Require Export prosa.model.priority.classes.
Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "_ + _" was already used in scope nat_scope.
[notation-overridden,parsing]
Notation "_ - _" was already used in scope nat_scope.
[notation-overridden,parsing]
Notation "_ <= _" was already used in scope nat_scope.
[notation-overridden,parsing]
Notation "_ < _" was already used in scope nat_scope.
[notation-overridden,parsing]
Notation "_ >= _" was already used in scope nat_scope.
[notation-overridden,parsing]
Notation "_ > _" was already used in scope nat_scope.
[notation-overridden,parsing]
Notation "_ <= _ <= _" was already used in scope
nat_scope. [notation-overridden,parsing]
Notation "_ < _ <= _" was already used in scope
nat_scope. [notation-overridden,parsing]
Notation "_ <= _ < _" was already used in scope
nat_scope. [notation-overridden,parsing]
Notation "_ < _ < _" was already used in scope
nat_scope. [notation-overridden,parsing]
Notation "_ * _" was already used in scope nat_scope.
[notation-overridden,parsing]

(** * 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. *)
Instance DM (Task : TaskType) `{TaskDeadline Task} : FP_policy Task :=
{
  hep_task (tsk1 tsk2 : Task) := task_deadline tsk1 <= task_deadline tsk2
}.
The default value for instance locality is currently
"local" in a section and "global" otherwise, but is
scheduled to change in a future release. For the time
being, adding instances outside of sections without
specifying an explicit locality attribute is therefore
deprecated. It is recommended to use "export" whenever
possible. Use the attributes #[local], #[global] and
#[export] depending on your choice. For example:
"#[export] Instance Foo : Bar := baz."
[deprecated-instance-without-locality,deprecated]

(** 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. *)
  Lemma DM_is_reflexive : reflexive_priorities.
Task: TaskType
H: TaskDeadline Task
Job: JobType
H0: JobTask Job Task
reflexive_priorities
  Proof.
Task: TaskType
H: TaskDeadline Task
Job: JobType
H0: JobTask Job Task
reflexive_priorities by move=> ?; rewrite /hep_job_at /JLFP_to_JLDP /hep_job /FP_to_JLFP /hep_task /DM. Qed.

  (** DM is transitive. *)
  Lemma DM_is_transitive : transitive_priorities.
Task: TaskType
H: TaskDeadline Task
Job: JobType
H0: JobTask Job Task
transitive_priorities
  Proof.
Task: TaskType
H: TaskDeadline Task
Job: JobType
H0: JobTask Job Task
transitive_priorities by intros t y x z; apply leq_trans. Qed.

  (** DM is total. *)
  Lemma DM_is_total : total_priorities.
Task: TaskType
H: TaskDeadline Task
Job: JobType
H0: JobTask Job Task
total_priorities
 Proof.
Task: TaskType
H: TaskDeadline Task
Job: JobType
H0: JobTask Job Task
total_priorities 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. *)
Global Hint Resolve
     DM_is_reflexive
     DM_is_transitive
     DM_is_total
  : basic_facts.