Library rt.implementation.apa.schedule

Require Import rt.util.all.
Require Import rt.model.priority.
Require Import rt.model.arrival.basic.arrival_sequence rt.model.arrival.basic.task.
Require Import
Require Import rt.model.schedule.apa.affinity rt.model.schedule.apa.platform.
Require Import

Module ConcreteScheduler.

  Import SporadicTaskset ArrivalSequence Schedule Platform Priority Affinity ScheduleConstruction.

  (* In this section, we implement a concrete weak APA scheduler. *)
  Section Implementation.

    Context {Job: eqType}.
    Context {sporadic_task: eqType}.

    Variable job_arrival: Job time.
    Variable job_cost: Job time.
    Variable job_task: Job sporadic_task.

    (* Let num_cpus denote the number of processors, ...*)
    Variable num_cpus: nat.

    (* ... and let arr_seq be any arrival sequence.*)
    Variable arr_seq: arrival_sequence Job.

    (* Let alpha be an affinity associated with each task. *)
    Variable alpha: task_affinity sporadic_task num_cpus.

    (* Assume 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 num_cpus.
      Variable cpu: processor num_cpus.
      Variable t: time.

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

      (* Consider the list of pending jobs at time t, ... *)
      Definition pending_jobs := [seq j <- actual_arrivals t | is_pending j t].

      (* ...which we sort by priority. *)
      Definition sorted_pending_jobs := sort (higher_eq_priority t) pending_jobs.

      (* Now we implement the algorithm that generates the APA schedule. *)

      (* Given a job j at time t, we first define a predicate that states
         whether j should preempt a mapping (cpu, x), where x is either Some j'
         that is currently mapped to cpu or None. *)

        Definition should_be_scheduled (j: Job) p :=
        let '(cpu, mapped_job) := p in
          if mapped_job is Some j' then (* If there is a job j', check the priority and affinity. *)
            (can_execute_on alpha (job_task j) cpu) &&
            ~~ (higher_eq_priority t j' j)
          else (* Else, if cpu is idle, check only the affinity. *)
            (can_execute_on alpha (job_task j) cpu).

      (* Next, using the "should_be_scheduled" predicate, we define a function
         that tries to schedule job j by updating a list of mappings.
         It does so by replacing the first pair (cpu, x) where j can be
         scheduled (if it exists). *)

      Definition update_available_cpu allocation j :=
        replace_first (should_be_scheduled j) (* search for processors that j can preempt *)
                      (set_pair_2nd (Some j)) (* replace the mapping in that processor with j *)
                      allocation. (* list of mappings *)

      (* Consider the empty mapping. *)
      Let empty_mapping : seq (processor num_cpus × option Job) :=
        (zip (enum (processor num_cpus)) (nseq num_cpus None)).

      (* Using the fuction "update_available_cpu", we now define an iteration
         that iteratively maps each pending job to a processor.

         Starting with an empty mapping,
         <(cpu0, None), (cpu1, None), (cpu2, None), ...>,
         it tries to schedule each job on some processor and yields an updated list: 
         <(cpu0, None), (cpu1, Some j5), (cpu2, Some j9), ...>. *)

      Definition schedule_jobs_from_list l :=
        foldl update_available_cpu empty_mapping l.

      (* To conclude, we take the list of pairs and convert to a function denoting
         the actual schedule. *)

      Definition apa_schedule :=
        pairs_to_function None (schedule_jobs_from_list sorted_pending_jobs) cpu.

    End ScheduleConstruction.

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

    Let empty_schedule : schedule Job num_cpus := fun cpu tNone.
    Definition scheduler :=
      build_schedule_from_prefixes num_cpus apa_schedule empty_schedule.

    (* Then, by showing that the construction function depends only on the prefix, ... *)
    Lemma scheduler_depends_only_on_prefix:
       sched1 sched2 cpu t,
        ( t0 cpu0, t0 < t sched1 cpu0 t0 = sched2 cpu0 t0)
        apa_schedule sched1 cpu t = apa_schedule sched2 cpu t.

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

  End Implementation.

  (* In this section, we prove several properties about the scheduling algorithm we
     implemented. For example, we show that it is APA work conserving. *)

  Section Proofs.

    Context {Job: eqType}.
    Context {sporadic_task: eqType}.

    Variable job_arrival: Job time.
    Variable job_cost: Job time.
    Variable job_task: Job sporadic_task.

    (* Assume a positive number of processors. *)
    Variable num_cpus: nat.
    Hypothesis H_at_least_one_cpu: num_cpus > 0.

    (* Let alpha be an affinity associated with each task. *)
    Variable alpha: task_affinity sporadic_task num_cpus.

