# Lemmas about Workload of Sets of Jobs

In this file, we establish basic facts about the workload of sets of jobs.

Consider any type of tasks ...

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

To begin with, we establish an auxiliary rewriting lemma that allows us to introduce a filter on the considered set of jobs, provided the filter predicate P2 is implied by the job-selection predicate P1.
(P1 P2 : pred Job) (jobs : seq Job),
( j, j \in jobs P1 j P2 j)
workload_of_jobs P1 jobs = workload_of_jobs P1 [seq j <- jobs | P2 j ].
Proof.
moveP1 P2 jobs IMPL.
rewrite /workload_of_jobs big_filter_cond big_seq_cond [RHS]big_seq_cond.
apply: eq_biglj.
case: (boolP (j \in jobs)) ⇒ // IN.
rewrite !andTb.
case: (boolP (P1 j)) ⇒ //= P1j; first by rewrite (IMPL j IN P1j).
by rewrite andbF.
Qed.

We establish that if the predicate P1 implies the predicate P2, then the cumulative workload of jobs that respect P1 is bounded by the cumulative workload of jobs that respect P2.
(P1 P2 : pred Job) (jobs : seq Job),
( j, P1 j P2 j)
Proof.
moveP1 P2 jobs IMPLIES; rewrite /workload_of_jobs.
apply: leq_sum_seq_predj' _.
by apply: IMPLIES.
Qed.

The cumulative workload of jobs from an empty sequence is always zero.
(P : pred Job), workload_of_jobs P [::] = 0.
Proof. by move ⇒ ?; rewrite /workload_of_jobs big_nil. Qed.

The workload of a set of jobs can be equivalently rewritten as sum over their tasks.
{P : pred Job} (Q : pred Task) {js : seq Job} (ts : seq Task),
{in js, j, (job_task j) \in ts}
{in js, j, P j Q (job_task j)}
uniq js
uniq ts
let P_and_job_of tsk_o j := P j && (job_task j == tsk_o) in
= \sum_(tsk_o <- ts | Q tsk_o ) workload_of_jobs (P_and_job_of tsk_o) js.
Proof.
moveP Q js ts IN_ts PQ UJ UT //=.
rewrite -big_filter {1}/workload_of_jobs //.
apply: sum_over_partitions_eq ⇒ // [j IN Px|]; last exact: filter_uniq.
rewrite mem_filter; apply/andP; split; last by apply: IN_ts.
by apply: PQ.
Qed.

In this section we state a lemma about splitting the workload among tasks of different priority relative to a job j.
Consider any JLFP policy.
Context `{JLFP_policy Job}.

Consider the workload of all the jobs that have priority higher-than-or-equal-to the priority of j. This workload can be split by task into the workload of higher-or-equal priority jobs from the task of j and higher-or-equal priority jobs from all tasks except for the task of j.
jobs j,
workload_of_jobs (another_hep_job^~j) jobs =
Proof.
movejobs j.
apply sum_split_exhaustive_mutually_exclusive_predsjo.
case (hep_job jo j) eqn: EQ1;
case (jo != j) eqn: EQ2;
case (job_task jo != job_task j) eqn: EQ3; try lia.
move : EQ2 ⇒ /eqP EQ2.
rewrite EQ2 in EQ3.
by apply /negP /negPn /eqP.
by case (hep_job jo j) eqn: EQ1; case (jo != j) eqn: EQ2;
case (job_task jo != job_task j) eqn: EQ3; try lia.
Qed.

Consider any arrival sequence with consistent arrivals.
In this section, we prove a few useful properties regarding the predicate of workload_of_jobs.
Section PredicateProperties.

Consider a sequence of jobs jobs.
Variable jobs : seq Job.

First, we show that workload of jobs for an unsatisfiable predicate is equal to 0.
workload_of_jobs pred0 jobs = 0.
Proof. by rewrite /workload_of_jobs; apply big_pred0. Qed.

Next, consider two arbitrary predicates P and P'.
Variable P P' : pred Job.

We show that workload_of_jobs conditioned on P can be split into two summands: (1) workload_of_jobs conditioned on P P' and (2) workload_of_jobs conditioned on P ~~ P'.
workload_of_jobs P jobs =
workload_of_jobs (fun jP j && P' j) jobs + workload_of_jobs (fun jP j && ~~ P' j) jobs.
Proof.
rewrite /workload_of_jobs !big_mkcond [in X in _ = X]big_mkcond
[in X in _ = _ + X]big_mkcond //= -big_split //=.
apply: eq_big_seqj' IN.
by destruct (P _), (P' _); simpl; lia.
Qed.

