Library prosa.analysis.facts.preemption.job.preemptive

Preemptions in Fully Preemptive Model

In this section, we prove that instantiation of predicate job_preemptable to the fully preemptive model indeed defines a valid preemption model.
We assume that jobs are fully preemptive.
  #[local] Existing Instance fully_preemptive_job_model.

Consider any type of jobs.
  Context {Job : JobType}.
  Context `{JobArrival Job}.
  Context `{JobCost Job}.

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.

Then, we prove that fully_preemptive_model is a valid preemption model.
  Lemma valid_fully_preemptive_model:
    valid_preemption_model arr_seq sched.
      by intros j ARR; repeat split; intros t CONTR.

We also prove that under the fully preemptive model job_max_nonpreemptive_segment j is equal to 0, when job_cost j = 0 ...
  Lemma job_max_nps_is_0:
      job_cost j = 0
      job_max_nonpreemptive_segment j = 0.
    movej H1.
    rewrite /job_max_nonpreemptive_segment /lengths_of_segments
    by rewrite H1.

... or ε when job_cost j > 0.
  Lemma job_max_nps_is_ε:
      job_cost j > 0
      job_max_nonpreemptive_segment j = ε.
    intros ? POS.
    rewrite /job_max_nonpreemptive_segment /lengths_of_segments
    rewrite /job_preemptable /fully_preemptive_job_model.
    rewrite filter_predT.
    apply max0_of_uniform_set.
    - rewrite /range /index_iota subn0.
      rewrite [size _]pred_Sn -[in X in _ X]addn1 -size_of_seq_of_distances size_iota.
      + by rewrite -pred_Sn.
      + by rewrite ltnS.
    - by apply distances_of_iota_ε.

End FullyPreemptiveModel.
Global Hint Resolve valid_fully_preemptive_model : basic_rt_facts.