SCHED' into SCHED.
move: (SCHED) ⇒ PENDING.
try ( apply scheduled_implies_pending with (job_arrival0 := job_arrival)
(job_cost0 := job_cost) in PENDING; try (by done) ) ||
apply scheduled_implies_pending with (job_arrival := job_arrival)
(job_cost := job_cost) in PENDING; try (by done).
destruct (ltnP (job_arrival j_other) (job_arrival j)) as [BEFOREother | BEFOREj].
{
move: (BEFOREother) ⇒ LT; rewrite -(ltn_add2r R) in LT.
specialize (PREV j_other ARRother SAMEtsk BEFOREother).
move: PENDING ⇒ /andP [_ /negP NOTCOMP]; apply NOTCOMP.
try ( apply completion_monotonic with (t0 := job_arrival j_other + R); try (by done) ) ||
apply completion_monotonic with (t := job_arrival j_other + R); try (by done).
apply leq_trans with (n := job_arrival j); last by done.
apply leq_trans with (n := job_arrival j_other + task_deadline tsk);
first by rewrite leq_add2l; apply NOMISS.
apply leq_trans with (n := job_arrival j_other + task_period tsk);
first by rewrite leq_add2l; apply CONSTR; rewrite -JOBtsk FROMTS.
rewrite -SAMEtsk; apply SPO; try (by done); [ | by rewrite JOBtsk | by apply ltnW].
by intro EQ; subst j_other; rewrite ltnn in BEFOREother.
}
{
move: PENDING ⇒ /andP [ARRIVED _].
exploit (SPO j j_other); try (by done); [ | by rewrite SAMEtsk | ]; last first.
{
apply/negP; rewrite -ltnNge.
apply leq_ltn_trans with (n := t); first by done.
apply leq_trans with (n := job_arrival j + R); first by done.
by rewrite leq_add2l; apply leq_trans with (n := task_deadline tsk);
[by apply NOMISS | by rewrite JOBtsk CONSTR // -JOBtsk FROMTS].
}
by red; intros EQtsk; subst j_other; rewrite /backlogged SCHED andbF in BACK.
}
Qed.
Let other_scheduled_task (t: time) (tsk_other: sporadic_task) :=
task_is_scheduled job_task sched tsk_other t &&
is_hp_task tsk_other.
Lemma bertogna_fp_all_cpus_are_busy:
∀ t,
job_arrival j ≤ t < job_arrival j + R →
backlogged job_arrival job_cost sched j t →
count (other_scheduled_task t) ts = num_cpus.
Proof.
rename H_valid_task_parameters into PARAMS,
H_all_jobs_from_taskset into FROMTS,
H_job_of_tsk into JOBtsk,
H_sporadic_tasks into SPO,
H_valid_job_parameters into JOBPARAMS,
H_constrained_deadlines into RESTR,
H_hp_bounds_has_interfering_tasks into HAS,
H_interfering_tasks_miss_no_deadlines into NOMISS,
H_response_time_of_interfering_tasks_is_known into PREV.
unfold sporadic_task_model, is_response_time_bound_of_task in ×.
move ⇒ t /andP [LEt LTt] BACK.
(try ( apply platform_fp_cpus_busy_with_interfering_tasks with (task_cost0 := task_cost)
(task_period0 := task_period) (task_deadline0 := task_deadline) (job_task0 := job_task)
(arr_seq0 := arr_seq) (ts0 := ts) (tsk0 := tsk) (higher_eq_priority0 := higher_eq_priority)
in BACK ) ||
apply platform_fp_cpus_busy_with_interfering_tasks with (task_cost := task_cost)
(task_period := task_period) (task_deadline := task_deadline) (job_task := job_task)
(arr_seq := arr_seq) (ts := ts) (tsk := tsk) (higher_eq_priority := higher_eq_priority)
in BACK); try (by done); first by apply PARAMS.
{
apply leq_trans with (n := job_arrival j + R); first by done.
rewrite leq_add2l.
by apply leq_trans with (n := task_deadline tsk); last by apply RESTR.
}
{
intros j_other tsk_other ARRother JOBother INTERF.
feed (HAS tsk_other); first by rewrite -JOBother FROMTS.
move: (HAS INTERF) ⇒ [R' IN].
