Library prosa.analysis.facts.model.workload

Lemmas about Workload of Sets of Jobs

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

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}.

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.
  Lemma workload_of_jobs_filter :
     (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 ].
    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.

Next, consider any job arrival sequence consistent with the arrival times of the jobs.
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.
  Lemma workload_of_jobs_nil_tail :
     {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).
    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.

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.
   Lemma workload_of_jobs_cat:
     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).
    movet t1 t2 P /andP [GE LE].
    rewrite /workload_of_jobs /arrivals_between.
    by rewrite (arrivals_between_cat _ _ t) // big_cat.

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. To define the workload of all jobs, since workload_of_jobs expects a predicate, we use predT, which is the always-true predicate.
  Lemma workload_minus_job_cost :
    workload_of_jobs (fun jhp : Jobjhp != j) jobs =
    workload_of_jobs predT jobs - job_cost j.
    rewrite /workload_of_jobs (big_rem j) //= eq_refl //= add0n.
    rewrite [in RHS](big_rem j) //= addnC -subnBA //= subnn subn0.
    rewrite [in LHS]big_seq_cond [in RHS]big_seq_cond.
    apply eq_biglj'.
    rewrite Bool.andb_true_r.
    destruct (j' \in rem (T:=Job) j jobs) eqn:INjobs; last by done.
    apply /negP ⇒ /eqP EQUAL.
    by rewrite EQUAL mem_rem_uniqF in INjobs.

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.
    Lemma workload_equal_subset :
      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)).
      clear H_jobs_uniq H_j_in_jobs H_t1_le_t2.
      rewrite /workload_of_jobs big_seq_cond.
      rewrite -[in X in X _]big_filter -[in X in _ X]big_filter.
      apply leq_sum_sub_uniq; first by apply filter_uniq, arrivals_uniq; rt_eauto.
      movej'; rewrite mem_filter ⇒ [/andP [/andP [A /andP [C D]] _]].
      rewrite mem_filter; apply/andP; split; first by done.
      apply job_in_arrivals_between; eauto.
      - by eapply in_arrivals_implies_arrived; eauto 2.
      - apply in_arrivals_implies_arrived_between in A; auto; move: A ⇒ /andP [A E].
        by unfold ε; apply/andP; split; lia.

  End Subset.

In this section, we prove a few useful properties regarding the predicate of workload_of_jobs.
  Section PredicateProperties.

First, we show that workload of jobs for an unsatisfiable predicate is equal to 0.
    Lemma workload_of_jobs_pred0 :
      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'.
    Lemma workload_of_jobs_case_on_pred :
      workload_of_jobs P jobs =
        workload_of_jobs (fun jP j && P' j) jobs + workload_of_jobs (fun jP j && ~~ P' j) jobs.
      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.

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'.
    Lemma workload_of_jobs_equiv_pred :
      {in jobs, P =1 P'}
      workload_of_jobs P jobs = workload_of_jobs P' jobs.
      intros × EQUIV.
      rewrite /workload_of_jobs !big_mkcond [in X in _ = X]big_mkcond //=.
      by apply: eq_big_seqj' IN; rewrite EQUIV.

  End PredicateProperties.

End WorkloadFacts.