Library prosa.implementation.refinements.FP.nonpreemptive_sched

Require Export prosa.analysis.facts.preemption.task.nonpreemptive.
Require Export prosa.analysis.facts.preemption.rtc_threshold.nonpreemptive.
Require Export prosa.analysis.facts.readiness.sequential.
Require Export prosa.analysis.definitions.tardiness.
Require Export prosa.implementation.facts.ideal_uni.prio_aware.
Require Export prosa.implementation.definitions.task.

Fully-Nonpreemptive Fixed-Priority Schedules

In this section, we prove some facts about the fully-nonpreemptive preemption policy under fixed-priority schedules.

Some lemmas in this file are not novel facts; they are used to uniform POET's certificates and minimize their verbosity.

Section Schedule.

In this file, we adopt the Prosa standard implementation of jobs and tasks.

Definition Task := [eqType of concrete_task ].
Definition Job := [eqType of concrete_job ].

Consider any arrival sequence with consistent arrivals, ...

Variable arr_seq : arrival_sequence Job.
Hypothesis H_arrival_times_are_consistent : consistent_arrival_times arr_seq.

... assume sequential readiness, ...

Instance sequential_ready_instance : JobReady Job (ideal.processor_state Job) :=
sequential_ready_instance arr_seq.

... and consider any fully-nonpreemptive, fixed-priority schedule.

  #[local] Existing Instance fully_nonpreemptive_job_model.
  #[local] Existing Instance NumericFPAscending.
  Definition sched := uni_schedule arr_seq.

First, we show that only ready jobs execute.

Lemma sched_jobs_must_be_ready_to_execute :
jobs_must_be_ready_to_execute sched.

Next, we remark that such a schedule is valid.

Remark sched_valid :
valid_schedule sched arr_seq.

Next, we show that an unfinished job scheduled at time t is always scheduled at time t+1 as well. Note that the validity of this fact also depends on which readiness model is used.

  Lemma sched_nonpreemptive_next:
    ∀ j t,
      scheduled_at sched j t →
      ~~ completed_by sched j t .+1 →
      scheduled_at sched j t .+1.

Using the lemma above, we show that the schedule is nonpreemptive.

Lemma sched_nonpreemptive :
nonpreemptive_schedule (uni_schedule arr_seq).

Finally, we show that the fixed-priority policy is respected at each preemption point.

Lemma respects_policy_at_preemption_point_np :
respects_FP_policy_at_preemption_point arr_seq sched (NumericFPAscending Task).

End Schedule.