Library rt.model.arrival.jitter.task_arrival

Require Import rt.util.all.
Require Import rt.model.arrival.basic.task rt.model.arrival.basic.job.
Require Import rt.model.arrival.basic.task_arrival.
Require Import rt.model.arrival.jitter.arrival_sequence.

(* In this file, we provide definitions and lemmas about task arrivals
   in jitter-aware schedules. *)

Module TaskArrivalWithJitter.

  Import ArrivalSequenceWithJitter SporadicTaskset.
  Export TaskArrival.

  (* First, we identify the arrivals of a particular task. *)
  Section NumberOfArrivals.

    Context {Task: eqType}.
    Context {Job: eqType}.
    Variable job_arrival: Job time.
    Variable job_jitter: Job time.
    Variable job_task: Job Task.

    (* Consider any job arrival sequence ...*)
    Variable arr_seq: arrival_sequence Job.

    (* ...and recall the list of jobs with actual arrival time in a given interval. *)
    Let arrivals_between := actual_arrivals_between job_arrival job_jitter arr_seq.

    (* Next, let tsk be any task. *)
    Variable tsk: Task.

    (* By checking the jobs spawned by tsk... *)
    Definition is_job_of_tsk := is_job_of_task job_task tsk.

    (* ...we identify the jobs of tsk that arrived in any interval [t1, t2) ... *)
    Definition actual_arrivals_of_task_between (t1 t2: time) :=
      [seq j <- arrivals_between t1 t2 | is_job_of_tsk j].

    (* ...and also count the number of such job arrivals. *)
    Definition num_actual_arrivals_of_task (t1 t2: time) :=
      size (actual_arrivals_of_task_between t1 t2).

  End NumberOfArrivals.

  (* Next, we prove properties about lists of actual arrivals
     that are sorted by arrival time. *)

  Section DistanceBetweenSporadicJobs.

    Context {Task: eqType}.
    Variable task_period: Task time.

    Context {Job: eqType}.
    Variable job_arrival: Job time.
    Variable job_jitter: Job time.
    Variable job_task: Job Task.

    (* Consider any arrival sequence with consistent, duplicate-free arrivals, ... *)
    Variable arr_seq: arrival_sequence Job.
    Hypothesis H_consistent_arrivals: arrival_times_are_consistent job_arrival arr_seq.
    Hypothesis H_no_duplicate_arrivals: arrival_sequence_is_a_set arr_seq.

    (* ...where jobs are sporadic. *)
    Hypothesis H_sporadic_jobs:
      sporadic_task_model task_period job_arrival job_task arr_seq.

    (* For simplicity, let's define some local names. *)
    Let actual_job_arrival := actual_arrival job_arrival job_jitter.

    (* Next, let tsk be any task to be scheduled. *)
    Variable tsk: Task.

    (* Consider any time interval [t1, t2)... *)
    Variable t1 t2: time.

    (* ...and the associated actual arrivals of task tsk. *)
    Let arriving_jobs := actual_arrivals_of_task_between job_arrival job_jitter
                                                         job_task arr_seq tsk t1 t2.
    Let num_arrivals := num_actual_arrivals_of_task job_arrival job_jitter job_task arr_seq tsk t1 t2.

    (* Then, consider the sequence of such jobs ordered by arrival times... *)
    Let by_arrival_time j j' := job_arrival j job_arrival j'.
    Let sorted_jobs := sort by_arrival_time arriving_jobs.

    (* ...and let (nth_job i) denote the i-th job in the sorted sequence. *)
    Variable elem: Job.
    Let nth_job := nth elem sorted_jobs.

    (* First, we recall some trivial properties about nth_job. *)
    Remark sorted_arrivals_properties_of_nth:
        idx < num_arrivals
        t1 actual_job_arrival (nth_job idx) < t2
        job_task (nth_job idx) = tsk
        arrives_in arr_seq (nth_job idx).

    (* Next, we conclude that consecutive jobs are different. *)
    Lemma sorted_arrivals_current_differs_from_next:
        idx < num_arrivals.-1
        nth_job idx nth_job idx.+1.

    (* Since the list is sorted, we prove that each job arrives at
       least (task_period tsk) time units after the previous job. *)

    Lemma sorted_arrivals_separated_by_period:
        idx < num_arrivals.-1
        job_arrival (nth_job idx.+1) job_arrival (nth_job idx) + task_period tsk.

    (* If the list of arrivals is not empty, we analyze the distance between
       the first and last jobs. *)

    Section FirstAndLastJobs.

      (* Suppose that there is at least one job, ... *)
      Hypothesis H_at_least_one_job:
        num_arrivals 1.

      (* which case we identify the first and last jobs and their
         respective arrival times (note that they could be the same job). *)

      Let j_first := nth_job 0.
      Let j_last := nth_job (num_arrivals.-1).
      Let a_first := job_arrival j_first.
      Let a_last := job_arrival j_last.

      (* By induction, we prove that that each job with index 'idx' arrives at
         least idx*(task_period tsk) units after the first job. *)

      Lemma sorted_arrivals_distance_from_first_job:
          idx < num_arrivals
          job_arrival (nth_job idx) a_first + idx × task_period tsk.

      (* Therefore, the first and last jobs are separated by at least
         (num_arrivals - 1) periods. *)

      Corollary sorted_arrivals_distance_between_first_and_last:
        a_last a_first + (num_arrivals-1) × task_period tsk.

    End FirstAndLastJobs.

  End DistanceBetweenSporadicJobs.

End TaskArrivalWithJitter.