Library prosa.analysis.facts.readiness.sequential

Throughout this file, we assume the sequential task readiness model, which means that a job is ready to execute only if all prior jobs of the same task have completed.
In this section, we show some useful properties of the sequential task readiness model.
Consider any type of job associated with any type of tasks ...
  Context {Job : JobType}.
  Context {Task : TaskType}.
  Context `{JobTask Job Task}.
  Context `{JobArrival Job}.
  Context `{JobCost Job}.

... and any kind of processor state.
  Context {PState : ProcessorState Job}.

Consider any arrival sequence with consistent arrivals.
Recall that we assume sequential tasks.
Consider any valid schedule of arr_seq.
  Variable sched : schedule PState.
  Hypothesis H_valid_schedule : valid_schedule sched arr_seq.

Consider an FP policy that indicates a reflexive higher-or-equal priority relation.
  Context `{FP_policy Task}.
  Hypothesis H_priority_is_reflexive : reflexive_priorities.

First, we show that the sequential readiness model is non-clairvoyant.
  Fact sequential_readiness_nonclairvoyance :
    nonclairvoyant_readiness sequential_readiness.
    intros sched1 sched2 ? ? ID ? LE; rewrite //=.
    erewrite identical_prefix_pending; eauto 2.
    destruct (boolP (pending sched2 j t)) as [_ | _] ⇒ //=.
    destruct (boolP (prior_jobs_complete arr_seq sched2 j t)) as [ALL | NOT_ALL]; apply/eqP.
    - rewrite eqb_id; apply/allP; intros ? IN.
      move: ALL ⇒ /allP ALL; specialize (ALL x IN).
      by erewrite identical_prefix_completed_by; eauto 2.
    - move: NOT_ALL ⇒ /allPn [x IN NCOMP].
      rewrite eqbF_neg; apply/allPn; x ⇒ //.
      by erewrite identical_prefix_completed_by; eauto 2.

Next, we show that the sequential readiness model ensures that tasks are sequential. That is, that jobs of the same task execute in order of their arrival.
  Lemma sequential_readiness_implies_sequential_tasks :
    sequential_tasks arr_seq sched.
    intros j1 j2 t ARR1 ARR2 SAME LT SCHED.
    destruct (boolP (job_ready sched j2 t)) as [READY | NREADY].
    - move: READY ⇒ /andP [PEND /allP ALL]; apply: ALL.
      rewrite mem_filter; apply/andP; split; first by done.
      by apply arrived_between_implies_in_arrivals ⇒ //.
    - exfalso; apply/(negP NREADY)/job_scheduled_implies_ready ⇒ //.
      exact: (valid_schedule_jobs_must_be_ready_to_execute sched arr_seq).

Finally, we show that the sequential readiness model is a work-bearing readiness model. That is, if a job j is pending but not ready at a time instant t, then there exists another (earlier) job of the same task that is pending and ready at time t.
  Lemma sequential_readiness_implies_work_bearing_readiness :
    work_bearing_readiness arr_seq sched.
    intros j.
    have EX: k, job_arrival j k by ( (job_arrival j)).
    destruct EX as [k LE]; move: j LE.
    induction k; intros ? ? ? ARR PEND.
    { destruct (boolP (job_ready sched j t)) as [READY | NREADY].
      { by j; repeat split; eauto using (H_priority_is_reflexive 0). }
      { move: NREADY; rewrite //= PEND Bool.andb_true_l ⇒ /allPn [jhp IN NCOMP].
        apply arrives_in_task_arrivals_before_implies_arrives_before in IN; last by done.
        by exfalso; move: LE; rewrite leqn0 ⇒ /eqP EQ; rewrite EQ in IN.
    { move: LE; rewrite leq_eqVlt ltnS ⇒ /orP [/eqP EQ | LE]; last by apply IHk.
      destruct (boolP (job_ready sched j t)) as [READY | NREADY].
      { by j; repeat split; eauto using (H_priority_is_reflexive 0). }
      { move: NREADY; rewrite //= PEND Bool.andb_true_l ⇒ /allPn [j' IN NCOMP].
        have LE' : job_arrival j' k.
        { by apply arrives_in_task_arrivals_before_implies_arrives_before in IN; rewrite // -ltnS -EQ. }
        have ARR' : arrives_in arr_seq j'.
        { by eapply arrives_in_task_arrivals_implies_arrived; eauto 2. }
        have PEND' : pending sched j' t.
        { apply/andP; split; last by done.
          move: PEND ⇒ /andP [LE _].
          by unfold has_arrived in *; lia.
        specialize (IHk j' LE' t ARR' PEND').
        destruct IHk as [j'' [ARR'' [READY'' HEP'']]].
         j''; repeat split; auto.
        clear EQ; apply arrives_in_task_arrivals_implies_job_task in IN; move: IN ⇒ /eqP EQ.
        by rewrite /hep_job /FP_to_JLFP -EQ.

End SequentialTasksReadiness.