Library prosa.analysis.facts.model.service_of_jobs
From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq fintype bigop.
Require Export prosa.model.aggregate.workload.
Require Export prosa.model.aggregate.service_of_jobs.
Require Export prosa.analysis.facts.behavior.completion.
Require Export prosa.analysis.facts.model.ideal_schedule.
Require Import prosa.model.processor.ideal.
Require Export prosa.model.aggregate.workload.
Require Export prosa.model.aggregate.service_of_jobs.
Require Export prosa.analysis.facts.behavior.completion.
Require Export prosa.analysis.facts.model.ideal_schedule.
Require Import prosa.model.processor.ideal.
Lemmas about Service Received by Sets of Jobs
In this file, we establish basic facts about the service received by sets of jobs.
Consider any type of tasks ...
... and any type of jobs associated with these tasks.
Context {Job : JobType}.
Context `{JobTask Job Task}.
Context `{JobArrival Job}.
Context `{JobCost Job}.
Context `{JobTask Job Task}.
Context `{JobArrival Job}.
Context `{JobCost Job}.
Consider any kind of processor state model, ...
... any job arrival sequence with consistent arrivals, ....
Variable arr_seq : arrival_sequence Job.
Hypothesis H_arrival_times_are_consistent : consistent_arrival_times arr_seq.
Hypothesis H_arrival_times_are_consistent : consistent_arrival_times arr_seq.
... and any schedule of this arrival sequence ...
... 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.
Hypothesis H_completed_jobs_dont_execute : completed_jobs_dont_execute sched.
Let P be any predicate over jobs.
We show that the total service of jobs released in a time interval
[t1,t2)
during [t1,t2) is equal to the sum of:
(1) the total service of jobs released in time interval [t1, t) during time [t1, t)
(2) the total service of jobs released in time interval [t1, t) during time [t, t2)
and (3) the total service of jobs released in time interval [t, t2) during time [t, t2).
Lemma service_of_jobs_cat_scheduling_interval :
∀ t1 t2 t,
t1 ≤ t ≤ t2 →
service_of_jobs sched P (arrivals_between arr_seq t1 t2) t1 t2
= service_of_jobs sched P (arrivals_between arr_seq t1 t) t1 t
+ service_of_jobs sched P (arrivals_between arr_seq t1 t) t t2
+ service_of_jobs sched P (arrivals_between arr_seq t t2) t t2.
Proof.
move ⇒ t1 t2 t /andP [GEt LEt].
rewrite (arrivals_between_cat _ _ t) //.
rewrite {1}/service_of_jobs big_cat //=.
rewrite exchange_big //= (@big_cat_nat _ _ _ t) //=;
rewrite [in X in X + _ + _]exchange_big //= [in X in _ + X + _]exchange_big //=.
apply/eqP; rewrite -!addnA eqn_add2l eqn_add2l.
rewrite exchange_big //= (@big_cat_nat _ _ _ t) //= [in X in _ + X]exchange_big //=.
rewrite -[service_of_jobs _ _ _ _ _]add0n /service_of_jobs eqn_add2r.
rewrite big_nat_cond big1 //.
move ⇒ x /andP [/andP [GEi LTi] _].
rewrite big_seq_cond big1 //.
move ⇒ j /andP [ARR Ps].
apply service_before_job_arrival_zero with H0; auto.
eapply in_arrivals_implies_arrived_between in ARR; eauto 2.
by move: ARR ⇒ /andP [N1 N2]; apply leq_trans with t.
Qed.
∀ t1 t2 t,
t1 ≤ t ≤ t2 →
service_of_jobs sched P (arrivals_between arr_seq t1 t2) t1 t2
= service_of_jobs sched P (arrivals_between arr_seq t1 t) t1 t
+ service_of_jobs sched P (arrivals_between arr_seq t1 t) t t2
+ service_of_jobs sched P (arrivals_between arr_seq t t2) t t2.
