# Priority Inversion in a Busy Interval

In this module, we reason about priority inversion that occurs during a busy interval due to non-preemptive sections.
Consider any type of tasks ...

... and any type of jobs associated with these tasks.
Context {Job : JobType}.
Context `{JobArrival Job}.
Context `{JobCost Job}.

Consider any valid arrival sequence ...
... and any uniprocessor schedule of this arrival sequence.
Context {PState : ProcessorState Job}.
Hypothesis H_uni : uniprocessor_model PState.
Variable sched : schedule PState.

Consider a JLFP policy that indicates a higher-or-equal priority relation, and assume that the relation is reflexive and transitive.
Consider a valid preemption model with known maximum non-preemptive segment lengths.
Further, allow for any work-bearing notion of job readiness.

We assume that the schedule is valid ...
... and that the schedule respects the scheduling policy at every preemption point.
Consider any job j of tsk with positive job cost.
Variable j : Job.
Hypothesis H_j_arrives : arrives_in arr_seq j.
Hypothesis H_job_cost_positive : job_cost_positive j.

Consider any busy interval prefix `[t1, t2)` of job j.
Variable t1 t2 : instant.
Hypothesis H_busy_interval_prefix :
busy_interval_prefix arr_seq sched j t1 t2.

## Lower Priority In Busy Intervals

First, we state some basic properties about a lower priority job executing in the busy interval of the job under consideration. From the definition of the busy interval it follows that a lower priority job can only be executing inside the busy interval as a result of priority inversion.
Consider a lower-priority job.
Variable jlp : Job.
Hypothesis H_jlp_lp : ~~hep_job jlp j.

Consider an instant t within the busy window of the job such that jlp is scheduled at t.
Variable t : instant.
Hypothesis H_t_in_busy : t1 t < t2.
Hypothesis H_jlp_scheduled_at_t : scheduled_at sched jlp t.

First, we prove that no time from t1 up to the instant t can be a preemption point.
Lemma lower_priority_job_scheduled_implies_no_preemption_time :
t',
t1 t' t
~~ preemption_time arr_seq sched t'.
Proof.
movet' /andP[LE1 GT1].
apply /negPPT.
move: H_jlp_lp ⇒ /negP LP; apply: LP.
have [ptst [IN1 [PTT STT]]] : ptst : nat,
t' ptst t preemption_time arr_seq sched ptst scheduled_at sched jlp ptst.
{ by apply: scheduling_of_any_segment_starts_with_preemption_time_continuously_sched ⇒ //=. }
apply: (scheduled_at_preemption_time_implies_higher_or_equal_priority arr_seq _ sched _ _ _ _ j _ _ t1 t2 _ ptst) ⇒ //.
by lia.
Qed.

Then it follows that the job must have been continuously scheduled from t1 up to t.
Lemma lower_priority_job_continuously_scheduled:
t',
t1 t' t
scheduled_at sched jlp t'.
Proof.
movet' IN.
move : IN ⇒ /andP [IN1' IN2'].
apply: neg_pt_scheduled_continuously_after ⇒ //.
move ⇒ ??.
apply lower_priority_job_scheduled_implies_no_preemption_time.
by lia.
Qed.

Any lower-priority jobs that are scheduled inside the busy-interval prefix `[t1,t2)` must arrive before that interval.
Lemma low_priority_job_arrives_before_busy_interval_prefix:
job_arrival jlp < t1.
Proof.
have SCHED1 : scheduled_at sched jlp t1.
{ apply lower_priority_job_continuously_scheduled ⇒ //. lia. }
have ARR1: job_arrival jlp t1 by apply: (has_arrived_scheduled sched jlp _ t1 SCHED1).
rewrite /has_arrived leq_eqVlt in ARR1.
move : ARR1 ⇒ /orP[/eqP EQ| ?]; last by done.
exfalso.
have PP := scheduling_of_any_segment_starts_with_preemption_time _
arr_seq H_valid_arrivals
sched H_sched_valid
H_valid_preemption_model jlp t H_jlp_scheduled_at_t.
feed PP ⇒ //.
move: PP ⇒ [pt [/andP [NEQ1 NEQ2] [PT FA]]].
apply /negP.
by apply (lower_priority_job_scheduled_implies_no_preemption_time ) ⇒ //=; lia.
Qed.

