# Facts about Request Bound Functions (RBFs)

In this file, we prove some lemmas about request bound functions.

## RBF is a Bound on Workload

First, we show that a task's RBF is indeed an upper bound on its workload.
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 arrival sequence with consistent, non-duplicate arrivals, ...
... any schedule corresponding to this arrival sequence, ...
... and an FP policy that indicates a higher-or-equal priority relation.

Further, consider a task set ts...

... and let tsk be any task in ts.
Hypothesis H_tsk_in_ts : tsk \in ts.

Assume that the job costs are no larger than the task costs ...
... and that all jobs come from the task set.
Let max_arrivals be any arrival bound for task-set ts.
Next, recall the notions of total workload of jobs...
... and the workload of jobs of the same task as job j.
Finally, let us define some local names for clarity.
In this section, we prove that the workload of all jobs is no larger than the request bound function.

Consider any time t and any interval of length Δ.
Variable t : instant.
Variable Δ : instant.

First, we show that workload of task tsk is bounded by the number of arrivals of the task times the cost of the task.
Proof.
by rewrite -big_filter !TASK !big_nil. }
{ rewrite //= big_filter big_seq_cond [in X in _ X]big_seq_cond.
apply leq_sum.
movej' /andP [IN TSKj'].
rewrite muln1.
move: TSKj' ⇒ /eqP TSKj'; rewrite -TSKj'.
apply H_valid_job_cost.
by apply in_arrivals_implies_arrived in IN. }
Qed.

As a corollary, we prove that workload of task is no larger the than task request bound function.
Proof.
rewrite leq_mul2l; apply/orP; right.
by apply H_is_arrival_bound; last rewrite leq_addr.
Qed.

Next, we prove that total workload of tasks is no larger than the total request bound function.
total_workload t (t + Δ) total_rbf Δ.
Proof.
set l := arrivals_between arr_seq t (t + Δ).
apply (@leq_trans (\sum_(tsk' <- ts) (\sum_(j0 <- l | job_task j0 == tsk') job_cost j0))).
have EXCHANGE := exchange_big_dep predT.
rewrite EXCHANGE //=; clear EXCHANGE.
rewrite /workload_of_jobs -/l big_seq_cond [X in _ X]big_seq_cond.
apply leq_sum; movej0 /andP [IN0 HP0].
rewrite big_mkcond (big_rem (job_task j0)) /=.
- by rewrite eq_refl; apply leq_addr.
- by apply in_arrivals_implies_arrived in IN0; apply H_all_jobs_from_taskset. }
apply leq_sum_seq; intros tsk0 INtsk0 HP0.
apply (@leq_trans (task_cost tsk0 × size (task_arrivals_between arr_seq tsk0 t (t + Δ)))).
{ rewrite -sum1_size big_distrr /= big_filter -/l /workload_of_jobs muln1.
apply leq_sum_seqj0 IN0 /eqP <-.
apply H_valid_job_cost.
by apply in_arrivals_implies_arrived in IN0. }
{ rewrite leq_mul2l; apply/orP; right.
by apply H_is_arrival_bound; last rewrite leq_addr. }
Qed.

Next, we consider any job j of tsk.
Variable j : Job.
Hypothesis H_job_of_tsk : job_of_task tsk j.

