Library rt.restructuring.analysis.basic_facts.ideal_schedule

From mathcomp Require Import all_ssreflect.
Require Export rt.restructuring.model.processor.platform_properties.

Throughout this file, we assume ideal uniprocessor schedules.

Require Import rt.restructuring.model.processor.ideal.

Note: we do not re-export the basic definitions to avoid littering the global namespace with type class instances.

In this section we estlish the classes to which an ideal schedule belongs.

Section ScheduleClass.

Local Transparent service_in 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 uniprocessor model.

  Lemma ideal_proc_model_is_a_uniprocessor_model:
    uniprocessor_model (processor_state Job).
  Proof.
    move⇒ j1 j2 sched t /existsP[[]/eqP E1] /existsP[[]/eqP E2].
    by move: E1 E2=>->[].
  Qed.

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).
  Proof.
    move⇒ j s /existsP[[]/eqP->]/=.
    by rewrite eqxx.
  Qed.

The ideal processor model is a unit-service model.

  Lemma ideal_proc_model_provides_unit_service:
    unit_service_proc_model (processor_state Job).
  Proof.
    move⇒ j s.
    rewrite /service_in/=/nat_of_bool.
    by case:ifP.
  Qed.

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

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

  Lemma service_in_is_scheduled_in (j : Job) s :
    service_in j s = scheduled_in j s.
  Proof.
    by rewrite /service_in scheduled_in_def/=.
  Qed.

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

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),
      is_idle sched t ∨ ∃ j, scheduled_at sched j t.
  Proof.
    intros.
    unfold is_idle; simpl; destruct (sched t) eqn:EQ.
    - by right; ∃ s; auto; rewrite scheduled_at_def EQ.
    - by left; auto.
  Qed.

End ScheduleClass.

Automation

We add the above lemmas into a "Hint Database" basic_facts, so Coq will be able to apply them automatically.

Hint Resolve ideal_proc_model_is_a_uniprocessor_model
ideal_proc_model_ensures_ideal_progress
ideal_proc_model_provides_unit_service : basic_facts.

We also provide tactics for case analysis on ideal processor state.

The first tactic generates two subgoals: 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].