# Library prosa.results.edf.rta.floating_nonpreemptive

Require Import prosa.model.readiness.basic.
Require Export prosa.results.edf.rta.bounded_nps.
Require Export prosa.analysis.facts.preemption.rtc_threshold.floating.
Require Export prosa.analysis.facts.readiness.sequential.
Require Import prosa.model.priority.edf.
Require Export prosa.analysis.definitions.blocking_bound.edf.

# RTA for EDF with Floating Non-Preemptive Regions

In this module we prove the RTA theorem for floating non-preemptive regions EDF model.

## Setup and Assumptions

Section RTAforModelWithFloatingNonpreemptiveRegionsWithArrivalCurves.
Consider any type of tasks ...
Context {Task : TaskType}.
Context `{TaskCost Task}.
Context `{TaskDeadline Task}.
... and any type of jobs associated with these tasks.
Context {Job : JobType}.
Context `{JobTask Job Task}.
Context `{JobArrival Job}.
Context `{JobCost Job}.
We assume the classic (i.e., Liu & Layland) model of readiness without jitter or self-suspensions, wherein pending jobs are always ready.
#[local] Existing Instance basic_ready_instance.
We assume that jobs are limited-preemptive.
#[local] Existing Instance limited_preemptive_job_model.
Consider any arrival sequence with consistent, non-duplicate arrivals.
Variable arr_seq : arrival_sequence Job.
Hypothesis H_valid_arrival_sequence : valid_arrival_sequence arr_seq.
Assume we have the model with floating non-preemptive regions. I.e., for each task only the length of the maximal non-preemptive segment is known and each job is divided into a number of non-preemptive segments by inserting preemption points.
Context `{JobPreemptionPoints Job}
`{TaskMaxNonpreemptiveSegment Task}.
Hypothesis H_valid_task_model_with_floating_nonpreemptive_regions :
valid_model_with_floating_nonpreemptive_regions arr_seq.
Consider an arbitrary task set ts, ...
Variable ts : list Task.
... assume that all jobs come from this task set, ...
Hypothesis H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts.
... and the cost of a job cannot be larger than the task cost.
Hypothesis H_valid_job_cost :
arrivals_have_valid_job_costs arr_seq.
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.
Context `{MaxArrivals Task}.
Hypothesis H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals.
Hypothesis H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts.
Let tsk be any task in ts that is to be analyzed.
Variable tsk : Task.
Hypothesis H_tsk_in_ts : tsk \in ts.
Next, consider any valid ideal uni-processor schedule with limited preemptions of this arrival sequence ...
Variable sched : schedule (ideal.processor_state Job).
Hypothesis H_sched_valid : valid_schedule sched arr_seq.
Hypothesis H_schedule_with_limited_preemptions :
schedule_respects_preemption_model arr_seq sched.
Next, we assume that the schedule is a work-conserving schedule...
Hypothesis H_work_conserving : work_conserving arr_seq sched.
... and the schedule respects the scheduling policy.
Hypothesis H_respects_policy : respects_JLFP_policy_at_preemption_point arr_seq sched (EDF Job).

## Total Workload and Length of Busy Interval

We introduce the abbreviation rbf for the task request bound function, which is defined as task_cost(T) × max_arrivals(T,Δ) for a task T.
Let rbf := task_request_bound_function.
Next, we introduce task_rbf as an abbreviation for the task request bound function of task tsk.
Let task_rbf := rbf tsk.
Using the sum of individual request bound functions, we define the request bound function of all tasks (total request bound function).
Let total_rbf := total_request_bound_function ts.
Let L be any positive fixed point of the busy interval recurrence.
Variable L : duration.
Hypothesis H_L_positive : L > 0.
Hypothesis H_fixed_point : L = total_rbf L.

## Response-Time Bound

To reduce the time complexity of the analysis, recall the notion of search space.
Let is_in_search_space := bounded_nps.is_in_search_space ts tsk L.
Consider any value R, and assume that for any given arrival offset A in the 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 ≥ blocking_bound ts tsk A + task_rbf (A + ε)
+ bound_on_athep_workload ts tsk A (A + F)
∧ R ≥ F.
Now, we can leverage the results for the abstract model with bounded nonpreemptive segments to establish a response-time bound for the more concrete model with floating nonpreemptive regions.
Theorem uniprocessor_response_time_bound_edf_with_floating_nonpreemptive_regions :
task_response_time_bound arr_seq sched tsk R.
Proof.
move: (H_valid_task_model_with_floating_nonpreemptive_regions) ⇒ [LIMJ JMLETM].
move: (LIMJ) ⇒ [BEG [END _]].
eapply uniprocessor_response_time_bound_edf_with_bounded_nonpreemptive_segments with (L := L) ⇒ //.
rewrite subnn.
intros A SP.
apply H_R_is_maximum in SP.
move: SP ⇒ [F [EQ LE]].
∃ F.
by rewrite subn0 addn0; split.
Qed.
End RTAforModelWithFloatingNonpreemptiveRegionsWithArrivalCurves.