Library rt.analysis.uni.basic.fp_rta_comp

Require Import rt.util.all.
Require Import rt.model.arrival.basic.task rt.model.arrival.basic.job rt.model.arrival.basic.arrival_sequence rt.model.priority
Require Import rt.model.schedule.uni.schedule rt.model.schedule.uni.schedulability rt.model.schedule.uni.response_time.
Require Import rt.model.schedule.uni.basic.platform.
Require Import rt.analysis.uni.basic.workload_bound_fp rt.analysis.uni.basic.fp_rta_theory.

Module ResponseTimeIterationFP.

  Import Job SporadicTaskset UniprocessorSchedule WorkloadBoundFP Priority
         ResponseTime Schedulability Platform TaskArrival ResponseTimeAnalysisFP.

  (* In this section, we define the response-time analysis for uniprocessor FP scheduling. *)
  Section Analysis.

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

    (* In the algorithm, we consider pairs of tasks and computed response-time bounds. *)
    Let task_with_response_time := (SporadicTask × time)%type.

    (* Assume a fixed-priority policy. *)
    Variable higher_eq_priority: FP_policy SporadicTask.

    (* We begin by defining the fixed-point iteration for computing the
       response-time bound of each task. *)

    (* First, to ensure that the algorithm converges, we will run the iteration
       on each task for at most (task_deadline tsk - task_cost tsk + 1) steps,
       i.e., the worst-case time complexity of the procedure. *)

    Definition max_steps (tsk: SporadicTask) :=
      task_deadline tsk - task_cost tsk + 1.

    (* Next, based on the workload bound for uniprocessor FP scheduling, ... *)
    Let W := total_workload_bound_fp task_cost task_period higher_eq_priority.

    (* ...we compute the response-time bound R of a single task as follows:

           R (step) =  task_cost tsk                if step = 0,
                       W (ts, tsk, R (step-1))      otherwise.       *)

    Definition per_task_rta ts tsk :=
      iter_fixpoint (W ts tsk) (max_steps tsk) (task_cost tsk).

    (* Then, to validate the computed response-time bound, we
       check (a) if the iteration returned some value and
       (b) if the value is no larger than the deadline of the task. *)

    Let is_valid_bound tsk_R :=
      if tsk_R is (tsk, Some R) then
        if R task_deadline tsk then
          Some (tsk, R)
        else None
      else None.

    (* At the end, the response-time bounds for the entire taskset
       can be computed using the fixed-point iteration on each task.
       If all values are no larger than the deadline, we return the
       pairs of tasks and response-time bounds, else we return None. *)

    Definition fp_claimed_bounds ts: option (seq task_with_response_time) :=
      let possible_bounds := [seq (tsk, per_task_rta ts tsk) | tsk <- ts] in
        if all is_valid_bound possible_bounds then
          Some (pmap is_valid_bound possible_bounds)
        else None.

    (* The schedulability test simply checks if we got a list of
       response-time bounds (i.e., if the computation did not fail). *)

    Definition fp_schedulable (ts: seq SporadicTask) :=
      fp_claimed_bounds ts != None.

    (* In this section, we prove some properties about the computed
       list of response-time bounds. *)

    Section Lemmas.

      (* Let ts be any taskset to be analyzed. *)
      Variable ts: seq SporadicTask.

      (* Assume that the response-time analysis does not fail.*)
      Variable rt_bounds: seq task_with_response_time.
      Hypothesis H_analysis_succeeds:
        fp_claimed_bounds ts = Some rt_bounds.

      (* First, we prove that a response-time bound exists for each task. *)
      Section BoundExists.

        (* Let tsk be any task in ts. *)
        Variable tsk: SporadicTask.
        Hypothesis H_tsk_in_ts: tsk \in ts.

        (* Since the analysis succeeded, there must be a corresponding
           response-time bound R for this task. *)

        Lemma fp_claimed_bounds_for_every_task:
           R, (tsk, R) \in rt_bounds.

      End BoundExists.

      (* Next, assuming that a bound exists, we prove some of its properties. *)
      Section PropertiesOfBound.

        (* Let tsk and R be any pair of task and response-time bound
           returned by the analysis. *)

        Variable tsk: SporadicTask.
        Variable R: time.
        Hypothesis H_tsk_R_computed: (tsk, R) \in rt_bounds.

        (* First, we show that tsk comes from task set ts. *)
        Lemma fp_claimed_bounds_from_taskset:
          tsk \in ts.

