# Library prosa.analysis.facts.readiness.basic

Require Export 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 : JLFP_policy Job}.
Hypothesis H_priority_is_reflexive : reflexive_job_priorities JLFP.
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 j ? ARR PEND.
∃ j; repeat split ⇒ //.
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.