# Library prosa.results.rta.ideal.edf.fully_preemptive

Require Import prosa.model.readiness.basic.
Require Import prosa.results.rta.ideal.edf.bounded_nps.
Require Export prosa.analysis.facts.preemption.task.preemptive.
Require Export prosa.analysis.facts.preemption.rtc_threshold.preemptive.
Require Export prosa.analysis.facts.readiness.basic.
Require Import prosa.model.task.preemption.fully_preemptive.

# RTA for Fully Preemptive EDF

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

## Setup and Assumptions

Section RTAforFullyPreemptiveEDFModelwithArrivalCurves.
We assume that jobs and tasks are fully preemptive.
#[local] Existing Instance fully_preemptive_job_model.
#[local] Existing Instance fully_preemptive_task_model.
#[local] Existing Instance fully_preemptive_rtc_threshold.
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.
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.
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 uniprocessor schedule of the arrival sequence.
Variable sched : schedule (ideal.processor_state Job).
Hypothesis H_sched_valid : valid_schedule sched arr_seq.
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.
Context {JLFP : JLFP_policy Job}.
Hypothesis H_policy_is_EDF : policy_is_EDF JLFP.
Hypothesis H_respects_policy :
respects_JLFP_policy_at_preemption_point arr_seq sched JLFP.

## 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.
If jobs are fully preemptive, lower priority jobs do not cause priority inversion. Hence, the blocking bound is always 0 for any A.
Let blocking_bound (A : duration) := 0.
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 ≥ 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 non-preemptive segments to establish a response-time bound for the more concrete model of fully preemptive scheduling.
Theorem uniprocessor_response_time_bound_fully_preemptive_edf :
task_response_time_bound arr_seq sched tsk R.
End RTAforFullyPreemptiveEDFModelwithArrivalCurves.