# Library prosa.analysis.abstract.restricted_supply.iw_readiness

Require Export prosa.analysis.abstract.abstract_rta.
Require Export prosa.analysis.abstract.IBF.supply_task.
Require Export prosa.analysis.facts.interference.
Require Export prosa.analysis.facts.busy_interval.service_inversion.
Require Export prosa.analysis.definitions.service_inversion.readiness_aware.
Require Export prosa.analysis.facts.readiness_interference.
Require Export prosa.analysis.facts.model.uniprocessor.
Require Export prosa.model.schedule.work_conserving.

# Readiness-Aware JLFP Instantiation of Interference and Interfering Workload Functions for Restricted-Supply Uniprocessors

Here we instantiate the interference and interfering workload functions, as required by abstract restricted-supply analysis (aRSA).
Section IWInstantiation.
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 a fully supply-consuming, unit-supply uniprocessor model.
Context `{PState : ProcessorState Job}.
Hypothesis H_uniprocessor_proc_model : uniprocessor_model PState.
Hypothesis H_unit_supply_proc_model : unit_supply_proc_model PState.
Hypothesis H_consumed_supply_proc_model : fully_consuming_proc_model PState.
Allow for any notion of job readiness.
Context `{!JobReady Job PState}.
Consider any valid arrival sequence ...
Variable arr_seq : arrival_sequence Job.
Hypothesis H_valid_arrival_sequence : valid_arrival_sequence arr_seq.
... and any valid schedule of this arrival sequence.
Variable sched : schedule PState.
Hypothesis H_valid_schedule : valid_schedule sched arr_seq.
Consider any JLFP policy and assume that the priority relation is reflexive and transitive.
Context {JLFP : JLFP_policy Job}.
Hypothesis H_priority_is_reflexive : reflexive_job_priorities JLFP.
Hypothesis H_priority_is_transitive : transitive_job_priorities JLFP.

# Interference and Interfering Workload

In the following, we introduce definitions of interference and interfering workload.

## Instantiation of Interference

We say that job j incurs interference at time t iff it cannot execute due to (1) the lack of supply at time t, (2) due to service inversion (i.e., a lower-priority job receiving service at t), (3) due to another higher-or-equal-priority job receiving service, (4) due to j and all other higher-or-equal-priority jobs becoming non-ready.
#[local] Instance rs_readiness_jlfp_interference : Interference Job :=
{
interference (j : Job) (t : instant) :=
is_blackout sched t
|| another_hep_job_interference arr_seq sched j t
|| service_inversion arr_seq sched j t
|| (has_supply sched t && ~~ some_hep_job_ready arr_seq sched j t)
}.

## Instantiation of Interfering Workload

The interfering workload, in turn, is defined as the sum of the indicator values of the predicates in the definition of interference.
#[local] Instance rs_readiness_jlfp_interfering_workload : InterferingWorkload Job :=
{
interfering_workload (j : Job) (t : instant) :=
is_blackout sched t
+ other_hep_jobs_interfering_workload arr_seq j t
+ service_inversion arr_seq sched j t
+ has_supply sched t && ~~ some_hep_job_ready arr_seq sched j t
}.
To simplify our proof, we define a notion of cumulative_readiness_interference that is disjoint from is_blackout. This will help us to split cumulative_interference and cumulative_interfering_workload into disjoint parts.
Let readiness_interference_during (j : Job) (t1 t2 : instant) :=
\sum_(t1 ≤ t < t2) has_supply sched t && ~~ some_hep_job_ready arr_seq sched j t.

## Equivalences

We prove that we can split cumulative interference into disjoint parts.
Lemma cumulative_interference_split :
∀ j t1 t2,
cumulative_interference j t1 t2
= blackout_during sched t1 t2
+ cumulative_another_hep_job_interference arr_seq sched j t1 t2
+ cumulative_service_inversion arr_seq sched j t1 t2
+ readiness_interference_during j t1 t2.
