Require Import prosa.classic.util.all.
Require Import prosa.classic.model.arrival.basic.job
Require Import prosa.classic.model.schedule.uni.service
Require Import
Require Import prosa.classic.model.arrival.curves.bounds.
Require Import prosa.classic.analysis.uni.arrival_curves.workload_bound.
Require Import

From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq path fintype bigop.

Abstract RTA for FP-schedulers

In this module we propose the abstract response-time analysis (RTA) for FP uniprocessor scheduling of real-time tasks with arbitrary arrival models.
Module AbstractRTAforFPwithArrivalCurves.

  Import Job ArrivalCurves TaskArrival Priority ScheduleOfTask
         UniprocessorSchedule Workload Service ResponseTime MaxArrivalsWorkloadBound
         LimitedPreemptionPlatform RBF BusyIntervalJLFP JLFPInstantiation.

  Section AbstractResponseTimeAnalysisForFP.

    Context {Task: eqType}.
    Variable task_cost: Task time.

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

    (* Consider any arrival sequence with consistent, non-duplicate arrivals. *)
    Variable arr_seq: arrival_sequence Job.
    Hypothesis H_arrival_times_are_consistent: arrival_times_are_consistent job_arrival arr_seq.
    Hypothesis H_arr_seq_is_a_set: arrival_sequence_is_a_set arr_seq.

    (* 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 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.

    (* Next, assume that the schedule is a work-conserving schedule. *)
    Hypothesis H_work_conserving: work_conserving job_arrival job_cost arr_seq sched.

    (* Assume we have sequential jobs, i.e, jobs from the 
       same task execute in the order of their arrival. *)

    Hypothesis H_sequential_jobs: sequential_jobs job_arrival job_cost sched job_task.

    (* Assume that a job cost cannot be larger than a task cost. *)
    Hypothesis H_job_cost_le_task_cost:
        task_cost job_cost job_task arr_seq.

    (* Consider an arbitrary task set ts. *)
    Variable ts: list Task.

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

   (* Let max_arrivals be a family of proper arrival curves, i.e., for any task tsk in ts 
      max_arrival tsk is (1) an arrival bound of tsk, and (2) it is a monotonic function 
      that equals 0 for the empty interval delta = 0. *)

    Variable max_arrivals: Task time nat.
    Hypothesis H_family_of_proper_arrival_curves:
      family_of_proper_arrival_curves job_task arr_seq max_arrivals ts.

    (* Let tsk be any task in ts that is to be analyzed. *)
    Variable tsk: Task.
    Hypothesis H_tsk_in_ts: tsk \in ts.

    (* Consider proper job lock-in service and task lock-in functions, i.e... *)
    Variable job_lock_in_service: Job time.
    Variable task_lock_in_service: Task time.

    (* ...we assume that for all jobs with positive cost in the arrival sequence the 
       lock-in service is (1) positive, (2) no bigger than costs of the corresponding
       jobs, and (3) a job becomes nonpreemptive after it reaches its lock-in service... *)

    Hypothesis H_proper_job_lock_in_service:
      proper_job_lock_in_service job_cost arr_seq sched job_lock_in_service.

    (* ...and that task_lock_in_service tsk is (1) no bigger than tsk's cost, (2) for any
       job of task tsk job_lock_in_service is bounded by task_lock_in_service. *)

    Hypothesis H_proper_task_lock_in_service:
        task_cost job_task arr_seq job_lock_in_service task_lock_in_service tsk.

    (* Consider an FP policy that indicates a higher-or-equal priority relation,
       and assume that the relation is reflexive. Note that we do not relate 
       the FP policy with the scheduler. However, we define functions for 
       Interference and Interfering Workload that actively use the concept of 
       priorities. We require the FP policy to be reflexive, so a job cannot 
       cause lower-priority interference (i.e. priority inversion) to itself. *)

    Variable higher_eq_priority: FP_policy Task.
    Hypothesis H_priority_is_reflexive: FP_is_reflexive higher_eq_priority.

    (* We also lift the FP priority relation to a corresponding JLFP priority relation. *)
     Let jlfp_higher_eq_priority j1 j2 := FP_to_JLFP job_task higher_eq_priority j1 j2.

