Library rt.analysis.global.parallel.bertogna_fp_theory

Require Import rt.util.all.
Require Import rt.model.arrival.basic.task rt.model.arrival.basic.job rt.model.arrival.basic.task_arrival
        rt.model.priority.
Require Import rt.model.schedule.global.workload rt.model.schedule.global.schedulability
               rt.model.schedule.global.response_time.
Require Import rt.model.schedule.global.basic.schedule rt.model.schedule.global.basic.platform
               rt.model.schedule.global.basic.constrained_deadlines rt.model.schedule.global.basic.interference.
Require Import rt.analysis.global.parallel.workload_bound rt.analysis.global.parallel.interference_bound_fp.

Module ResponseTimeAnalysisFP.

  Export Job SporadicTaskset ScheduleOfSporadicTask Workload Interference
         InterferenceBoundFP Platform Schedulability ResponseTime
         Priority TaskArrival WorkloadBound ConstrainedDeadlines.

  (* In this section, we prove that any fixed point in Bertogna and
     Cirinei's RTA for FP scheduling modified to consider (potentially)
     parallel jobs yields a safe response-time bound. This is an extension
     of the analysis found in Chapter 18.2 of Baruah et al.'s book
     Multiprocessor Scheduling for Real-time Systems. *)

  Section ResponseTimeBound.

    Context {sporadic_task: eqType}.
    Variable task_cost: sporadic_task time.
    Variable task_period: sporadic_task time.
    Variable task_deadline: sporadic_task time.

    Context {Job: eqType}.
    Variable job_arrival: Job time.
    Variable job_cost: Job time.
    Variable job_deadline: Job time.
    Variable job_task: Job sporadic_task.

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

    (* ... in which jobs arrive sporadically and have valid parameters. *)
    Hypothesis H_sporadic_tasks:
      sporadic_task_model task_period job_arrival job_task arr_seq.
    Hypothesis H_valid_job_parameters:
       j,
        arrives_in arr_seq j
        valid_sporadic_job task_cost task_deadline job_cost job_deadline job_task j.

    (* Assume that we have a task set where all tasks have valid
       parameters and constrained deadlines, ... *)

    Variable ts: taskset_of sporadic_task.
    Hypothesis H_valid_task_parameters:
      valid_sporadic_taskset task_cost task_period task_deadline ts.
    Hypothesis H_constrained_deadlines:
       tsk, tsk \in ts task_deadline tsk task_period tsk.

    (* ... and that all jobs in the arrival sequence come from the task set. *)
    Hypothesis H_all_jobs_from_taskset:
       j, arrives_in arr_seq j job_task j \in ts.

    (* Next, consider any schedule such that...*)
    Variable num_cpus: nat.
    Variable sched: schedule Job num_cpus.
    Hypothesis H_jobs_come_from_arrival_sequence:
      jobs_come_from_arrival_sequence sched arr_seq.

    (* ...jobs do not execute before their arrival times nor longer
       than their execution costs. *)

    Hypothesis H_jobs_must_arrive_to_execute:
      jobs_must_arrive_to_execute job_arrival sched.
    Hypothesis H_completed_jobs_dont_execute:
      completed_jobs_dont_execute job_cost sched.

    (* Consider a given FP policy, ... *)
    Variable higher_eq_priority: FP_policy sporadic_task.

    (* ... and assume that the schedule is an APA work-conserving
       schedule that respects this policy. *)

    Hypothesis H_work_conserving: work_conserving job_arrival job_cost arr_seq sched.
    Hypothesis H_respects_FP_policy:
      respects_FP_policy job_arrival job_cost job_task arr_seq sched higher_eq_priority.

    (* Assume that there exists at least one processor. *)
    Hypothesis H_at_least_one_cpu: num_cpus > 0.

    (* Let's define some local names to avoid passing many parameters. *)
    Let no_deadline_is_missed_by_tsk (tsk: sporadic_task) :=
      task_misses_no_deadline job_arrival job_cost job_deadline job_task arr_seq sched tsk.
    Let response_time_bounded_by (tsk: sporadic_task) :=
      is_response_time_bound_of_task job_arrival job_cost job_task arr_seq sched tsk.

    (* Next, we consider the response-time recurrence.
       Let tsk be a task in ts that is to be analyzed. *)

    Variable tsk: sporadic_task.
    Hypothesis task_in_ts: tsk \in ts.

    (* Let is_hp_task denote whether a task is a higher-priority task
       (with respect to tsk). *)

    Let is_hp_task := higher_priority_task higher_eq_priority tsk.

    (* Assume a response-time bound is known... *)
    Let task_with_response_time := (sporadic_task × time)%type.
    Variable hp_bounds: seq task_with_response_time.
    Hypothesis H_response_time_of_interfering_tasks_is_known:
       hp_tsk R,
        (hp_tsk, R) \in hp_bounds
        response_time_bounded_by hp_tsk R.

