Library prosa.analysis.facts.readiness.basic

Require Import prosa.model.readiness.basic.
Require Export prosa.analysis.facts.behavior.completion.
Require Export prosa.analysis.definitions.readiness.
Require Export prosa.analysis.definitions.work_bearing_readiness.

Section LiuAndLaylandReadiness.

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 kind of jobs ...
  Context {Job : JobType}.

... and any kind of processor state.
  Context {PState : ProcessorState Job}.

Suppose jobs have an arrival time and a cost.
  Context `{JobArrival Job} `{JobCost Job}.

The Liu & Layland readiness model is trivially non-clairvoyant.
  Fact basic_readiness_nonclairvoyance :
    nonclairvoyant_readiness basic_ready_instance.
  Proof.
    move⇒ sched sched' j h PREFIX t IN.
    rewrite /job_ready /basic_ready_instance.
    now apply (identical_prefix_pending _ _ h).
  Qed.

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

... and any schedule of these jobs.
  Variable sched : schedule PState.

In the basic Liu & Layland model, a schedule satisfies that only ready jobs execute as long as jobs must arrive to execute and completed jobs don't execute, which we note with the following theorem.
  Lemma basic_readiness_compliance :
    jobs_must_arrive_to_execute sched →
    completed_jobs_dont_execute sched →
    jobs_must_be_ready_to_execute sched.
  Proof.
    move⇒ ARR COMP.
    rewrite /jobs_must_be_ready_to_execute ⇒ j t SCHED.
    rewrite /job_ready /basic_ready_instance /pending.
    apply /andP; split.
    - by apply ARR.
    - rewrite -less_service_than_cost_is_incomplete.
      by apply COMP.
  Qed.

Consider a JLFP policy that indicates a reflexive higher-or-equal priority relation.
  Context `{JLFP_policy Job}.
  Hypothesis H_priority_is_reflexive : reflexive_priorities.

We show that the basic readiness model is a work-bearing readiness model. That is, at any time instant t, if a job j is pending, then there exists a job (namely j itself) with higher-or-equal priority that is ready at time t.
  Fact basic_readiness_is_work_bearing_readiness :
    work_bearing_readiness arr_seq sched.
  Proof.
    intros ? ? ARR PEND.
    ∃ j; repeat split ⇒ //.
    by eapply (H_priority_is_reflexive 0).
  Qed.

End LiuAndLaylandReadiness.

We add the above lemma into a "Hint Database" basic_rt_facts, so Coq will be able to apply it automatically.
Global Hint Resolve basic_readiness_is_work_bearing_readiness : basic_rt_facts.