    (* For clarity, let's define some local names. *)
    Let job_pending_at := pending job_arrival job_cost sched.
    Let job_scheduled_at := scheduled_at sched.
    Let job_completed_by := completed_by job_cost sched.
    Let job_backlogged_at := backlogged job_arrival job_cost sched.
    Let arrivals_between := jobs_arrived_between arr_seq.
    Let response_time_bounded_by := is_response_time_bound_of_task job_arrival job_cost job_task arr_seq sched.
    Let quiet_time := quiet_time job_arrival job_cost arr_seq sched jlfp_higher_eq_priority.
    Let busy_interval := busy_interval job_arrival job_cost arr_seq sched jlfp_higher_eq_priority.
    Let task_scheduled_at := task_scheduled_at job_task sched.
    Let cumulative_task_interference :=
      AbstractSeqRTA.cumul_task_interference job_task arr_seq sched.

    (* We introduce task_rbf as an abbreviation of the task request bound function,
       which is defined as task_cost(tsk) × max_arrivals(tsk,Δ). *)

    Let task_rbf := task_request_bound_function task_cost max_arrivals tsk.

    (* Using the sum of individual request bound functions, we define the request bound 
       function of all tasks with higher-or-equal priority (with respect to tsk). *)

    Let total_hep_rbf :=
      total_hep_request_bound_function_FP task_cost higher_eq_priority max_arrivals ts tsk.

    (* Similarly, we define the request bound function of all tasks other 
       than tsk with higher-or-equal priority (with respect to tsk). *)

    Let total_ohep_rbf :=
      total_ohep_request_bound_function_FP task_cost higher_eq_priority max_arrivals ts tsk.

    (* Assume that there eixsts a constant priority_inversion_bound that bounds 
       the length of any priority inversion experienced by any job of tsk. 
       Sinse we analyze only task tsk, we ignore the lengths of priority 
       inversions incurred by any other tasks. *)

    Variable priority_inversion_bound: time.
    Hypothesis H_priority_inversion_is_bounded:
        job_arrival job_cost job_task arr_seq sched jlfp_higher_eq_priority tsk priority_inversion_bound.

    (* Let L be any positive fixed point of the busy interval recurrence. *)
    Variable L: time.
    Hypothesis H_L_positive: L > 0.
    Hypothesis H_fixed_point: L = priority_inversion_bound + total_hep_rbf L.

    (* To reduce the time complexity of the analysis, recall the notion of search space.
       Intuitively, this corresponds to all "interesting" arrival offsets that the job under
       analysis might have with regard to the beginning of its busy-window. *)

    Let is_in_search_space A := (A < L) && (task_rbf A != task_rbf (A + ε)).

    (* Let R be a value that upper-bounds the solution of each response-time recurrence, 
       i.e., for any relative arrival time A in the search space, there exists a corresponding 
       solution F such that F + (task cost - task lock-in service) R. *)

    Variable R: time.
    Hypothesis H_R_is_maximum:
        is_in_search_space A
          A + F = priority_inversion_bound
                  + (task_rbf (A + ε) - (task_cost tsk - task_lock_in_service tsk))
                  + total_ohep_rbf (A + F)
          F + (task_cost tsk - task_lock_in_service tsk) R.

    (* Instantiation of Interference *)
    (* We say that job j incurs interference at time t iff it cannot execute due to 
       a higher-or-equal-priority job being scheduled, or if it incurs a priority inversion. 
       (for more details see file limited.jlfp_instantiation.v) *)

    Let interference (j: Job) (t: time) :=
      interference sched jlfp_higher_eq_priority j t.

    (* Instantiation of Interfering Workload *)
    (* The interfering workload, in turn, is defined as the sum of the priority inversion 
       function and interfering workload of jobs with higher or equal priority.
       (for more details see file *)

    Let interfering_workload (j: Job) (t: time) :=
      interfering_workload job_cost arr_seq sched jlfp_higher_eq_priority j t.

    (* Finally, we define the interference bound function as the sum of the priority 
       interference bound and the higher-or-equal-priority workload. *)

    Let IBF R := priority_inversion_bound + total_ohep_rbf R.

