Library prosa.analysis.facts.model.ideal.schedule

Note: we do not re-export the basic definitions to avoid littering the global namespace with type class instances.
In this section we establish the classes to which an ideal schedule belongs.
Section ScheduleClass.

We assume ideal uni-processor schedules.
  #[local] Existing Instance ideal.processor_state.

  Local Transparent scheduled_in scheduled_on.
Consider any job type and the ideal processor model.
  Context {Job: JobType}.
  Context `{JobArrival Job}.
  Context `{JobCost Job}.

We note that the ideal processor model is indeed a uni-processor model.
  Lemma ideal_proc_model_is_a_uniprocessor_model:
    uniprocessor_model (processor_state Job).
    movej1 j2 sched t /existsP[[]/eqP E1] /existsP[[]/eqP E2].
    by move: E1 E2 =>->[].

By definition, service_in is the sum of the service received in total across all cores. In the ideal uniprocessor model, however, there is only one "core," which is expressed by using unit as the type of cores. The type unit only has a single member tt, which serves as a placeholder. Consequently, the definition of service_in simplifies to a single term of the sum, the service on the one core, which we note with the following lemma that relates service_in to service_on.
  Lemma service_in_service_on (j : Job) s :
    service_in j s = service_on j s tt.
    by rewrite /service_in sum_unit1.

Furthermore, since the ideal uniprocessor state is represented by the option Job type, service_in further simplifies to a simple equality comparison, which we note next.
  Lemma service_in_def (j : Job) (s : processor_state Job) :
    service_in j s = (s == Some j).
    by rewrite service_in_service_on.

We observe that the ideal processor model falls into the category of ideal-progress models, i.e., a scheduled job always receives service.
  Lemma ideal_proc_model_ensures_ideal_progress:
    ideal_progress_proc_model (processor_state Job).
    movej s /existsP[[]/eqP->] /=.
    by rewrite service_in_def /= eqxx /nat_of_bool.

The ideal processor model is a unit-service model.
  Lemma ideal_proc_model_provides_unit_service:
    unit_service_proc_model (processor_state Job).
    movej s.
    rewrite service_in_def /= /nat_of_bool.
    by case:ifP.

The ideal processor model is a unit-supply model.
  Lemma ideal_proc_model_provides_unit_supply :
    unit_supply_proc_model (processor_state Job).
    by moves; rewrite /supply_in /index_enum sum_unit1.

  Lemma scheduled_in_def (j : Job) s :
    scheduled_in j s = (s == Some j).
    rewrite /scheduled_in/scheduled_on/=.
    case: existsP=>[[_->]//|].
    case: (s == Some j)=>//=[[]].
    by .

  Lemma scheduled_at_def sched (j : Job) t :
    scheduled_at sched j t = (sched t == Some j).
      by rewrite /scheduled_at scheduled_in_def.

The ideal processor model is a fully supply-consuming processor model.
  Lemma ideal_proc_model_fully_consuming :
    fully_consuming_proc_model (processor_state Job).
    movej S t SCHED.
    rewrite /service_at /supply_at /supply_in service_in_def.
    move: SCHED; rewrite scheduled_at_def ⇒ →.
    by rewrite sum_unit1.

  Lemma service_in_is_scheduled_in (j : Job) s :
    service_in j s = scheduled_in j s.
    by rewrite service_in_def scheduled_in_def.

  Lemma service_at_is_scheduled_at sched (j : Job) t :
    service_at sched j t = scheduled_at sched j t.
      by rewrite /service_at service_in_is_scheduled_in.

  Lemma service_on_def (j : Job) (s : processor_state Job) c :
    service_on j s c = (s == Some j).
  Proof. by []. Qed.

  Lemma service_at_def sched (j : Job) t :
    service_at sched j t = (sched t == Some j).
    by rewrite /service_at service_in_def.

The ideal uniprocessor always has supply.
  Lemma ideal_proc_has_supply :
     (sched : schedule (ideal.processor_state Job)) (t : instant),
      has_supply sched t.
    movesched t; rewrite /has_supply /supply_at /supply_in //=.
    by rewrite sum_unit1.

