Library prosa.analysis.facts.preemption.rtc_threshold.preemptive

Require Export prosa.analysis.facts.preemption.rtc_threshold.job_preemptable.
Require Export prosa.model.preemption.fully_preemptive.
Require Export prosa.model.task.preemption.fully_preemptive.

Task's Run to Completion Threshold

In this section, we prove that the instantiation of the task run-to-completion threshold for the fully preemptive model is valid.
Section TaskRTCThresholdFullyPreemptiveModel.

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 `{JobCost Job}.

Assume that jobs and tasks are fully preemptive.
  #[local] Existing Instance fully_preemptive_job_model.
  #[local] Existing Instance fully_preemptive_task_model.
  #[local] Existing Instance fully_preemptive_rtc_threshold.

Next, consider any arrival sequence ...
  Variable arr_seq : arrival_sequence Job.

... and 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 task_rtct function defines a valid task's run to completion threshold.
  Lemma fully_preemptive_valid_task_run_to_completion_threshold:
    ∀ tsk, valid_task_run_to_completion_threshold arr_seq tsk.
  Proof.
    intros; split.
    - by rewrite /task_rtc_bounded_by_cost.
    - intros j ARR TSK.
      apply leq_trans with (job_cost j) ⇒ //.
      by move: TSK ⇒ /eqP <-; apply H_valid_job_cost.
  Qed.

End TaskRTCThresholdFullyPreemptiveModel.
Global Hint Resolve fully_preemptive_valid_task_run_to_completion_threshold : basic_rt_facts.