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

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 prosa.classic.model.arrival.basic.task_arrival.
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.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
               prosa.classic.analysis.uni.susp.dynamic.jitter.jitter_schedule_properties
               prosa.classic.analysis.uni.susp.dynamic.jitter.jitter_schedule_service
               prosa.classic.analysis.uni.susp.dynamic.jitter.jitter_taskset_generation.
From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq fintype bigop.

Module RTAByReduction.

  Import TaskArrival SporadicTaskset Suspension Priority Workload Service Schedulability
         UniprocessorScheduleWithJitter ResponseTime SuspensionIntervals ValidSuspensionAwareSchedule.

  Module susp_aware := PlatformWithSuspensions.
  Module reduction := JitterScheduleConstruction.
  Module reduction_prop := JitterScheduleProperties.
  Module reduction_serv := JitterScheduleService.
  Module ts_gen := JitterTaskSetGeneration.

  Section ComparingResponseTimeBounds.

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

    Variable ts: seq Task.
    Hypothesis H_constrained_deadlines:
      constrained_deadline_model task_period task_deadline ts.

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

    Hypothesis H_sporadic_arrivals:
      sporadic_task_model task_period job_arrival job_task arr_seq.

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

    Hypothesis H_job_deadlines_equal_task_deadlines:
      ∀ j, arrives_in arr_seq j → job_deadline j = task_deadline (job_task j).

    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 task_suspension_bound: Task → time.

    Hypothesis H_positive_costs:
      ∀ j, arrives_in arr_seq j → job_cost j > 0.

    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.

    Variable tsk: Task.
    Hypothesis H_tsk_in_ts: tsk \in ts.

    Let other_hep_task tsk_other :=
      higher_eq_priority tsk_other tsk && (tsk_other != tsk).
    Let task_response_time_in_sched_susp_bounded_by :=
      is_response_time_bound_of_task job_arrival job_cost job_task arr_seq sched_susp.
    Let job_response_time_in_sched_susp_bounded_by :=
      is_response_time_bound_of_job job_arrival job_cost sched_susp.
    Let completed_in_sched_susp_by := completed_by job_cost sched_susp.
    Let job_misses_no_deadline_in_sched_susp :=
      job_misses_no_deadline job_arrival job_cost job_deadline sched_susp.

    Variable R: Task → time.

    Hypothesis H_valid_response_time_bound_of_hp_tasks:
      ∀ tsk_hp,
        tsk_hp \in ts →
        other_hep_task tsk_hp →
        task_response_time_in_sched_susp_bounded_by tsk_hp (R tsk_hp).

    Definition actual_response_time (j_hp: Job) : time :=
      [pick-min r ≤ R (job_task j_hp) |
         job_response_time_in_sched_susp_bounded_by j_hp r].

    Variable j: Job.
    Hypothesis H_j_arrives: arrives_in arr_seq j.
    Hypothesis H_job_of_tsk: job_task j = tsk.

    Hypothesis H_no_deadline_misses_for_previous_jobs:
      ∀ j0,
        arrives_in arr_seq j0 →
        job_arrival j0 < job_arrival j →
        job_task j0 = job_task j →
        job_misses_no_deadline_in_sched_susp j0.

    Let inflated_task_cost := ts_gen.inflated_task_cost task_cost task_suspension_bound tsk.
    Let task_jitter := ts_gen.task_jitter task_cost higher_eq_priority tsk.

    Let sched_jitter := reduction.sched_jitter job_arrival job_task arr_seq higher_eq_priority
                        job_cost job_suspension_duration j actual_response_time.

    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
                                           actual_response_time.

    Let job_response_time_in_sched_jitter_bounded_by :=
      is_response_time_bound_of_job job_arrival inflated_job_cost sched_jitter.

    Hypothesis H_valid_response_time_bound_in_sched_jitter:
      job_response_time_in_sched_jitter_bounded_by j (R tsk).

    Theorem valid_response_time_bound_in_sched_susp:
      job_response_time_in_sched_susp_bounded_by j (R tsk).

  End ComparingResponseTimeBounds.

End RTAByReduction.