( try ( apply completion_monotonic with (t0 := job_arrival j_other + R'); try (by done) ) ||
apply completion_monotonic with (t := job_arrival j_other + R'); try (by done));
last by apply PREV with (hp_tsk := tsk_other).
{
rewrite leq_add2l.
apply leq_trans with (n := task_deadline tsk_other); first by apply NOMISS.
by apply RESTR; rewrite -JOBother FROMTS.
}
}
{
ins; ( try ( apply completion_monotonic with (t0 := job_arrival j0 + R) ) ||
apply completion_monotonic with (t := job_arrival j0 + R)); try (by done);
last by apply H_previous_jobs_of_tsk_completed.
rewrite leq_add2l.
by apply leq_trans with (n := task_deadline tsk); last by apply RESTR.
}
Qed.
Lemma bertogna_fp_interference_on_all_cpus :
\sum_(tsk_k <- hp_tasks) x tsk_k = X × num_cpus.
Proof.
have DIFFTASK := bertogna_fp_interference_by_different_tasks.
rename H_work_conserving into WORK, H_respects_FP_policy into FP,
H_jobs_come_from_arrival_sequence into SEQ,
H_all_jobs_from_taskset into FROMTS, H_job_of_tsk into JOBtsk.
unfold sporadic_task_model in ×.
unfold x, X, total_interference, task_interference.
rewrite -big_mkcond -exchange_big big_distrl /= mul1n.
rewrite [\sum_(_ ≤ _ < _ | backlogged _ _ _ _ _) _]big_mkcond.
apply eq_big_nat; move ⇒ t /andP [GEt LTt].
destruct (backlogged job_arrival job_cost sched j t) eqn:BACK;
last by rewrite big1 //; ins; rewrite big1.
rewrite big_mkcond /=.
rewrite exchange_big /=.
apply eq_trans with (y := \sum_(cpu < num_cpus) 1); last by simpl_sum_const.
apply eq_bigr; intros cpu _.
move: (WORK j t H_j_arrives BACK cpu) ⇒ [j_other /eqP SCHED]; unfold scheduled_on in ×.
rewrite (bigD1_seq (job_task j_other)) /=; last by rewrite filter_uniq; destruct ts.
{
rewrite (eq_bigr (fun i ⇒ 0));
last by intros i DIFF; rewrite /task_scheduled_on SCHED;apply/eqP;rewrite eqb0 eq_sym.
rewrite big_const_seq iter_addn mul0n 2!addn0; apply/eqP; rewrite eqb1.
by unfold task_scheduled_on; rewrite SCHED.
}
have ARRother: arrives_in arr_seq j_other.
by apply (SEQ j_other t); apply/existsP; ∃ cpu; apply/eqP.
rewrite mem_filter; apply/andP; split; last by apply FROMTS.
apply/andP; split.
{
rewrite -JOBtsk; apply FP with (t := t); try by done.
by apply/existsP; ∃ cpu; apply/eqP.
}
apply DIFFTASK with (t := t); try (by done); first by auto.
by apply/existsP; ∃ cpu; apply/eqP.
Qed.
Let num_tasks_exceeding delta := count (fun i ⇒ x i ≥ delta) (hp_tasks).
Lemma bertogna_fp_interference_in_non_full_processors :
∀ delta,
0 < num_tasks_exceeding delta < num_cpus →
\sum_(i <- hp_tasks | x i < delta) x i ≥ delta × (num_cpus - num_tasks_exceeding delta).
Proof.
have INV := bertogna_fp_all_cpus_are_busy.
rename H_all_jobs_from_taskset into FROMTS, H_jobs_come_from_arrival_sequence into FROMSEQ,
H_valid_task_parameters into PARAMS, H_job_of_tsk into JOBtsk,
H_sporadic_tasks into SPO, H_previous_jobs_of_tsk_completed into BEFOREok,
H_response_time_no_larger_than_deadline into NOMISS,
H_constrained_deadlines into CONSTR, H_sequential_jobs into SEQ,
H_respects_FP_policy into FP, H_hp_bounds_has_interfering_tasks into HASHP,
H_interfering_tasks_miss_no_deadlines into NOMISSHP.
unfold sporadic_task_model in ×.
move ⇒ delta /andP [HAS LT].
rewrite -has_count in HAS.
set some_interference_A := fun t ⇒
has (fun tsk_k ⇒ backlogged job_arrival job_cost sched j t &&
(x tsk_k ≥ delta) &&
task_is_scheduled job_task sched tsk_k t) hp_tasks.
set total_interference_B := fun t ⇒
backlogged job_arrival job_cost sched j t ×
count (fun tsk_k ⇒ (x tsk_k < delta) &&
task_is_scheduled job_task sched tsk_k t) hp_tasks.
apply leq_trans with ((\sum_(job_arrival j ≤ t < job_arrival j + R)
some_interference_A t) × (num_cpus - num_tasks_exceeding delta)).
