Built with Alectryon, running Coq+SerAPI v8.15.0+0.15.0. Bubbles () indicate interactive fragments: hover for details, tap to reveal contents. Use Ctrl+↑ Ctrl+↓ to navigate, Ctrl+🖱️ to focus. On Mac, use ⌘ instead of Ctrl.
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]


(** * FIFO Priority Policy *)

(** We define the basic FIFO priority policy, under which jobs are prioritized
    in order of their arrival times. The FIFO policy belongs to the class of
    JLFP policies. *)
#[export] Instance FIFO (Job : JobType) `{JobArrival Job} : JLFP_policy Job :=
{
  hep_job (j1 j2 : Job) := job_arrival j1 <= job_arrival j2
}.

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

  (**  Consider any type of jobs with arrival times. *)
  Context {Job : JobType}.
  Context `{JobArrival Job}.

  (** FIFO is reflexive. *)
  
Job: JobType
H: JobArrival Job
reflexive_priorities

  
Job: JobType
H: JobArrival Job
reflexive_priorities
 by move=> t j; apply: leqnn. Qed.

  (** FIFO is transitive. *)
  
Job: JobType
H: JobArrival Job
transitive_priorities

  
Job: JobType
H: JobArrival Job
transitive_priorities
 by move=> t y x z; apply: leq_trans. Qed.

  (** FIFO is total. *)
  
Job: JobType
H: JobArrival Job
total_priorities

  
Job: JobType
H: JobArrival Job
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
     FIFO_is_reflexive
     FIFO_is_transitive
     FIFO_is_total
  : basic_rt_facts.