Finally, we show that lower-priority jobs that are scheduled inside the busy-interval prefix `[t1,t2)` must also be scheduled before the interval.
Lemma low_priority_job_scheduled_before_busy_interval_prefix:
t', t' < t1 scheduled_at sched jlp t'.
Proof.
move: H_t_in_busy ⇒ /andP [GE LT].
have ARR := low_priority_job_arrives_before_busy_interval_prefix .
t1.-1; split.
{ by rewrite prednK; last apply leq_ltn_trans with (job_arrival jlp). }
eapply neg_pt_scheduled_at ⇒ //.
- rewrite prednK; last by apply leq_ltn_trans with (job_arrival jlp).
apply lower_priority_job_continuously_scheduled ⇒ //. lia.
- rewrite prednK; last by apply leq_ltn_trans with (job_arrival jlp).
apply lower_priority_job_scheduled_implies_no_preemption_time.
by lia.
Qed.

End LowerPriorityJobScheduled.

In this section, we prove that priority inversion only occurs at the start of the busy window and occurs due to only one job.
Section SingleJob.

Suppose job j incurs priority inversion at a time t_pi in its busy window.
Variable t_pi : instant.
Hypothesis H_from_t1_before_t2 : t1 t_pi < t2.
Hypothesis H_PI_occurs : priority_inversion arr_seq sched j t_pi.

First, we show that there is no preemption time in the interval `[t1,t_pi]`.
Lemma no_preemption_time_before_pi :
t,
t1 t t_pi
~~ preemption_time arr_seq sched t.
Proof.
moveppt intl.
move : H_PI_occurs ⇒ /uni_priority_inversion_P PI.
feed_n 5 PI ⇒ //=.
move : PI ⇒ [jlp SCHED NHEP].
by apply (lower_priority_job_scheduled_implies_no_preemption_time jlp NHEP t_pi).
Qed.

Next, we show that the same job will be scheduled from the start of the busy interval to the priority inversion time t_pi.
Lemma pi_job_remains_scheduled :
jlp,
scheduled_at sched jlp t_pi
t,
t1 t t_pi scheduled_at sched jlp t.
Proof.
move : H_PI_occurs ⇒ /uni_priority_inversion_P PI.
feed_n 5 PI ⇒ //=.
move : PI ⇒ [jlp SCHED NHEP].
movejlp1 SCHED3 t IN.
apply: lower_priority_job_continuously_scheduled ⇒ //.
by have → : jlp1 = jlp.
Qed.

Thus, priority inversion takes place from the start of the busy interval to the instant t_pi, i.e., priority inversion takes place continuously.
Lemma pi_continuous :
t,
t1 t t_pi
priority_inversion arr_seq sched j t.
Proof.
move: (H_PI_occurs) ⇒ /andP[j_nsched_pi /hasP[jlp jlp_sched_pi nHEPj]] t INTL.
apply /uni_priority_inversion_P ⇒ // ; jlp ⇒ //.
apply: pi_job_remains_scheduled ⇒ //.
by rewrite -(scheduled_jobs_at_iff arr_seq).
Qed.

End SingleJob.

As a simple corollary to the lemmas proved in the previous section, we show that for any two jobs j1 and j2 that cause priority inversion to job j, it is the case that j1 = j2.
Section SingleJobEq.

Consider a time instant ts1 in `[t1, t2)` ...
Variable ts1 : instant.
Hypothesis H_ts1_in_busy_prefix : t1 ts1 < t2.

... and a lower-priority (w.r.t. job j) job j1 that is scheduled at time ts1.
Variable j1 : Job.
Hypothesis H_j1_sched : scheduled_at sched j1 ts1.
Hypothesis H_j1_lower_prio : ~~ hep_job j1 j.

Similarly, consider a time instant ts2 in `[t1, t2)` ...
Variable ts2 : instant.
Hypothesis H_ts2_in_busy_prefix : t1 ts2 < t2.

... and a lower-priority job j2 that is scheduled at time ts2.
Variable j2 : Job.
Hypothesis H_j2_sched : scheduled_at sched j2 ts2.
Hypothesis H_j2_lower_prio : ~~ hep_job j2 j.

