# Library prosa.results.fixed_priority.rta.bounded_pi

Require Export prosa.model.schedule.priority_driven.

Require Export prosa.analysis.abstract.ideal.iw_instantiation.

Require Export prosa.analysis.facts.busy_interval.busy_interval.

Require Export prosa.analysis.abstract.ideal.iw_instantiation.

Require Export prosa.analysis.facts.busy_interval.busy_interval.

# Abstract RTA for FP-schedulers with Bounded Priority Inversion

In this module we instantiate the Abstract Response-Time analysis (aRTA) to FP-schedulers for ideal uni-processor model of real-time tasks with arbitrary arrival models.
Consider any type of tasks ...

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

Context {Job : JobType}.

Context `{JobTask Job Task}.

Context {Arrival : JobArrival Job}.

Context {Cost : JobCost Job}.

Context `{JobPreemptable Job}.

Context `{JobTask Job Task}.

Context {Arrival : JobArrival Job}.

Context {Cost : JobCost Job}.

Context `{JobPreemptable Job}.

Consider an FP policy that indicates a higher-or-equal priority relation,
and assume that the relation is reflexive. Note that we do not relate
the FP policy with the scheduler. However, we define functions for
Interference and Interfering Workload that actively use the concept of
priorities. We require the FP policy to be reflexive, so a job cannot
cause lower-priority interference (i.e. priority inversion) to itself.

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.

Hypothesis H_valid_arrival_sequence : valid_arrival_sequence arr_seq.

Next, consider any ideal uni-processor schedule of this arrival sequence, ...

... allow for any work-bearing notion of job readiness, ...

Context `{@JobReady Job (ideal.processor_state Job) Cost Arrival}.

Hypothesis H_job_ready : work_bearing_readiness arr_seq sched.

Hypothesis H_job_ready : work_bearing_readiness arr_seq sched.

... and assume that the schedule is valid.

## Instantiation of Interference

We say that job j incurs interference at time t iff it cannot execute due to a higher-or-equal-priority job being scheduled, or if it incurs a priority inversion.
#[local] Instance ideal_jlfp_interference : Interference Job :=

ideal_jlfp_interference arr_seq sched.

ideal_jlfp_interference arr_seq sched.

## Instantiation of Interfering Workload

The interfering workload, in turn, is defined as the sum of the priority inversion function and interfering workload of jobs with higher or equal priority.
#[local] Instance ideal_jlfp_interfering_workload : InterferingWorkload Job :=

ideal_jlfp_interfering_workload arr_seq sched.

ideal_jlfp_interfering_workload arr_seq sched.

Note that we differentiate between abstract and classical
notions of work conserving schedule.

Let work_conserving_ab := definitions.work_conserving arr_seq sched.

Let work_conserving_cl := work_conserving.work_conserving arr_seq sched.

Let work_conserving_cl := work_conserving.work_conserving arr_seq sched.

We assume that the schedule is a work-conserving schedule in the

*classical*sense, and later prove that the hypothesis about abstract work-conservation also holds.
Assume we have sequential tasks, i.e, jobs from the
same task execute in the order of their arrival.

Assume that a job cost cannot be larger than a task cost.

Consider an arbitrary task set ts.

Next, we assume that all jobs come from the task set.

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.

Consider a valid preemption model...

...and a valid task run-to-completion threshold function. That
is, task_rtct tsk is (1) no bigger than tsk's cost, (2) for
any job of task tsk job_rtct is bounded by task_rtct.

Hypothesis H_valid_run_to_completion_threshold :

valid_task_run_to_completion_threshold arr_seq tsk.

valid_task_run_to_completion_threshold arr_seq tsk.

For clarity, let's define some local names.

Let job_pending_at := pending sched.

Let job_scheduled_at := scheduled_at sched.

Let job_completed_by := completed_by sched.

Let job_backlogged_at := backlogged sched.

Let response_time_bounded_by := task_response_time_bound arr_seq sched.

Let job_scheduled_at := scheduled_at sched.

Let job_completed_by := completed_by sched.

Let job_backlogged_at := backlogged sched.