Proof.
rewrite /cumulative_interference /cumul_cond_interference /rs_readiness_jlfp_interference
/cond_interference /interference.
move⇒ j t1 t2; rewrite -big_split //= -big_split -big_split //=.
apply/eqP; rewrite eqn_leq; apply/andP; split; rewrite leq_sum//; move⇒ t _.
{ by case is_blackout, another_hep_job_interference, service_inversion, some_hep_job_ready. }
{ have [/negPf BL|SUP] := blackout_or_supply sched t.
{ rewrite /is_blackout BL //= addn0 -addnA addnC addn1 ltnS leqn0 addn_eq0.
move: BL ⇒ /idPn BL.
by apply/andP; split;
[ rewrite eqb0 no_hep_job_interference_without_supply
| rewrite eqb0 /service_inversion; apply /nandP;
right; by rewrite blackout_implies_no_service_inversion ]. }
{ rewrite /is_blackout SUP //= add0n.
have [NO_HEP_READY | SOME_HEP_READY] := boolP(some_hep_job_ready arr_seq sched j t); last first.
{ rewrite orbT //= addn1 ltnS leqn0 addn_eq0.
by apply /andP; split;
[rewrite eqb0 no_hep_ready_implies_no_another_hep_interference
| rewrite eqb0 no_hep_ready_implies_no_service_inversion]. }
{ rewrite orbF addn0.
have A: ∀ (a b : bool), (a → ~~ b) → a + b ≤ a || b by lia.
apply A ⇒
/hasP [jhp /served_at_and_receives_service_consistent SERVjo /andP[HEPjo _]].
have SCHEDjo : scheduled_at sched jhp t by apply service_at_implies_scheduled_at.
rewrite /service_inversion NO_HEP_READY //=.
apply /negP ⇒
/andP[_ /hasP[jlp /served_at_and_receives_service_consistent SERVjlp NOTHEPjlp]].
rewrite hep_job_at_jlfp in NOTHEPjlp.
have SCHEDjlp : scheduled_at sched jlp t by apply service_at_implies_scheduled_at.
have NEQ : jlp != jhp by apply /negP ⇒ /eqP ?; subst jlp; lia.
apply scheduled_job_at_neq with (j' := jhp) in SCHEDjlp ⇒ //=.
by lia.
}
}
}
Qed.
Similarly, we prove that we can split cumulative interfering workload into disjoint parts.
Lemma cumulative_interfering_workload_split :
∀ j t1 t2,
cumulative_interfering_workload j t1 t2
= blackout_during sched t1 t2
+ cumulative_other_hep_jobs_interfering_workload arr_seq j t1 t2
+ cumulative_service_inversion arr_seq sched j t1 t2
+ readiness_interference_during j t1 t2.
Proof.
by move⇒ j t1 t2; rewrite -big_split //= -big_split -big_split //=.
Qed.
Now, let tsk be any given task.
Variable tsk : Task.
Let `[t1, t2)` be a time interval and let j be any job of task tsk that is not completed by time t2. Then we prove a bound on the cumulative interference received due to jobs of other tasks executing.
Lemma cumulative_task_interference_split :
∀ j t1 t2,
arrives_in arr_seq j →
job_of_task tsk j →
~~ completed_by sched j t2 →
cumul_cond_interference (nonself_intra arr_seq sched) j t1 t2
≤ cumulative_another_task_hep_job_interference arr_seq sched j t1 t2
+ cumulative_service_inversion arr_seq sched j t1 t2
+ readiness_interference_during j t1 t2.
Proof.
move⇒ j t1 R ARR TSK NCOMPL.
rewrite /cumul_task_interference /cumul_cond_interference.
rewrite -big_split -big_split //= big_seq_cond [leqRHS]big_seq_cond.
apply leq_sum ⇒ t /andP [IN _].
rewrite /cond_interference /nonself_intra /nonself /interference
/rs_readiness_jlfp_interference /nonself_intra.
have [SUP|NOSUP] := boolP (has_supply sched t); last first.