We prove that the sum of job cost of jobs whose corresponding task satisfies a predicate pred is bounded by the RBF of these tasks.
Lemma sum_of_jobs_le_sum_rbf :
\sum_(j' <- arrivals_between arr_seq t (t + Δ) | pred (job_task j'))
job_cost j'
\sum_(tsk' <- ts| pred tsk')
Proof.
movepred.
apply (@leq_trans (\sum_(tsk' <- filter pred ts)
(\sum_(j' <- arrivals_between arr_seq t (t + Δ)
| job_task j' == tsk') job_cost j'))).
- move: (H_job_of_tsk) ⇒ /eqP TSK.
rewrite [X in _ X]big_filter.
set P := fun xpred (job_task x).
rewrite (exchange_big_dep P) //=; last by rewrite /P; move ⇒ ???/eqP→.
rewrite /P /workload_of_jobs big_seq_cond [X in _ X]big_seq_cond.
apply leq_sumj0 /andP [IN0 HP0].
+ by rewrite HP0 andTb eq_refl; apply leq_addr.
+ by apply in_arrivals_implies_arrived in IN0; apply H_all_jobs_from_taskset.
- rewrite big_filter.
apply leq_sum_seqtsk0 INtsk0 HP0.
apply (@leq_trans (task_cost tsk0 × size (task_arrivals_between arr_seq tsk0 t (t + Δ)))).
+ rewrite -sum1_size big_distrr /= big_filter /workload_of_jobs.
rewrite muln1 /arrivals_between /arrival_sequence.arrivals_between.
apply leq_sum_seq; movej0 IN0 /eqP EQ.
by rewrite -EQ; apply H_valid_job_cost; apply in_arrivals_implies_arrived in IN0.
+ rewrite leq_mul2l; apply/orP; right.
by apply H_is_arrival_bound; last by rewrite leq_addr.
Qed.

Using lemma sum_of_jobs_le_sum_rbf, we prove that the workload of higher-or-equal priority jobs (w.r.t. task tsk) is no larger than the total request-bound function of higher-or-equal priority tasks.
workload_of_hep_jobs arr_seq j t (t + Δ) total_hep_rbf tsk Δ.
Proof.
last by movej'; rewrite /pred_task; move: H_job_of_tsk ⇒ /eqP →.
erewrite (eq_big pred_task); [|by done|by movetsk'; eauto].
by apply: sum_of_jobs_le_sum_rbf.
Qed.

We next prove that the higher-or-equal-priority workload of tasks different from tsk is bounded by total_ohep_rbf.
The athep_workload_is_bounded predicate allows the workload bound to depend on two arguments: the relative offset A (w.r.t. the beginning of the corresponding busy interval) of a job to be analyzed and the length of an interval Δ. In the case of FP and total_ohep_rbf function, the relative offset (A) does not play a role and is therefore ignored.
arr_seq sched tsk (fun (A Δ : duration) ⇒ total_ohep_rbf tsk Δ).
Proof.
movej t1 Δ POS TSK _.
last by movej'; rewrite /pred_task; move: TSK ⇒ /eqP →.
by eapply sum_of_jobs_le_sum_rbf ⇒ //.
Qed.

In this section, we show that total RBF is a bound on higher-or-equal priority workload under any JLFP policy.
Section TotalRBFBound.

Consider any type of tasks ...

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

Consider a JLFP policy that indicates a higher-or-equal priority relation ...
Context `{JLFP_policy Job}.

... and any valid arrival sequence.
Further, consider a task set ts.

Assume that the job costs are no larger than the task costs ...
... and that all jobs come from the task set.
Let max_arrivals be any arrival bound for task set ts.
Consider any time t and any interval of length Δ.
Variable t : instant.
Variable Δ : duration.

Next, we consider any job j.
Variable j : Job.

A simple consequence of lemma hep_workload_le_total_hep_rbf is that the workload of higher-or-equal priority jobs is bounded by the total request-bound function.
workload_of_hep_jobs arr_seq j t (t + Δ)
total_request_bound_function ts Δ.
Proof.
rewrite /workload_of_hep_jobs (leqRW (workload_of_jobs_weaken _ predT _ _ )); last by done.
Qed.

End TotalRBFBound.

## RBF Properties

In this section, we prove simple properties and identities of RBFs.
Consider any type of tasks ...

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

Consider any arrival sequence.

Let max_arrivals be a family of valid arrival curves, i.e., for any task tsk in ts max_arrival tsk is (1) an arrival bound of tsk, and (2) it is a monotonic function that equals 0 for the empty interval Δ = 0.
Let's define some local names for clarity.
We prove that task_rbf 0 is equal to 0.
Proof.
apply/eqP; rewrite muln_eq0; apply/orP; right; apply/eqP.
by move: H_valid_arrival_curve ⇒ [T1 T2].
Qed.

