Library prosa.classic.analysis.uni.basic.fp_rta_theory

Require Import prosa.classic.util.all.
Require Import prosa.classic.model.arrival.basic.job prosa.classic.model.arrival.basic.task prosa.classic.model.priority prosa.classic.model.arrival.basic.task_arrival
               prosa.classic.model.arrival.basic.arrival_bounds.
Require Import prosa.classic.model.schedule.uni.schedule_of_task prosa.classic.model.schedule.uni.workload
               prosa.classic.model.schedule.uni.schedulability prosa.classic.model.schedule.uni.response_time
               prosa.classic.model.schedule.uni.service.
Require Import prosa.classic.model.schedule.uni.limited.busy_interval prosa.classic.model.schedule.uni.basic.platform.
Require Import prosa.classic.analysis.uni.basic.workload_bound_fp.
From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq fintype bigop.

Module ResponseTimeAnalysisFP.

  Import Job ScheduleOfTask SporadicTaskset Priority ResponseTime
         TaskArrival ArrivalBounds WorkloadBoundFP Platform Schedulability
         BusyIntervalJLFP Workload Service.

  Section ResponseTimeBound.

    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.

    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.

    Hypothesis H_sporadic_tasks:
      sporadic_task_model task_period job_arrival job_task arr_seq.
    Hypothesis H_valid_job_parameters:
      ∀ j,
        arrives_in arr_seq j →
        valid_sporadic_job task_cost task_deadline job_cost job_deadline job_task j.

    Variable ts: seq SporadicTask.
    Hypothesis H_valid_task_parameters:
      valid_sporadic_taskset task_cost task_period task_deadline ts.

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

    Variable sched: schedule Job.
    Hypothesis H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence sched arr_seq.

    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.

    Variable higher_eq_priority: FP_policy SporadicTask.
    Hypothesis H_priority_is_reflexive: FP_is_reflexive higher_eq_priority.
    Hypothesis H_priority_is_transitive: FP_is_transitive higher_eq_priority.

    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.

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

    Let response_time_bounded_by :=
      is_response_time_bound_of_task job_arrival job_cost job_task arr_seq sched.
    Let W := total_workload_bound_fp task_cost task_period higher_eq_priority ts tsk.

    Variable R: time.
    Hypothesis H_R_positive: R > 0.
    Hypothesis H_response_time_is_fixed_point: R = W R.

    Theorem uniprocessor_response_time_bound_fp:
      response_time_bounded_by tsk R.

  End ResponseTimeBound.

End ResponseTimeAnalysisFP.