{ by move: (NOSUP); rewrite /is_blackout ⇒ → //=; rewrite andbT andbF //. }
{ move: (SUP); rewrite /is_blackout ⇒ ->; rewrite andbT //=.
have [NO_HEP_READY | SOME_HEP_READY] := boolP(some_hep_job_ready arr_seq sched j t); try by lia.
rewrite orbF addn0.
rewrite /service_inversion NO_HEP_READY.
have [IDLE | [j' SCHED]]:= scheduled_at_cases arr_seq ltac:(done) sched ltac:(done) ltac:(done) t.
{ rewrite_neg idle_implies_no_service_inversion;
rewrite_neg no_hep_job_interference_when_idle;
rewrite_neg no_hep_task_interference_when_idle.
by lia. }
{ case SERVtsk : task_served_at ⇒ //=.
case: pred.service_inversion; try by lia.
move: SERVtsk ⇒ /negbT.
erewrite task_served_at_eq_job_of_task ⇒ //.
erewrite interference_ahep_def ⇒ //.
erewrite interference_athep_def ⇒ //.
rewrite /another_hep_job /another_task_hep_job.
rewrite orbF addn0 /job_of_task ⇒ →.
by lia.
} }
Qed.
Next, consider cumulative intra-supply interference, which accounts for any interference that occurs while the processor offers supply.
To this end, let's first define a predicate to express intra-supply interference.
Let intra_supply (_j : Job) (t : instant) := has_supply sched t.
The cumulative interference conditioned on intra_supply can be split into three disjoint parts.
Lemma cumulative_intra_interference_split :
∀ j t1 t2,
cumul_cond_interference intra_supply j t1 t2
≤ cumulative_another_hep_job_interference arr_seq sched j t1 t2
+ cumulative_service_inversion arr_seq sched j t1 t2
+ readiness_interference_during j t1 t2.
Proof.
move⇒ j t1 t2.
rewrite /cumul_cond_interference -big_split -big_split //= big_seq_cond [leqRHS]big_seq_cond.
apply leq_sum ⇒ t /andP [IN _].
rewrite /cond_interference /nonself /interference
/rs_readiness_jlfp_interference /intra_supply.
have [SUP|NOSUP] := boolP (has_supply sched t); last first.
{ by move: (NOSUP); rewrite /is_blackout ⇒ → //=; rewrite andbT andbF //. }
{ move: (SUP); rewrite /is_blackout ⇒ ->; rewrite andTb //=.
have L : ∀ (a b c : bool), a || b || c ≤ a + b + c by clear ⇒ [] [] [] [].
by apply L. }
Qed.
In this section, we prove that the (abstract) cumulative interfering workload due to other higher-or-equal priority jobs is equal to the conventional workload (from other higher-or-equal priority jobs).
Section InstantiatedWorkloadEquivalence.
Let `[t1, t2)` be any time interval.
Variables t1 t2 : instant.
Consider any job j of tsk in the given arrival sequence.
Variable j : Job.
Hypothesis H_j_arrives : arrives_in arr_seq j.
Hypothesis H_job_of_tsk : job_of_task tsk j.
The cumulative interfering workload (w.r.t. j) due to other higher-or-equal priority jobs is equal to the conventional workload from other higher-or-equal priority jobs.
Lemma cumulative_iw_hep_eq_workload_of_ohep :
cumulative_other_hep_jobs_interfering_workload arr_seq j t1 t2
= workload_of_other_hep_jobs arr_seq j t1 t2.
Proof.
rewrite /cumulative_other_hep_jobs_interfering_workload /workload_of_other_hep_jobs.
case NEQ: (t1 < t2); last first.
{ move: NEQ ⇒ /negP /negP; rewrite -leqNgt ⇒ NEQ.
rewrite big_geq // /arrivals_between big_geq //.
by rewrite /workload_of_jobs big_nil. }
{ interval_to_duration t1 t2 k.
elim: k ⇒ [|k IHk].
- rewrite !addn0 big_geq// /arrivals_between big_geq//.
by rewrite /workload_of_jobs big_nil.
- rewrite addnS big_nat_recr //=; last by rewrite leq_addr.
rewrite IHk /arrivals_between big_nat_recr //=.
+ by rewrite /workload_of_jobs big_cat.
+ by rewrite leq_addr. }
Qed.
End InstantiatedWorkloadEquivalence.