    (* ... for every higher-priority task. *)
    Hypothesis H_hp_bounds_has_interfering_tasks:
       hp_tsk,
        hp_tsk \in ts
        is_hp_task hp_tsk
           R, (hp_tsk, R) \in hp_bounds.

    (* Let R be the fixed point of Bertogna and Cirinei's recurrence, ...*)
    Variable R: time.
    Hypothesis H_response_time_recurrence_holds :
      R = task_cost tsk +
          div_floor
            (total_interference_bound_fp task_cost task_period hp_bounds R)
            num_cpus.

    (* ... and assume that R is no larger than the deadline of tsk.*)
    Hypothesis H_response_time_no_larger_than_deadline:
      R task_deadline tsk.

    (* In order to prove that R is a response-time bound, we first provide some lemmas. *)
    Section Lemmas.

      (* Consider any job j of tsk. *)
      Variable j: Job.
      Hypothesis H_j_arrives: arrives_in arr_seq j.
      Hypothesis H_job_of_tsk: job_task j = tsk.

      (* Assume that job j is the first job of tsk not to complete by the response time bound. *)
      Hypothesis H_j_not_completed: ~~ completed job_cost sched j (job_arrival j + R).
      Hypothesis H_previous_jobs_of_tsk_completed :
         j0,
          arrives_in arr_seq j0
          job_task j0 = tsk
          job_arrival j0 < job_arrival j
          completed job_cost sched j0 (job_arrival j0 + R).

      (* Let's call x the interference incurred by job j due to tsk_other, ...*)
      Let x (tsk_other: sporadic_task) :=
        task_interference job_arrival job_cost job_task sched j
                          tsk_other (job_arrival j) (job_arrival j + R).

      (* ...and X the total interference incurred by job j due to any task. *)
      Let X := total_interference job_arrival job_cost sched j (job_arrival j) (job_arrival j + R).

      (* Recall Bertogna and Cirinei's workload bound. *)
      Let workload_bound (tsk_other: sporadic_task) (R_other: time) :=
        W task_cost task_period tsk_other R_other R.

      (* Let hp_tasks denote the set of higher-priority tasks. *)
      Let hp_tasks := [seq tsk_other <- ts | is_hp_task tsk_other].

      (* Now we establish results about the higher-priority tasks. *)
      Section LemmasAboutHPTasks.

        (* Let (tsk_other, R_other) be any pair of higher-priority task and
           response-time bound computed in previous iterations. *)

        Variable tsk_other: sporadic_task.
        Variable R_other: time.
        Hypothesis H_response_time_of_tsk_other: (tsk_other, R_other) \in hp_bounds.

        (* Since tsk_other cannot interfere more than it executes, we show that
           the interference caused by tsk_other is no larger than workload bound W. *)

        Lemma bertogna_fp_workload_bounds_interference :
          x tsk_other workload_bound tsk_other R_other.

      End LemmasAboutHPTasks.

      (* Next we prove some lemmas that help to derive a contradiction.*)
      Section DerivingContradiction.

        (* 0) Since job j did not complete by its response time bound, it follows that
              the total interference X >= R - e_k + 1. *)

        Lemma bertogna_fp_too_much_interference : X R - task_cost tsk + 1.

        (* Let's define a predicate to identify the other tasks that are scheduled. *)
        Let other_scheduled_task (t: time) (tsk_other: sporadic_task) :=
          task_is_scheduled job_task sched tsk_other t &&
          is_hp_task tsk_other.

        (* 1) Now we prove that, at all times that j is backlogged, the number
              of tasks other than tsk that are scheduled is exactly the number
              of processors in the system. This is required to prove lemma (2). *)

        Lemma bertogna_fp_all_cpus_are_busy:
          \sum_(tsk_k <- hp_tasks) x tsk_k = X × num_cpus.

        (* 2) Next, using lemmas (0) and (1) we prove that the reduction-based
              interference bound is not enough to cover the sum of the minima over all tasks
              (artifact of the proof by contradiction). *)

        Lemma bertogna_fp_sum_exceeds_total_interference:
          \sum_((tsk_k, R_k) <- hp_bounds)
           x tsk_k > total_interference_bound_fp task_cost task_period hp_bounds R.

        (* 3) After concluding that the sum of the minima exceeds (R - e_i + 1),
              we prove that there exists a tuple (tsk_k, R_k) that satisfies
              min (x_k, R - e_i + 1) > min (W_k, R - e_i + 1).
              This implies that x_k > W_k, which is of course a contradiction,
              since W_k is a valid task interference bound. *)

        Lemma bertogna_fp_exists_task_that_exceeds_bound :
           tsk_k R_k,
            (tsk_k, R_k) \in hp_bounds
            x tsk_k > workload_bound tsk_k R_k.

      End DerivingContradiction.

    End Lemmas.

    (* Using the lemmas above, we prove that R bounds the response time of task tsk. *)
    Theorem bertogna_cirinei_response_time_bound_fp :
      response_time_bounded_by tsk R.

  End ResponseTimeBound.

End ResponseTimeAnalysisFP.