We show that if P is indistinguishable from P' on set jobs, then workload_of_jobs conditioned on P is equal to workload_of_jobs conditioned on P'.
{in jobs, P =1 P'}
workload_of_jobs P jobs = workload_of_jobs P' jobs.
Proof.
intros × EQUIV.
rewrite /workload_of_jobs !big_mkcond [in X in _ = X]big_mkcond //=.
by apply: eq_big_seqj' IN; rewrite EQUIV.
Qed.

End PredicateProperties.

In this section, we bound the workload of jobs of a particular task by the task's RBF.

Consider an arbitrary task.
Variable tsk : Task.

Consider a valid arrival curve that is respected by the task tsk.
Suppose all arrivals have WCET-compliant job costs.
Consider an instant t1 and a duration Δ.
Variable t1 : instant.
Variable Δ : duration.

We prove that the workload of jobs of a task tsk in any interval is bound by the request bound function of the task in that interval.
workload_of_jobs (job_of_task tsk) (arrivals_between arr_seq t1 (t1 + Δ))
Proof.
apply: (@leq_trans (task_cost tsk × number_of_task_arrivals arr_seq tsk t1 (t1 + Δ))).
apply: sum_majorant_constantj IN TSK.
have: valid_job_cost j; last by rewrite /valid_job_cost; move: TSK ⇒ /eqP →.
exact/H_valid_job_cost/in_arrivals_implies_arrived. }
{ rewrite leq_mul2l; apply/orP; right.
by apply H_task_repsects_max_arrivals; lia. }
Qed.

For convenience, we combine the preceding bound with workload_of_jobs_weaken, as the two are often used together.
P,
workload_of_jobs (fun j(P j) && (job_task j == tsk))
(arrivals_between arr_seq t1 (t1 + Δ))
Proof.
moveP.
have LEQ: ar, workload_of_jobs (fun j : JobP j && (job_task j == tsk)) ar
by movear; apply: workload_of_jobs_weakenj /andP [_ +].
by apply/(leq_trans (LEQ _))/workload_le_rbf.
Qed.

In this section, we prove one equality about the workload of a job.

Assume there are no duplicates in the arrival sequence.
We prove that the workload of a job in an interval ```[t1, ``` t2)>> is equal to the cost of the job if the job's arrival is in the interval and 0 otherwise.
(j : Job) (t1 t2 : instant),
arrives_in arr_seq j
workload_of_job arr_seq j t1 t2
= if t1 job_arrival j < t2 then job_cost j else 0.
Proof.
movej t1 t2 ARR; case NEQ: (_ _ < _).
erewrite big_pred1_seq; first reflexivity.
- by apply arrived_between_implies_in_arrivals ⇒ //=.
- by done.
- by movej'; rewrite /pred1 //=.
}
{ apply big1_seqj' /andP [/eqP EQ IN]; subst j'; exfalso.
by apply job_arrival_between in IN ⇒ //.
}
Qed.

In the following section, we relate three types of workload: workload of a job j, workload of higher-or-equal priority jobs distinct from j, and workload of higher-or-equal priority jobs.

Consider a JLFP policy that indicates a higher-or-equal priority relation and assume that the relation is reflexive.
Context {JLFP : JLFP_policy Job}.
Hypothesis H_priority_is_reflexive : reflexive_job_priorities JLFP.

We prove that the sum of the workload of a job j and the workload of higher-or-equal priority jobs distinct from j is equal to the workload of higher-or-equal priority jobs.
(j : Job) (t1 t2 : instant),
workload_of_job arr_seq j t1 t2 + workload_of_other_hep_jobs arr_seq j t1 t2
= workload_of_hep_jobs arr_seq j t1 t2.
Proof.
movej t1 t2.
rewrite [RHS](workload_of_jobs_case_on_pred _ _ (fun jhpjhp != j)).
have EQ: a b c d, a = b c = d a + c = b + d by lia.
rewrite addnC; apply EQ; first by reflexivity.
clear EQ; apply workload_of_jobs_equiv_predjo IN.
have [EQ|NEQ] := eqVneq j jo.
{ by subst; rewrite andbT; symmetry; apply H_priority_is_reflexive. }
by rewrite andbF.
Qed.

