Library prosa.analysis.facts.priority.fifo

Require Import prosa.model.readiness.basic.
Require Export prosa.model.task.sequentiality.
Require Export prosa.model.priority.fifo.
Require Export prosa.model.schedule.work_conserving.
Require Export prosa.analysis.definitions.priority_inversion.
Require Export prosa.analysis.facts.priority.sequential.
Require Export prosa.analysis.facts.readiness.basic.
Require Export prosa.analysis.facts.busy_interval.all.
Require Export prosa.analysis.facts.preemption.job.nonpreemptive.
Require Export prosa.analysis.facts.priority.inversion.
Require Export prosa.analysis.facts.busy_interval.service_inversion.

We first make some trivial observations about the FIFO priority policy to avoid having to re-reason these steps repeatedly in the subsequent proofs.
Section PriorityFacts.

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

Under FIFO scheduling, hep_job is simply a statement about arrival times.
  Fact hep_job_arrival_FIFO :
    ∀ j j',
      hep_job j j' = (job_arrival j ≤ job_arrival j').

Similarly, ~~ hep_job implies a strict inequality on arrival times.
  Fact not_hep_job_arrival_FIFO :
    ∀ j j',
      ~~ hep_job j j' = (job_arrival j' < job_arrival j).

Combining the above two facts, we get that, trivially, ~~ hep_job j j' implies hep_job j' j, ...
  Fact not_hep_job_FIFO :
    ∀ j j',
      ~~ hep_job j j' → hep_job j' j.

... from which we can infer always_higher_priority.
  Fact not_hep_job_always_higher_priority_FIFO :
    ∀ j j',
      ~~ hep_job j j' → always_higher_priority j' j.

End PriorityFacts.

In this section, we prove some fundamental properties of the FIFO policy.
Section BasicLemmas.

We assume the basic (i.e., Liu & Layland) readiness model under which any pending job is ready.
  #[local] Existing Instance basic_ready_instance.

Consider any type of jobs with arrival times and execution costs.
  Context `{Job : JobType} {Arrival : JobArrival Job} {Cost : JobCost Job}.

Consider any valid arrival sequence of such jobs ...
  Variable arr_seq : arrival_sequence Job.
  Hypothesis H_valid_arrivals : valid_arrival_sequence arr_seq.

... and the resulting uniprocessor schedule.
  Context {PState : ProcessorState Job}.
  Hypothesis H_uniproc : uniprocessor_model PState.
  Variable sched : schedule PState.
We assume that the schedule is valid and work-conserving.
  Hypothesis H_schedule_is_valid : valid_schedule sched arr_seq.
  Hypothesis H_work_conservation : work_conserving arr_seq sched.

Suppose jobs have preemption points ...
  Context `{JobPreemptable Job}.

...and that the preemption model is valid.
  Hypothesis H_valid_preemption_model :
    valid_preemption_model arr_seq sched.

Assume that the schedule respects the FIFO scheduling policy whenever jobs are preemptable.
  Hypothesis H_respects_policy : respects_JLFP_policy_at_preemption_point arr_seq sched (FIFO Job).

We observe that there is no priority inversion in a FIFO-compliant schedule.
  Lemma FIFO_implies_no_priority_inversion :
    ∀ j t,
      arrives_in arr_seq j →
      pending sched j t →
      ~~ priority_inversion arr_seq sched j t.

We prove that in a FIFO-compliant schedule, if a job j is scheduled, then all jobs with higher priority than j have been completed.
  Lemma scheduled_implies_higher_priority_completed :
    ∀ j t,
      scheduled_at sched j t →
      ∀ j_hp,
        arrives_in arr_seq j_hp →
        ~~ hep_job j j_hp →
        completed_by sched j_hp t.

In this section, we prove the cumulative priority inversion for any task is bounded by 0.
  Section PriorityInversionBounded.

Consider any kind of tasks.
    Context `{Task : TaskType} `{JobTask Job Task}.

Consider a task tsk.
    Variable tsk : Task.

Assume the arrival times are consistent.
    Hypothesis H_consistent_arrival_times : consistent_arrival_times arr_seq.

Assume that the schedule follows the FIFO policy at preemption time.
    Hypothesis H_respects_policy_at_preemption_point :
      respects_JLFP_policy_at_preemption_point arr_seq sched (FIFO Job).

Assume the schedule is valid.
    Hypothesis H_valid_schedule : valid_schedule sched arr_seq.

Assume there are no duplicates in the arrival sequence.
    Hypothesis H_arrival_sequence_is_a_set : arrival_sequence_uniq arr_seq.

Then we prove that the amount of priority inversion is bounded by 0.
    Lemma FIFO_implies_no_pi :
      priority_inversion_is_bounded_by arr_seq sched tsk (constant 0).
As a corollary, FIFO implies the absence of service inversion.
    Corollary FIFO_implies_no_service_inversion :
      service_inversion_is_bounded_by arr_seq sched tsk (constant 0).

  End PriorityInversionBounded.

The next lemma considers FIFO schedules in the context of tasks.
  Section SequentialTasks.

If the scheduled jobs stem from a set of tasks, ...
    Context {Task : TaskType}.
    Context `{JobTask Job Task}.

... then the tasks in a FIFO-compliant schedule necessarily execute sequentially.
    Lemma tasks_execute_sequentially : sequential_tasks arr_seq sched.

We also note that the FIFO policy respects sequential tasks.
    Fact fifo_respects_sequential_tasks : policy_respects_sequential_tasks (FIFO Job).

  End SequentialTasks.

Finally, let us further assume that there are no needless preemptions among jobs of equal priority.
  Hypothesis H_no_superfluous_preemptions : no_superfluous_preemptions sched.

In the absence of superfluous preemptions and under assumption of the basic readiness model, there are no preemptions at all in a FIFO-compliant schedule.
  Lemma no_preemptions_under_FIFO :
    ∀ j t,
      ~~ preempted_at sched j t.
It immediately follows that FIFO schedules are non-preemptive.
  Corollary FIFO_is_nonpreemptive : nonpreemptive_schedule sched.

End BasicLemmas.

We add the following lemmas to the basic facts database
Global Hint Resolve
  fifo_respects_sequential_tasks
  tasks_execute_sequentially
  : basic_rt_facts.