# Library prosa.analysis.facts.priority.gel

In this file we state and prove some basic facts about the GEL scheduling policy.
Section GELBasicFacts.

Consider any type of tasks and jobs.

Under GEL, hep_job is a statement about absolute priority points.
Fact hep_job_priority_point :
j j',
Proof. by movej j'; rewrite /hep_job/GEL/job_priority_point. Qed.

If we are looking at two jobs of the same task, then hep_job is a statement about their respective arrival times.
Fact hep_job_arrival_gel :
j j',
hep_job j j' = (job_arrival j job_arrival j').
Proof.
movej j' /eqP SAME.
by rewrite hep_job_priority_point SAME; lia.
Qed.

Section HEPJobArrival.
Consider a job j...
Variable j : Job.

... and a higher or equal priority job j'.
Variable j' : Job.
Hypothesis H_j'_hep : hep_job j' j.

The arrival time of j' is bounded as follows.
Lemma hep_job_arrives_before :
((job_arrival j')%:R
(job_arrival j)%:R +
Proof. by move : H_j'_hep; rewrite hep_job_priority_point; lia. Qed.

Using the above lemma, we prove that for any higher-or-equal priority job j', the term job_arrival j + task_priority_point j) - task_priority_point j') is positive.
Corollary hep_job_arrives_after_zero :
(0 (job_arrival j)%:R +
Proof. exact: le_trans hep_job_arrives_before. Qed.

End HEPJobArrival.

Next, we prove that the GEL policy respects sequential tasks.
Proof. by movej1 j2 TSK ARR; rewrite hep_job_arrival_gel. Qed.

In this section, we prove that in a schedule following the GEL policy, tasks are always sequential.

Consider any arrival sequence.
Variable arr_seq : arrival_sequence Job.

Allow for any uniprocessor model.
Context {PState : ProcessorState Job}.
Hypothesis H_uniproc : uniprocessor_model PState.

Next, consider any schedule of the arrival sequence, ...
Variable sched : schedule PState.

Context `{JobCost Job}.
Hypothesis H_valid_arrivals : valid_arrival_sequence arr_seq.

... allow for any work-bearing notion of job readiness, ...
Context `{@JobReady Job PState _ Arrival}.

... and assume that the schedule is valid.
Hypothesis H_sched_valid : valid_schedule sched arr_seq.

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

... and assume that it defines a valid preemption model.
Next, we assume that the schedule respects the scheduling policy at every preemption point.
To prove sequentiality, we use the lemma early_hep_job_is_scheduled. Clearly, under the GEL priority policy, jobs satisfy the conditions described by the lemma (i.e., given two jobs j1 and j2 from the same task, if j1 arrives earlier than j2, then j1 always has a higher priority than job j2, and hence completes before j2); therefore GEL implies the sequential_tasks property.
Proof.
movej1 j2 t ARR1 ARR2 SAME LT.
apply: early_hep_job_is_scheduled ⇒ //.
rewrite always_higher_priority_jlfp !hep_job_arrival_gel //.
- by rewrite -ltnNge; apply/andP; split ⇒ //.
Qed.

End GELBasicFacts.

We add the following lemma to the basic facts database.
Global Hint Resolve