Library prosa.analysis.facts.preemption.task.preemptive

Require Export prosa.analysis.facts.preemption.job.preemptive.
Require Export prosa.model.task.preemption.fully_preemptive.

Platform for Fully Preemptive Model

In this section, we prove that the instantiations of the functions job_preemptable and task_max_nonpreemptive_segment for the fully preemptive model indeed defines a valid preemption model with bounded non-preemptive regions.

Section FullyPreemptiveModel.

Consider any type of tasks ...

Context {Task : TaskType}.
Context `{TaskCost Task}.

... and any type of jobs associated with these tasks.

  Context {Job : JobType}.
  Context `{JobTask Job Task}.
  Context `{JobArrival Job}.
  Context `{JobCost Job}.

Assume that jobs and tasks are fully preemptive.

  #[local] Existing Instance fully_preemptive_job_model.
  #[local] Existing Instance fully_preemptive_task_model.
  #[local] Existing Instance fully_preemptive_rtc_threshold.

Consider any kind of processor state model, ...

Context {PState : ProcessorState Job}.

... any job arrival sequence, ...

Variable arr_seq : arrival_sequence Job.

... and any given schedule.

Variable sched : schedule PState.

We prove that the fully_preemptive_model function defines a model with bounded non-preemptive regions.

  Lemma fully_preemptive_model_is_model_with_bounded_nonpreemptive_regions:
    model_with_bounded_nonpreemptive_segments arr_seq.
  Proof.
    intros j ARR; split.
    - case: (posnP (job_cost j)) ⇒ [ZERO|POS].
      + by rewrite /job_respects_max_nonpreemptive_segment job_max_nps_is_0.
      + by rewrite /job_respects_max_nonpreemptive_segment job_max_nps_is_ε.
    - intros t; ∃ t; split.
      + by apply/andP; split; [ done | rewrite leq_addr].
      + by done.
  Qed.

Which together with lemma valid_fully_preemptive_model gives us the fact that fully_preemptive_model defined a valid preemption model with bounded non-preemptive regions.

  Corollary fully_preemptive_model_is_valid_model_with_bounded_nonpreemptive_segments:
    valid_model_with_bounded_nonpreemptive_segments arr_seq sched.
  Proof.
    split.
    apply valid_fully_preemptive_model.
    apply fully_preemptive_model_is_model_with_bounded_nonpreemptive_regions.
  Qed.

End FullyPreemptiveModel.

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

Global Hint Resolve
     valid_fully_preemptive_model
     fully_preemptive_model_is_model_with_bounded_nonpreemptive_regions
     fully_preemptive_model_is_valid_model_with_bounded_nonpreemptive_segments : basic_rt_facts.