Proof.
move ⇒ t1 t2 t /andP [GEt LEt].
rewrite (arrivals_between_cat _ _ t) //.
rewrite {1}/service_of_jobs big_cat //=.
rewrite exchange_big //= (@big_cat_nat _ _ _ t) //=;
rewrite [in X in X + _ + _]exchange_big //= [in X in _ + X + _]exchange_big //=.
apply/eqP; rewrite -!addnA eqn_add2l eqn_add2l.
rewrite exchange_big //= (@big_cat_nat _ _ _ t) //= [in X in _ + X]exchange_big //=.
rewrite -[service_of_jobs _ _ _ _ _]add0n /service_of_jobs eqn_add2r.
rewrite big_nat_cond big1 //.
move ⇒ x /andP [/andP [GEi LTi] _].
rewrite big_seq_cond big1 //.
move ⇒ j /andP [ARR Ps].
apply service_before_job_arrival_zero with H0; auto.
eapply in_arrivals_implies_arrived_between in ARR; eauto 2.
by move: ARR ⇒ /andP [N1 N2]; apply leq_trans with t.
Qed.
We show that the total service of jobs released in a time interval
[t1, t2)
during [t, t2) is equal to the sum of:
(1) the total service of jobs released in a time interval [t1, t) during [t, t2)
and (2) the total service of jobs released in a time interval [t, t2) during [t, t2).
Lemma service_of_jobs_cat_arrival_interval :
∀ t1 t2 t,
t1 ≤ t ≤ t2 →
service_of_jobs sched P (arrivals_between arr_seq t1 t2) t t2 =
service_of_jobs sched P (arrivals_between arr_seq t1 t) t t2 +
service_of_jobs sched P (arrivals_between arr_seq t t2) t t2.
Proof.
move ⇒ t1 t2 t /andP [GEt LEt].
apply/eqP;rewrite eq_sym; apply/eqP.
rewrite [in X in _ = X](arrivals_between_cat _ _ t) //.
by rewrite {3}/service_of_jobs -big_cat //=.
Qed.
End GenericModelLemmas.
∀ t1 t2 t,
t1 ≤ t ≤ t2 →
service_of_jobs sched P (arrivals_between arr_seq t1 t2) t t2 =
service_of_jobs sched P (arrivals_between arr_seq t1 t) t t2 +
service_of_jobs sched P (arrivals_between arr_seq t t2) t t2.
Proof.
move ⇒ t1 t2 t /andP [GEt LEt].
apply/eqP;rewrite eq_sym; apply/eqP.
rewrite [in X in _ = X](arrivals_between_cat _ _ t) //.
by rewrite {3}/service_of_jobs -big_cat //=.
Qed.
End GenericModelLemmas.
In this section, we prove some properties about service
of sets of jobs for unit-service processor models.
Consider any type of tasks ...
... and any type of jobs associated with these tasks.
Context {Job : JobType}.
Context `{JobTask Job Task}.
Context `{JobArrival Job}.
Context `{JobCost Job}.
Context `{JobTask Job Task}.
Context `{JobArrival Job}.
Context `{JobCost Job}.
Consider any kind of unit-service processor state model.
Context {PState : Type}.
Context `{ProcessorState Job PState}.
Hypothesis H_unit_service_proc_model : unit_service_proc_model PState.
Context `{ProcessorState Job PState}.
Hypothesis H_unit_service_proc_model : unit_service_proc_model PState.
Consider any arrival sequence with consistent arrivals.
Variable arr_seq : arrival_sequence Job.
Hypothesis H_arrival_times_are_consistent : consistent_arrival_times arr_seq.
Hypothesis H_arrival_times_are_consistent : consistent_arrival_times arr_seq.
Next, consider any unit-service schedule of this arrival sequence ...
Variable sched : schedule PState.
Hypothesis H_jobs_come_from_arrival_sequence:
jobs_come_from_arrival_sequence sched arr_seq.
Hypothesis H_jobs_come_from_arrival_sequence:
jobs_come_from_arrival_sequence sched arr_seq.
... 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.
Hypothesis H_completed_jobs_dont_execute : completed_jobs_dont_execute sched.
Let P be any predicate over jobs.
First, we prove that the service received by any set of jobs is
upper-bounded by the corresponding workload.
Lemma service_of_jobs_le_workload:
∀ (jobs : seq Job) (t1 t2 : instant),
service_of_jobs sched P jobs t1 t2 ≤ workload_of_jobs P jobs.