In this section we prove that the abstract definition of busy interval is equivalent to the conventional, concrete definition of busy interval for JLFP scheduling.
Section BusyIntervalEquivalence.
In order to avoid confusion, we denote the notion of a quiet time in the classical sense as quiet_time_cl, and the notion of quiet time in the abstract sense as quiet_time_ab.
Let quiet_time_cl := classical.quiet_time arr_seq sched.
Let quiet_time_ab := abstract.definitions.quiet_time sched.
We similarly define local names for the two notions of a busy interval prefix ...
Let busy_interval_prefix_cl := classical.busy_interval_prefix arr_seq sched.
Let busy_interval_prefix_ab := abstract.definitions.busy_interval_prefix sched.
... and the two notions of a busy interval.
Let busy_interval_cl := classical.busy_interval arr_seq sched.
Let busy_interval_ab := abstract.definitions.busy_interval sched.
Consider any job j of tsk.
Variable j : Job.
Hypothesis H_j_arrives : arrives_in arr_seq j.
To prove that the two notions of busy intervals are equivalent, we first show that the respective notions of quiet times are also equivalent.
First, we show that the classical notion of quiet time implies the abstract notion of quiet time.
Lemma quiet_time_cl_implies_quiet_time_ab :
∀ t, quiet_time_cl j t → quiet_time_ab j t.
Proof.
have zero_is_quiet_time: ∀ j, quiet_time_cl j 0.
{ by move ⇒ jhp ARR HP AB; move: AB; rewrite /arrived_before ltn0. }
move⇒ t QT; apply/andP; split; last first.
{ rewrite negb_and negbK; apply/orP.
by case ARR: (arrived_before j t); [right | left]; [apply QT | ]. }
{ erewrite cumulative_interference_split, cumulative_interfering_workload_split.
rewrite -!addnA eqn_add2l eqn_add2r.
rewrite cumulative_i_ohep_eq_service_of_ohep //.
rewrite //= cumulative_iw_hep_eq_workload_of_ohep eq_sym; apply/eqP.
apply all_jobs_have_completed_equiv_workload_eq_service ⇒ //.
move ⇒ j0 IN HEP; apply QT.
- by apply in_arrivals_implies_arrived in IN.
- by move: HEP ⇒ /andP [HEP HP].
- by apply in_arrivals_implies_arrived_between in IN.
}
Qed.
And vice versa, the abstract notion of quiet time implies the classical notion of quiet time.
Lemma quiet_time_ab_implies_quiet_time_cl :
∀ t, quiet_time_ab j t → quiet_time_cl j t.
Proof.
have zero_is_quiet_time: ∀ j, quiet_time_cl j 0.
{ by move ⇒ jhp ARR HP AB; move: AB; rewrite /arrived_before ltn0. }
move ⇒ t /andP [T0 T1] jhp ARR HP ARB.
eapply all_jobs_have_completed_equiv_workload_eq_service with
(P := fun jhp ⇒ hep_job jhp j) (t1 := 0) (t2 := t) ⇒ //.
erewrite service_of_jobs_case_on_pred with (P2 := fun j' ⇒ j' != j).
erewrite workload_of_jobs_case_on_pred with (P' := fun j' ⇒ j' != j) ⇒ //.
replace ((fun j0 : Job ⇒ hep_job j0 j && (j0 != j))) with (another_hep_job^~j); last by rewrite /another_hep_job.
rewrite -/(service_of_other_hep_jobs arr_seq sched j 0 t) -cumulative_i_ohep_eq_service_of_ohep //.
rewrite -/(workload_of_other_hep_jobs arr_seq j 0 t) -cumulative_iw_hep_eq_workload_of_ohep //.
move: T1; rewrite negb_and ⇒ /orP [NA | /negPn COMP].
{ have PRED__eq: {in arrivals_between arr_seq 0 t, (fun j__copy : Job ⇒ hep_job j__copy j && ~~ (j__copy != j)) =1 pred0}.