Then, j1 is equal to j2.
Corollary only_one_pi_job :
j1 = j2.
Proof.
have [NEQ|NEQ] := leqP ts1 ts2.
{ apply: H_uni; first by apply H_j1_sched.
apply: pi_job_remains_scheduled; try apply H_j2_sched; try lia.
by erewrite priority_inversion_hep_job.
}
{ apply: H_uni; last by apply H_j2_sched.
apply: pi_job_remains_scheduled; try apply H_j1_sched; try lia.
by erewrite priority_inversion_hep_job; try apply H_j1_sched.
}
Qed.

End SingleJobEq.

From the above lemmas, it follows that either job j incurs no priority inversion at all or certainly at time t1, i.e., the beginning of its busy interval.
Lemma busy_interval_pi_cases :
cumulative_priority_inversion arr_seq sched j t1 t2 = 0
priority_inversion arr_seq sched j t1.
Proof.
case: (posnP (cumulative_priority_inversion arr_seq sched j t1 t2)); first by left.
rewrite sum_nat_gt0 // ⇒ /hasP[pi].
rewrite mem_filter /= mem_index_iota lt0bINTL PI_pi.
by right; apply: pi_continuous =>//; lia.
Qed.

Next, we use the above facts to establish bounds on the maximum priority inversion that can be incurred in a busy interval.

# Priority Inversion due to Non-Preemptive Sections

First, we introduce the notion of the maximum length of a nonpreemptive segment among all lower priority jobs (w.r.t. a given job j) arrived so far.
Note that any bound on the max_lp_nonpreemptive_segment function is also be a bound on the maximum priority inversion (assuming there are no other mechanisms that could cause priority inversion). This bound may be different for different scheduler and/or task models. Thus, we don't define such a bound in this module.

First, assuming proper non-preemptive sections, ...
... we observe that the maximum non-preemptive segment length of any task that releases a job with lower priority (w.r.t. a given job j) and non-zero execution cost upper-bounds the maximum possible non-preemptive segment length of any lower-priority job.
max_lp_nonpreemptive_segment j t1
\max_(j_lp <- arrivals_between arr_seq 0 t1 | (~~ hep_job j_lp j)
&& (job_cost j_lp > 0))
Proof.
rewrite /max_lp_nonpreemptive_segment.
apply: leq_big_maxj' JINB NOTHEP.
rewrite leq_sub2r //.
apply in_arrivals_implies_arrived in JINB.
by apply H_valid_nps.
Qed.

Next, we prove that the function max_lp_nonpreemptive_segment indeed upper-bounds the priority inversion length.
Section PreemptionTimeExists.

In this section, we require the jobs to have valid bounded non-preemptive segments.
First, we prove that, if a job with higher-or-equal priority is scheduled at a quiet time t+1, then this is the first time when this job is scheduled.
Lemma hp_job_not_scheduled_before_quiet_time :
jhp t,
quiet_time arr_seq sched j t.+1
scheduled_at sched jhp t.+1
hep_job jhp j
~~ scheduled_at sched jhp t.
Proof.
intros jhp t QT SCHED1 HP.
apply/negP; intros SCHED2.
specialize (QT jhp).
feed_n 3 QT ⇒ //.
- have MATE: jobs_must_arrive_to_execute sched by [].
by have HA: has_arrived jhp t by exact: MATE.
- apply completed_implies_not_scheduled in QT ⇒ //.
by move: QT ⇒ /negP NSCHED; apply: NSCHED.
Qed.

Thus, there must be a preemption time in the interval t1, t1 + max_lp_nonpreemptive_segment j t1. That is, if a job with higher-or-equal priority is scheduled at time instant t1, then t1 is a preemption time. Otherwise, if a job with lower priority is scheduled at time t1, then this job also should be scheduled before the beginning of the busy interval. So, the next preemption time will be no more than max_lp_nonpreemptive_segment j t1 time units later.
We proceed by doing a case analysis.
Section CaseAnalysis.

(1) Case when the schedule is idle at time t1.
Section Case1.

Assume that the schedule is idle at time t1.
Hypothesis H_is_idle : is_idle arr_seq sched t1.