Proof.
intros jobs t1 t2.
apply leq_sum; intros j _.
by apply cumulative_service_le_job_cost.
Qed.
∀ (jobs : seq Job) (t1 t2 : instant),
service_of_jobs sched P jobs t1 t2 ≤ workload_of_jobs P jobs.
Proof.
intros jobs t1 t2.
apply leq_sum; intros j _.
by apply cumulative_service_le_job_cost.
Qed.
In this section, we introduce a connection between the cumulative
service, cumulative workload, and completion of jobs.
Consider an arbitrary time interval
[t1, t2).
Let jobs be a set of all jobs arrived during
[t1, t2).
Next, we consider some time instant t_compl.
And state the proposition that all jobs are completed by time
t_compl. Next we show that this proposition is equivalent to
the fact that workload of jobs = service of jobs.
First, we prove that if the workload of jobs is equal to the
service of jobs, then any job in jobs is completed by time
t_compl.
Lemma workload_eq_service_impl_all_jobs_have_completed :
workload_of_jobs P jobs = service_of_jobs sched P jobs t1 t_compl →
all_jobs_completed_by t_compl.
Proof.
unfold jobs; intros EQ j ARR Pj; move: (ARR) ⇒ ARR2.
apply in_arrivals_implies_arrived_between in ARR; last by done.
move: ARR ⇒ /andP [T1 T2].
have F1: ∀ a b, (a < b) || (a ≥ b).
{ by intros; destruct (a < b) eqn:EQU; apply/orP;
[by left |right]; apply negbT in EQU; rewrite leqNgt. }
move: (F1 t_compl t1) ⇒ /orP [LT | GT].
- rewrite /service_of_jobs /service_during in EQ.
rewrite exchange_big big_geq //= in EQ; last by rewrite ltnW.
rewrite /workload_of_jobs in EQ.
rewrite (big_rem j) ?Pj //= in EQ.
move: EQ ⇒ /eqP; rewrite addn_eq0; move ⇒ /andP [CZ _].
unfold completed_by, service.completed_by.
by move: CZ ⇒ /eqP CZ; rewrite CZ.
- unfold workload_of_jobs, service_of_jobs in EQ; unfold completed_by, service.completed_by.
rewrite /service -(service_during_cat _ _ _ t1); last by apply/andP; split.
rewrite cumulative_service_before_job_arrival_zero // add0n.
rewrite <- sum_majorant_eqn with (E1 := fun j ⇒ service_during sched j t1 t_compl)
(xs := arrivals_between arr_seq t1 t2) (P := P); try done.
by intros; apply cumulative_service_le_job_cost; auto using ideal_proc_model_provides_unit_service.
Qed.
workload_of_jobs P jobs = service_of_jobs sched P jobs t1 t_compl →
all_jobs_completed_by t_compl.
Proof.
unfold jobs; intros EQ j ARR Pj; move: (ARR) ⇒ ARR2.
apply in_arrivals_implies_arrived_between in ARR; last by done.
move: ARR ⇒ /andP [T1 T2].
have F1: ∀ a b, (a < b) || (a ≥ b).
{ by intros; destruct (a < b) eqn:EQU; apply/orP;
[by left |right]; apply negbT in EQU; rewrite leqNgt. }
move: (F1 t_compl t1) ⇒ /orP [LT | GT].
- rewrite /service_of_jobs /service_during in EQ.
rewrite exchange_big big_geq //= in EQ; last by rewrite ltnW.
rewrite /workload_of_jobs in EQ.
rewrite (big_rem j) ?Pj //= in EQ.
move: EQ ⇒ /eqP; rewrite addn_eq0; move ⇒ /andP [CZ _].
unfold completed_by, service.completed_by.
by move: CZ ⇒ /eqP CZ; rewrite CZ.
- unfold workload_of_jobs, service_of_jobs in EQ; unfold completed_by, service.completed_by.
rewrite /service -(service_during_cat _ _ _ t1); last by apply/andP; split.
rewrite cumulative_service_before_job_arrival_zero // add0n.
rewrite <- sum_majorant_eqn with (E1 := fun j ⇒ service_during sched j t1 t_compl)
(xs := arrivals_between arr_seq t1 t2) (P := P); try done.
by intros; apply cumulative_service_le_job_cost; auto using ideal_proc_model_provides_unit_service.
