Library prosa.analysis.facts.preemption.task.nonpreemptive

Require Export prosa.analysis.facts.preemption.job.nonpreemptive.
Require Export prosa.model.task.preemption.fully_nonpreemptive.

Platform for Fully Non-Preemptive Model

In this section, we prove that instantiation of functions job_preemptable and task_max_nonpreemptive_segment to the fully non-preemptive model indeed defines a valid preemption model with bounded non-preemptive regions.
Section FullyNonPreemptiveModel.

Consider any type of tasks ...
  Context {Task : TaskType}.
  Context `{TaskCost Task}.

... and any type of jobs associated with these tasks.
  Context {Job : JobType}.
  Context `{JobTask Job Task}.
  Context `{JobArrival Job}.
  Context `{JobCost Job}.

Assume a fully non-preemptive task model.
  #[local] Existing Instance fully_nonpreemptive_job_model.
  #[local] Existing Instance fully_nonpreemptive_task_model.
  #[local] Existing Instance fully_nonpreemptive_rtc_threshold.

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

Next, consider any non-preemptive unit-service schedule of the arrival sequence ...
  Context {PState : ProcessorState Job}.
  Hypothesis H_unit_service: unit_service_proc_model PState.
  Variable sched : schedule PState.
  Hypothesis H_nonpreemptive_sched : nonpreemptive_schedule sched.

... where jobs do not execute before their arrival or after completion.
  Hypothesis H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched.
  Hypothesis H_completed_jobs_dont_execute : completed_jobs_dont_execute sched.

Assume that a job cost cannot be larger than a task cost.
  Hypothesis H_valid_job_cost:
    arrivals_have_valid_job_costs arr_seq.

Then we prove that the fully_nonpreemptive_model function defines a model with bounded non-preemptive regions.
  Lemma fully_nonpreemptive_model_is_model_with_bounded_nonpreemptive_regions:
    model_with_bounded_nonpreemptive_segments arr_seq.

Which together with lemma valid_fully_nonpreemptive_model gives us the fact that fully_nonpreemptive_model defined a valid preemption model with bounded non-preemptive regions.
  Corollary fully_nonpreemptive_model_is_valid_model_with_bounded_nonpreemptive_regions:
    valid_model_with_bounded_nonpreemptive_segments arr_seq sched.

End FullyNonPreemptiveModel.

We add the above lemma into a "Hint Database" basic_rt_facts, so Coq will be able to apply them automatically.
Global Hint Resolve
     valid_fully_nonpreemptive_model
     fully_nonpreemptive_model_is_model_with_bounded_nonpreemptive_regions
     fully_nonpreemptive_model_is_valid_model_with_bounded_nonpreemptive_regions : basic_rt_facts.