Library prosa.analysis.facts.model.scheduled

Require Export prosa.model.schedule.scheduled.
Require Export prosa.model.processor.platform_properties.
Require Export prosa.analysis.facts.behavior.arrivals.

Correctness of the Scheduled Job(s)

In this file, we establish the correctness of the computable definition of the (set of) scheduled job(s) and a useful case-analysis lemma.

Section ScheduledJobs.

Consider any type of jobs with arrival times.

Context {Job : JobType}.
Context `{JobArrival Job}.

Next, consider any kind of processor state model, ...

Context {PState : ProcessorState Job}.

... any valid arrival sequence, ...

Variable arr_seq : arrival_sequence Job.
Hypothesis H_valid_arrivals : valid_arrival_sequence arr_seq.

... and any schedule.

Variable sched : schedule PState.

Now we establish the validity conditions under which these definitions yield the expected results: all jobs must come from the arrival sequence and do not execute before their arrival.

NB: We do not use valid_schedule here to avoid introducing a dependency on a readiness mode.

  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 sched.

The Set of Jobs Scheduled at a Given Time

Under these assumptions, scheduled_jobs_at is correct: a job is included in the sequence if and only if it scheduled.

  Lemma scheduled_jobs_at_iff :
    ∀ j t,
      j \in scheduled_jobs_at arr_seq sched t = scheduled_at sched j t.
  Proof.
    move⇒ j t; rewrite mem_filter; apply: andb_idr ⇒ SCHED.
    by apply: arrivals_before_scheduled_at; rt_eauto.
  Qed.

Conversely, if no jobs are in the sequence, then no job is scheduled.

  Lemma scheduled_jobs_at_nil :
    ∀ t,
      nilp (scheduled_jobs_at arr_seq sched t)
      ↔ (∀ j, ~~ scheduled_at sched j t ).
  Proof.
    move⇒ t; split ⇒ [/nilP NIL j|NS]
    ; first by apply: contraT ⇒ /negbNE
               ; rewrite -scheduled_jobs_at_iff NIL in_nil.
    apply /nilP.
    case NN: (scheduled_jobs_at _) ⇒ [//|j js].
    exfalso.
    have: scheduled_at sched j t; last by move: (NS j) ⇒ /negP.
    rewrite -scheduled_jobs_at_iff NN.
    exact: mem_head.
  Qed.

The Job Scheduled on a Uniprocessor

For convenience, we note similar rewriting rules for uniprocessors.

Section Uniprocessors.

Suppose we are looking at a uniprocessor.

Hypothesis H_uni : uniprocessor_model PState.

Then clearly there is at most one job scheduled at any time t.

    Corollary scheduled_jobs_at_seq1 :
      ∀ t,
        size (scheduled_jobs_at arr_seq sched t) ≤ 1.
    Proof.
      move⇒ t; set sja := scheduled_jobs_at _ _ _; rewrite leqNgt.
      have uniq_sja : uniq sja by apply/filter_uniq/arrivals_uniq; rt_eauto.
      apply/negP ⇒ /exists_two/(_ uniq_sja)[j1 [j2 [j1j2 [j1sja j2sja]]]].
      by apply/j1j2/(H_uni j1 j2 sched t); rewrite -scheduled_jobs_at_iff.
    Qed.

Then scheduled_job_at t is correct: it yields some job j if and only if j is scheduled at time t.

    Corollary scheduled_job_at_iff :
      ∀ j t,
        scheduled_job_at arr_seq sched t = Some j ↔ scheduled_at sched j t.
    Proof.
      move⇒ j t; rewrite -scheduled_jobs_at_iff /scheduled_job_at.
      have := scheduled_jobs_at_seq1 t.
      case: scheduled_jobs_at ⇒ [//|j' [|//]] _ /=.
      by rewrite in_cons in_nil orbF; split ⇒ [[]->|/eqP ->].
    Qed.

Conversely, if scheduled_job_at t returns None, then no job is scheduled at time t.

    Corollary scheduled_job_at_none :
      ∀ t,
        scheduled_job_at arr_seq sched t = None
        ↔ (∀ j, ~~ scheduled_at sched j t ).
    Proof.
      move⇒ t. rewrite /scheduled_job_at -scheduled_jobs_at_nil.
      by case: (scheduled_jobs_at arr_seq sched t).
    Qed.

  End Uniprocessors.

Case Analysis: a Scheduled Job Exists or no Job is Scheduled

Last but not least, we note a case analysis that results from the above considerations: at any point in time, either some job is scheduled, or it holds that all jobs are not scheduled.

  Lemma scheduled_at_dec :
    ∀ t,
      {∃ j , scheduled_at sched j t } + {∀ j, ~~ scheduled_at sched j t }.
  Proof.
    move⇒ t.
    case SJA: (scheduled_jobs_at arr_seq sched t) ⇒ [|j js].
    - by right; rewrite -scheduled_jobs_at_nil SJA.
    - left; ∃ j.
      rewrite -scheduled_jobs_at_iff SJA.
      exact: mem_head.
  Qed.

End ScheduledJobs.