Library prosa.analysis.facts.model.sequential

Consider any type of job associated with any type of tasks...

  Context {Job: JobType}.
  Context {Task: TaskType}.
  Context `{JobTask Job Task}.

... with arrival times and costs ...

Context `{JobArrival Job}.
Context `{JobCost Job}.

... and any kind of processor state model.

Trivially, same_task is symmetric.

  Remark same_task_sym :
    ∀ j1 j2,
      same_task j1 j2 = same_task j2 j1.
  Proof. by move⇒ j1 j2; rewrite /same_task eq_sym. Qed.

Consider any arrival sequence ...

... and any schedule of this arrival sequence ...

Variable sched : schedule PState.

... in which the sequential tasks hypothesis holds.

A simple corollary of this hypothesis is that the scheduler executes a job with the earliest arrival time.

  Corollary scheduler_executes_job_with_earliest_arrival:
    ∀ j1 j2 t,
      arrives_in arr_seq j1 →
      arrives_in arr_seq j2 →
      same_task j1 j2 →
      ~~ completed_by sched j2 t →
      scheduled_at sched j1 t →
      job_arrival j1 ≤ job_arrival j2.
  Proof.
    move⇒ j1 j2 t ARR1 ARR2 TSK NCOMPL SCHED.
    have {}TSK := eqbLR (same_task_sym _ _) TSK.
    have SEQ := H_sequential_tasks j2 j1 t ARR2 ARR1 TSK.
    rewrite leqNgt; apply/negP ⇒ ARR.
    exact/(negP NCOMPL)/SEQ.
  Qed.

Likewise, if we see an earlier-arrived incomplete job j1 while another job j2 is scheduled, then j1 and j2 must stem from different tasks.

  Corollary sequential_tasks_different_tasks :
    ∀ j1 j2 t,
      arrives_in arr_seq j1 →
      arrives_in arr_seq j2 →
      job_arrival j1 < job_arrival j2 →
      ~~ completed_by sched j1 t →
      scheduled_at sched j2 t →
      ~~ same_task j1 j2.
  Proof.
    move⇒ j1 j2 t IN1 IN2 LT_ARR INCOMP SCHED.
    apply: contraL INCOMP ⇒ SAME.
    by apply/negPn/H_sequential_tasks; eauto.
  Qed.

End ExecutionOrder.