We prove that task_rbf is monotone.
Proof.
rewrite /monotone; intros ? ? LE.
apply/orP; right.
by move: H_valid_arrival_curve ⇒ [_ T]; apply T.
Qed.

In the following, we assume that tsk has a positive cost ...
Hypothesis H_positive_cost : 0 < task_cost tsk.

... and max_arrivals tsk ε is positive.
Then we prove that task_rbf at ε is greater than or equal to the task's WCET.
Proof.
have ALT: n, n = 0 n > 0 by clear; intros n; destruct n; [left | right].
specialize (ALT (task_cost tsk)); destruct ALT as [Z | POS]; first by rewrite Z.
by rewrite leq_pmul2l //=.
Qed.

As a corollary, we prove that the task_rbf at any point A greater than 0 is no less than the task's WCET.
A,
A > 0
Proof.
case ⇒ // A GEQ.
Qed.

Then, we prove that task_rbf at ε is greater than 0.
Proof.
apply leq_trans with (task_cost tsk) ⇒ [//|].
Qed.

Hypothesis H_tsk_in_ts : tsk \in ts.

Next, we prove that cost of tsk is less than or equal to the total_request_bound_function.
t,
t > 0
Proof.
case⇒ [//|t] GE.
rewrite /total_request_bound_function.
erewrite big_rem; last by exact H_tsk_in_ts.
Qed.

End RequestBoundFunctions.

## Monotonicity of the Total RBF

In the following section, we note some trivial facts about the monotonicity of various total RBF variants.
Consider a set of tasks characterized by WCETs and arrival curves.
We observe that the total RBF is monotonically increasing.
Lemma total_rbf_monotone :
monotone leq (total_request_bound_function ts).
Proof. by apply: sum_leq_monotsk IN; apply: task_rbf_monotone. Qed.

Furthermore, for any fixed-priority policy, ...

... the total RBF of higher- or equal-priority tasks is also monotonic, ...
Lemma total_hep_rbf_monotone :
tsk,
monotone leq (total_hep_request_bound_function_FP ts tsk).
Proof.
movetsk.
apply: sum_leq_monotsk' IN.
Qed.

... as is the variant that excludes the reference task.
Lemma total_ohep_rbf_monotone :
tsk,
monotone leq (total_ohep_request_bound_function_FP ts tsk).
Proof.
movetsk.
apply: sum_leq_monotsk' IN.
Qed.

End TotalRBFMonotonic.

## RBFs Equal to Zero for Duration ε

In the following section, we derive simple properties that follow in the pathological case of an RBF that yields zero for duration ε.
Consider a set of tasks characterized by WCETs and arrival curves ...

... and any consistent arrival sequence of valid jobs of these tasks.
Suppose the arrival curves are correct.
Consider any valid schedule corresponding to this arrival sequence.
First, we observe that, if a task's RBF is zero for a duration ε, then it trivially has a response-time bound of zero.
Lemma pathological_rbf_response_time_bound :
tsk,
tsk \in ts
Proof.
movetsk IN ZERO j ARR TASK.
rewrite /job_response_time_bound/completed_by.
move: ZERO. rewrite /task_request_bound_function ⇒ /eqP.
rewrite muln_eq0 ⇒ /orP [/eqP COST|/eqP NEVER].
{ apply: leq_trans.
- by apply: H_valid_job_cost.
by rewrite COST. }
{ exfalso.
have: 0 < max_arrivals tsk ε
by apply: (non_pathological_max_arrivals tsk arr_seq _ j).
by rewrite NEVER. }
Qed.

Second, given a fixed-priority policy with reflexive priorities, ...

... if the total RBF of all equal- and higher-priority tasks is zero, then the reference task's response-time bound is also trivially zero.
Lemma pathological_total_hep_rbf_response_time_bound :
tsk,
tsk \in ts
total_hep_request_bound_function_FP ts tsk ε = 0
Proof.
movetsk IN ZERO j ARR TASK.
apply: pathological_rbf_response_time_bound; eauto.
apply /eqP.
move: ZERO ⇒ /eqP; rewrite sum_nat_eq0_nat ⇒ /allP; apply.
rewrite mem_filter; apply /andP; split ⇒ //.
Qed.