        (* Next, we prove that R is computed using the per-task
           fixed-point iteration, ... *)

        Lemma fp_claimed_bounds_computes_iteration:
          per_task_rta ts tsk = Some R.

        (* ...which implies that R is also a fixed point of the recurrence. *)
        Lemma fp_claimed_bounds_yields_fixed_point :
          R = W ts tsk R.

        (* Since the analysis validates the computed values, it follows
           that R is no larger than the deadline of tsk. *)

        Lemma fp_claimed_bounds_le_deadline:
          R task_deadline tsk.

        (* Using the monotonicity of the workload bound, we also prove that
           the computed response-time bound is positive. This ensures that
           the busy interval to be analyzed is not empty. *)

        Section BoundPositive.

          (* Assume that the priority relation is reflexive. *)
          Hypothesis H_priority_is_reflexive:
            FP_is_reflexive higher_eq_priority.

          (* Assume that tasks have positive costs and periods. *)
          Hypothesis H_cost_positive: task_cost tsk > 0.
          Hypothesis H_period_positive:
             tsk, tsk \in ts task_period tsk > 0.

          (* Then, we prove that the fixed-point R is positive. *)
          Lemma fp_claimed_bounds_gt_zero :
            R > 0.

        End BoundPositive.

      End PropertiesOfBound.

    End Lemmas.

  End Analysis.

  (* In this section, we prove the correctness of the RTA. *)
  Section ProvingCorrectness.

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

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

    (* Consider a task set ts... *)
    Variable ts: taskset_of SporadicTask.

    (* ...where tasks have valid parameters. *)
    Hypothesis H_valid_task_parameters:
      valid_sporadic_taskset task_cost task_period task_deadline ts.

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

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

    (* have valid parameters...*)
    Hypothesis H_valid_job_parameters:
        arrives_in arr_seq j
        valid_sporadic_job task_cost task_deadline job_cost job_deadline job_task j.

    (* ... and jobs satisfy the sporadic task model.*)
    Hypothesis H_sporadic_tasks:
      sporadic_task_model task_period job_arrival job_task arr_seq.

    (* Assume any fixed-priority policy... *)
    Variable higher_eq_priority: FP_policy SporadicTask.

    (* ...that is reflexive and transitive, i.e., indicating higher-or-equal task priority. *)
    Hypothesis H_priority_reflexive: FP_is_reflexive higher_eq_priority.
    Hypothesis H_priority_transitive: FP_is_transitive higher_eq_priority.

    (* Next, consider any uniprocessor schedule of this arrival sequence...*)
    Variable sched: schedule Job.
    Hypothesis H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence sched arr_seq.

    (* ...where jobs do not execute before their arrival times nor after completion. *)
    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.

    (* Also assume that the scheduler is work-conserving and respects the FP 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.

    (* For simplicity, let's define some local names. *)
    Let no_deadline_missed_by_task :=
      task_misses_no_deadline job_arrival job_cost job_deadline job_task arr_seq sched.
    Let no_deadline_missed_by_job :=
      job_misses_no_deadline job_arrival job_cost job_deadline sched.
    Let response_time_bounded_by :=
      is_response_time_bound_of_task job_arrival job_cost job_task arr_seq sched.

    (* Recall the iteration for the response-time analysis and the corresponding
       schedulability test. *)

    Let RTA_claimed_bounds :=
      fp_claimed_bounds task_cost task_period task_deadline higher_eq_priority ts.
    Let claimed_to_be_schedulable :=
      fp_schedulable task_cost task_period task_deadline higher_eq_priority ts.

    (* First, we prove that the RTA yields valid response-time bounds. *)
    Theorem fp_analysis_yields_response_time_bounds :
       tsk R,
        (tsk, R) \In RTA_claimed_bounds
        response_time_bounded_by tsk R.

    (* Next, we show that the RTA is a sufficient schedulability analysis. *)
    Section AnalysisIsSufficient.

      (* If the schedulability test suceeds, ...*)
      Hypothesis H_test_succeeds: claimed_to_be_schedulable.

      (* ...then no task misses its deadline. *)
      Theorem taskset_schedulable_by_fp_rta :
         tsk, tsk \in ts no_deadline_missed_by_task tsk.

      (* Since all jobs of the arrival sequence are spawned by the task set,
         we also conclude that no job in the schedule misses its deadline. *)

      Theorem jobs_schedulable_by_fp_rta :
          arrives_in arr_seq j
          no_deadline_missed_by_job j.

    End AnalysisIsSufficient.

  End ProvingCorrectness.

End ResponseTimeIterationFP.