Library prosa.model.priority.numeric_fixed_priority

Require Export prosa.model.priority.classes.

Numeric Fixed Task Priorities

We define the notion of arbitrary numeric fixed task priorities, i.e., tasks are prioritized in order of user-provided numeric priority values, where numerically smaller values indicate lower priorities (as for instance it is the case in Linux).

First, we define a new task parameter task_priority that maps each task to a numeric priority value.

Class TaskPriority (Task : TaskType) := task_priority : Task → nat.

Based on this parameter, we define the corresponding FP policy.

Instance NumericFP (Task : TaskType) `{TaskPriority Task} : FP_policy Task :=
{
hep_task (tsk1 tsk2 : Task) := task_priority tsk1 ≥ task_priority tsk2
}.

In this section, we prove a few basic properties of numeric fixed priorities.

Section Properties.

Consider any kind of tasks with specified priorities...

Context {Task : TaskType}.
Context `{TaskPriority Task}.

...and jobs stemming from these tasks.

Context {Job : JobType}.
Context `{JobTask Job Task}.

The resulting priority policy is reflexive.

Lemma NFP_is_reflexive : reflexive_priorities.
Proof. by move⇒ ?; rewrite /hep_job_at /JLFP_to_JLDP /hep_job /FP_to_JLFP /hep_task /NumericFP. Qed.

The resulting priority policy is transitive.

  Lemma NFP_is_transitive : transitive_priorities.
  Proof.
    move⇒ t y x z.
    rewrite /hep_job_at /JLFP_to_JLDP /hep_job /FP_to_JLFP /hep_task /NumericFP.
    by move⇒ PRIO_yx PRIO_zy; apply leq_trans with (n := task_priority (job_task y)).
  Qed.

The resulting priority policy is total.

Lemma NFP_is_total : total_priorities.
Proof. by move⇒ t j1 j2; apply leq_total. Qed.

End Properties.

We add the above lemmas into a "Hint Database" basic_facts, so Coq will be able to apply them automatically.

Hint Resolve
     NFP_is_reflexive
     NFP_is_transitive
     NFP_is_total
  : basic_facts.