Qed.
And vice versa, the fact that any job in jobs is completed by time t_compl
implies that the workload of jobs is equal to the service of jobs.
Lemma all_jobs_have_completed_impl_workload_eq_service :
all_jobs_completed_by t_compl →
workload_of_jobs P jobs =
service_of_jobs sched P jobs t1 t_compl.
Proof.
unfold jobs; intros COMPL.
have F: ∀ j t, t ≤ t1 → j \in arrivals_between arr_seq t1 t2 → service_during sched j 0 t = 0.
{ intros j t LE ARR.
eapply in_arrivals_implies_arrived_between in ARR; eauto 2; move: ARR ⇒ /andP [GE LT].
eapply cumulative_service_before_job_arrival_zero; eauto 2.
by apply leq_trans with t1.
}
destruct (t_compl ≤ t1) eqn:EQ.
- unfold service_of_jobs. unfold service_during.
rewrite exchange_big //=.
rewrite big_geq; last by done.
rewrite /workload_of_jobs big1_seq //.
move ⇒ j /andP [Pj ARR].
specialize (COMPL _ ARR Pj).
rewrite <- F with (j := j) (t := t_compl); try done.
apply/eqP; rewrite eqn_leq; apply/andP; split.
+ by apply COMPL.
+ by apply service_at_most_cost.
- apply/eqP; rewrite eqn_leq; apply/andP; split; first last.
+ by apply service_of_jobs_le_workload.
+ rewrite /workload_of_jobs big_seq_cond [X in _ ≤ X]big_seq_cond.
rewrite leq_sum // ⇒ j /andP [ARR Pj].
specialize (COMPL _ ARR Pj).
rewrite -[service_during _ _ _ _ ]add0n -(F j t1) //= -(big_cat_nat) //=.
move: EQ =>/negP /negP; rewrite -ltnNge ⇒ EQ.
by apply ltnW.
Qed.
all_jobs_completed_by t_compl →
workload_of_jobs P jobs =
service_of_jobs sched P jobs t1 t_compl.
Proof.
unfold jobs; intros COMPL.
have F: ∀ j t, t ≤ t1 → j \in arrivals_between arr_seq t1 t2 → service_during sched j 0 t = 0.
{ intros j t LE ARR.
eapply in_arrivals_implies_arrived_between in ARR; eauto 2; move: ARR ⇒ /andP [GE LT].
eapply cumulative_service_before_job_arrival_zero; eauto 2.
by apply leq_trans with t1.
}
destruct (t_compl ≤ t1) eqn:EQ.
- unfold service_of_jobs. unfold service_during.
rewrite exchange_big //=.
rewrite big_geq; last by done.
rewrite /workload_of_jobs big1_seq //.
move ⇒ j /andP [Pj ARR].
specialize (COMPL _ ARR Pj).
rewrite <- F with (j := j) (t := t_compl); try done.
apply/eqP; rewrite eqn_leq; apply/andP; split.
+ by apply COMPL.
+ by apply service_at_most_cost.
- apply/eqP; rewrite eqn_leq; apply/andP; split; first last.
+ by apply service_of_jobs_le_workload.
+ rewrite /workload_of_jobs big_seq_cond [X in _ ≤ X]big_seq_cond.
rewrite leq_sum // ⇒ j /andP [ARR Pj].
specialize (COMPL _ ARR Pj).
rewrite -[service_during _ _ _ _ ]add0n -(F j t1) //= -(big_cat_nat) //=.
move: EQ =>/negP /negP; rewrite -ltnNge ⇒ EQ.
by apply ltnW.
Qed.
Using the lemmas above, we prove the equivalence.
Corollary all_jobs_have_completed_equiv_workload_eq_service :
all_jobs_completed_by t_compl ↔
workload_of_jobs P jobs = service_of_jobs sched P jobs t1 t_compl.
Proof.
split.
- by apply all_jobs_have_completed_impl_workload_eq_service.
- by apply workload_eq_service_impl_all_jobs_have_completed.
Qed.
End WorkloadServiceAndCompletion.