{ move ⇒ j__copy IN; apply negbTE.
rewrite negb_and; apply/orP; right; apply/negPn/eqP ⇒ EQ; subst j__copy.
move: NA ⇒ /negP CONTR; apply: CONTR.
by apply in_arrivals_implies_arrived_between in IN. }
erewrite service_of_jobs_equiv_pred with (P2 := pred0) ⇒ [|//].
erewrite workload_of_jobs_equiv_pred with (P' := pred0) ⇒ [|//].
move: T0; erewrite cumulative_interference_split, cumulative_interfering_workload_split.
rewrite -!addnA eqn_add2l eqn_add2r ⇒ /eqP EQ.
rewrite EQ; clear EQ; apply/eqP; rewrite eqn_add2l.
by erewrite workload_of_jobs_pred0, service_of_jobs_pred0.
}
{ have PRED__eq: {in arrivals_between arr_seq 0 t, (fun j0 : Job ⇒ hep_job j0 j && ~~ (j0 != j)) =1 eq_op j}.
{ move ⇒ j__copy IN.
replace (~~ (j__copy != j)) with (j__copy == j); last by case: (j__copy == j).
rewrite eq_sym; destruct (j == j__copy) eqn:EQ; last by rewrite Bool.andb_false_r.
by move: EQ ⇒ /eqP EQ; rewrite Bool.andb_true_r; apply/eqP; rewrite eqb_id; subst. }
erewrite service_of_jobs_equiv_pred with (P2 := eq_op j) ⇒ [|//].
erewrite workload_of_jobs_equiv_pred with (P' := eq_op j) ⇒ [|//].
move: T0; erewrite cumulative_interference_split, cumulative_interfering_workload_split.
rewrite -!addnA eqn_add2l eqn_add2r ⇒ /eqP EQ.
rewrite EQ; clear EQ; apply/eqP; rewrite eqn_add2l.
apply/eqP; eapply all_jobs_have_completed_equiv_workload_eq_service with
(P := eq_op j) (t1 := 0) (t2 := t) ⇒ //.
{ by move ⇒ j__copy _ /eqP EQ; subst j__copy. }
}
Qed.
The equivalence trivially follows from the lemmas above.
Corollary instantiated_quiet_time_equivalent_quiet_time :
∀ t,
quiet_time_cl j t ↔ quiet_time_ab j t.
Proof.
move ⇒ ?; split.
- by apply quiet_time_cl_implies_quiet_time_ab.
- by apply quiet_time_ab_implies_quiet_time_cl.
Qed.
Based on that, we prove that the concept of a busy-interval prefix obtained by instantiating the abstract definition of busy-interval prefix coincides with the conventional definition of busy-interval prefix.
Lemma instantiated_busy_interval_prefix_equivalent_busy_interval_prefix :
∀ t1 t2,
busy_interval_prefix_cl j t1 t2 ↔ busy_interval_prefix_ab j t1 t2.
Proof.
move ⇒ t1 t2; split.
{ move ⇒ [NEQ [QTt1 [NQT REL]]].
split⇒ [//|]; split.
- by apply instantiated_quiet_time_equivalent_quiet_time in QTt1.
- by move ⇒ t NE QT; eapply NQT; eauto 2; apply instantiated_quiet_time_equivalent_quiet_time.
}
{ move ⇒ [/andP [NEQ1 NEQ2] [QTt1 NQT]].
split; [ | split; [ |split] ].
- by apply leq_ltn_trans with (job_arrival j).
- by eapply instantiated_quiet_time_equivalent_quiet_time.
- by move ⇒ t NEQ QT; eapply NQT; eauto 2; eapply instantiated_quiet_time_equivalent_quiet_time in QT.
- by apply/andP.
}
Qed.
Similarly, we prove that the concept of busy interval obtained by instantiating the abstract definition of busy interval coincides with the conventional definition of busy interval.
Lemma instantiated_busy_interval_equivalent_busy_interval :
∀ t1 t2,
busy_interval_cl j t1 t2 ↔ busy_interval_ab j t1 t2.
Proof.
move ⇒ t1 t2; split.
{ move ⇒ [PREF QTt2]; split.
- by apply instantiated_busy_interval_prefix_equivalent_busy_interval_prefix.
