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# Library prosa.implementation.refinements.FP.nonpreemptive_sched

Require Export prosa.analysis.facts.preemption.task.nonpreemptive.
Require Export prosa.analysis.facts.preemption.rtc_threshold.nonpreemptive.
Require Export prosa.analysis.facts.readiness.sequential.
Require Export prosa.analysis.definitions.tardiness.
Require Export prosa.implementation.facts.ideal_uni.prio_aware.
Require Export prosa.implementation.definitions.task.

## Fully-Nonpreemptive Fixed-Priority Schedules

In this section, we prove some facts about the fully-nonpreemptive preemption policy under fixed-priority schedules.
Some lemmas in this file are not novel facts; they are used to uniform POET's certificates and minimize their verbosity.
Section Schedule.
In this file, we adopt the Prosa standard implementation of jobs and tasks.
Definition Task := concrete_task : eqType.
Definition Job := concrete_job : eqType.
Consider any valid arrival sequence, ...
Variable arr_seq : arrival_sequence Job.
Hypothesis H_valid_arrivals : valid_arrival_sequence arr_seq.
... assume sequential readiness, ...
Instance sequential_ready_instance : JobReady Job (ideal.processor_state Job) :=
sequential_ready_instance arr_seq.
... and consider any fully-nonpreemptive, fixed-priority schedule.
#[local] Existing Instance fully_nonpreemptive_job_model.
#[local] Existing Instance numeric_fp_ascending.
Definition sched := uni_schedule arr_seq.
First, we show that only ready jobs execute.
Lemma sched_jobs_must_be_ready_to_execute :
jobs_must_be_ready_to_execute sched.
Proof.
move⇒ j t SCHED.
apply jobs_must_be_ready ⇒ //.
- move⇒ tt ss jj IN.
rewrite /choose_highest_prio_job in IN.
by apply supremum_in in IN.
- by apply sequential_readiness_nonclairvoyance.
Qed.
Next, we remark that such a schedule is valid.
Remark sched_valid :
valid_schedule sched arr_seq.
Proof.
split.
- apply np_schedule_jobs_from_arrival_sequence.
by move⇒ t ???; eapply (supremum_in (hep_job_at t)).
- by apply sched_jobs_must_be_ready_to_execute.
Qed.
Next, we show that an unfinished job scheduled at time t is always scheduled at time t+1 as well. Note that the validity of this fact also depends on which readiness model is used.
Lemma sched_nonpreemptive_next :
∀ j t,
scheduled_at sched j t →
~~ completed_by sched j t.+1 →
scheduled_at sched j t.+1.
Proof.
move⇒ j t SCHED NCOMP.
have READY: job_ready sched j t by apply sched_jobs_must_be_ready_to_execute.
move: READY ⇒ /andP[/andP[ARR NC] /allP PRIOR].
have SZ: 0 < service sched j t.+1.
{ apply scheduled_implies_nonzero_service; last by simpl; ∃ t; split.
by apply ideal_proc_model_ensures_ideal_progress. }
have SC: service sched j t.+1 < job_cost j by move: NCOMP; rewrite /completed_by -ltnNge.
move: SCHED NCOMP; rewrite !scheduled_at_def /sched /uni_schedule /pmc_uni_schedule
/generic_schedule ⇒ /eqP SCHED NCOMP.
rewrite schedule_up_to_def {1}/allocation_at ifT; first by rewrite SCHED.
rewrite /prev_job_nonpreemptive SCHED /job_preemptable /fully_nonpreemptive_job_model.
move: SC SZ; rewrite /uni_schedule /pmc_uni_schedule /generic_schedule.
set chp:= choose_highest_prio_job; set sut:= schedule_up_to.
replace (_ (_ (_ _ _) _ t) j t.+1) with (service sched j t.+1); last first.
{ rewrite /uni_schedule /pmc_uni_schedule /generic_schedule /service
/service_during /service_at; apply eq_big_nat ⇒ i H.
replace (_ (_ _ _) _ t i) with (sut (allocation_at arr_seq chp) None i i) ⇒ //.
by apply schedule_up_to_prefix_inclusion. }
move⇒ SC SZ; apply /andP.
split; last by apply /norP; split; [rewrite -lt0n|move:SC; rewrite ltn_neqAle⇒ /andP[??]].
apply /andP; split; [apply /andP; split; [by unfold has_arrived in *; lia|]|].
{ unfold completed_by in *; rewrite <- ltnNge in ×.
replace (_ _ j _) with (service (fun t⇒ sut (allocation_at arr_seq chp) None t t) j t.+1)=> //.
rewrite /service /service_during //=; apply eq_big_nat.
move⇒ t' /andP[GEQ0 LEQt]; rewrite /service_at.
replace (_ _ _ _ _) with (sut (allocation_at arr_seq chp) None t t')=> //.
by symmetry; apply schedule_up_to_prefix_inclusion; lia. }
{ apply /allP ⇒ j' IN; apply (leq_trans (PRIOR j' IN)).
rewrite /service /service_during big_nat_recr //= -(addn0 (\sum_(0 ≤ _ < t) _)).
apply leq_add ⇒ //; apply eq_leq, eq_big_nat.
move⇒ t' /andP[GEQ0 LEQt].
rewrite /service_at /sched /uni_schedule /pmc_uni_schedule /generic_schedule.
replace (_ (_ _ _) _ t' t') with (sut (allocation_at arr_seq chp) None t t') ⇒ //.
by symmetry; apply schedule_up_to_prefix_inclusion; lia. }
Qed.
Using the lemma above, we show that the schedule is nonpreemptive.
Lemma sched_nonpreemptive :
nonpreemptive_schedule (uni_schedule arr_seq).
Proof.
rewrite /nonpreemptive_schedule.
move⇒ j t; elim⇒ [|t' IHt']; first by move⇒ t'; have->: t = 0 by lia.
move⇒ LEQ SCHED NCOMP.
destruct (ltngtP t t'.+1) as [LT | _ | EQ] ⇒ //.
feed_n 3 IHt'⇒ //.
{ specialize (completion_monotonic sched j t' t'.+1) ⇒ MONO.
feed MONO; first by lia.
by apply contra in MONO. }
by apply sched_nonpreemptive_next.
Qed.
Finally, we show that the fixed-priority policy is respected at each preemption point.
Lemma respects_policy_at_preemption_point_np :
respects_FP_policy_at_preemption_point arr_seq sched (numeric_fp_ascending Task).
Proof.
apply schedule_respects_policy ⇒ //.
by apply sequential_readiness_nonclairvoyance.
Qed.
End Schedule.