End UnitServiceModelLemmas.
all_jobs_completed_by t_compl ↔
workload_of_jobs P jobs = service_of_jobs sched P jobs t1 t_compl.
Proof.
split.
- by apply all_jobs_have_completed_impl_workload_eq_service.
- by apply workload_eq_service_impl_all_jobs_have_completed.
Qed.
End WorkloadServiceAndCompletion.
End UnitServiceModelLemmas.
In this section, we prove some properties about service
of sets of jobs for unit-service uniprocessor models.
Consider any type of tasks ...
... and any type of jobs associated with these tasks.
Context {Job : JobType}.
Context `{JobTask Job Task}.
Context `{JobArrival Job}.
Context `{JobCost Job}.
Context `{JobTask Job Task}.
Context `{JobArrival Job}.
Context `{JobCost Job}.
Consider any kind of unit-service uniprocessor state model.
Context {PState : Type}.
Context `{ProcessorState Job PState}.
Hypothesis H_unit_service_proc_model : unit_service_proc_model PState.
Hypothesis H_uniprocessor_model : uniprocessor_model PState.
Context `{ProcessorState Job PState}.
Hypothesis H_unit_service_proc_model : unit_service_proc_model PState.
Hypothesis H_uniprocessor_model : uniprocessor_model PState.
Consider any arrival sequence with consistent arrivals.
Variable arr_seq : arrival_sequence Job.
Hypothesis H_arrival_times_are_consistent : consistent_arrival_times arr_seq.
Hypothesis H_arrival_times_are_consistent : consistent_arrival_times arr_seq.
Next, consider any unit-service uni-processor schedule of this arrival sequence ...
Variable sched : schedule PState.
Hypothesis H_jobs_come_from_arrival_sequence :
jobs_come_from_arrival_sequence sched arr_seq.
Hypothesis H_jobs_come_from_arrival_sequence :
jobs_come_from_arrival_sequence sched arr_seq.
... 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.
Hypothesis H_completed_jobs_dont_execute : completed_jobs_dont_execute sched.
Let P be any predicate over jobs.
In this section, we prove that the service received by any set of jobs
is upper-bounded by the corresponding interval length.
Let jobs denote any set of jobs.
We prove that the overall service of jobs at a single time instant is at most 1.
Lemma service_of_jobs_le_1:
∀ t, \sum_(j <- jobs | P j) service_at sched j t ≤ 1.
Proof.
intros t.
eapply leq_trans.
{ by apply leq_sum_seq_pred with (P2 := predT); intros. }
simpl; induction jobs as [ | j js].
- by rewrite big_nil.
- feed IHjs; first by move: H_no_duplicate_jobs; rewrite cons_uniq ⇒ /andP [_ U].
rewrite big_cons.
destruct (service_is_zero_or_one H_unit_service_proc_model sched j t) as [Z | O].
+ by rewrite Z (leqRW IHjs).
+ have → : \sum_(j <- js) service_in j (sched t) = 0; last by rewrite O.
have POS: service_at sched j t > 0 by rewrite O.
apply service_at_implies_scheduled_at in POS.
apply/eqP; rewrite big_seq big1 //; intros j' IN.
apply/eqP; rewrite -leqn0 leqNgt.
eapply contra with (b := scheduled_at sched j' t);
first apply service_at_implies_scheduled_at.
apply/negP; intros SCHED.
specialize (H_uniprocessor_model _ _ _ _ POS SCHED); subst j'.
by move: H_no_duplicate_jobs; rewrite cons_uniq ⇒ /andP [/negP NIN _].
Qed.
∀ t, \sum_(j <- jobs | P j) service_at sched j t ≤ 1.
Proof.
intros t.
eapply leq_trans.
{ by apply leq_sum_seq_pred with (P2 := predT); intros. }
simpl; induction jobs as [ | j js].
- by rewrite big_nil.
- feed IHjs; first by move: H_no_duplicate_jobs; rewrite cons_uniq ⇒ /andP [_ U].
rewrite big_cons.
destruct (service_is_zero_or_one H_unit_service_proc_model sched j t) as [Z | O].
+ by rewrite Z (leqRW IHjs).