- by eapply instantiated_quiet_time_equivalent_quiet_time in QTt2.
}
{ move ⇒ [PREF QTt2]; split.
- by apply instantiated_busy_interval_prefix_equivalent_busy_interval_prefix.
- by eapply instantiated_quiet_time_equivalent_quiet_time.
}
Qed.
Next, we prove that abstract busy window implies the existence of a classic quiet time.
Fact abstract_busy_interval_classic_quiet_time :
∀ t1 t2,
busy_interval_ab j t1 t2 → quiet_time_cl j t1.
Proof.
by move⇒ ? ? /instantiated_busy_interval_equivalent_busy_interval [[_ [? _]] _].
Qed.
Also, we note a similar fact about classic busy-window prefixes.
Fact abstract_busy_interval_classic_busy_interval_prefix :
∀ t1 t2,
busy_interval_ab j t1 t2 → busy_interval_prefix_cl j t1 t2.
Proof. by move⇒ ? ? /instantiated_busy_interval_equivalent_busy_interval [+ _]. Qed.
End BusyIntervalEquivalence.
In this section we prove some properties about the interference and interfering workload as defined in this file.
Section I_IW_correctness.
Note that we differentiate between abstract and classical notions of work-conserving schedules.
Let work_conserving_ab := definitions.work_conserving arr_seq sched.
Let work_conserving_cl := work_conserving.work_conserving arr_seq sched.
Recall the notion of an abstract busy-interval prefix.
Let busy_interval_prefix_ab := definitions.busy_interval_prefix sched.
We assume that the schedule is a work-conserving schedule in the classical sense, and later prove that this implies that abstract work-conservation also holds.
Hypothesis H_work_conserving : work_conserving_cl.
In this section, we prove the correctness of interference inside the busy interval, i.e., we prove that if interference for a job is false then the job is scheduled and vice versa. This property is referred to as abstract work conservation.
Section Abstract_Work_Conservation.
Consider an arriving job j with positive cost.
Variable j : Job.
Hypothesis H_arrives : arrives_in arr_seq j.
Hypothesis H_job_cost_positive : 0 < job_cost j.
Let `[t1, t2)` be a busy-interval prefix of the job.
Variable t1 t2 : instant.
Hypothesis H_busy_interval_prefix : busy_interval_prefix_ab j t1 t2.
Consider a time t inside the busy-interval prefix of the job.
Variable t : instant.
Hypothesis H_t_in_busy_interval : t1 ≤ t < t2.
We first prove that, inside the busy interval, there always exists a pending higher-or-equal-priority job. To prove this, we make use of the obtained result that, for the given interference and interfering workload functions, the notions of abstract and classical busy intervals are equivalent.
Lemma pending_hep_job_exists_inside_busy_interval :
∃ jhp,
arrives_in arr_seq jhp
∧ pending sched jhp t
∧ hep_job jhp j.
Proof.
move: H_t_in_busy_interval ⇒ /andP [GE LT].
move: H_busy_interval_prefix ⇒
/instantiated_busy_interval_prefix_equivalent_busy_interval_prefix BUSY_cl.
move: (BUSY_cl H_arrives) ⇒ [_ [QTt [NQT REL]]].
move: (ltngtP t1.+1 t2) ⇒ [GT|CONTR|EQ]; first last.
- subst t2; rewrite ltnS in LT.
have EQ: t1 = t by apply/eqP; rewrite eqn_leq; apply/andP; split.
subst t1; clear GE LT; ∃ j; repeat split⇒ //.
+ move: REL; rewrite ltnS -eqn_leq eq_sym ⇒ /eqP REL;
by rewrite -REL; eapply job_pending_at_arrival; eauto 2.
- by exfalso; move_neq_down CONTR; eapply leq_ltn_trans; eauto 2.
- have EX:
∃ hp__seq: seq Job, ∀ j__hp,
j__hp \in hp__seq ↔ arrives_in arr_seq j__hp ∧ pending sched j__hp t ∧ hep_job j__hp j.
