# RTA for Fully Preemptive FP Model

In this section we prove the RTA theorem for the fully preemptive FP model

## Setup and Assumptions

We assume ideal uni-processor schedules.
#[local] Existing Instance ideal.processor_state.

Consider any type of tasks ...

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

We assume that jobs and tasks are fully preemptive.
#[local] Existing Instance fully_preemptive_job_model.
#[local] Existing Instance fully_preemptive_rtc_threshold.

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.
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.

Recall that we assume sequential readiness.

Next, consider any ideal uniprocessor schedule of this arrival sequence ...
Consider an FP policy that indicates a higher-or-equal priority relation, and assume that the relation is reflexive and transitive.
Next, we assume that the schedule is a work-conserving schedule...
... and the schedule respects the scheduling policy.

## 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.
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 = 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.
Variable R : duration.
Hypothesis H_R_is_maximum:
(A : duration),
is_in_search_space A
(F : duration),
A + F task_rbf (A + ε) + total_ohep_rbf (A + F)
R F.

Now, we can leverage the results for the abstract model with bounded non-preemptive segments to establish a response-time bound for the more concrete model of fully preemptive scheduling.

Let response_time_bounded_by := task_response_time_bound arr_seq sched.

Theorem uniprocessor_response_time_bound_fully_preemptive_fp:
response_time_bounded_by tsk R.
Proof.
have BLOCK: blocking_bound ts tsk = 0.
eapply uniprocessor_response_time_bound_fp_with_bounded_nonpreemptive_segments with (L:=L) ⇒ //
.