{
rewrite leq_mul2r; apply/orP; right.
move: HAS ⇒ /hasP HAS; destruct HAS as [tsk_a INa LEa].
apply leq_trans with (n := x tsk_a); first by apply LEa.
unfold x, task_interference, some_interference_A.
apply leq_sum_nat; move ⇒ t /andP [GEt LTt] _.
destruct (backlogged job_arrival job_cost sched j t) eqn:BACK;
last by rewrite (eq_bigr (fun x ⇒ 0)); [by simpl_sum_const | by ins].
destruct ([∃ cpu, task_scheduled_on job_task sched tsk_a cpu t]) eqn:SCHED;
last first.
{
apply negbT in SCHED; rewrite negb_exists in SCHED; move: SCHED ⇒ /forallP ALL.
rewrite (eq_bigr (fun x ⇒ 0)); first by simpl_sum_const.
by intros cpu _; specialize (ALL cpu); apply negbTE in ALL; rewrite ALL.
}
move: SCHED ⇒ /existsP [cpu SCHED].
apply leq_trans with (n := 1); last first.
{
rewrite lt0b; apply/hasP; ∃ tsk_a; first by done.
by rewrite LEa 2!andTb; apply/existsP; ∃ cpu.
}
rewrite (bigD1 cpu) /= // SCHED.
rewrite (eq_bigr (fun x ⇒ 0)); first by simpl_sum_const; rewrite leq_b1.
intros cpu' DIFF.
apply/eqP; rewrite eqb0; apply/negP.
intros SCHED'.
move: DIFF ⇒ /negP DIFF; apply DIFF; apply/eqP.
unfold task_scheduled_on in ×.
destruct (sched cpu t) as [j1|] eqn:SCHED1; last by done.
destruct (sched cpu' t) as [j2|] eqn:SCHED2; last by done.
move: SCHED SCHED' ⇒ /eqP JOB /eqP JOB'.
subst tsk_a; symmetry in JOB'.
have ARR1: arrives_in arr_seq j1.
by apply (FROMSEQ j1 t); apply/existsP; ∃ cpu; apply/eqP.
have ARR2: arrives_in arr_seq j2.
by apply (FROMSEQ j2 t); apply/existsP; ∃ cpu'; apply/eqP.
assert (PENDING1: pending job_arrival job_cost sched j1 t).
{
apply scheduled_implies_pending; try by done.
by apply/existsP; ∃ cpu; apply/eqP.
}
assert (PENDING2: pending job_arrival job_cost sched j2 t).
{
apply scheduled_implies_pending; try by done.
by apply/existsP; ∃ cpu'; apply/eqP.
}
assert (BUG: j1 = j2).
{
destruct (job_task j1 == tsk) eqn:SAMEtsk.
{
move: SAMEtsk ⇒ /eqP SAMEtsk.
move: (PENDING1) ⇒ SAMEjob.
( try ( apply platform_fp_no_multiple_jobs_of_tsk with (task_cost0 := task_cost)
(arr_seq0 := arr_seq) (task_period0 := task_period) (task_deadline0 := task_deadline)
(job_task0 := job_task) (tsk0 := tsk) (j0 := j) in SAMEjob ) ||
apply platform_fp_no_multiple_jobs_of_tsk with (task_cost := task_cost)
(arr_seq := arr_seq) (task_period := task_period) (task_deadline := task_deadline)
(job_task := job_task) (tsk := tsk) (j := j) in SAMEjob); try (by done);
[ | by apply PARAMS | |]; last 2 first.
{
apply (leq_trans LTt); rewrite leq_add2l.
by apply leq_trans with (n := task_deadline tsk); last by apply CONSTR.
}
{
intros j0 ARR0 JOB0 LT0.
( try ( apply completion_monotonic with (t0 := job_arrival j0 + R) ) ||
apply completion_monotonic with (t := job_arrival j0 + R)); try (by done);
last by apply BEFOREok.
rewrite leq_add2l.
by apply leq_trans with (n := task_deadline tsk); last by apply CONSTR.