+ have → : \sum_(j <- js) service_in j (sched t) = 0; last by rewrite O.
have POS: service_at sched j t > 0 by rewrite O.
apply service_at_implies_scheduled_at in POS.
apply/eqP; rewrite big_seq big1 //; intros j' IN.
apply/eqP; rewrite -leqn0 leqNgt.
eapply contra with (b := scheduled_at sched j' t);
first apply service_at_implies_scheduled_at.
apply/negP; intros SCHED.
specialize (H_uniprocessor_model _ _ _ _ POS SCHED); subst j'.
by move: H_no_duplicate_jobs; rewrite cons_uniq ⇒ /andP [/negP NIN _].
Qed.
Next, we prove that the service received by those jobs is no larger
than their workload.
Corollary service_of_jobs_le_length_of_interval:
∀ (t : instant) (Δ : duration),
service_of_jobs sched P jobs t (t + Δ) ≤ Δ.
Proof.
intros.
have EQ: \sum_(t ≤ x < t + Δ) 1 = Δ.
{ by rewrite big_const_nat iter_addn mul1n addn0 -{2}[t]addn0 subnDl subn0. }
rewrite -{2}EQ {EQ}.
rewrite /service_of_jobs/service_during/service_at exchange_big //=.
rewrite leq_sum //.
move ⇒ t' _.
by apply service_of_jobs_le_1.
Qed.
∀ (t : instant) (Δ : duration),
service_of_jobs sched P jobs t (t + Δ) ≤ Δ.
Proof.
intros.
have EQ: \sum_(t ≤ x < t + Δ) 1 = Δ.
{ by rewrite big_const_nat iter_addn mul1n addn0 -{2}[t]addn0 subnDl subn0. }
rewrite -{2}EQ {EQ}.
rewrite /service_of_jobs/service_during/service_at exchange_big //=.
rewrite leq_sum //.
move ⇒ t' _.
by apply service_of_jobs_le_1.
Qed.
The same holds for two time instants.
Corollary service_of_jobs_le_length_of_interval':
∀ (t1 t2 : instant),
service_of_jobs sched P jobs t1 t2 ≤ t2 - t1.
Proof.
intros.
have <-: \sum_(t1 ≤ x < t2) 1 = t2 - t1.
{ by rewrite big_const_nat iter_addn mul1n addn0. }
rewrite /service_of_jobs exchange_big //=.
rewrite leq_sum //.
move ⇒ t' _.
have SE := service_of_jobs_le_1 t'.
eapply leq_trans; last by eassumption.
by rewrite leq_sum.
Qed.
End ServiceOfJobsIsBoundedByLength.
End UnitServiceUniProcessorModelLemmas.
∀ (t1 t2 : instant),
service_of_jobs sched P jobs t1 t2 ≤ t2 - t1.
Proof.
intros.
have <-: \sum_(t1 ≤ x < t2) 1 = t2 - t1.
{ by rewrite big_const_nat iter_addn mul1n addn0. }
rewrite /service_of_jobs exchange_big //=.
rewrite leq_sum //.
move ⇒ t' _.
have SE := service_of_jobs_le_1 t'.
eapply leq_trans; last by eassumption.
by rewrite leq_sum.
Qed.
End ServiceOfJobsIsBoundedByLength.
End UnitServiceUniProcessorModelLemmas.
In this section, we prove some properties about service
of sets of jobs for ideal uni-processor model.
Consider any type of tasks ...
... and any type of jobs associated with these tasks.
Context {Job : JobType}.
Context `{JobTask Job Task}.
Context `{JobArrival Job}.
Context `{JobCost Job}.
Context `{JobTask Job Task}.
Context `{JobArrival Job}.
Context `{JobCost Job}.
Consider any arrival sequence with consistent arrivals.
Variable arr_seq : arrival_sequence Job.
Hypothesis H_arrival_times_are_consistent : consistent_arrival_times arr_seq.
Hypothesis H_arrival_times_are_consistent : consistent_arrival_times arr_seq.
Next, consider any ideal uni-processor schedule of this arrival sequence ...
Variable sched : schedule (ideal.processor_state Job).
Hypothesis H_jobs_come_from_arrival_sequence:
jobs_come_from_arrival_sequence sched arr_seq.
Hypothesis H_jobs_come_from_arrival_sequence:
jobs_come_from_arrival_sequence sched arr_seq.