Thus we we can prove any response-time bound from such a pathological case, which is useful to eliminate this case in higher-level analyses.
Corollary pathological_total_hep_rbf_any_bound :
tsk,
tsk \in ts
total_hep_request_bound_function_FP ts tsk ε = 0
R,
Proof.
movetsk IN ZERO R.
move: (pathological_total_hep_rbf_response_time_bound tsk IN ZERO).
apply: completion_monotonic; last by apply: COMP.
by lia.
Qed.

End DegenerateTotalRBFs.

Consider any type of tasks ...

... and any type of jobs associated with these tasks, where each task has a cost and an associated arrival curve.

Consider an FP policy that indicates a higher-or-equal priority relation.

...and let tsk be any task that serves as the reference point for "higher or equal priority" (usually, but not necessarily, from ts).

We establish that the bound on the total workload due to higher-or-equal-priority tasks can be partitioned task-wise. In other words, it is equal to the sum of the bound on the total workload due to higher-priority tasks and the bound on the total workload due to equal- priority tasks.
L,
total_hep_request_bound_function_FP ts tsk L
= total_hp_request_bound_function_FP ts tsk L
+ total_ep_request_bound_function_FP ts tsk L.
Proof.
moveL; apply sum_split_exhaustive_mutually_exclusive_predst.
- by rewrite -andb_orr orNb Bool.andb_true_r.
- rewrite !negb_and; case: (hep_task _ _) =>//=.
Qed.

Now, assume that the priorities are reflexive.
If the task set does not contain duplicates, then the total higher-or-equal-priority RBF for any task can be split as the sum of the total other higher-or-equal-priority workload and the RBF of the task itself.
Lemma split_hep_rbf :
Δ,
tsk \in ts
uniq ts
total_hep_request_bound_function_FP ts tsk Δ
= total_ohep_request_bound_function_FP ts tsk Δ
Proof.
moveΔ IN UNIQ.
rewrite /total_hep_request_bound_function_FP /total_ohep_request_bound_function_FP.
rewrite (bigID_idem _ _ (fun tskotsko != tsk)) //=.
rewrite (eq_bigl (fun ii == tsk)); last first.
- movetsko.
case (tsko == tsk) eqn: EQ; last by lia.
move : EQ ⇒ /eqP →.
by rewrite H_priority_is_reflexive //=.
- rewrite (big_rem tsk) //= eq_refl.
rewrite big_seq_cond big_pred0; first by rewrite addn0 //=.
movetsko.
case (tsko == tsk) eqn: EQ; last by lia.
move : EQ ⇒ /eqP →.
rewrite andbT.
by apply mem_rem_uniqF ⇒ //=.
Qed.

If the task set may contain duplicates, then the we can only say that the sum of other higher-or-equal-priority RBF and task tsk's RBF is at most the total higher-or-equal-priority workload.
Lemma split_hep_rbf_weaken:
Δ,
tsk \in ts
total_ohep_request_bound_function_FP ts tsk Δ + task_request_bound_function tsk Δ
total_hep_request_bound_function_FP ts tsk Δ.
Proof.
moveΔ IN.
rewrite /total_hep_request_bound_function_FP /total_ohep_request_bound_function_FP.
rewrite [leqRHS](bigID_idem _ _ (fun tskotsko != tsk)) //=.
rewrite (eq_bigl (fun tskotsko == tsk)); last first.
- movetsko.
case (tsko ==tsk) eqn: TSKEQ; last by lia.
move : TSKEQ ⇒ /eqP →.
by rewrite (H_priority_is_reflexive tsk) //=.
- rewrite (big_rem tsk) //= eq_refl.
Qed.

End FP_RBF_partitioning.

In this section, we state a few facts for RBFs in the context of a fixed-priority policy.
Section RBFFOrFP.