Then time instant t1 is a preemption time.
Lemma preemption_time_exists_case1:
pr_t,
preemption_time arr_seq sched pr_t
t1 pr_t t1 + max_lp_nonpreemptive_segment j t1.
Proof.
set (service := service sched).
move: (H_valid_model_with_bounded_nonpreemptive_segments) ⇒ CORR.
move: (H_busy_interval_prefix) ⇒ [NEM [QT1 [NQT HPJ]]].
t1; split.
- exact: idle_time_is_pt.
- by apply/andP; split; last rewrite leq_addr.
Qed.

End Case1.

(2) Case when a job with higher-or-equal priority is scheduled at time t1.
Section Case2.

Assume that a job jhp with higher-or-equal priority is scheduled at time t1.
Variable jhp : Job.
Hypothesis H_jhp_is_scheduled : scheduled_at sched jhp t1.
Hypothesis H_jhp_hep_priority : hep_job jhp j.

Then time instant t1 is a preemption time.
Lemma preemption_time_exists_case2:
pr_t,
preemption_time arr_seq sched pr_t
t1 pr_t t1 + max_lp_nonpreemptive_segment j t1.
Proof.
set (service := service sched).
move : (H_valid_model_with_bounded_nonpreemptive_segments) ⇒ [VALID BOUNDED].
move: (H_valid_model_with_bounded_nonpreemptive_segments) ⇒ CORR.
move: (H_busy_interval_prefix) ⇒ [NEM [QT1 [NQT HPJ]]].
t1; split; last by apply/andP; split; [|rewrite leq_addr].
destruct t1.
- exact: zero_is_pt.
- apply: first_moment_is_pt H_jhp_is_scheduled ⇒ //.
exact: hp_job_not_scheduled_before_quiet_time.
Qed.

End Case2.

The following argument requires a unit-service assumption.
Hypothesis H_unit : unit_service_proc_model PState.

(3) Case when a job with lower priority is scheduled at time t1.
Section Case3.

Assume that a job jhp with lower priority is scheduled at time t1.
Variable jlp : Job.
Hypothesis H_jlp_is_scheduled : scheduled_at sched jlp t1.
Hypothesis H_jlp_low_priority : ~~ hep_job jlp j.

To prove the lemma in this case we need a few auxiliary facts about the first preemption point of job jlp.
Section FirstPreemptionPointOfjlp.

Let's denote the progress of job jlp at time t1 as progr_t1.
Consider the first preemption point of job jlp after progr_t1.
For correctness, we also assume that fpt does not exceed the length of the maximum non-preemptive segment.
First we show that fpt is indeed the first preemption point after progr_t1.
Lemma no_intermediate_preemption_point:
ρ,
progr_t1 ρ < progr_t1 + fpt
~~ job_preemptable jlp ρ.
Proof.
moveprog /andP [GE LT].
apply/negP; intros PPJ.
move: H_fpt_is_first_preemption_pointK; specialize (K prog).
feed_n 2 K ⇒ [|//|]; first (apply/andP; split⇒ //).
{ apply leq_trans with (service sched jlp t1 + fpt).
+ exact: ltnW.
+ by rewrite leq_add2l; apply H_progr_le_max_nonp_segment.
}
by move: K; rewrite leqNgt; move ⇒ /negP NLT; apply: NLT.
Qed.

Thanks to the fact that the scheduler respects the notion of preemption points we show that jlp is continuously scheduled in time interval `[t1, t1 + fpt)`.
Lemma continuously_scheduled_between_preemption_points:
t',
t1 t' < t1 + fpt
scheduled_at sched jlp t'.
Proof.
move: (H_valid_model_with_bounded_nonpreemptive_segments) ⇒ CORR.
have ARRs : arrives_in arr_seq jlp by [].
movet' /andP [GE LT].
have Fact: Δ, t' = t1 + Δ.
{ by (t' - t1); apply/eqP; rewrite eq_sym; apply/eqP; rewrite subnKC. }
move: Fact ⇒ [Δ EQ]; subst t'.
have NPPJ := @no_intermediate_preemption_point (@service _ _ sched jlp (t1 + Δ)).
apply proj1 in CORR; specialize (CORR jlp ARRs).
move: CORR ⇒ [_ [_ [T _] ]].
apply T; apply: NPPJ; apply/andP; split.
{ by apply service_monotonic; rewrite leq_addr. }
rewrite /service -(service_during_cat _ _ _ t1).
apply leq_ltn_trans with Δ ⇒ [|//].
rewrite -{2}(sum_of_ones t1 Δ).
by rewrite leq_sum. }
{ by apply/andP; split; [|rewrite leq_addr]. }
Qed.