Let response_time_bounded_by := task_response_time_bound arr_seq sched.

We introduce task_rbf as an abbreviation of the task request
bound function, which is defined as task_cost(tsk) ×
max_arrivals(tsk,Δ).

Using the sum of individual request bound functions, we define
the request bound function of all tasks with higher-or-equal
priority (with respect to tsk).

Similarly, we define the request bound function of all tasks
other than tsk with higher-or-equal priority (with respect to
tsk).

Assume that there exists a constant priority_inversion_bound
that bounds the length of any priority inversion experienced by
any job of tsk. Since we analyze only task tsk, we ignore
the lengths of priority inversions incurred by any other
tasks.

Variable priority_inversion_bound : duration.

Hypothesis H_priority_inversion_is_bounded :

priority_inversion_is_bounded_by_constant

arr_seq sched tsk priority_inversion_bound.

Hypothesis H_priority_inversion_is_bounded :

priority_inversion_is_bounded_by_constant

arr_seq sched tsk priority_inversion_bound.

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 = priority_inversion_bound + total_hep_rbf L.

Hypothesis H_L_positive : L > 0.

Hypothesis H_fixed_point :

L = priority_inversion_bound + total_hep_rbf L.

To reduce the time complexity of the analysis, recall the notion
of search space. Intuitively, this corresponds to all
"interesting" arrival offsets that the job under analysis might
have with regard to the beginning of its busy-window.

Let R be a value that upper-bounds the solution of each
response-time recurrence, i.e., for any relative arrival time
A in the search space, there exists a corresponding solution
F such that R ≥ F + (task cost - task lock-in service).

Variable R : duration.

Hypothesis H_R_is_maximum :

∀ (A : duration),

is_in_search_space A →

∃ (F : duration),

A + F ≥ priority_inversion_bound

+ (task_rbf (A + ε) - (task_cost tsk - task_rtct tsk))

+ total_ohep_rbf (A + F) ∧

R ≥ F + (task_cost tsk - task_rtct tsk).

Hypothesis H_R_is_maximum :

∀ (A : duration),

is_in_search_space A →

∃ (F : duration),

A + F ≥ priority_inversion_bound

+ (task_rbf (A + ε) - (task_cost tsk - task_rtct tsk))

+ total_ohep_rbf (A + F) ∧

R ≥ F + (task_cost tsk - task_rtct tsk).

Finally, we define the interference bound function
(IBF_other). IBF_other bounds the interference if tasks are
sequential. Since tasks are sequential, we exclude interference
from other jobs of the same task. For FP, we define IBF_other
as the sum of the priority interference bound and the
higher-or-equal-priority workload.

## Filling Out Hypotheses Of Abstract RTA Theorem

In this section we prove that all preconditions necessary to use the abstract theorem are satisfied.
Recall that L is assumed to be a fixed point of the busy
interval recurrence. Thanks to this fact, we can prove that
every busy interval (according to the concrete definition) is
bounded. In addition, we know that the conventional concept of
busy interval and the one obtained from the abstract
definition (with the interference and interfering workload)
coincide. Thus, it follows that any busy interval (in the
abstract sense) is bounded.

Lemma instantiated_busy_intervals_are_bounded :

busy_intervals_are_bounded_by arr_seq sched tsk L.

Proof.

move ⇒ j ARR TSK POS.

edestruct (exists_busy_interval) with (delta := L) as [t1 [t2 [T1 [T2 BI]]]]; rt_eauto.

{ intros; rewrite {2}H_fixed_point leq_add //.

rewrite /workload_of_higher_or_equal_priority_jobs /total_hep_rbf

/total_hep_request_bound_function_FP

/workload_of_jobs /hep_job /FP_to_JLFP.

move: (TSK) ⇒ /eqP →.

by apply: sum_of_jobs_le_sum_rbf; eauto. }

∃ t1, t2; split; first by done. split; first by done.

eapply instantiated_busy_interval_equivalent_busy_interval; rt_eauto.

Qed.

busy_intervals_are_bounded_by arr_seq sched tsk L.

