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

Require Export prosa.results.rta.ideal.fp.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.sequential.
Require Export prosa.model.task.preemption.fully_preemptive.

# RTA for Fully Preemptive FP Model

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

## Setup and Assumptions

Section RTAforFullyPreemptiveFPModelwithArrivalCurves.
We assume ideal uni-processor schedules.
#[local] Existing Instance ideal.processor_state.
Consider any type of tasks ...
Context {Task : TaskType}.
Context `{TaskCost 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 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 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 the 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_arrivals 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.
Recall that we assume sequential readiness.
#[local] Instance sequential_readiness : JobReady _ _ :=
sequential_ready_instance arr_seq.
Next, consider any ideal uniprocessor schedule of this arrival sequence.
Variable sched : schedule (ideal.processor_state Job).
Hypothesis H_sched_valid : valid_schedule sched arr_seq.
Consider an FP policy that indicates a higher-or-equal priority relation, and assume that the relation is reflexive and transitive.
Context {FP : FP_policy Task}.
Hypothesis H_priority_is_reflexive : reflexive_task_priorities FP.
Hypothesis H_priority_is_transitive : transitive_task_priorities FP.
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_FP_policy_at_preemption_point arr_seq sched FP.

## 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 with higher priority ...
Let total_hep_rbf := total_hep_request_bound_function_FP ts tsk.
... and the request bound function of all tasks with higher priority other than task tsk.
Let total_ohep_rbf := total_ohep_request_bound_function_FP ts 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 the search space.
Let is_in_search_space := is_in_search_space tsk L.
Next, consider any value R, and assume that for any given arrival A from 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) + 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.
Theorem uniprocessor_response_time_bound_fully_preemptive_fp :
task_response_time_bound arr_seq sched tsk R.
End RTAforFullyPreemptiveFPModelwithArrivalCurves.