Thus, assuming an ideal-progress processor model, job jlp reaches its preemption point at time instant t1 + fpt, which implies that time instant t1 + fpt is a preemption time.
Lemma first_preemption_time :
ideal_progress_proc_model PState
preemption_time arr_seq sched (t1 + fpt).
Proof.
moveH_progress.
have [IDLE|[j' SCHED']] :=
scheduled_at_cases _ H_valid_arrivals sched ltac:(by []) ltac:(by []) (t1 + fpt);
first exact: idle_time_is_pt.
have [EQ|NEQ] := (eqVneq jlp j').
{ move: (SCHED'); rewrite -(scheduled_job_at_scheduled_at arr_seq) // -EQ /preemption_time ⇒ /eqP →.
rewrite /service -(service_during_cat _ _ _ t1); last first.
{ by apply/andP; split; last rewrite leq_addr. }
have ->: service_during sched jlp t1 (t1 + fpt) = fpt ⇒ //.
{ rewrite -{2}(sum_of_ones t1 fpt) /service_during.
apply/eqP; rewrite eqn_leq //; apply/andP; split.
+ by rewrite leq_sum.
+ rewrite big_nat_cond [in X in _ X]big_nat_cond.
rewrite leq_sum //.
movex /andP [HYP _].
exact/H_progress/continuously_scheduled_between_preemption_points. } }
{ case: (posnP fpt) ⇒ [ZERO|POS].
{ subst fpt; rewrite addn0 in SCHED'.
exfalso; move: NEQ ⇒ /negP; apply; apply/eqP.
exact: H_uni. }
{ have [sm EQ2]: sm, sm.+1 = fpt by fpt.-1; rewrite prednK.
apply: first_moment_is_pt (SCHED') ⇒ //.
apply: scheduled_job_at_neq ⇒ //.
apply: (continuously_scheduled_between_preemption_points (t1 + sm)).
Qed.

Lemma preemption_time_le_max_len_of_np_segment :
t1 t1 + fpt t1 + max_lp_nonpreemptive_segment j t1.
Proof.
have ARRs : arrives_in arr_seq jlp by [].
apply/andP; split; first by rewrite leq_addr.
unfold max_lp_nonpreemptive_segment.
rewrite (big_rem jlp) //=.
{ rewrite H_jlp_low_priority //=.
have NZ: service sched jlp t1 < job_cost jlp by exact: service_lt_cost.
rewrite ifT; last lia.
apply leq_trans with (job_max_nonpreemptive_segment jlp - ε).
- by apply H_progr_le_max_nonp_segment.
- by rewrite leq_maxl.
}
apply: arrived_between_implies_in_arrivals ⇒ [//|//|].
apply/andP; split⇒ [//|].
eapply low_priority_job_arrives_before_busy_interval_prefix with t1; eauto 2.
by move: (H_busy_interval_prefix) ⇒ [NEM [QT1 [NQT HPJ]]]; apply/andP.
Qed.

End FirstPreemptionPointOfjlp.

For the next step, we assume an ideal-progress processor.
Hypothesis H_progress : ideal_progress_proc_model PState.

Next, we combine the above facts to conclude the lemma.
Lemma preemption_time_exists_case3:
pr_t,
preemption_time arr_seq sched pr_t
t1 pr_t t1 + max_lp_nonpreemptive_segment j t1.
Proof.
set (service := service sched).
have EX: pt,
((service jlp t1) pt (service jlp t1) + (job_max_nonpreemptive_segment jlp - 1)) && job_preemptable jlp pt.
{ have ARRs: arrives_in arr_seq jlp by [].
move: (proj2 (H_valid_model_with_bounded_nonpreemptive_segments) jlp ARRs) =>[_ EXPP].
destruct H_sched_valid as [A B].
specialize (EXPP (service jlp t1)).
feed EXPP.
{ apply/andP; split⇒ [//|].
exact: service_at_most_cost.
}
move: EXPP ⇒ [pt [NEQ PP]].
by pt; apply/andP.
}
move: (ex_minnP EX) ⇒ [sm_pt /andP [NEQ PP] MIN]; clear EX.
have Fact: Δ, sm_pt = service jlp t1 + Δ.
{ (sm_pt - service jlp t1).
apply/eqP; rewrite eq_sym; apply/eqP; rewrite subnKC //.
by move: NEQ ⇒ /andP [T _]. }
move: Fact ⇒ [Δ EQ]; subst sm_pt; rename Δ into sm_pt.
(t1 + sm_pt); split.
{ apply first_preemption_time; rewrite /service.service//.
+ by intros; apply MIN; apply/andP; split.
+ by lia.
}
apply: preemption_time_le_max_len_of_np_segment ⇒ //.
by lia.
Qed.

