Library prosa.analysis.facts.readiness.sequential

Require Export prosa.analysis.definitions.readiness.
Require Export prosa.analysis.definitions.work_bearing_readiness.
Require Export prosa.analysis.facts.behavior.completion.
Require Export prosa.analysis.facts.model.task_arrivals.

Throughout this file, we assume the sequential task readiness model.
Require Export prosa.model.readiness.sequential.

In this section, we show some useful properties of the sequential task readiness model.
Section SequentialTasksReadiness.

Consider any type of job associated with any type of tasks ...
  Context {Job : JobType}.
  Context {Task : TaskType}.
  Context `{JobTask Job Task}.
  Context `{JobArrival Job}.
  Context `{JobCost Job}.

... and any kind of processor state.
  Context {PState : Type}.
  Context `{ProcessorState Job PState}.

Consider any arrival sequence with consistent arrivals.
  Variable arr_seq : arrival_sequence Job.
  Hypothesis H_arrival_times_are_consistent : consistent_arrival_times arr_seq.

  (* As it was mentioned in the beginning of this file, 
     we assume the sequential task readiness model. *)
  Instance sequential_ready_instance : JobReady Job PState :=
    sequential.sequential_ready_instance arr_seq.

  (* Consider any valid schedule of arr_seq. *)
  Variable sched : schedule PState.
  Hypothesis H_valid_schedule : valid_schedule sched arr_seq.

Consider an FP policy that indicates a reflexive higher-or-equal priority relation.
  Context `{FP_policy Task}.
  Hypothesis H_priority_is_reflexive : reflexive_priorities.

First, we show that the sequential readiness model is non-clairvoyant.
  Fact sequential_readiness_nonclairvoyance :
    nonclairvoyant_readiness sequential_ready_instance.

Next, we show that the sequential readiness model ensures that tasks are sequential. That is, that jobs of the same task execute in order of their arrival.
  Lemma sequential_readiness_implies_sequential_tasks :
    sequential_tasks arr_seq sched.

  (* Finally, we show that the sequential readiness model is a
     work-bearing readiness model. That is, if a job j is pending
     but not ready at a time instant t, then there exists another
     (earlier) job of the same task that is pending and ready at time
     t. *)
  Lemma sequential_readiness_implies_work_bearing_readiness :
    work_bearing_readiness arr_seq sched.

End SequentialTasksReadiness.