Require Import prosa.classic.util.all.
Require Import prosa.classic.model.arrival.basic.task prosa.classic.model.priority prosa.classic.model.arrival.basic.task_arrival.
Require Import
Require Import
Require Import
From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq fintype bigop div path.

Module ResponseTimeAnalysisFP.

  Export JobWithJitter SporadicTaskset ScheduleOfSporadicTaskWithJitter
         Workload Interference Platform ConstrainedDeadlines Schedulability
         ResponseTime Priority TaskArrival WorkloadBoundJitter
         Interference InterferenceBoundFP.

  (* In this section, we prove that any fixed point in Bertogna and
     Cirinei's RTA for FP scheduling modified to account for jitter
     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.
    Variable task_jitter: 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.
    Variable job_jitter: Job time.

    (* 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 job_arrival job_task arr_seq.
    Hypothesis H_valid_job_parameters:
        arrives_in arr_seq j
        valid_sporadic_job_with_jitter task_cost task_deadline task_jitter job_cost
                                       job_deadline job_task job_jitter 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:
        arrives_in arr_seq j job_task j \in ts.

    (* Next, consider any schedule of this arrival sequence 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.

    (* are sequential, do not execute before the jitter
       has passed and nor longer than their execution costs. *)

    Hypothesis H_sequential_jobs: sequential_jobs sched.
    Hypothesis H_jobs_execute_after_jitter:
      jobs_execute_after_jitter job_arrival job_jitter 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.

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

    (* ...and assume that the schedule is work-conserving and respects this policy. *)
    Hypothesis H_work_conserving: work_conserving job_arrival job_cost job_jitter arr_seq sched.
    Hypothesis H_respects_priority:
      respects_FP_policy job_arrival job_cost job_task job_jitter arr_seq sched higher_eq_priority.

    (* 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 (task_jitter hp_tsk + R).

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

    (* Assume that the response-time bounds are larger than task costs. *)
    Hypothesis H_response_time_bounds_ge_cost:
       hp_tsk R,
        (hp_tsk, R) \in hp_bounds R task_cost hp_tsk.

    (* Assume that no deadline is missed by any higher-priority task. *)
    Hypothesis H_interfering_tasks_miss_no_deadlines:
       hp_tsk R,
        (hp_tsk, R) \in hp_bounds
        task_jitter hp_tsk + R task_deadline hp_tsk.

    (* 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 +
            (total_interference_bound_fp task_cost task_period task_jitter
                                         tsk hp_bounds R)

    (* ... and assume that jitter + R is no larger than the deadline of tsk.*)
    Hypothesis H_response_time_no_larger_than_deadline:
      task_jitter tsk + 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_job_arrives: arrives_in arr_seq j.
      Hypothesis H_job_of_tsk: job_task j = tsk.

      (* Let t1 be the first point in time where j can actually be scheduled. *)
      Let t1 := job_arrival j + job_jitter j.

      (* 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 (t1 + R).
      Hypothesis H_previous_jobs_of_tsk_completed :
          arrives_in arr_seq j0
          job_task j0 = tsk
          job_arrival j0 < job_arrival j
          completed job_cost sched j0 (job_arrival j0 + task_jitter tsk + 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 job_jitter sched j tsk_other t1 (t1 + R).

      (* ...and X the total interference incurred by job j due to any task. *)
      Let X := total_interference job_arrival job_cost job_jitter sched j t1 (t1 + R).

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

        (* 1) Next, we prove that during the scheduling window of j, any job that is
              scheduled while j is backlogged comes from a different task.
              This follows from the fact that j is the first job not to complete
              by its response-time bound, so previous jobs of j's task must have
              completed by their periods and cannot be pending. *)

        Lemma bertogna_fp_interference_by_different_tasks :
           t j_other,
            t1 t < t1 + R
            arrives_in arr_seq j_other
            backlogged job_arrival job_cost job_jitter sched j t
            scheduled sched j_other t
            job_task j_other != tsk.

        (* 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.

        (* 2) 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 (3). *)

        Lemma bertogna_fp_all_cpus_are_busy:
            t1 t < t1 + R
            backlogged job_arrival job_cost job_jitter sched j t
            count (other_scheduled_task t) ts = num_cpus.

        (* 3) 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 (4). *)

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

        (* Before stating the next lemma, let (num_tasks_exceeding delta) be the
           number of interfering tasks whose interference x is larger than delta. *)

        Let num_tasks_exceeding delta := count (fun ix i delta) (hp_tasks).

        (* 4) Now we prove that, for any delta, if (num_task_exceeding delta > 0), then the
              cumulative interference caused by the complementary set of interfering tasks fills
              the remaining, not-completely-full (num_cpus - num_tasks_exceeding delta)
              processors. *)

        Lemma bertogna_fp_interference_in_non_full_processors :
            0 < num_tasks_exceeding delta < num_cpus
            \sum_(i <- hp_tasks | x i < delta) x i delta × (num_cpus - num_tasks_exceeding delta).

        (* 5) Based on lemma (4), we prove that, for any interval delta, if the sum of per-task
              interference exceeds (delta * num_cpus), the same applies for the
              sum of the minimum of the interference and delta. *)

        Lemma bertogna_fp_minimum_exceeds_interference :
            \sum_(tsk_k <- hp_tasks) x tsk_k delta × num_cpus
               \sum_(tsk_k <- hp_tasks) minn (x tsk_k) delta
               delta × num_cpus.

        (* 6) Next, using lemmas (0), (3) and (5) 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)
            minn (x tsk_k) (R - task_cost tsk + 1) >
          total_interference_bound_fp task_cost task_period task_jitter tsk hp_bounds R.

        (* 7) 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
            (minn (x tsk_k) (R - task_cost tsk + 1) >
              minn (workload_bound tsk_k R_k) (R - task_cost tsk + 1)).

      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 (task_jitter tsk + R).

  End ResponseTimeBound.

End ResponseTimeAnalysisFP.