# RTA for FP-schedulers with Fixed Preemption Points

In this module we prove the RTA theorem for FP-schedulers with fixed preemption points.
Throughout this file, we assume the FP priority policy, ideal uni-processor schedules, and the basic (i.e., Liu & Layland) readiness model.
Furthermore, we assume the task model with fixed preemption points.

## Setup and Assumptions

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.
Consider an arbitrary task set ts, ...

... assume that all jobs come from the task set, ...
... and the cost of a job cannot be larger than the task cost.
First, we assume we have the model with fixed preemption points. I.e., each task is divided into a number of non-preemptive segments by inserting statically predefined preemption points.
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 delta = 0.
Let tsk be any task in ts that is to be analyzed.
Hypothesis H_tsk_in_ts : tsk \in ts.

Next, consider any ideal uni-processor schedule with limited preemptions of this arrival sequence ...
... where jobs do not execute before their arrival or after completion.
Consider an FP policy that indicates a higher-or-equal priority relation, and assume that the relation is reflexive and transitive.
Assume we have sequential tasks, i.e, jobs from the same task execute in the order of their arrival.
Next, we assume that the schedule is a work-conserving schedule...
... and the schedule respects the policy defined by the job_preemptable function (i.e., jobs have bounded non-preemptive segments).

## Total Workload and Length of Busy Interval

We introduce the abbreviation rbf for the task request bound function, which is defined as × for a task T.

Using the sum of individual request bound functions, we define the request bound function of all tasks with higher priority ...
... and the request bound function of all tasks with higher priority other than task tsk.
Next, we define a bound for the priority inversion caused by tasks of lower priority.
Let L be any positive fixed point of the busy interval recurrence, determined by the sum of blocking and higher-or-equal-priority workload.
Variable L : duration.
Hypothesis H_L_positive : L > 0.
Hypothesis H_fixed_point : L = blocking_bound + total_hep_rbf L.

## Response-Time Bound

To reduce the time complexity of the analysis, recall the notion of search space.
Next, consider any value R, and assume that for any given arrival A from search space there is a solution of the response-time bound recurrence which is bounded by R.
Now, we can reuse the results for the abstract model with bounded non-preemptive segments to establish a response-time bound for the more concrete model of fixed preemption points.

Let response_time_bounded_by := task_response_time_bound arr_seq sched.

Theorem uniprocessor_response_time_bound_fp_with_fixed_preemption_points:
response_time_bounded_by tsk R.
Proof.
move: (H_valid_model_with_fixed_preemption_points) ⇒ [MLP [BEG [END [INCR [HYP1 [HYP2 HYP3]]]]]].
move: (MLP) ⇒ [BEGj [ENDj _]].
edestruct (posnP (task_cost tsk)) as [ZERO|POSt].
{ intros j ARR TSK.
move: (H_valid_job_cost _ ARR) ⇒ POSt.
move: POSt; rewrite /valid_job_cost TSK ZERO leqn0; move ⇒ /eqP Z.
by rewrite /job_response_time_bound /completed_by Z.
}
eapply uniprocessor_response_time_bound_fp_with_bounded_nonpreemptive_segments
with (L0 := L).
all: eauto 2 with basic_facts.
intros A SP.
destruct (H_R_is_maximum _ SP) as[FF [EQ1 EQ2]].
FF; rewrite subKn; first by done.
- rewrite /last0 -nth_last.
apply HYP3; try by done.