Filling Out Hypotheses Of Abstract RTA Theorem

In this section we prove that all preconditions necessary to use the abstract theorem are satisfied.
    Section FillingOutHypothesesOfAbstractRTATheorem.

      (* First, we prove that in the instantiation of interference and interfering workload, 
         we really take into account everything that can interfere with tsk's jobs, and thus, 
         the scheduler satisfies the abstract notion of work conserving schedule. *)

      Lemma instantiated_i_and_w_are_consistent_with_schedule:
          job_arrival job_cost job_task arr_seq sched tsk interference interfering_workload.

      (* Next, we prove that the interference and interfering workload 
         functions are consistent with sequential jobs. *)

      Lemma instantiated_interference_and_workload_consistent_with_sequential_jobs:
          job_arrival job_cost job_task arr_seq sched tsk interference interfering_workload.

      (* Recall that L is assumed to be a fixed point of the busy interval recurrence. Thanks to
         this fact, we can prove that every busy interval (according to the concrete definition) 
         is bounded. In addition, we know that the conventional concept of busy interval and the 
         one obtained from the abstract definition (with the interference and interfering 
         workload) coincide. Thus, it follows that any busy interval (in the abstract sense) 
         is bounded. *)

      Lemma instantiated_busy_intervals_are_bounded:
          job_arrival job_cost job_task arr_seq sched tsk interference interfering_workload L.

      (* Next, we prove that IBF is indeed an interference bound.

         Recall that in module abstract_seq_RTA hypothesis task_interference_is_bounded_by expects 
         to receive a function that maps some task t, the relative arrival time of a job j of task t, 
         and the length of the interval to the maximum amount of interference (for more details see 
         files limited.abstract_RTA.definitions and limited.abstract_RTA.abstract_seq_rta).

         However, in this module we analyze only one task -- tsk, therefore it is “hardcoded” 
         inside the interference bound function IBF. Moreover, in case of a model with fixed 
         priorities, interference that some job j incurs from higher-or-equal priority jobs does not
         depend on the relative arrival time of job j. Therefore, in order for the IBF signature to
         match the required signature in module abstract_seq_RTA, we wrap the IBF function in a 
         function that accepts, but simply ignores, the task and the relative arrival time. *)

      Lemma instantiated_task_interference_is_bounded:
          job_arrival job_cost job_task arr_seq sched tsk interference interfering_workload
          (fun t A RIBF R).

      (* Finally, we show that there exists a solution for the response-time recurrence. *)
      Section SolutionOfResponseTimeReccurenceExists.

        (* 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.
        Hypothesis H_job_cost_positive: job_cost_positive job_cost j.

        (* Given any job j of task tsk that arrives exactly A units after the beginning of 
           the busy interval, the bound of the total interference incurred by j within an 
           interval of length Δ is equal to task_rbf (A + ε) - task_cost tsk + IBF Δ. *)

        Let total_interference_bound tsk A Δ :=
          task_rbf (A + ε) - task_cost tsk + IBF Δ.

        (* Next, consider any A from the search space (in the abstract sence). *)
        Variable A: time.
        Hypothesis H_A_is_in_abstract_search_space:
          AbstractRTAReduction.is_in_search_space tsk L total_interference_bound A.

        (* We prove that A is also in the concrete search space. *)
        Lemma A_is_in_concrete_search_space:
          is_in_search_space A.

        (* Then, there exists a solution for the response-time recurrence (in the abstract sense). *)
        Corollary correct_search_space:
            A + F = task_rbf (A + ε) - (task_cost tsk - task_lock_in_service tsk) + IBF (A + F)
            F + (task_cost tsk - task_lock_in_service tsk) R.

      End SolutionOfResponseTimeReccurenceExists.

    End FillingOutHypothesesOfAbstractRTATheorem.

Final Theorem

    (* Based on the properties established above, we apply the abstract analysis 
       framework to infer that R is a response-time bound for tsk. *)

    Theorem uniprocessor_response_time_bound_fp:
      response_time_bounded_by tsk R.

  End AbstractResponseTimeAnalysisForFP.

End AbstractRTAforFPwithArrivalCurves.