... 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.
Hypothesis H_completed_jobs_dont_execute : completed_jobs_dont_execute sched.
Let P be any predicate over jobs.
We prove that if the total service within some time interval
[t1,t2)
is smaller than t2 - t1, then there is an idle time instant t ∈ [t1,t2)].
Lemma low_service_implies_existence_of_idle_time :
∀ t1 t2,
service_of_jobs sched predT (arrivals_between arr_seq 0 t2) t1 t2 < t2 - t1 →
∃ t, t1 ≤ t < t2 ∧ is_idle sched t.
Proof.
intros ? ? SERV.
destruct (t1 ≤ t2) eqn:LE; last first.
{ move: LE ⇒ /negP/negP; rewrite -ltnNge.
move ⇒ LT; apply ltnW in LT; rewrite -subn_eq0 in LT.
by move: LT ⇒ /eqP → in SERV; rewrite ltn0 in SERV.
}
have EX: ∃ δ, t2 = t1 + δ.
{ by ∃ (t2 - t1); rewrite subnKC // ltnW. }
move: EX ⇒ [δ EQ]; subst t2; clear LE.
rewrite {3}[t1 + δ]addnC -addnBA // subnn addn0 in SERV.
rewrite /service_of_jobs exchange_big //= in SERV.
apply sum_le_summation_range in SERV.
move: SERV ⇒ [x [/andP [GEx LEx] L]].
∃ x; split; first by apply/andP; split.
apply/negPn; apply/negP; intros NIDLE.
unfold is_idle in NIDLE.
destruct(sched x) eqn:SCHED; last by done.
move: SCHED ⇒ /eqP SCHED; clear NIDLE; rewrite -scheduled_at_def/= in SCHED.
move: L ⇒ /eqP; rewrite -sum_nat_eq0_nat; move ⇒ /allP L.
specialize (L s); feed L.
{ unfold arrivals_between.
eapply arrived_between_implies_in_arrivals; eauto 2.
by apply H_jobs_must_arrive_to_execute in SCHED; apply leq_ltn_trans with x.
}
move: SCHED L ⇒ /=.
rewrite scheduled_at_def service_at_def ⇒ /eqP→ /eqP.
by rewrite eqxx.
Qed.
End IdealModelLemmas.
∀ t1 t2,
service_of_jobs sched predT (arrivals_between arr_seq 0 t2) t1 t2 < t2 - t1 →
∃ t, t1 ≤ t < t2 ∧ is_idle sched t.
Proof.
intros ? ? SERV.
destruct (t1 ≤ t2) eqn:LE; last first.
{ move: LE ⇒ /negP/negP; rewrite -ltnNge.
move ⇒ LT; apply ltnW in LT; rewrite -subn_eq0 in LT.
by move: LT ⇒ /eqP → in SERV; rewrite ltn0 in SERV.
}
have EX: ∃ δ, t2 = t1 + δ.
{ by ∃ (t2 - t1); rewrite subnKC // ltnW. }
move: EX ⇒ [δ EQ]; subst t2; clear LE.
rewrite {3}[t1 + δ]addnC -addnBA // subnn addn0 in SERV.
rewrite /service_of_jobs exchange_big //= in SERV.
apply sum_le_summation_range in SERV.
move: SERV ⇒ [x [/andP [GEx LEx] L]].
∃ x; split; first by apply/andP; split.
apply/negPn; apply/negP; intros NIDLE.
unfold is_idle in NIDLE.
destruct(sched x) eqn:SCHED; last by done.
move: SCHED ⇒ /eqP SCHED; clear NIDLE; rewrite -scheduled_at_def/= in SCHED.
move: L ⇒ /eqP; rewrite -sum_nat_eq0_nat; move ⇒ /allP L.
specialize (L s); feed L.
{ unfold arrivals_between.
eapply arrived_between_implies_in_arrivals; eauto 2.
by apply H_jobs_must_arrive_to_execute in SCHED; apply leq_ltn_trans with x.
}
move: SCHED L ⇒ /=.
rewrite scheduled_at_def service_at_def ⇒ /eqP→ /eqP.
by rewrite eqxx.
Qed.
End IdealModelLemmas.