{ ∃ (filter (fun jo ⇒ (pending sched jo t)
&& (hep_job jo j)) (arrivals_between arr_seq 0 t.+1)).
intros; split; intros T.
- move: T; rewrite mem_filter ⇒ /andP [/andP [PEN HP] IN].
repeat split; eauto using in_arrivals_implies_arrived.
- move: T ⇒ [ARR [PEN HP]].
rewrite mem_filter; apply/andP; split; first (apply/andP; split⇒ //).
apply: arrived_between_implies_in_arrivals ⇒ //.
by apply/andP; split; last rewrite ltnS; move: PEN ⇒ /andP [T _]. }
move: EX ⇒ [hp__seq SE]; case FL: (hp__seq) ⇒ [ | jhp jhps].
+ subst hp__seq; exfalso; move: GE; rewrite leq_eqVlt ⇒ /orP [/eqP EQ| GE].
× subst t; apply NQT with t1.+1; first by apply/andP; split.
intros jhp ARR HP ARRB; apply negbNE; apply/negP; intros NCOMP.
move: (SE jhp) ⇒ [_ SE2].
rewrite in_nil in SE2; feed SE2⇒ [|//]; clear SE2.
repeat split⇒ //; first apply/andP; split⇒ //.
apply/negP; intros COMLP; move: NCOMP ⇒ /negP NCOMP; apply: NCOMP.
by apply completion_monotonic with t1.
× apply NQT with t; first by apply/andP; split.
intros jhp ARR HP ARRB; apply negbNE; apply/negP; intros NCOMP.
move: (SE jhp) ⇒ [_ SE2].
rewrite in_nil in SE2; feed SE2 ⇒ [|//]; clear SE2.
by repeat split; auto; apply/andP; split; first apply ltnW.
+ move: (SE jhp)=> [SE1 _]; subst; clear SE.
by ∃ jhp; apply SE1; rewrite in_cons; apply/orP; left.
Qed.
We now prove that, if interference is false at a time t, then the job is scheduled at time t.
Lemma not_interference_implies_scheduled :
~~ interference j t → receives_service_at sched j t.
Proof.
rewrite !negb_or ⇒ /andP[/andP[/andP[HYP1 HYP2] HYP3] HYP4].
move: HYP1; rewrite /is_blackout negbK ⇒ SUP; apply ideal_progress_inside_supplies ⇒ //=.
move: HYP2 ⇒ /hasPn HYP2.
move: HYP4; rewrite negb_and SUP //= ⇒ /negPn /hasP [jo INjo READYjo].
move: INjo; rewrite mem_filter ⇒ /andP[HEPjo ARRjo].
move: HYP3 ⇒ /nandP [/hasPn NO_HEP_READY | /nandP[/negPn SERVj | /hasPn NO_LP_SERV]].
{ exfalso.
have INjo: jo \in [seq j' <- arrivals_up_to arr_seq t | hep_job j' j]
by rewrite mem_filter; apply/andP; split.
by move: (NO_HEP_READY jo INjo); rewrite READYjo. }
{ apply service_at_implies_scheduled_at.
by apply: served_at_and_receives_service_consistent. }
{ have [BACK | NOTBACK] := boolP(backlogged sched jo t).
{ have ARRjo' : arrives_in arr_seq jo
by move: ARRjo; rewrite /arrivals_up_to; apply in_arrivals_implies_arrived.
move: (H_work_conserving jo t ARRjo' BACK) ⇒ [j' SCHED].
have SERV: receives_service_at sched j' t by apply ideal_progress_inside_supplies.
have IN_SERV: j' \in served_jobs_at arr_seq sched t by apply receives_service_and_served_at_consistent.
move: (NO_LP_SERV j' IN_SERV) ⇒ /negPn /[!hep_job_at_jlfp] HEPj'.
move: (HYP2 j' IN_SERV); rewrite /another_hep_job.
rewrite HEPj' andTb ⇒ /negPn /eqP ⇒ EQ.
by subst j. }
{ move: NOTBACK; rewrite /backlogged READYjo andTb negbK ⇒ SCHED.
have SERV: receives_service_at sched jo t by apply ideal_progress_inside_supplies.
have IN_SERV: jo \in served_jobs_at arr_seq sched t by apply receives_service_and_served_at_consistent.
move: (HYP2 jo IN_SERV); rewrite /another_hep_job HEPjo andTb ⇒ /negPn /eqP EQ.
by subst j. } }
Qed.