    (* Let arr_seq be 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_arrival_sequence_is_a_set: arrival_sequence_is_a_set arr_seq.

    (* Consider any JLDP policy higher_eq_priority that is transitive and total. *)
    Variable higher_eq_priority: JLDP_policy Job.
    Hypothesis H_priority_transitive: JLDP_is_transitive higher_eq_priority.
    Hypothesis H_priority_total: t, total (higher_eq_priority t).

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

    (* Next, we provide some helper lemmas about the scheduler construction. *)
    Section HelperLemmas.

      (* To avoid many parameters, let's also rename the scheduling function.
         We make a generic version (for any list of jobs l), ... *)

      Let schedule_jobs t l := schedule_jobs_from_list job_task num_cpus alpha higher_eq_priority t l.
      (* ... and a specific version (for the pending jobs in sched). *)
      Let schedule_pending_jobs t :=
        schedule_jobs t (sorted_pending_jobs job_arrival job_cost num_cpus arr_seq
                                             higher_eq_priority sched t).

      (* Next, we show that there are no duplicate cpus in the mapping. *)
      Lemma scheduler_uniq_cpus :
         t l,
          uniq (unzip1 (schedule_jobs t l)).

      (* Next, we show that if a job j is in the mapping, then j must come from the list
         of jobs l used in the construction. *)

      Lemma scheduler_job_in_mapping :
         l j t cpu,
          (cpu, Some j) \in schedule_jobs t l j \in l.

      (* Next, we prove that if a pair (cpu, j) is in the mapping, then
         cpu must be part of j's affinity. *)

      Lemma scheduler_mapping_respects_affinity :
         j t cpu,
          (cpu, Some j) \in schedule_pending_jobs t
          can_execute_on alpha (job_task j) cpu.

      (* Next, we show that the mapping does not schedule the same job j in two
         different cpus. *)

      Lemma scheduler_has_no_duplicate_jobs :
         j t cpu1 cpu2,
          (cpu1, Some j) \in schedule_pending_jobs t
          (cpu2, Some j) \in schedule_pending_jobs t
          cpu1 = cpu2.

      (* Based on the definition of the schedule, a job j is scheduled on cpu
         iff (cpu, Some j) is the final mapping. *)

      Lemma scheduler_scheduled_on :
         j cpu t,
          scheduled_on sched j cpu t = ((cpu, Some j) \in schedule_pending_jobs t).

      (* Now we show that for every cpu, there is always a pair in the mapping. *)
      Lemma scheduler_has_cpus :
         cpu t l,
            (cpu, x) \in schedule_jobs t l.

      (* Next, consider a list of jobs l that is sorted by priority and does not have
         We prove that for any job j in l, if j is not scheduled at time t,
         then every cpu in j's affinity has some job mapped at time t.  *)

      Lemma scheduler_mapping_is_work_conserving :
         j cpu t l,
          j \in l
          sorted (higher_eq_priority t) l
          uniq l
          ( cpu, (cpu, Some j) \notin schedule_jobs t l)
          can_execute_on alpha (job_task j) cpu
            (cpu, Some j_other) \in schedule_jobs t l.

      (* Next, we prove that the mapping respects priority. *)
      Lemma scheduler_priority :
         j j_hp cpu t,
          arrives_in arr_seq j
          backlogged job_arrival job_cost sched j t
          can_execute_on alpha (job_task j) cpu
          scheduled_on sched j_hp cpu t
          higher_eq_priority t j_hp j.

    End HelperLemmas.

    (* Now, we prove the important properties about the 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.

    (* Jobs do not execute before they arrive, ...*)
    Theorem scheduler_jobs_must_arrive_to_execute:
      jobs_must_arrive_to_execute job_arrival sched.

    (* ..., jobs are sequential, ... *)
    Theorem scheduler_sequential_jobs: sequential_jobs sched.

    (* ... and jobs do not execute after completion. *)
    Theorem scheduler_completed_jobs_dont_execute:
      completed_jobs_dont_execute job_cost sched.

    (* In addition, the scheduler is APA work conserving, ... *)
    Theorem scheduler_apa_work_conserving:
      apa_work_conserving job_arrival job_cost job_task arr_seq sched alpha.

    (* ..., respects affinities, ... *)
    Theorem scheduler_respects_affinity:
      respects_affinity job_task sched alpha.

    (* ... and respects the JLDP policy under weak APA scheduling. *)
    Theorem scheduler_respects_policy :
      respects_JLDP_policy_under_weak_APA job_arrival job_cost job_task arr_seq
                                          sched alpha higher_eq_priority.

  End Proofs.

End ConcreteScheduler.