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# Library prosa.analysis.facts.preemption.task.preemptive

Require Export prosa.analysis.facts.preemption.job.preemptive.
Require Export prosa.model.task.preemption.fully_preemptive.

# Platform for Fully Preemptive Model

In this section, we prove that the instantiations of the functions job_preemptable and task_max_nonpreemptive_segment for the fully preemptive model indeed defines a valid preemption model with bounded non-preemptive regions.
Section FullyPreemptiveModel.
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 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.
Consider any kind of processor state model, ...
Context {PState : ProcessorState Job}.
... any job arrival sequence, ...
Variable arr_seq : arrival_sequence Job.
... and any given schedule.
Variable sched : schedule PState.
We prove that the fully_preemptive_model function defines a model with bounded non-preemptive regions.
Lemma fully_preemptive_model_is_model_with_bounded_nonpreemptive_regions :
model_with_bounded_nonpreemptive_segments arr_seq.
Proof.
intros j ARR; split.
- case: (posnP (job_cost j)) ⇒ [ZERO|POS].
+ by rewrite /job_respects_max_nonpreemptive_segment job_max_nps_is_0.
+ by rewrite /job_respects_max_nonpreemptive_segment job_max_nps_is_ε.
- intros t; ∃ t; split⇒ [|//].
by apply/andP; split; [|rewrite leq_addr].
Qed.
Which together with lemma valid_fully_preemptive_model gives us the fact that fully_preemptive_model defined a valid preemption model with bounded non-preemptive regions.
Corollary fully_preemptive_model_is_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched.
Proof.
split.
- by apply valid_fully_preemptive_model.
- by apply fully_preemptive_model_is_model_with_bounded_nonpreemptive_regions.
Qed.
End FullyPreemptiveModel.
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_preemptive_model
fully_preemptive_model_is_model_with_bounded_nonpreemptive_regions
fully_preemptive_model_is_valid_model_with_bounded_nonpreemptive_segments : basic_rt_facts.