Library prosa.analysis.facts.preemption.task.floating

Require Export prosa.analysis.facts.preemption.job.limited.

Furthermore, we assume the task model with floating non-preemptive regions.

Require Import prosa.model.preemption.limited_preemptive.
Require Import prosa.model.task.preemption.floating_nonpreemptive.

Platform for Floating Non-Preemptive Regions Model

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

Section FloatingNonPreemptiveRegionsModel.

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}.

In addition, we assume the existence of a function mapping a task to its maximal non-preemptive segment ...

Context `{TaskMaxNonpreemptiveSegment Task}.

.. and the existence of functions mapping a job to the sequence of its preemption points.

Context `{JobPreemptionPoints Job}.

Consider any arrival sequence.

Variable arr_seq : arrival_sequence Job.

Next, consider any ideal uni-processor preemption-aware schedule of this arrival sequence ...

  Variable sched : schedule (ideal.processor_state Job).
  Hypothesis H_preemption_aware_schedule:
    schedule_respects_preemption_model arr_seq sched.

... where jobs do not execute before their arrival or after completion.

Hypothesis H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched.
Hypothesis H_completed_jobs_dont_execute : completed_jobs_dont_execute sched.

Next, we assume that preemption points are defined by the model with floating non-preemptive regions.

Hypothesis H_valid_model_with_floating_nonpreemptive_regions:
valid_model_with_floating_nonpreemptive_regions arr_seq.

Then, we prove that the job_preemptable and task_max_nonpreemptive_segment functions define a model with bounded non-preemptive regions.

  Lemma floating_preemption_points_model_is_model_with_bounded_nonpreemptive_regions:
    model_with_bounded_nonpreemptive_segments arr_seq.
  Proof.
    intros j ARR.
    move: (H_valid_model_with_floating_nonpreemptive_regions) ⇒ LIM; move: LIM (LIM) ⇒ [LIM L] [[BEG [END NDEC]] MAX].
    case: (posnP (job_cost j)) ⇒ [ZERO|POS].
    - split.
      rewrite /job_respects_max_nonpreemptive_segment /job_max_nonpreemptive_segment
              /lengths_of_segments /parameter.job_preemption_points; rewrite ZERO; simpl.
      rewrite /job_preemptable /limited_preemptions_model; erewrite zero_in_preemption_points; eauto 2.
      + move ⇒ progr; rewrite ZERO leqn0; move ⇒ /andP [_ /eqP LE].
        ∃ 0; rewrite LE; split; first by apply/andP; split.
          by eapply zero_in_preemption_points; eauto 2.
    - split; last (move ⇒ progr /andP [_ LE]; destruct (progr \in job_preemption_points j) eqn:NotIN).
      + by apply MAX.
      + ∃ progr; split; first apply/andP; first split; rewrite ?leq_addr; by done.
      + move: NotIN ⇒ /eqP; rewrite eqbF_neg; move ⇒ NotIN.
        edestruct (work_belongs_to_some_nonpreemptive_segment arr_seq) as [x [SIZE2 N]]; eauto 2. move: N ⇒ /andP [N1 N2].
        set ptl := nth 0 (job_preemption_points j) x.
        set ptr := nth 0 (job_preemption_points j) x.+1.
        ∃ ptr; split; first last.
        × by unfold job_preemptable, limited_preemptions_model; apply mem_nth.
        × apply/andP; split; first by apply ltnW.
          apply leq_trans with (ptl + (job_max_nonpreemptive_segment j - ε ) + 1); first last.
          -- rewrite addn1 ltn_add2r; apply N1.
          -- unfold job_max_nonpreemptive_segment.
             rewrite -addnA -leq_subLR -(leq_add2r 1).
             rewrite [in X in _ ≤ X]addnC -leq_subLR.
             rewrite !subn1 !addn1 prednK.
             { rewrite -[_.+1 .-1]pred_Sn. rewrite /lengths_of_segments.
               erewrite job_parameters_max_np_to_job_limited; eauto.
                 by apply distance_between_neighboring_elements_le_max_distance_in_seq. }
             { rewrite /lengths_of_segments; erewrite job_parameters_max_np_to_job_limited; eauto.
               apply max_distance_in_nontrivial_seq_is_positive; first by eauto 2.
               ∃ 0, (job_cost j); repeat split.
               - by eapply zero_in_preemption_points; eauto.
               - by eapply job_cost_in_nonpreemptive_points; eauto.
               - by apply/eqP; rewrite eq_sym -lt0n; apply POS.
             }
  Qed.

Which together with lemma valid_fixed_preemption_points_model gives us the fact that functions job_preemptable and task_max_nonpreemptive_segment define a valid preemption model with bounded non-preemptive regions.

  Corollary floating_preemption_points_model_is_valid_model_with_bounded_nonpreemptive_regions:
    valid_model_with_bounded_nonpreemptive_segments arr_seq sched.
  Proof.
    split.
    - apply valid_fixed_preemption_points_model_lemma; auto.
        by apply H_valid_model_with_floating_nonpreemptive_regions.
    - apply floating_preemption_points_model_is_model_with_bounded_nonpreemptive_regions.
  Qed.

End FloatingNonPreemptiveRegionsModel.

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

Hint Resolve
     valid_fixed_preemption_points_model_lemma
     floating_preemption_points_model_is_model_with_bounded_nonpreemptive_regions
     floating_preemption_points_model_is_valid_model_with_bounded_nonpreemptive_regions : basic_facts.