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]
Require Export prosa.model.task.arrival.sporadic.
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 "[ 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]


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

(** We define the notion of rate-monotonic task priorities for sporadic tasks,
    i.e., the classic FP policy in which sporadic tasks are prioritized in
    order of their minimum inter-arrival times (or periods). *)
#[export] Instance RM (Task : TaskType) `{SporadicModel Task} : FP_policy Task :=
{
  hep_task (tsk1 tsk2 : Task) := task_min_inter_arrival_time tsk1 <= task_min_inter_arrival_time tsk2
}.

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

  (**  Consider sporadic tasks... *)
  Context {Task : TaskType}.
  Context `{SporadicModel Task}.

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

  (** RM is reflexive. *)
  Lemma RM_is_reflexive : reflexive_priorities.
Task: TaskType
H: SporadicModel Task
Job: JobType
H0: JobTask Job Task
reflexive_priorities
  Proof.
Task: TaskType
H: SporadicModel 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 /RM. Qed.

  (** RM is transitive. *)
  Lemma RM_is_transitive : transitive_priorities.
Task: TaskType
H: SporadicModel Task
Job: JobType
H0: JobTask Job Task
transitive_priorities
  Proof.
Task: TaskType
H: SporadicModel Task
Job: JobType
H0: JobTask Job Task
transitive_priorities by move=> t y x z; apply: leq_trans. Qed.

  (** RM is total. *)
  Lemma RM_is_total : total_priorities.
Task: TaskType
H: SporadicModel Task
Job: JobType
H0: JobTask Job Task
total_priorities
  Proof.
Task: TaskType
H: SporadicModel 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_rt_facts, so Coq
    will be able to apply them automatically. *)
Global Hint Resolve
     RM_is_reflexive
     RM_is_transitive
     RM_is_total
  : basic_rt_facts.