Conversely, if the job is scheduled at time t, then interference is false.
Lemma scheduled_implies_no_interference :
receives_service_at sched j t → ~~ interference j t.
Proof.
move⇒ RSERV; apply/negP ⇒ /orP[/orP[/orP[BL|INT]|INV]|/andP[_ /hasPn NO_HEP_READY]].
- by apply/negP; first apply: no_blackout_when_service_received.
- move: INT; rewrite_neg @no_ahep_interference_when_served.
by apply: receives_service_implies_has_supply.
- move: INV; rewrite /service_inversion ⇒ /andP[_ /andP[/negP NOTSERVj _]].
by move: NOTSERVj; rewrite receives_service_and_served_at_consistent.
- have HEPj: hep_job j j by done.
have SCHEDj: scheduled_at sched j t by apply service_at_implies_scheduled_at.
have READYj: job_ready sched j t by done.
have ARRj: j \in served_jobs_at arr_seq sched t by apply receives_service_and_served_at_consistent.
move: ARRj; rewrite mem_filter ⇒ /andP [_ INj].
have IN: j \in [seq j0 <- arrivals_up_to arr_seq t | hep_job j0 j]
by rewrite mem_filter; apply /andP; split ⇒ //=.
move: (NO_HEP_READY j IN) ⇒ NOREADYj.
by rewrite READYj in NOREADYj.
Qed.
End Abstract_Work_Conservation.
Using the above two lemmas, we can prove that abstract work conservation always holds for these instantiations of interference (I) and interfering workload (W).
Corollary instantiated_i_and_w_are_coherent_with_schedule :
work_conserving_ab.
Proof.
move ⇒ j t1 t2 t ARR POS BUSY NEQ; split.
- by move⇒ G; apply: (not_interference_implies_scheduled j ARR POS); eauto 2; apply/negP.
- by move⇒ SERV; apply/negP; apply: scheduled_implies_no_interference; eauto 2.
Qed.
Next, in order to prove that these definitions of interference and interfering workload are consistent with sequential tasks, we need to assume that the policy under consideration respects sequential tasks.
Hypothesis H_policy_respects_sequential_tasks :
policy_respects_sequential_tasks JLFP.
We prove that these definitions of interference and interfering workload are consistent with sequential tasks.
Lemma instantiated_interference_and_workload_consistent_with_sequential_tasks :
interference_and_workload_consistent_with_sequential_tasks arr_seq sched tsk.
Proof.
move ⇒ j t1 t2 ARR /eqP TSK POS BUSY.
eapply instantiated_busy_interval_equivalent_busy_interval in BUSY ⇒ //.
eapply all_jobs_have_completed_equiv_workload_eq_service ⇒ //.
move ⇒ s INs /eqP TSKs.
move: (INs) ⇒ NEQ.
eapply in_arrivals_implies_arrived_between in NEQ ⇒ //.
move: NEQ ⇒ /andP [_ JAs].
move: (BUSY) ⇒ [[ _ [QT [_ /andP [JAj _]]] _]].
apply QT ⇒ //; first exact: in_arrivals_implies_arrived.
apply H_policy_respects_sequential_tasks; first by rewrite TSK TSKs.
by apply leq_trans with t1; [lia |].
Qed.
Finally, we show that cumulative interference (I) never exceeds the cumulative interfering workload (W).
Lemma instantiated_i_and_w_no_speculative_execution :
no_speculative_execution.
Proof.
move⇒ j t1.
rewrite cumulative_interference_split cumulative_interfering_workload_split.
rewrite -!addnA leq_add2l leq_add2r cumulative_i_ohep_eq_service_of_ohep //=.
rewrite cumulative_iw_hep_eq_workload_of_ohep.
by apply service_of_jobs_le_workload ⇒ //.
Qed.
End I_IW_correctness.
End IWInstantiation.