}
move: BACK ⇒ /andP [_ /negP NOTSCHED]; exfalso; apply NOTSCHED.
by rewrite -SAMEjob; apply/existsP; ∃ cpu; apply/eqP.
}
{
assert (INTERF: is_hp_task (job_task j1)).
{
apply/andP; split; last by rewrite SAMEtsk.
rewrite -JOBtsk; apply FP with (t := t); try (by done).
by apply/existsP; ∃ cpu; apply/eqP.
}
( try ( apply platform_fp_no_multiple_jobs_of_interfering_tasks with
(job_arrival0 := job_arrival) (arr_seq0 := arr_seq) (task_period0 := task_period)
(tsk0 := tsk) (higher_eq_priority0 := higher_eq_priority)
(job_cost0 := job_cost) (job_task0 := job_task) (sched0 := sched) (t0 := t) ) ||
apply platform_fp_no_multiple_jobs_of_interfering_tasks with
(job_arrival := job_arrival) (arr_seq := arr_seq) (task_period := task_period)
(tsk := tsk) (higher_eq_priority := higher_eq_priority)
(job_cost := job_cost) (job_task := job_task) (sched := sched) (t := t));
rewrite ?JOBtsk ?SAMEtsk //.
{
intros j0 tsk0 ARR0 JOB0 INTERF0.
feed (HASHP tsk0); first by rewrite -JOB0 FROMTS.
move: (HASHP INTERF0) ⇒ [R0 IN0].
( try ( apply completion_monotonic with (t0 := job_arrival j0 + R0); try (by done) ) ||
apply completion_monotonic with (t := job_arrival j0 + R0); try (by done));
last by eapply H_response_time_of_interfering_tasks_is_known; first by apply IN0.
rewrite leq_add2l.
by apply leq_trans with (n := task_deadline tsk0);
[by apply NOMISSHP | by apply CONSTR; rewrite -JOB0 FROMTS].
}
}
}
by subst j2; apply SEQ with (j := j1) (t := t).
}
apply leq_trans with (\sum_(job_arrival j ≤ t < job_arrival j + R)
total_interference_B t).
{
rewrite big_distrl /=.
apply leq_sum_nat; move ⇒ t LEt _.
unfold some_interference_A, total_interference_B.
destruct (backlogged job_arrival job_cost sched j t) eqn:BACK;
[rewrite mul1n /= | by rewrite has_pred0 //].
destruct (has (fun tsk_k : sporadic_task ⇒ (delta ≤ x tsk_k) &&
task_is_scheduled job_task sched tsk_k t) hp_tasks) eqn:HAS';
last by done.
rewrite mul1n; move: HAS ⇒ /hasP [tsk_k INk LEk].
unfold num_tasks_exceeding.
apply leq_trans with (n := num_cpus -
count (fun i ⇒ (x i ≥ delta) &&
task_is_scheduled job_task sched i t) hp_tasks).
{
apply leq_sub2l.
rewrite -2!sum1_count big_mkcond /=.
rewrite [\sum_(_ <- _ | _ ≤ _)_]big_mkcond /=.
apply leq_sum; intros i _.
by destruct (task_is_scheduled job_task sched i t);
[by rewrite andbT | by rewrite andbF].
}
rewrite -count_filter -[count _ hp_tasks]count_filter.
eapply leq_trans with (n := count (predC (fun tsk ⇒ delta ≤ x tsk)) _);
last by apply eq_leq, eq_in_count; red; ins; rewrite ltnNge.
rewrite leq_subLR count_predC size_filter.
apply leq_trans with (n := count (other_scheduled_task t) ts);
[by rewrite INV | by rewrite count_filter].
}
{
unfold x at 2, total_interference_B.
rewrite exchange_big /=; apply leq_sum; intros t _.
destruct (backlogged job_arrival job_cost sched j t) eqn:BACK; last by ins.
rewrite mul1n -sum1_count.
rewrite big_mkcond [\sum_(i <- hp_tasks | _ < _) _]big_mkcond /=.
apply leq_sum_seq; move ⇒ tsk_k IN _.
destruct (x tsk_k < delta); [rewrite andTb | by rewrite andFb].
destruct (task_is_scheduled job_task sched tsk_k t) eqn:SCHED; last by done.
move: SCHED ⇒ /existsP [cpu SCHED].
by rewrite (bigD1 cpu) /= // SCHED.
