Library prosa.analysis.facts.model.sequential

Require Export prosa.model.task.sequentiality.

Section ExecutionOrder.

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.
  Context {PState: ProcessorState Job}.

Consider any arrival sequence ...
  Variable arr_seq : arrival_sequence Job.

... and any schedule of this arrival sequence ...
  Variable sched : schedule PState.

... in which the sequential tasks hypothesis holds.
  Hypothesis H_sequential_tasks: sequential_tasks arr_seq sched.

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.
    intros ? ? t ARR1 ARR2 TSK NCOMPL SCHED.
    rewrite /same_task eq_sym in TSK.
    have SEQ := H_sequential_tasks j2 j1 t ARR2 ARR1 TSK.
    rewrite leqNgt; apply/negP; intros ARR.
    move: NCOMPL ⇒ /negP NCOMPL; apply: NCOMPL.
    by apply SEQ.
  Qed.

End ExecutionOrder.