Library prosa.classic.analysis.uni.susp.dynamic.jitter.jitter_schedule_properties

Require Import prosa.classic.util.all.
Require Import prosa.classic.model.priority prosa.classic.model.suspension.
Require Import prosa.classic.model.arrival.basic.job prosa.classic.model.arrival.basic.task
               prosa.classic.model.arrival.basic.arrival_sequence.
Require Import prosa.classic.model.arrival.jitter.job.
Require Import prosa.classic.model.schedule.uni.schedulability prosa.classic.model.schedule.uni.service
               prosa.classic.model.schedule.uni.workload
               prosa.classic.model.schedule.uni.response_time.
Require Import prosa.classic.model.schedule.uni.jitter.schedule
               prosa.classic.model.schedule.uni.jitter.valid_schedule
               prosa.classic.model.schedule.uni.jitter.platform.
Require Import prosa.classic.model.schedule.uni.susp.suspension_intervals
               prosa.classic.model.schedule.uni.susp.schedule
               prosa.classic.model.schedule.uni.susp.valid_schedule
               prosa.classic.model.schedule.uni.susp.platform.
Require Import prosa.classic.analysis.uni.susp.dynamic.jitter.jitter_schedule.
Require Import prosa.classic.model.schedule.uni.transformation.construction.
From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq fintype bigop path.

Module JitterScheduleProperties.

  Import Job SporadicTaskset Suspension Priority SuspensionIntervals Workload Service
         UniprocessorScheduleWithJitter Schedulability ResponseTime
         ScheduleConstruction ValidSuspensionAwareSchedule ValidJitterAwareSchedule.

  Module basic := schedule.UniprocessorSchedule.
  Module susp := ScheduleWithSuspensions.
  Module jitter_aware := Platform.
  Module susp_aware := PlatformWithSuspensions.
  Module job_jitter := JobWithJitter.
  Module reduction := JitterScheduleConstruction.

  Section ProvingScheduleProperties.

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

    Variable ts: seq Task.

    Variable arr_seq: arrival_sequence Job.
    Hypothesis H_arrival_times_are_consistent:
      arrival_times_are_consistent job_arrival arr_seq.

    Hypothesis H_jobs_from_taskset:
      ∀ j, arrives_in arr_seq j → job_task j \in ts.

    Variable higher_eq_priority: FP_policy Task.
    Hypothesis H_priority_is_reflexive: FP_is_reflexive higher_eq_priority.
    Hypothesis H_priority_is_transitive: FP_is_transitive higher_eq_priority.
    Hypothesis H_priority_is_total: FP_is_total_over_task_set higher_eq_priority ts.
    Let job_higher_eq_priority := FP_to_JLDP job_task higher_eq_priority.

    Variable job_cost: Job → time.
    Variable task_cost: Task → time.

    Variable job_suspension_duration: job_suspension Job.

    Variable sched_susp: schedule Job.
    Hypothesis H_valid_schedule:
      valid_suspension_aware_schedule job_arrival arr_seq job_higher_eq_priority
                                      job_suspension_duration job_cost sched_susp.

    Let job_response_time_in_sched_susp_bounded_by :=
      is_response_time_bound_of_job job_arrival job_cost sched_susp.

    Variable j: Job.
    Hypothesis H_from_arrival_sequence: arrives_in arr_seq j.

    Let arr_j := job_arrival j.
    Let task_of_j := job_task j.

    Let other_hep_task tsk_other :=
      higher_eq_priority tsk_other task_of_j && (tsk_other != task_of_j).

    Variable R: Job → time.

    Let sched_jitter := reduction.sched_jitter job_arrival job_task arr_seq higher_eq_priority
                                               job_cost job_suspension_duration j R.
    Let inflated_job_cost := reduction.inflated_job_cost job_cost job_suspension_duration j.
    Let job_jitter := reduction.job_jitter job_arrival job_task higher_eq_priority job_cost j R.

    Section PropertiesOfScheduleConstruction.

      Let build_schedule := reduction.build_schedule job_arrival job_task arr_seq higher_eq_priority
                                                     job_cost job_suspension_duration j R.

      Lemma sched_jitter_depends_only_on_service:
        ∀ sched1 sched2 t,
          (∀ j, service sched1 j t = service sched2 j t) →
          build_schedule sched1 t = build_schedule sched2 t.

      Corollary sched_jitter_uses_construction_function:
        ∀ t,
          sched_jitter t = build_schedule sched_jitter t.

    End PropertiesOfScheduleConstruction.

    Section ScheduleIsValid.

      Let pending_jobs_other_than_j :=
        reduction.pending_jobs_other_than_j job_arrival job_task arr_seq higher_eq_priority
                                            job_cost job_suspension_duration j R sched_jitter.
      Let hp_job_other_than_j :=
        reduction.highest_priority_job_other_than_j job_arrival job_task arr_seq higher_eq_priority
                                                    job_cost job_suspension_duration j R sched_jitter.

      Let is_valid_jitter_aware_schedule :=
        valid_jitter_aware_schedule job_arrival arr_seq job_higher_eq_priority
                                    inflated_job_cost job_jitter.

      Lemma sched_jitter_jobs_come_from_arrival_sequence:
        jobs_come_from_arrival_sequence sched_jitter arr_seq.

      Lemma sched_jitter_jobs_execute_after_jitter:
        jobs_execute_after_jitter job_arrival job_jitter sched_jitter.

      Lemma sched_jitter_completed_jobs_dont_execute:
        completed_jobs_dont_execute inflated_job_cost sched_jitter.

      Lemma sched_jitter_work_conserving:
        jitter_aware.work_conserving job_arrival inflated_job_cost job_jitter arr_seq sched_jitter.

      Lemma sched_jitter_respects_policy:
        jitter_aware.respects_FP_policy job_arrival inflated_job_cost job_jitter
                                        job_task arr_seq sched_jitter higher_eq_priority.

      Corollary sched_jitter_is_valid: is_valid_jitter_aware_schedule sched_jitter.

      Lemma sched_jitter_does_not_pick_j:
        ∀ j_hp t,
          arrives_in arr_seq j_hp →
          j_hp != j →
          pending job_arrival inflated_job_cost job_jitter sched_jitter j_hp t →
          higher_eq_priority (job_task j_hp) (job_task j) →
          ~~ scheduled_at sched_jitter j t.

    End ScheduleIsValid.

  End ProvingScheduleProperties.

End JitterScheduleProperties.