}
Qed.
Lemma bertogna_fp_minimum_exceeds_interference :
∀ delta,
\sum_(tsk_k <- hp_tasks) x tsk_k ≥ delta × num_cpus →
\sum_(tsk_k <- hp_tasks) minn (x tsk_k) delta ≥
delta × num_cpus.
Proof.
intros delta SUMLESS.
set more_interf := fun tsk_k ⇒ x tsk_k ≥ delta.
rewrite [\sum_(_ <- _) minn _ _](bigID more_interf) /=.
unfold more_interf, minn.
rewrite [\sum_(_ <- _ | delta ≤ _)_](eq_bigr (fun i ⇒ delta));
last by intros i COND; rewrite leqNgt in COND; destruct (delta > x i).
rewrite [\sum_(_ <- _ | ~~_)_](eq_big (fun i ⇒ x i < delta)
(fun i ⇒ x i));
[| by red; ins; rewrite ltnNge
| by intros i COND; rewrite -ltnNge in COND; rewrite COND].
destruct (~~ has (fun i ⇒ delta ≤ x i) hp_tasks) eqn:HASa.
{
rewrite [\sum_(_ <- _ | _ ≤ _) _]big_hasC; last by apply HASa.
rewrite big_seq_cond; move: HASa ⇒ /hasPn HASa.
rewrite add0n (eq_bigl (fun i ⇒ (i \in hp_tasks) && true));
last by red; intros tsk_k; destruct (tsk_k \in hp_tasks) eqn:INk;
[by rewrite andTb ltnNge; apply HASa | by rewrite andFb].
by rewrite -big_seq_cond.
} apply negbFE in HASa.
destruct (num_tasks_exceeding delta ≥ num_cpus) eqn:CARD.
{
apply leq_trans with (delta × num_tasks_exceeding delta);
first by rewrite leq_mul2l; apply/orP; right.
unfold num_tasks_exceeding; rewrite -sum1_count big_distrr /=.
rewrite -[\sum_(_ <- _ | _) _]addn0.
by apply leq_add; [by apply leq_sum; ins; rewrite muln1|by ins].
} apply negbT in CARD; rewrite -ltnNge in CARD.
rewrite big_const_seq iter_addn addn0; fold num_tasks_exceeding.
apply leq_trans with (n := delta × num_tasks_exceeding delta +
delta × (num_cpus - num_tasks_exceeding delta));
first by rewrite -mulnDr subnKC //; apply ltnW.
rewrite leq_add2l; apply bertogna_fp_interference_in_non_full_processors.
by apply/andP; split; first by rewrite -has_count.
Qed.
Lemma bertogna_fp_sum_exceeds_total_interference:
\sum_((tsk_k, R_k) <- hp_bounds)
minn (x tsk_k) (R - task_cost tsk + 1) >
total_interference_bound_fp task_cost task_period tsk hp_bounds R.
Proof.
have EXCEEDS := bertogna_fp_minimum_exceeds_interference.
have ALLBUSY := bertogna_fp_interference_on_all_cpus.
have TOOMUCH := bertogna_fp_too_much_interference.
rename H_hp_bounds_has_interfering_tasks into HAS,
H_response_time_recurrence_holds into REC.
apply leq_trans with (n := \sum_(tsk_k <- hp_tasks) minn (x tsk_k) (R - task_cost tsk + 1));
last first.
{
rewrite (eq_bigr (fun i ⇒ minn (x (fst i)) (R - task_cost tsk + 1)));
last by ins; destruct i.
have MAP := @big_map _ 0 addn _ _ (fun x ⇒ fst x) hp_bounds (fun x ⇒ true) (fun y ⇒ minn (x y) (R - task_cost tsk + 1)).
rewrite -MAP.
apply leq_sum_sub_uniq; first by apply filter_uniq; destruct ts.
red; move ⇒ tsk0 IN0.
rewrite mem_filter in IN0; move: IN0 ⇒ /andP [INTERF0 IN0].
apply/mapP.
feed (HAS tsk0); first by done.
move: (HAS INTERF0) ⇒ [R0 IN].
by ∃ (tsk0, R0).
