Library rt.implementation.apa.arrival_sequence

(* We can reuse the.apa definition of periodic arrival sequence. *)
Require Import rt.util.all.
Require Import rt.model.arrival.basic.arrival_sequence rt.model.arrival.basic.job rt.model.arrival.basic.task rt.model.arrival.basic.task_arrival.
Require Import rt.implementation.apa.task rt.implementation.apa.job.
From mathcomp Require Import ssreflect ssrbool ssrfun ssrnat eqtype seq div.

Module ConcreteArrivalSequence.

  Import Job ArrivalSequence ConcreteTask ConcreteJob SporadicTaskset TaskArrival.

  Section PeriodicArrivals.

    Context {num_cpus: nat}.
    Variable ts: concrete_taskset num_cpus.

    (* At any time t, we release Some job of tsk if t is a multiple of the period,
       otherwise we release None. *)

    Definition add_job (arr: time) (tsk: @concrete_task num_cpus) : option (@concrete_job _) :=
      if task_period tsk %| arr then
        Some (Build_concrete_job (arr %/ task_period tsk) arr (task_cost tsk) (task_deadline tsk) tsk)

    (* The arrival sequence at any time t is simply the partial map of add_job. *)
    Definition periodic_arrival_sequence (t: time) := pmap (add_job t) ts.

  End PeriodicArrivals.

  Section Proofs.

    Context {num_cpus: nat}.

    (* Let ts be any concrete task set with valid parameters. *)
    Variable ts: concrete_taskset num_cpus.
    Hypothesis H_valid_task_parameters:
      valid_sporadic_taskset task_cost task_period task_deadline ts.

    (* Regarding the periodic arrival sequence built from ts, we prove that...*)
    Let arr_seq := periodic_arrival_sequence ts.

    (* ... arrival times are consistent, ... *)
    Theorem periodic_arrivals_are_consistent:
      arrival_times_are_consistent job_arrival arr_seq.
      movej t ARRj.
      rewrite /arrives_at mem_pmap in ARRj.
      move: ARRj ⇒ /mapP ARRj; destruct ARRj as [tsk IN SOME].
      by unfold add_job in *; desf.

    (* ... every job comes from the task set, ... *)
    Theorem periodic_arrivals_all_jobs_from_taskset:
        arrives_in arr_seq j
        job_task j \in ts.
      movej [t ARRj].
      rewrite mem_pmap in ARRj.
      move: ARRj ⇒ /mapP ARRj; destruct ARRj as [tsk IN SOME].
      by unfold add_job in *; desf.

    (* ..., jobs have valid parameters, ... *)
    Theorem periodic_arrivals_valid_job_parameters:
        arrives_in arr_seq j
        valid_sporadic_job task_cost task_deadline job_cost job_deadline job_task j.
      rename H_valid_task_parameters into PARAMS.
      unfold valid_sporadic_taskset, is_valid_sporadic_task in ×.
      movej [t ARRj].
      rewrite mem_pmap in ARRj; move: ARRj ⇒ /mapP [tsk IN SOME].
      unfold add_job in SOME; desf.
      specialize (PARAMS tsk IN); des.
      unfold valid_sporadic_job, valid_realtime_job, job_cost_positive.
      by repeat split; try (by done); apply leqnn.

    (* ... job arrivals satisfy the sporadic task model, ... *)
    Theorem periodic_arrivals_are_sporadic:
      sporadic_task_model task_period job_arrival job_task arr_seq.
      movej j' /eqP DIFF [arr ARR] [arr' ARR'] SAMEtsk LE.
      rewrite eqE /= /job_eqdef negb_and /= SAMEtsk eq_refl orbF in DIFF.
      rewrite 2!mem_pmap in ARR ARR'.
      move: ARR ARR' ⇒ /mapP [tsk_j INj SOMEj] /mapP [tsk_j' INj' SOMEj'].
      unfold add_job in SOMEj, SOMEj'; desf; simpl in *;
      move: Heq0 Heq ⇒ /dvdnP [k DIV] /dvdnP [k' DIV'].
        rewrite DIV DIV' -mulSnr.
        rewrite leq_eqVlt in LE; move: LE ⇒ /orP [/eqP EQ | LESS].
          exfalso; move: DIFF ⇒ /negP DIFF; apply DIFF.
          by subst; rewrite EQ !eq_refl.
        subst; rewrite leq_mul2r; apply/orP; right.
        by rewrite ltn_mul2r in LESS; move: LESS ⇒ /andP [_ LT].

    (* ... and the arrival sequence has no duplicate jobs. *)
    Theorem periodic_arrivals_is_a_set:
      arrival_sequence_is_a_set arr_seq.
      intros t.
      unfold arr_seq, periodic_arrival_sequence.
      apply (pmap_uniq) with (g := job_task); last by destruct ts.
      by unfold add_job, ocancel; intro tsk; desf.

  End Proofs.

End ConcreteArrivalSequence.