If at some point in time t the predicate P by which we select jobs from the set of arrivals in an interval `[t1, t2)` becomes certainly false, then we may disregard all jobs arriving at time t or later.
{P t1 t2 t},
t t2
( j, j \in (arrivals_between arr_seq t1 t2) job_arrival j t ~~ P j)
workload_of_jobs P (arrivals_between arr_seq t1 t2)
= workload_of_jobs P (arrivals_between arr_seq t1 t).
Proof.
moveP t1 t2 t LE IMPL.
have → : arrivals_between arr_seq t1 t = [seq j <- (arrivals_between arr_seq t1 t2) | job_arrival j < t]
by apply: arrivals_between_filter.
rewrite (workload_of_jobs_filter _ (fun jjob_arrival j < t)) // ⇒ j IN Pj.
case: (leqP t (job_arrival j)) ⇒ // TAIL.
by move: (IMPL j IN TAIL) ⇒ /negP.
Qed.

For simplicity, let's define a local name.
We observe that the cumulative workload of all jobs arriving in a time interval `[t1, t2)` and respecting a predicate P can be split into two parts.
t t1 t2 P,
t1 t t2
workload_of_jobs P (arrivals_between t1 t2) =
workload_of_jobs P (arrivals_between t1 t) + workload_of_jobs P (arrivals_between t t2).
Proof.
movet t1 t2 P /andP [GE LE].
by rewrite (arrivals_between_cat _ _ t) // big_cat.
Qed.

As a corollary, we prove that the workload in any range `[t1,t3)` always bounds the workload in any sub-range `[t1,t2)`.
t1 t2 t3 P,
t1 t2
t2 t3
workload_of_jobs P (arrivals_between t1 t2)
workload_of_jobs P (arrivals_between t1 t3).
Proof.
movet1 t2 t3 P ??.
rewrite (workload_of_jobs_cat t2 t1 t3 P _ ) //=; [| apply /andP; split; done].
Qed.

Consider a job j ...
Variable j : Job.

... and a duplicate-free sequence of jobs jobs.
Variable jobs : seq Job.
Hypothesis H_jobs_uniq : uniq jobs.

Further, assume that j is contained in jobs.
Hypothesis H_j_in_jobs : j \in jobs.

To help with rewriting, we prove that the workload of jobs minus the job cost of j is equal to the workload of all jobs except j.
P,
workload_of_jobs (fun jhp : JobP jhp && (jhp != j)) jobs
= workload_of_jobs P jobs - (if P j then job_cost j else 0).
Proof.
moveP.
rewrite [in X in _ = X - _](bigID_idem _ _ (fun jo ⇒ (jo != j))) //=.
have → : \sum_(j0 <- jobs | P j0 && ~~ (j0 != j)) job_cost j0 =(if P j then job_cost j else 0);
last by lia.
rewrite (big_rem j) //=.
rewrite negbK eq_refl andbT.
have → : \sum_(y <- rem (T:=Job) j jobs | P y && ~~ (y != j)) job_cost y= 0;
last by rewrite addn0.
rewrite big_seq_cond.
apply big_pred0jo.
rewrite negbK andbA andbC andbA.
case (P jo); [rewrite andbT | lia].
case (jo == j) eqn: JJ; [rewrite andTb| lia].
move : JJ ⇒ /eqP →.
by apply mem_rem_uniqF ⇒ //=.
Qed.

Next, we specialize the above lemma to the trivial predicate predT.
workload_of_jobs (fun jhp : Jobjhp != j) jobs =
workload_of_jobs predT jobs - job_cost j.
Proof.
by rewrite (workload_minus_job_cost' predT) //=.
Qed.

In this section, we prove the relation between two different ways of constraining workload_of_jobs to only those jobs that arrive prior to a given time.
Section Subset.

Assume that arrival times are consistent and that arrivals are unique.
Consider a time interval `[t1, t2)` and a time instant t.
Variable t1 t2 t : instant.
Hypothesis H_t1_le_t2 : t1 t2.

Let P be an arbitrary predicate on jobs.
Variable P : pred Job.

Consider the window `[t1,t2)`. We prove that the total workload of the jobs arriving in this window before some t is the same as the workload of the jobs arriving in `[t1,t)`. Note that we only require t1 to be less-or-equal than t2. Consequently, the interval `[t1,t)` may be empty.
workload_of_jobs (fun j(job_arrival j t) && P j) (arrivals_between t1 t2)
workload_of_jobs (fun jP j) (arrivals_between t1 (t + ε)).
Proof.
clear H_jobs_uniq H_j_in_jobs H_t1_le_t2.