Proof.

move ⇒ j ARR TSK POS.

edestruct (exists_busy_interval) with (delta := L) as [t1 [t2 [T1 [T2 BI]]]]; rt_eauto.

{ intros; rewrite {2}H_fixed_point leq_add //.

rewrite /workload_of_higher_or_equal_priority_jobs /total_hep_rbf

/total_hep_request_bound_function_FP

/workload_of_jobs /hep_job /FP_to_JLFP.

move: (TSK) ⇒ /eqP →.

by apply: sum_of_jobs_le_sum_rbf; eauto. }

∃ t1, t2; split; first by done. split; first by done.

eapply instantiated_busy_interval_equivalent_busy_interval; rt_eauto.

Qed.

Next, we prove that IBF_other is indeed an interference
bound.
Recall that in module abstract_seq_RTA hypothesis
task_interference_is_bounded_by expects to receive a
function that maps some task tsk, the relative arrival time of
a job j of task tsk, and the length of the interval to the
maximum amount of interference.
However, in this module we analyze only one task -- tsk,
therefore it is “hard-coded” inside the interference bound
function IBF_other. Moreover, in case of a model with fixed
priorities, interference that some job j incurs from
higher-or-equal priority jobs does not depend on the relative
arrival time of job j. Therefore, in order for the
IBF_other signature to match the required signature in
module abstract_seq_RTA, we wrap the IBF_other function in
a function that accepts, but simply ignores, the task and the
relative arrival time.

Lemma instantiated_task_interference_is_bounded :

task_interference_is_bounded_by arr_seq sched tsk (fun t A R ⇒ IBF_other R).

Proof.

intros ? ? ? ? ARR TSK ? NCOMPL BUSY; simpl.

move: (posnP (@job_cost _ Cost j)) ⇒ [ZERO|POS].

{ by exfalso; rewrite /completed_by ZERO in NCOMPL. }

eapply instantiated_busy_interval_equivalent_busy_interval in BUSY; rt_eauto.

rewrite /ideal_jlfp_interference; erewrite cumulative_task_interference_split; rt_eauto; last first.

{ by move: BUSY ⇒ [[_ [_ [_ /andP [GE LT]]]] _]; eapply arrived_between_implies_in_arrivals; rt_eauto. }

rewrite /IBF_other leq_add; try done.

{ apply leq_trans with (cumulative_priority_inversion arr_seq sched j t1 (t1 + R0)); first by done.

apply leq_trans with (cumulative_priority_inversion arr_seq sched j t1 t2); last first.

{ by apply H_priority_inversion_is_bounded; rt_eauto; move: BUSY ⇒ [PREF QT2]. }

rewrite [X in _ ≤ X](@big_cat_nat _ _ _ (t1 + R0)) //=.

- by rewrite leq_addr.

- by rewrite leq_addr.

- by rewrite ltnW.

}

{ erewrite cumulative_i_thep_eq_service_of_othep; rt_eauto;

last by unfold quiet_time; move: BUSY ⇒ [[_ [T1 T2]] _].

apply: leq_trans.

{ apply service_of_jobs_le_workload; first apply ideal_proc_model_provides_unit_service.

by apply (valid_schedule_implies_completed_jobs_dont_execute sched arr_seq). }

{ rewrite /workload_of_jobs /total_ohep_rbf /total_ohep_request_bound_function_FP.

rewrite /another_task_hep_job /hep_job /FP_to_JLFP.

set (pred_task tsk_other := hep_task tsk_other tsk && (tsk_other != tsk)).

rewrite (eq_big (fun j⇒ pred_task (job_task j)) job_cost) //;

last by move⇒ j'; rewrite /pred_task; move: TSK ⇒ /eqP →.

erewrite (eq_big pred_task); [|by done|by move⇒ tsk'; eauto].

by apply: sum_of_jobs_le_sum_rbf; eauto. } }

Qed.

task_interference_is_bounded_by arr_seq sched tsk (fun t A R ⇒ IBF_other R).

Proof.

intros ? ? ? ? ARR TSK ? NCOMPL BUSY; simpl.

move: (posnP (@job_cost _ Cost j)) ⇒ [ZERO|POS].

