# 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.
As a corollary, we prove that workload of task is no larger the than task request bound function.

Next, we prove that total workload of tasks is no larger than the total request bound function.
Next, we consider any job j of tsk.
Variable j : Job.
Hypothesis H_job_of_tsk : job_of_task tsk j.

Consider any general predicate defined on tasks.

We prove that the sum of job cost of jobs whose corresponding task satisfies the predicate pred is bound by the RBF of these tasks.

## 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.

We prove that task_rbf is monotone.
Consider any job j of tsk. This guarantees that there exists at least one job of task tsk.
Variable j : Job.
Hypothesis H_j_arrives : arrives_in arr_seq j.
Hypothesis H_job_of_tsk : job_of_task tsk j.

Then we prove that task_rbf at ε is greater than or equal to the task's WCET.
As a corollary, we prove that the task_rbf at any point A greater than 0 is no less than the task's WCET.
Assume that tsk has a positive cost.
Hypothesis H_positive_cost : 0 < task_cost tsk.

Then, we prove that task_rbf at ε is greater than 0.
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.

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

... the total RBF of higher- or equal-priority tasks is also monotonic, ...
... as is the variant that excludes the reference task.

## 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 ...