Next we prove a lemma which helps us to do a case analysis on the state of an ideal schedule.
  Lemma ideal_proc_model_sched_case_analysis:
     (sched : schedule (ideal.processor_state Job)) (t : instant),
      ideal_is_idle sched t j, scheduled_at sched j t.
    movesched t.
    unfold ideal_is_idle; simpl; destruct (sched t) as [s|] eqn:EQ.
    - by right; s; auto; rewrite scheduled_at_def EQ.
    - by left; auto.

We prove that if a job j is scheduled at a time instant t, then the scheduler is not idle at t.
  Lemma ideal_sched_implies_not_idle sched (j : Job) t :
    scheduled_at sched j t
    ¬ ideal_is_idle sched t.
    rewrite scheduled_at_def ⇒ /eqP SCHED /eqP IDLE.
    by rewrite IDLE in SCHED; inversion SCHED.

On a similar note, if a scheduler is idle at a time instant t, then no job can receive service at t.
  Lemma ideal_not_idle_implies_sched sched (j : Job) t :
    ideal_is_idle sched t
    service_at sched j t = 0.
  Proof. by rewrite service_at_is_scheduled_at scheduled_at_def ⇒ /eqP →. Qed.

In the following, we relate the ideal uniprocessor state to the generic definition job_scheduled_at. Specifically, the two notions are equivalent. To show this, require an arrival sequence in context.
Consider any arrival sequence ...
    Variable arr_seq : arrival_sequence Job.
    Context `{JobArrival Job}.

... and any ideal uni-processor schedule of this arrival sequence.
    Variable sched : schedule (ideal.processor_state Job).

Suppose all jobs come from the arrival sequence and do not execute before their arrival time (which must be consistent with the arrival sequence).
The generic notion scheduled_job_at coincides with our notion of ideal processor state. This observation allows cutting out the generic notion in proofs specific to ideal uniprocessor schedules.
    Lemma scheduled_job_at_def :
        scheduled_job_at arr_seq sched t = sched t.
      case: (scheduled_at_dec arr_seq _ sched _ _ t) ⇒ //[[j SCHED]|NS].
      { move: (SCHED); rewrite scheduled_at_def ⇒ /eqP →.
        by apply/eqP; rewrite scheduled_job_at_scheduled_at
           ; auto using ideal_proc_model_is_a_uniprocessor_model. }
      { move: (NS); rewrite -scheduled_job_at_none; eautoNONE.
        case SCHED: (sched t) ⇒ [j|]; last by rewrite NONE.
        move: SCHED ⇒ /eqP; rewrite -scheduled_at_def.
        by move: (NS j) ⇒ /negP. }

Similarly, the generic and specific notions of idle instants coincide, too.
    Lemma is_idle_def :
        is_idle arr_seq sched t = ideal_is_idle sched t.
      rewrite /is_idle/ideal_is_idle -scheduled_job_at_def /scheduled_job_at.
      by case: (scheduled_jobs_at _ _ _).

  End RelationToGenericScheduledJob.

End ScheduleClass.


We add the above lemmas into a "Hint Database" basic_rt_facts, so Coq will be able to apply them automatically.
Global Hint Resolve ideal_proc_model_is_a_uniprocessor_model
     ideal_proc_has_supply : basic_rt_facts.

We also provide tactics for case analysis on ideal processor state.
The first tactic generates two sub-goals: one with idle processor and the other with processor executing a job named JobName.
Ltac ideal_proc_model_sched_case_analysis sched t JobName :=
  let Idle := fresh "Idle" in
  let Sched := fresh "Sched_" JobName in
  destruct (ideal_proc_model_sched_case_analysis sched t) as [Idle | [JobName Sched]].

The second tactic is similar to the first, but it additionally generates two equalities: sched t = None and sched t = Some j.
Ltac ideal_proc_model_sched_case_analysis_eq sched t JobName :=
  let Idle := fresh "Idle" in
  let IdleEq := fresh "Eq" Idle in
  let Sched := fresh "Sched_" JobName in
  let SchedEq := fresh "Eq" Sched in
  destruct (ideal_proc_model_sched_case_analysis sched t) as [Idle | [JobName Sched]];
  [move: (Idle) ⇒ /eqP IdleEq; rewrite ?IdleEq
  | move: (Sched); simpl; move ⇒ /eqP SchedEq; rewrite ?SchedEq].