# 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.

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.

Consider any kind of jobs ...

... and any kind of processor state.

Suppose jobs have an arrival time and a cost.

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.

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 ...

... and any schedule of these jobs.

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.

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.

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.

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.