End Case3.

End CaseAnalysis.

As Case 3 depends on unit-service and ideal-progress assumptions, we require the same here.
Hypothesis H_unit : unit_service_proc_model PState.
Hypothesis H_progress : ideal_progress_proc_model PState.

By doing the case analysis, we show that indeed there is a preemption time in the time interval [t1, t1 + max_lp_nonpreemptive_segment j t1].
Lemma preemption_time_exists :
pr_t,
preemption_time arr_seq sched pr_t
t1 pr_t t1 + max_lp_nonpreemptive_segment j t1.
Proof.
have [Idle|[s Sched_s]] :=
scheduled_at_cases _ H_valid_arrivals sched ltac:(by []) ltac:(by []) t1.
- by apply preemption_time_exists_case1.
- destruct (hep_job s j) eqn:PRIO.
+ exact: preemption_time_exists_case2.
+ apply: preemption_time_exists_case3 ⇒ //.
by rewrite -eqbF_neg; apply /eqP.
Qed.

End PreemptionTimeExists.

In this section we prove that if a preemption point ppt exists in a job's busy window, it suffers no priority inversion after ppt. Equivalently the cumulative_priority_inversion of the job in the busy window t1,t2 is bounded by the cumulative_priority_inversion of the job in the time window t1,[ppt]).
Consider the preemption point ppt.
Variable ppt: instant.
Hypothesis H_preemption_point : preemption_time arr_seq sched ppt.
Hypothesis H_after_t1 : t1 ppt.

We first establish the aforementioned result by showing that j cannot suffer priority inversion after the preemption time ppt ...
Lemma no_priority_inversion_after_preemption_point :
t,
ppt t < t2
~~ priority_inversion arr_seq sched j t.
Proof.
movet /andP [pptt tt2].
apply /negPPI.
move: PI ⇒ /andP[j_nsched_pi /hasP[jlp jlp_sched_pi nHEPj]].
have [/eqP SCHED1 //|[j' /eqP SCHED2 //=]] :=
scheduled_jobs_at_uni_cases arr_seq ltac:(done) sched ltac:(done)
ltac:(done) ltac:(done) t; first by rewrite SCHED1 in jlp_sched_pi.
rewrite SCHED2 mem_seq1 in jlp_sched_pi.
move : jlp_sched_pi ⇒ /eqP HJLP.
replace j' with jlp in *; clear HJLP.
move : SCHED2 ⇒ /eqP SCHED2.
rewrite scheduled_jobs_at_scheduled_at in SCHED2 ⇒ //=.
have [ptst [IN1 [PTT STT]]] : ptst : nat,
ppt ptst t preemption_time arr_seq sched ptst scheduled_at sched jlp ptst.
{ by apply: scheduling_of_any_segment_starts_with_preemption_time_continuously_sched. }
apply /negP /negPn.
apply: scheduled_at_preemption_time_implies_higher_or_equal_priority ⇒ //.
by lia.
Qed.

... and then lift this fact to cumulative priority inversion.
Lemma priority_inversion_occurs_only_till_preemption_point :
cumulative_priority_inversion arr_seq sched j t1 t2
cumulative_priority_inversion arr_seq sched j t1 ppt.
Proof.
have [LEQ|LT_t1t2] := leqP t1 t2;
last by rewrite /cumulative_priority_inversion big_geq //; exact: ltnW.
have [LEQ_t2ppt|LT] := leqP t2 ppt;
first by rewrite (cumulative_priority_inversion_cat _ _ _ t2 t1 ppt) //