# 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.