{ by exfalso; rewrite /completed_by ZERO in NCOMPL. }

eapply instantiated_busy_interval_equivalent_busy_interval in BUSY; rt_eauto.

rewrite /ideal_jlfp_interference; erewrite cumulative_task_interference_split; rt_eauto; last first.

{ by move: BUSY ⇒ [[_ [_ [_ /andP [GE LT]]]] _]; eapply arrived_between_implies_in_arrivals; rt_eauto. }

rewrite /IBF_other leq_add; try done.

{ apply leq_trans with (cumulative_priority_inversion arr_seq sched j t1 (t1 + R0)); first by done.

apply leq_trans with (cumulative_priority_inversion arr_seq sched j t1 t2); last first.

{ by apply H_priority_inversion_is_bounded; rt_eauto; move: BUSY ⇒ [PREF QT2]. }

rewrite [X in _ ≤ X](@big_cat_nat _ _ _ (t1 + R0)) //=.

- by rewrite leq_addr.

- by rewrite leq_addr.

- by rewrite ltnW.

}

{ erewrite cumulative_i_thep_eq_service_of_othep; rt_eauto;

last by unfold quiet_time; move: BUSY ⇒ [[_ [T1 T2]] _].

apply: leq_trans.

{ apply service_of_jobs_le_workload; first apply ideal_proc_model_provides_unit_service.

by apply (valid_schedule_implies_completed_jobs_dont_execute sched arr_seq). }

{ rewrite /workload_of_jobs /total_ohep_rbf /total_ohep_request_bound_function_FP.

rewrite /another_task_hep_job /hep_job /FP_to_JLFP.

set (pred_task tsk_other := hep_task tsk_other tsk && (tsk_other != tsk)).

rewrite (eq_big (fun j⇒ pred_task (job_task j)) job_cost) //;

last by move⇒ j'; rewrite /pred_task; move: TSK ⇒ /eqP →.

erewrite (eq_big pred_task); [|by done|by move⇒ tsk'; eauto].

by apply: sum_of_jobs_le_sum_rbf; eauto. } }

Qed.

Finally, we show that there exists a solution for the
response-time recurrence.

Variable j : Job.

Hypothesis H_j_arrives : arrives_in arr_seq j.

Hypothesis H_job_of_tsk : job_of_task tsk j.

Hypothesis H_job_cost_positive: job_cost_positive j.

Hypothesis H_j_arrives : arrives_in arr_seq j.

Hypothesis H_job_of_tsk : job_of_task tsk j.

Hypothesis H_job_cost_positive: job_cost_positive j.

Given any job j of task tsk that arrives exactly A
units after the beginning of the busy interval, the bound of
the total interference incurred by j within an interval of
length Δ is equal to task_rbf (A + ε) - task_cost tsk +
IBF_other Δ.

Next, consider any A from the search space (in the
abstract sense).

Variable A : duration.

Hypothesis H_A_is_in_abstract_search_space :

search_space.is_in_search_space tsk L total_interference_bound A.

Hypothesis H_A_is_in_abstract_search_space :

search_space.is_in_search_space tsk L total_interference_bound A.

We prove that A is also in the concrete search space.

Lemma A_is_in_concrete_search_space :

is_in_search_space A.

Proof.

move: H_A_is_in_abstract_search_space ⇒ [INSP | [/andP [POSA LTL] [x [LTx INSP2]]]].

- rewrite INSP.

apply/andP; split; first by done.

rewrite neq_ltn; apply/orP; left.

rewrite {1}/task_rbf; erewrite task_rbf_0_zero; eauto 2; try done.

rewrite add0n /task_rbf; apply leq_trans with (task_cost tsk).

+ by apply leq_trans with (job_cost j); eauto 2; move: (H_job_of_tsk) ⇒ /eqP <-; eauto 2.

+ by eapply task_rbf_1_ge_task_cost; rt_eauto.

- apply/andP; split; first by done.

apply/negP; intros EQ; move: EQ ⇒ /eqP EQ.

apply INSP2.

unfold total_interference_bound in ×.

rewrite subn1 addn1 prednK; last by done.

by rewrite -EQ.

