# Library prosa.results.fixed_priority.rta.floating_nonpreemptive

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

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

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

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

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

# RTA for Model with Floating Non-Preemptive Regions

In this module we prove the RTA theorem for floating non-preemptive regions FP model.
Furthermore, we assume the task model with floating non-preemptive regions.

Require Import prosa.model.preemption.limited_preemptive.

Require Import prosa.model.task.preemption.floating_nonpreemptive.

Require Import prosa.model.task.preemption.floating_nonpreemptive.

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.

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 level 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.

`{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, ...

... 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 uni-processor schedule with limited preemptions 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_schedule_with_limited_preemptions:

schedule_respects_preemption_model arr_seq sched.

Hypothesis H_jobs_come_from_arrival_sequence:

jobs_come_from_arrival_sequence sched arr_seq.

Hypothesis H_schedule_with_limited_preemptions:

schedule_respects_preemption_model arr_seq sched.

... 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, jobs 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.

Next, we define a bound for the priority inversion caused by tasks of lower priority.

Let blocking_bound :=

\max_(tsk_other <- ts | ~~ hep_task tsk_other tsk)

(task_max_nonpreemptive_segment tsk_other - ε).

\max_(tsk_other <- ts | ~~ hep_task tsk_other tsk)

(task_max_nonpreemptive_segment tsk_other - ε).

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 = blocking_bound + total_hep_rbf L.

Hypothesis H_L_positive : L > 0.

Hypothesis H_fixed_point : L = blocking_bound + 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 = blocking_bound + 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 = blocking_bound + task_rbf (A + ε) + total_ohep_rbf (A + F) ∧

F ≤ R.

Now, we can reuse 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.

Let response_time_bounded_by := task_response_time_bound arr_seq sched.

Theorem uniprocessor_response_time_bound_fp_with_floating_nonpreemptive_regions:

response_time_bounded_by tsk R.

Proof.

move: (H_valid_task_model_with_floating_nonpreemptive_regions) ⇒ [LIMJ JMLETM].

move: (LIMJ) ⇒ [BEG [END _]].

eapply uniprocessor_response_time_bound_fp_with_bounded_nonpreemptive_segments.

all: eauto 2 with basic_facts.

intros A SP.

rewrite subnn subn0.

destruct (H_R_is_maximum _ SP) as [F [EQ LE]].

by ∃ F; rewrite addn0; split.

Qed.

End RTAforFloatingModelwithArrivalCurves.