Consider a set of tasks characterized by WCETs and arrival curves.
For any fixed-priority policy, ...

... total_ohep_request_bound_function_FP at 0 is always 0.
Lemma total_ohep_rbf0 :
total_ohep_request_bound_function_FP ts tsk 0 = 0.
Proof.
rewrite /total_ohep_request_bound_function_FPtsk.
apply /eqP.
rewrite sum_nat_eq0_nat.
apply /allPtsk' IN.
apply /eqP.
apply H_valid_arrival_curve.
rewrite mem_filter in IN.
by move : IN ⇒ /andP[_ ->].
Qed.

Next we show how total_ohep_request_bound_function_FP can bound the workload of jobs in a given interval.
Consider any types of jobs.

Consider any arrival sequence that only has jobs from the task set and where all arrivals have a valid job cost.
Assume there exists an arrival curve and that the arrival sequence respects this curve.
Consider any task tsk and any job j of the task tsk.
Variable j : Job.

For any interval `[t1, t1 + Δ)`, the workload of jobs that have higher task priority than the task priority of j is bounded by total_ohep_request_bound_function_FP for the duration Δ.
Δ t1,
total_ohep_request_bound_function_FP ts tsk Δ.
Proof.
moveΔ t1.
erewrite (eq_big pred_task); [|by done|by movetsk'; eauto].
by apply: sum_of_jobs_le_sum_rbf.
Qed.

End RBFFOrFP.

We know that the workload of a task in any interval must be bounded by the task's RBF in that interval. However, in the proofs of several lemmas, we are required to reason about the workload of a task in an interval excluding the cost of a particular job (usually the job under analysis). Such a workload can be tightly bounded by the task's RBF for the interval excluding the cost of one task.
Notice, however, that this is not a trivial result since a naive approach to proving it would fail. Suppose we want to prove that some quantity A - B is upper bounded by some quantity C - D. This usually requires us to prove that A is upper bounded by C and D is upper bounded by B. In our case, this would be equivalent to proving that the task cost is upper-bounded by the job cost, which of course is not true.
So, a different approach is needed, which we show in this section.

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 arrival sequence ...
... and assume that WCETs are respected.
Let tsk be any task ...

... characterized by a valid arrival curve.
Consider any job j of tsk ...
Variable j : Job.

... that arrives in the given arrival sequence.
Hypothesis H_j_arrives_in : arrives_in arr_seq j.

Consider any time instant t1 and duration Δ such that j arrives before t1 + Δ.
Variables (t1 : instant) (Δ : duration).
Hypothesis H_job_arrival_lt : job_arrival j < t1 + Δ.

As a preparatory step, we restrict our attention to the sub-interval containing the job's arrival. We know that the job's arrival necessarily happens in the interval (`[job_arrival j], t1 + Δ`). This allows us to show that the task workload excluding the task cost can be bounded by the cost of the arrivals in the interval as follows.
Proof.
rewrite mulnDr mulnC muln1 addnK mulnC.
apply sum_majorant_constantj' ARR' /eqP TSK2.
rewrite -TSK2; apply H_arrivals_have_valid_job_costs.
apply rem_in in ARR'.
by eapply in_arrivals_implies_arrived ⇒ //=.
Qed.

To use the above lemma in our final theorem, we require that the arrival of the job under analysis necessarily happens in the interval we are considering.
Hypothesis H_j_arrives_after_t : t1 job_arrival j.

Under the above assumption, we can finally establish the desired bound.
Proof.
apply leq_trans with
- rewrite leq_sub2r // leq_mul2l; apply/orP ⇒ //=; right.
have POSE: Δ = (t1 + Δ - t1) by lia.
rewrite [in leqRHS]POSE.
eapply (H_is_arrival_curve t1 (t1 + Δ)).
by lia.
- rewrite (@num_arrivals_of_task_cat _ _ _ _ _ (job_arrival j)); last by apply /andP; split.
rewrite mulnDr.
last by apply/andP; split; lia.
+ rewrite -{1}[task_cost tsk]muln1 leq_mul2l; apply/orP; right.