Qed.

is_in_search_space A.

Proof.

move: H_A_is_in_abstract_search_space ⇒ [INSP | [/andP [POSA LTL] [x [LTx INSP2]]]].

- rewrite INSP.

apply/andP; split; first by done.

rewrite neq_ltn; apply/orP; left.

rewrite {1}/task_rbf; erewrite task_rbf_0_zero; eauto 2; try done.

rewrite add0n /task_rbf; apply leq_trans with (task_cost tsk).

+ by apply leq_trans with (job_cost j); eauto 2; move: (H_job_of_tsk) ⇒ /eqP <-; eauto 2.

+ by eapply task_rbf_1_ge_task_cost; rt_eauto.

- apply/andP; split; first by done.

apply/negP; intros EQ; move: EQ ⇒ /eqP EQ.

apply INSP2.

unfold total_interference_bound in ×.

rewrite subn1 addn1 prednK; last by done.

by rewrite -EQ.

Qed.

Then, there exists a solution for the response-time
recurrence (in the abstract sense).

Corollary correct_search_space :

∃ (F : duration),

A + F ≥ task_rbf (A + ε) - (task_cost tsk - task_rtct tsk) + IBF_other (A + F) ∧

R ≥ F + (task_cost tsk - task_rtct tsk).

Proof.

move: (H_R_is_maximum A) ⇒ FIX.

feed FIX; first by apply A_is_in_concrete_search_space.

move: FIX ⇒ [F [FIX NEQ]].

∃ F; split; last by done.

rewrite -{2}(leqRW FIX).

by rewrite addnA [_ + priority_inversion_bound]addnC -!addnA.

Qed.

End SolutionOfResponseTimeRecurrenceExists.

End FillingOutHypothesesOfAbstractRTATheorem.

∃ (F : duration),

A + F ≥ task_rbf (A + ε) - (task_cost tsk - task_rtct tsk) + IBF_other (A + F) ∧

R ≥ F + (task_cost tsk - task_rtct tsk).

Proof.

move: (H_R_is_maximum A) ⇒ FIX.

feed FIX; first by apply A_is_in_concrete_search_space.

move: FIX ⇒ [F [FIX NEQ]].

∃ F; split; last by done.

rewrite -{2}(leqRW FIX).

by rewrite addnA [_ + priority_inversion_bound]addnC -!addnA.

Qed.

End SolutionOfResponseTimeRecurrenceExists.

End FillingOutHypothesesOfAbstractRTATheorem.

## Final Theorem

Based on the properties established above, we apply the abstract analysis framework to infer that R is a response-time bound for tsk.
Theorem uniprocessor_response_time_bound_fp :

response_time_bounded_by tsk R.

Proof.

intros js ARRs TSKs.

move: (posnP (@job_cost _ Cost js)) ⇒ [ZERO|POS].

{ by rewrite /job_response_time_bound /completed_by ZERO. }

eapply uniprocessor_response_time_bound_seq; rt_eauto.

- by eapply instantiated_i_and_w_are_coherent_with_schedule; rt_eauto.

- by eapply instantiated_interference_and_workload_consistent_with_sequential_tasks; rt_eauto.

- by apply instantiated_busy_intervals_are_bounded.

- by apply instantiated_task_interference_is_bounded.

- by eapply correct_search_space; eauto 2.

Qed.

End AbstractRTAforFPwithArrivalCurves.

response_time_bounded_by tsk R.

Proof.

intros js ARRs TSKs.

move: (posnP (@job_cost _ Cost js)) ⇒ [ZERO|POS].

{ by rewrite /job_response_time_bound /completed_by ZERO. }

eapply uniprocessor_response_time_bound_seq; rt_eauto.

- by eapply instantiated_i_and_w_are_coherent_with_schedule; rt_eauto.

- by eapply instantiated_interference_and_workload_consistent_with_sequential_tasks; rt_eauto.

- by apply instantiated_busy_intervals_are_bounded.

- by apply instantiated_task_interference_is_bounded.

- by eapply correct_search_space; eauto 2.

Qed.

End AbstractRTAforFPwithArrivalCurves.