# Library prosa.results.fixed_priority.rta.fully_preemptive

Require Export prosa.results.fixed_priority.rta.bounded_nps.

Require Export prosa.analysis.facts.preemption.task.preemptive.

Require Export prosa.analysis.facts.preemption.rtc_threshold.preemptive.

From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq path fintype bigop.

Require Export prosa.analysis.facts.preemption.task.preemptive.

Require Export prosa.analysis.facts.preemption.rtc_threshold.preemptive.

From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq path fintype bigop.

# RTA for Fully Preemptive FP Model

In this section we prove the RTA theorem for the fully preemptive FP model
Furthermore, we assume the fully preemptive task model.

Consider any type of tasks ...

... and any type of jobs associated with these tasks.

Context {Job : JobType}.

Context `{JobTask Job Task}.

Context `{JobArrival Job}.

Context `{JobCost Job}.

Context `{JobTask Job Task}.

Context `{JobArrival Job}.

Context `{JobCost Job}.

Consider any arrival sequence with consistent, non-duplicate arrivals.

Variable arr_seq : arrival_sequence Job.

Hypothesis H_arrival_times_are_consistent : consistent_arrival_times arr_seq.

Hypothesis H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq.

Hypothesis H_arrival_times_are_consistent : consistent_arrival_times arr_seq.

Hypothesis H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq.

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.

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.

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.

Next, consider any ideal uniprocessor schedule of this arrival sequence ...

Variable sched : schedule (ideal.processor_state Job).

Hypothesis H_jobs_come_from_arrival_sequence:

jobs_come_from_arrival_sequence sched arr_seq.

Hypothesis H_jobs_come_from_arrival_sequence:

jobs_come_from_arrival_sequence sched arr_seq.

... where jobs do not execute before their arrival or after completion.

Hypothesis H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched.

Hypothesis H_completed_jobs_dont_execute : completed_jobs_dont_execute sched.

Hypothesis H_completed_jobs_dont_execute : completed_jobs_dont_execute sched.

Consider an FP policy that indicates a higher-or-equal priority relation,
and assume that the relation is reflexive and transitive.

Context `{FP_policy Task}.

Hypothesis H_priority_is_reflexive : reflexive_priorities.

Hypothesis H_priority_is_transitive : transitive_priorities.

Hypothesis H_priority_is_reflexive : reflexive_priorities.

Hypothesis H_priority_is_transitive : transitive_priorities.

Assume we have sequential tasks, i.e, tasks 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

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.

Hypothesis H_L_positive : L > 0.

Hypothesis H_fixed_point : L = total_hep_rbf L.

## Response-Time Bound

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) ∧

F ≤ R.

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) ∧

F ≤ R.

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.

{ by rewrite /blocking_bound /parameters.task_max_nonpreemptive_segment

/fully_preemptive.fully_preemptive_model subnn big1_eq. }

eapply uniprocessor_response_time_bound_fp_with_bounded_nonpreemptive_segments.

all: eauto 2 with basic_facts.

- by rewrite BLOCK add0n.

- move ⇒ A /andP [LT NEQ].

edestruct H_R_is_maximum as [F [FIX BOUND]].

{ by apply/andP; split; eauto 2. }

∃ F; split.

+ by rewrite BLOCK add0n subnn subn0.

+ by rewrite subnn addn0.

Qed.

End RTAforFullyPreemptiveFPModelwithArrivalCurves.