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

Require Export prosa.model.preemption.limited_preemptive.
Require Export prosa.model.task.preemption.floating_nonpreemptive.
Require Export prosa.analysis.facts.preemption.job.limited.

# Platform for Floating Non-Preemptive Regions Model

In this section, we prove that instantiation of functions job_preemptable and task_max_nonpreemptive_segment for the model with floating non-preemptive regions indeed defines a valid preemption model with bounded non-preemptive regions.
Section FloatingNonPreemptiveRegionsModel.
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}.
In addition, we assume the existence of a function mapping a task to its maximal non-preemptive segment ...
Context `{TaskMaxNonpreemptiveSegment Task}.
.. and the existence of functions mapping a job to the sequence of its preemption points, ...
Context `{JobPreemptionPoints Job}.
... i.e., we assume limited-preemptive jobs.
#[local] Existing Instance limited_preemptive_job_model.
Consider any arrival sequence.
Variable arr_seq : arrival_sequence Job.
Next, consider any preemption-aware schedule of this arrival sequence...
Context {PState : ProcessorState Job}.
Variable sched : schedule PState.
Hypothesis H_preemption_aware_schedule :
schedule_respects_preemption_model arr_seq 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.
Consider an arbitrary task set ts.
Variable ts : seq Task.
Next, we assume that preemption points are defined by the model with floating non-preemptive regions.
Hypothesis H_valid_model_with_floating_nonpreemptive_regions :
valid_model_with_floating_nonpreemptive_regions arr_seq ts.
Then, we prove that the job_preemptable and task_max_nonpreemptive_segment functions define a model with bounded non-preemptive regions.
Lemma floating_preemption_points_model_is_model_with_bounded_nonpreemptive_regions :
model_with_bounded_nonpreemptive_segments arr_seq.
Proof.
move: (H_valid_model_with_floating_nonpreemptive_regions) (H_valid_model_with_floating_nonpreemptive_regions)
⇒ [LIM [L _]] [[BEG [END NDEC]] [MAX _]] j ARR.
case: (posnP (job_cost j)) ⇒ [ZERO|POS].
- split.
+ rewrite /job_respects_max_nonpreemptive_segment /job_max_nonpreemptive_segment
/lengths_of_segments /job_preemption_points; rewrite ZERO; simpl.
by rewrite /job_preemptable /limited_preemptive_job_model; erewrite zero_in_preemption_points; eauto 2.
+ move ⇒ progr; rewrite ZERO leqn0 ⇒ /andP [_ /eqP LE].
∃ 0; rewrite LE; split; first by apply/andP; split.
by eapply zero_in_preemption_points; eauto 2.
- split; last (move ⇒ progr /andP [_ LE]; destruct (progr \in job_preemptive_points j) eqn:NotIN).
+ by apply MAX.
+ by ∃ progr; split; first apply/andP; first split; rewrite ?leq_addr // conversion_preserves_equivalence.
+ move: NotIN ⇒ /eqP; rewrite eqbF_neg ⇒ NotIN.
edestruct (work_belongs_to_some_nonpreemptive_segment arr_seq) as [x [SIZE2 N]]; eauto 2. move: N ⇒ /andP [N1 N2].
set ptl := nth 0 (job_preemptive_points j) x.
set ptr := nth 0 (job_preemptive_points j) x.+1.
∃ ptr; split; first last.
× by unfold job_preemptable, limited_preemptive_job_model; apply mem_nth.
× apply/andP; split; first by apply ltnW.
apply leq_trans with (ptl + (job_max_nonpreemptive_segment j - ε) + 1); first last.
-- rewrite addn1 ltn_add2r; apply N1.
-- unfold job_max_nonpreemptive_segment.
rewrite -addnA -leq_subLR -(leq_add2r 1).
rewrite [in X in _ ≤ X]addnC -leq_subLR.
rewrite !subn1 !addn1 prednK.
{ rewrite -[_.+1.-1]pred_Sn. rewrite /lengths_of_segments.
erewrite job_parameters_max_np_to_job_limited; eauto.
by apply distance_between_neighboring_elements_le_max_distance_in_seq. }
{ rewrite /lengths_of_segments; erewrite job_parameters_max_np_to_job_limited; eauto.
apply max_distance_in_nontrivial_seq_is_positive; first by eauto 2.
∃ 0, (job_cost j); repeat split.
- by eapply zero_in_preemption_points; eauto.
- by eapply job_cost_in_nonpreemptive_points; eauto.
- by apply/eqP; rewrite eq_sym -lt0n; apply POS.
}
Qed.
Which together with lemma valid_fixed_preemption_points_model gives us the fact that functions job_preemptable and task_max_nonpreemptive_segment define a valid preemption model with bounded non-preemptive regions.
Corollary floating_preemption_points_model_is_valid_model_with_bounded_nonpreemptive_regions :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched.
Proof.
split.
- apply valid_fixed_preemption_points_model_lemma; auto.
by apply H_valid_model_with_floating_nonpreemptive_regions.
- by apply floating_preemption_points_model_is_model_with_bounded_nonpreemptive_regions.
Qed.
End FloatingNonPreemptiveRegionsModel.
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_fixed_preemption_points_model_lemma
floating_preemption_points_model_is_model_with_bounded_nonpreemptive_regions
floating_preemption_points_model_is_valid_model_with_bounded_nonpreemptive_regions : basic_rt_facts.