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(** * Deadline-Monotonic Fixed-Priority Policy *)

(** We define the notion of deadline-monotonic task priorities, i.e., the
    classic FP policy in which tasks are prioritized in order of their relative
    deadlines. *)
#[export] Instance DM (Task : TaskType) `{TaskDeadline Task} : FP_policy Task :=
{
  hep_task (tsk1 tsk2 : Task) := task_deadline tsk1 <= task_deadline tsk2
}.

(** In this section, we prove a few basic properties of the DM policy. *)
Section Properties.

  (**  Consider any kind of tasks with relative deadlines... *)
  Context {Task : TaskType}.
  Context `{TaskDeadline Task}.

  (** ...and jobs stemming from these tasks. *)
  Context {Job : JobType}.
  Context `{JobTask Job Task}.

  (** DM is reflexive. *)
  
Task: TaskType
H: TaskDeadline Task
Job: JobType
H0: JobTask Job Task
reflexive_task_priorities (DM Task)

  
Task: TaskType
H: TaskDeadline Task
Job: JobType
H0: JobTask Job Task
reflexive_task_priorities (DM Task)
 by move=> ?; rewrite /hep_task /DM. Qed.

  (** DM is transitive. *)
  
Task: TaskType
H: TaskDeadline Task
Job: JobType
H0: JobTask Job Task
transitive_task_priorities (DM Task)

  
Task: TaskType
H: TaskDeadline Task
Job: JobType
H0: JobTask Job Task
transitive_task_priorities (DM Task)
 by move=> t y x z; apply: leq_trans. Qed.

  (** DM is total. *)
  
Task: TaskType
H: TaskDeadline Task
Job: JobType
H0: JobTask Job Task
total_task_priorities (DM Task)

  
Task: TaskType
H: TaskDeadline Task
Job: JobType
H0: JobTask Job Task
total_task_priorities (DM Task)
 by move=> ? ?; apply: leq_total. Qed.

End Properties.

(** We add the above lemmas into a "Hint Database" basic_rt_facts, so Coq
    will be able to apply them automatically. *)
Global Hint Resolve
     DM_is_reflexive
     DM_is_transitive
     DM_is_total
  : basic_rt_facts.