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

Furthermore, we assume the fully preemptive task model.

Task's Run to Completion Threshold

In this section, we prove that instantiation of function task run to completion threshold to the fully preemptive model indeed defines a valid run-to-completion threshold function.
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}.

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

... and assume that a job cost cannot be larger than a task cost.
Then, we prove that task_run_to_completion_threshold 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.
    intros; split.
    - by rewrite /task_rtc_bounded_by_cost.
    - intros j ARR TSK.
      apply leq_trans with (job_cost j); eauto 2 with basic_facts.
        by rewrite-TSK; apply H_valid_job_cost.

End TaskRTCThresholdFullyPreemptiveModel.
Hint Resolve fully_preemptive_valid_task_run_to_completion_threshold : basic_facts.