}
apply ltn_div_trunc with (d := num_cpus);
first by apply H_at_least_one_cpu.
rewrite -(ltn_add2l (task_cost tsk)) -REC.
rewrite -addn1 -leq_subLR.
rewrite -[R + 1 - _]addnBAC; last by rewrite REC; apply leq_addr.
rewrite leq_divRL; last by apply H_at_least_one_cpu.
apply EXCEEDS.
apply leq_trans with (n := X × num_cpus); last by rewrite ALLBUSY.
by rewrite leq_mul2r; apply/orP; right; apply TOOMUCH.
Qed.
Lemma bertogna_fp_exists_task_that_exceeds_bound :
∃ tsk_k R_k,
(tsk_k, R_k) \in hp_bounds ∧
(minn (x tsk_k) (R - task_cost tsk + 1) >
minn (workload_bound tsk_k R_k) (R - task_cost tsk + 1)).
Proof.
have SUM := bertogna_fp_sum_exceeds_total_interference.
rename H_hp_bounds_has_interfering_tasks into HASHP.
assert (HAS: has (fun tup : task_with_response_time ⇒
let (tsk_k, R_k) := tup in
(minn (x tsk_k) (R - task_cost tsk + 1) >
minn (workload_bound tsk_k R_k)(R - task_cost tsk + 1)))
hp_bounds).
{
apply/negP; unfold not; intro NOTHAS.
move: NOTHAS ⇒ /negP /hasPn ALL.
rewrite -[_ < _]negbK in SUM.
move: SUM ⇒ /negP SUM; apply SUM; rewrite -leqNgt.
rewrite (eq_bigr (fun i ⇒ minn (x (fst i)) (R - task_cost tsk + 1)));
last by ins; destruct i.
unfold total_interference_bound_fp.
rewrite big_seq_cond.
rewrite [\sum_(_ <- _ | true)_]big_seq_cond.
apply leq_sum.
intros p; rewrite andbT; intros IN.
by specialize (ALL p IN); destruct p; rewrite leqNgt.
}
move: HAS ⇒ /hasP HAS; destruct HAS as [[tsk_k R_k] HPk MINk]; ∃ tsk_k, R_k.
by repeat split.
Qed.
End DerivingContradiction.
End Lemmas.
Theorem bertogna_cirinei_response_time_bound_fp :
response_time_bounded_by tsk R.
Proof.
have WORKLOAD := bertogna_fp_workload_bounds_interference.
have EX := bertogna_fp_exists_task_that_exceeds_bound.
rename H_response_time_bounds_ge_cost into GE_COST,
H_interfering_tasks_miss_no_deadlines into NOMISS,
H_response_time_recurrence_holds into REC,
H_response_time_of_interfering_tasks_is_known into RESP,
H_hp_bounds_has_interfering_tasks into HAS,
H_response_time_no_larger_than_deadline into LEdl.
intros j ARRj JOBtsk.
remember (job_arrival j + R) as ctime.
generalize dependent j.
induction ctime as [ctime IH] using strong_ind.
intros j ARRj JOBtsk EQc; subst ctime.
assert (BEFOREok: ∀ j0,
arrives_in arr_seq j0 →
job_task j0 = tsk →
job_arrival j0 < job_arrival j →
service sched j0 (job_arrival j0 + R) ≥ job_cost j0).
{
by ins; apply IH; try (by done); rewrite ltn_add2r.
} clear IH.
unfold response_time_bounded_by, is_response_time_bound_of_task,
completed, completed_jobs_dont_execute, valid_sporadic_job in ×.
destruct (completed job_cost sched j (job_arrival j + R)) eqn:NOTCOMP;
first by done.
apply negbT in NOTCOMP; exfalso.
specialize (EX j ARRj JOBtsk NOTCOMP BEFOREok).
destruct EX as [tsk_k [R_k [HPk LTmin]]].
unfold minn at 1 in LTmin.
specialize (WORKLOAD j tsk_k R_k HPk).
destruct (W task_cost task_period tsk_k R_k R < R - task_cost tsk + 1); rewrite leq_min in LTmin;
last by move: LTmin ⇒ /andP [_ BUG]; rewrite ltnn in BUG.
move: LTmin ⇒ /andP [BUG _]; des.
apply leq_trans with (p := W task_cost task_period tsk_k R_k R) in BUG; last by done.
by rewrite ltnn in BUG.
Qed.
End ResponseTimeBound.
End ResponseTimeAnalysisFP.