Library prosa.analysis.facts.priority.sequential

In this section, we prove that, given two jobs j1 and j2, if job j1 arrives earlier than job j2 and j1 always has higher priority than j2, then j2 is scheduled only after j1 is completed.
Section SequentialJLFP.

Consider any type of jobs.
  Context {Job : JobType}.
  Context `{Arrival : JobArrival Job}.
  Context `{Cost : JobCost Job}.

Consider any arrival sequence with consistent arrival times.
Consider a JLFP-policy that indicates a higher-or-equal priority relation, and assume that this relation is transitive.
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.

... allow for any work-bearing notion of job readiness, ...
... and assume that the schedule is valid.
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....
... and that jobs must arrive to execute.
We show that, given two jobs j1 and j2, if job j1 arrives earlier than job j2 and j1 always has higher priority than j2, then j2 is scheduled only after j1 is completed.
  Lemma early_hep_job_is_scheduled :
     j1 j2,
      arrives_in arr_seq j1
      job_arrival j1 < job_arrival j2
      always_higher_priority j1 j2
       t,
        scheduled_at sched j2 t
        completed_by sched j1 t.
  Proof.
    movej1 j2 ARR LE AHP t SCHED.
    apply/negPn/negP; intros NCOMPL.
    edestruct scheduling_of_any_segment_starts_with_preemption_time
      as [pt [LET [PT ALL_SCHED]]] ⇒ //.
    move: LET ⇒ /andP [LE1 LE2].
    specialize (ALL_SCHED pt); feed ALL_SCHED; first by apply/andP; split.
    have PEND1: pending sched j1 pt.
    { apply/andP; split.
      - by rewrite /has_arrived; lia.
      - by move: NCOMPL; apply contra, completion_monotonic. }
    have [j3 [ARR3 [READY3 HEP3]]] := (H_job_ready _ _ ARR PEND1).
    move: (AHP pt) ⇒ /andP[HEP /negP NHEP]; eapply NHEP.
    apply: H_priority_is_transitive; last exact: HEP3.
    apply: H_respects_policy ⇒ //.
    apply/andP; split ⇒ //.
    apply/negPSCHED2.
    have EQ: j2 = j3 by apply: H_uniproc; eauto.
    by subst j2.
  Qed.

End SequentialJLFP.