Library rt.analysis.parallel.bertogna_edf_theory

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

Module ResponseTimeAnalysisEDF.

  Export Job SporadicTaskset Schedule ScheduleOfSporadicTask Workload Schedulability ResponseTime
         Priority SporadicTaskArrival WorkloadBound InterferenceBoundEDF
         Interference Platform.

  (* In this section, we prove that any fixed point in Bertogna and
     Cirinei's RTA for EDF scheduling modified to consider (potentially)
     parallel jobs yields a safe response-time bound. This is an extension
     of the analysis found in Chapter 17.1.2 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_cost: Job time.
    Variable job_deadline: Job time.
    Variable job_task: Job sporadic_task.

    (* Assume any job arrival sequence... *)
    Context {arr_seq: arrival_sequence Job}.

    (* ... in which jobs arrive sporadically and have valid parameters. *)
    Hypothesis H_sporadic_tasks:
      sporadic_task_model task_period arr_seq job_task.
    Hypothesis H_valid_job_parameters:
       (j: JobIn arr_seq),
        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 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: JobIn arr_seq), job_task j ts.

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

    (* 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 sched.
    Hypothesis H_completed_jobs_dont_execute:
      completed_jobs_dont_execute job_cost sched.

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

    (* Assume that the schedule is a work-conserving EDF schedule. *)
    Hypothesis H_work_conserving: work_conserving job_cost sched.
    Hypothesis H_edf_policy: enforces_JLDP_policy job_cost sched (EDF job_deadline).

    (* 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_cost job_deadline job_task sched tsk.
    Let response_time_bounded_by (tsk: sporadic_task) :=
      is_response_time_bound_of_task job_cost job_task tsk sched.

    (* Next we consider the response-time recurrence.
       Assume that a response-time bound R is known...  *)

    Let task_with_response_time := (sporadic_task × time)%type.
    Variable rt_bounds: seq task_with_response_time.

    (* ...for any task in the task set, ... *)
    Hypothesis H_rt_bounds_contains_all_tasks: unzip1 rt_bounds = ts.

    (* ...where R is a fixed-point of the response-time recurrence, ... *)
    Let I (tsk: sporadic_task) (delta: time) :=
      total_interference_bound_edf task_cost task_period task_deadline tsk rt_bounds delta.
    Hypothesis H_response_time_is_fixed_point :
       tsk R,
        (tsk, R) rt_bounds
        R = task_cost tsk + div_floor (I tsk R) num_cpus.

    (* ..., and R is no larger than the deadline. *)
    Hypothesis H_tasks_miss_no_deadlines:
       tsk_other R,
        (tsk_other, R) rt_bounds R task_deadline tsk_other.

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

      (* Let (tsk, R) be any task to be analyzed, with its response-time bound R. *)
      Variable tsk: sporadic_task.
      Variable R: time.
      Hypothesis H_tsk_R_in_rt_bounds: (tsk, R) rt_bounds.

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

      (* Assume that job j did not complete on time, ... *)
      Hypothesis H_j_not_completed: ¬ completed job_cost sched j (job_arrival j + R).

      (* ...and that it is the first job not to satisfy its response-time bound. *)
      Hypothesis H_all_previous_jobs_completed_on_time :
         (j_other: JobIn arr_seq) tsk_other R_other,
          job_task j_other = tsk_other
          (tsk_other, R_other) rt_bounds
          job_arrival j_other + R_other < job_arrival j + R
          completed job_cost sched j_other (job_arrival j_other + R_other).

      (* Let's call x the interference incurred by job j due to tsk_other, ...*)
      Let x (tsk_other: sporadic_task) :=
        task_interference 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_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.

      (*... and the EDF-specific bound, ... *)
      Let edf_specific_bound (tsk_other: sporadic_task) (R_other: time) :=
        edf_specific_interference_bound task_cost task_period task_deadline tsk tsk_other R_other.

      (* ... which combined form the interference bound. *)
      Let interference_bound (tsk_other: sporadic_task) (R_other: time) :=
        interference_bound_edf task_cost task_period task_deadline tsk R (tsk_other, R_other).

      (* Based on the definition of a different task, ... *)
      Let other_task := different_task tsk.

      (* ...let other_tasks denote the set of tasks that are different from tsk. *)
      Let other_tasks :=
        [seq tsk_other <- ts | other_task tsk_other].

      (* Now we establish results the interfering tasks. *)
      Section LemmasAboutInterferingTasks.

        (* 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) rt_bounds.

        (* Note that tsk_other is in task set ts ...*)
        Lemma bertogna_edf_tsk_other_in_ts: tsk_other ts.

        (* Also, R_other is larger than the cost of tsk_other. *)
        Lemma bertogna_edf_R_other_ge_cost :
          R_other task_cost tsk_other.

        (* 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_edf_workload_bounds_interference :
          x tsk_other workload_bound tsk_other R_other.

        (* Recall that the edf-specific interference bound also holds. *)
        Lemma bertogna_edf_specific_bound_holds :
          x tsk_other edf_specific_bound tsk_other R_other.

      End LemmasAboutInterferingTasks.

      (* 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_edf_too_much_interference : X R - task_cost tsk + 1.

      (* 1) Next, we prove that the sum of the interference of each task is equal
            to the total interference multiplied by the number of processors. This
            holds because interference only occurs when all processors are busy.
            With this lemma we can relate per-task interference with the total
            interference incurred by j (backlogged time). *)

      Lemma bertogna_edf_interference_on_all_cpus :
        \sum_(tsk_k <- other_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_edf_sum_exceeds_total_interference:
        \sum_((tsk_other, R_other) <- rt_bounds | other_task tsk_other)
          x tsk_other > I tsk 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, I_edf, R - e_i + 1).
            This implies that either x_k > W_k or x_k > I_edf, which is a contradiction,
            since both W_k and I_edf are valid task interference bounds. *)

      Lemma bertogna_edf_exists_task_that_exceeds_bound :
         tsk_other R_other,
          (tsk_other, R_other) rt_bounds
          x tsk_other > interference_bound tsk_other R_other.

      End DerivingContradiction.

    End Lemmas.

    Section MainProof.

      (* Let (tsk, R) be any task to be analyzed, with its response-time bound R. *)
      Variable tsk: sporadic_task.
      Variable R: time.
      Hypothesis H_tsk_R_in_rt_bounds: (tsk, R) rt_bounds.

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

    End MainProof.

  End ResponseTimeBound.

End ResponseTimeAnalysisEDF.