# Run-to-Completion Threshold of a job

In this module, we provide a sufficient condition under which a job receives enough service to become non-preemptive. Previously we defined the notion of run-to-completion threshold (see file abstract.run_to_completion_threshold.v). Run-to-completion threshold is the amount of service after which a job cannot be preempted until its completion. In this section we prove that if cumulative interference inside a busy interval is bounded by a certain constant then a job executes long enough to reach its run-to-completion threshold and become non-preemptive.
Consider any type of jobs.
Context {Job : JobType}.
Context `{JobArrival Job}.
Context `{JobCost Job}.

In addition, we assume existence of a function mapping jobs to their preemption points.
Context `{JobPreemptable Job}.

Consider any kind of uni-service ideal processor state model.
Consider any arrival sequence with consistent arrivals ...
... and any schedule of this arrival sequence.
Variable sched : schedule PState.

Assume we are provided with abstract functions for interference and interfering workload.
For simplicity, let's define some local names.
We assume that the schedule is work-conserving.
Let j be any job of task tsk with positive cost.
Variable j : Job.
Hypothesis H_j_arrives : arrives_in arr_seq j.
Hypothesis H_job_cost_positive : job_cost_positive j.

Next, consider any busy interval `[t1, t2)` of job j.
Variable t1 t2 : instant.
Hypothesis H_busy_interval : busy_interval j t1 t2.

First, we prove that job j completes by the end of the busy interval. Note that the busy interval contains the execution of job j, in addition time instant t2 is a quiet time. Thus by the definition of a quiet time the job should be completed before time t2.
Lemma job_completes_within_busy_interval:
completed_by sched j t2.
Proof.
move: (H_busy_interval) ⇒ [[/andP [_ LT] [_ _]] [_ QT2]].
unfold pending, has_arrived in QT2.
move: QT2; rewrite /pending negb_and; move ⇒ /orP [QT2|QT2].
{ by move: QT2 ⇒ /negP QT2; exfalso; apply QT2, ltnW. }
by rewrite Bool.negb_involutive in QT2.
Qed.

In this section we show that the cumulative interference is a complement to the total time where job j is scheduled inside the busy interval.
Consider any sub-interval `[t, t + delta)` inside the busy interval t1, t2).
Variables (t : instant) (delta : duration).
Hypothesis H_greater_than_or_equal : t1 t.
Hypothesis H_less_or_equal: t + delta t2.

We prove that sum of cumulative service and cumulative interference in the interval `[t, t + delta)` is equal to delta.
Lemma interference_is_complement_to_schedule:
service_during sched j t (t + delta) + cumul_interference j t (t + delta) = delta.
Proof.
rewrite /service_during /cumul_interference/service_at.
rewrite -big_split //=.
rewrite -{2}(sum_of_ones t delta).
rewrite big_nat [in RHS]big_nat.
apply: eq_bigrx /andP[Lo Hi].
move: (H_work_conserving j t1 t2 x) ⇒ Workj.
feed_n 4 Workj; try done.
{ by apply/andP; split; eapply leq_trans; eauto 2. }
destruct interference.
- replace (service_in _ _) with 0; auto; symmetry.
apply no_service_not_scheduled; auto.
now apply/negP; intros SCHED; apply Workj in SCHED; apply SCHED.
- replace (service_in _ _) with 1; auto; symmetry.
apply/eqP; rewrite eqn_leq; apply/andP; split.
+ apply H_unit_service_proc_model.
+ now apply H_ideal_progress_proc_model, Workj.
Qed.

End InterferenceIsComplement.

In this section, we prove a sufficient condition under which job j receives enough service.
Let progress_of_job be the desired service of job j.
Variable progress_of_job : duration.
Hypothesis H_progress_le_job_cost : progress_of_job job_cost j.

Assume that for some delta, the sum of desired progress and cumulative interference is bounded by delta (i.e., the supply).
Variable delta : duration.
progress_of_job + cumul_interference j t1 (t1 + delta) delta.

Then, it must be the case that the job has received no less service than progress_of_job.
service sched j (t1 + delta) progress_of_job.
Proof.
case NEQ: (t1 + delta t2); last first.
{ intros.
have L8 := job_completes_within_busy_interval.
apply leq_trans with (job_cost j); first by done.
rewrite /service.
rewrite -(service_during_cat _ _ _ t2).
apply leq_trans with (service_during sched j 0 t2); [by done | by rewrite leq_addr].
by apply/andP; split; last (apply negbT in NEQ; apply ltnW; rewrite ltnNge).
}
apply leq_subRL_impl in BOUND.
apply leq_trans with (delta - cumul_interference j t1 (t1 + delta)); first by done.
apply leq_trans with (service_during sched j t1 (t1 + delta)).
{ rewrite -{1}[delta](interference_is_complement_to_schedule t1) //.
}
{ rewrite /service -[X in _ X](service_during_cat _ _ _ t1).
by apply/andP; split; last rewrite leq_addr.
}
}
Qed.

End InterferenceBoundedImpliesEnoughService.

In this section we prove a simple lemma about completion of a job after is reaches run-to-completion threshold.
Assume that completed jobs do not execute ...
.. and the preemption model is valid.
Then, job j must complete in job_cost j - job_rtct j time units after it reaches run-to-completion threshold.
Lemma job_completes_after_reaching_run_to_completion_threshold:
t,
job_rtct j service sched j t
completed_by sched j (t + (job_cost j - job_rtct j)).
Proof.
movet ES.
set (job_cost j - job_rtct j) as job_last.
have LSNP := @job_nonpreemptive_after_run_to_completion_threshold
Job H0 H1 _ arr_seq sched _ j _ t.
apply negbNE; apply/negP; intros CONTR.
have SCHED: t', t t' t + job_last scheduled_at sched j t'.
{ movet' /andP [GE LT].
rewrite -[t'](@subnKC t) //.
eapply LSNP; eauto 2; first by rewrite leq_addr.
rewrite subnKC //.
apply/negP; intros COMPL.
move: CONTR ⇒ /negP Temp; apply: Temp.
exact: completion_monotonic COMPL.
}
have SERV: job_last + 1 \sum_(t t' < t + (job_last + 1)) service_at sched j t'.
rewrite big_nat_cond [in X in _ X]big_nat_cond.
rewrite leq_sum //.
movet' /andP [NEQ _].
apply H_ideal_progress_proc_model; apply SCHED.
}
move: (service_at_most_cost
_ H_completed_jobs_dont_execute j H_unit_service_proc_model
(t + job_last.+1)).
rewrite leqNgt; move ⇒ /negP T; apply: T.
rewrite /service -(service_during_cat _ _ _ t); last by (apply/andP; split; last rewrite leq_addr).
apply leq_trans with (job_rtct j + service_during sched j t (t + job_last.+1));