Library rt.implementation.uni.jitter.schedule

Require Import rt.util.all.
Require Import rt.model.arrival.basic.job rt.model.arrival.basic.arrival_sequence rt.model.priority.
Require Import rt.model.schedule.uni.jitter.schedule
Require Import

Module ConcreteScheduler.

  Import Job ArrivalSequence UniprocessorScheduleWithJitter Platform Priority

  (* In this section, we implement a jitter-aware priority-based uniprocessor scheduler *)
  Section Implementation.

    Context {Job: eqType}.
    Variable job_arrival: Job time.
    Variable job_cost: Job time.
    Variable job_jitter: Job time.

    (* Let arr_seq be any job arrival sequence. *)
    Variable arr_seq: arrival_sequence Job.

    (* Also, assume that a JLDP policy is given. *)
    Variable higher_eq_priority: JLDP_policy Job.

    (* Next, we show how to recursively construct the schedule. *)
    Section ScheduleConstruction.

      (* For any time t, suppose that we have generated the schedule prefix in the
         interval [0, t). Then, we must define what should be scheduled at time t. *)

      Variable sched_prefix: schedule Job.
      Variable t: time.

      (* For simplicity, let's use some local names. *)
      Let is_pending := pending job_arrival job_cost job_jitter sched_prefix.

      (* Consider the list of pending jobs with actual arrivals up to t. *)
      Definition pending_jobs :=
        [seq j <- actual_arrivals_up_to job_arrival job_jitter arr_seq t | is_pending j t].

      (* To make the scheduling decision, we just pick one of the highest-priority jobs
         that are pending (if it exists). *)

      Definition highest_priority_job :=
        seq_min (higher_eq_priority t) pending_jobs.

    End ScheduleConstruction.

    (* Starting from the empty schedule, the final schedule is obtained by iteratively
       picking the highest-priority job. *)

    Let empty_schedule : schedule Job := fun tNone.
    Definition scheduler :=
      build_schedule_from_prefixes highest_priority_job empty_schedule.

    (* Then, by showing that the construction function depends only on the prefix, ... *)
    Lemma scheduler_depends_only_on_prefix:
       sched1 sched2 t,
        ( t0, t0 < t sched1 t0 = sched2 t0)
        highest_priority_job sched1 t = highest_priority_job sched2 t.

    (* ...we infer that the generated schedule is indeed based on the construction function. *)
    Corollary scheduler_uses_construction_function:
       t, scheduler t = highest_priority_job scheduler t.

  End Implementation.

  (* In this section, we prove the properties of the scheduler that are used
     as hypotheses in the analyses. *)

  Section Proofs.

    Context {Job: eqType}.
    Variable job_arrival: Job time.
    Variable job_cost: Job time.
    Variable job_jitter: Job time.

    (* 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.

    (* Consider any JLDP policy that is reflexive, transitive and total. *)
    Variable higher_eq_priority: JLDP_policy Job.
    Hypothesis H_priority_is_reflexive: JLDP_is_reflexive higher_eq_priority.
    Hypothesis H_priority_is_transitive: JLDP_is_transitive higher_eq_priority.
    Hypothesis H_priority_is_total: JLDP_is_total arr_seq higher_eq_priority.

    (* Let sched denote our concrete scheduler implementation. *)
    Let sched := scheduler job_arrival job_cost job_jitter arr_seq higher_eq_priority.

    (* To conclude, we prove the important properties of the scheduler implementation. *)

    (* First, we show that scheduled jobs come from the arrival sequence. *)
      Lemma scheduler_jobs_come_from_arrival_sequence:
        jobs_come_from_arrival_sequence sched arr_seq.

    (* Next, we show that jobs do not execute before they arrive...*)
    Theorem scheduler_jobs_execute_after_jitter:
      jobs_execute_after_jitter job_arrival job_jitter sched.

    (* ... nor after completion. *)
    Theorem scheduler_completed_jobs_dont_execute:
      completed_jobs_dont_execute job_cost sched.

    (* In addition, the scheduler is work conserving... *)
    Theorem scheduler_work_conserving:
      work_conserving job_arrival job_cost job_jitter arr_seq sched.

    (* ... and respects the JLDP policy. *)
    Theorem scheduler_respects_policy :
      respects_JLDP_policy job_arrival job_cost job_jitter arr_seq sched higher_eq_priority.

  End Proofs.

End ConcreteScheduler.