# Library prosa.analysis.facts.busy_interval.ideal.inequalities

In this file, we prove some inequalities that always
hold inside the busy interval of a job. Throughout this file we
assume the ideal uniprocessor model.

Consider any kind of tasks and their jobs, each characterized by
an arrival time, a cost, and an arbitrary notion of readiness.

Context {Task : TaskType} {Job : JobType} `{JobTask Job Task}.

Context `{JobArrival Job} `{JobCost Job} {JR :JobReady Job (ideal.processor_state Job)}.

Context `{JobArrival Job} `{JobCost Job} {JR :JobReady Job (ideal.processor_state Job)}.

Consider a JLFP policy that is reflexive and respects sequential tasks.

Context {JLFP : JLFP_policy Job}.

Hypothesis H_policy_is_reflexive : reflexive_job_priorities JLFP.

Hypothesis H_policy_respecsts_sequential_tasks : policy_respects_sequential_tasks JLFP.

Hypothesis H_policy_is_reflexive : reflexive_job_priorities JLFP.

Hypothesis H_policy_respecsts_sequential_tasks : policy_respects_sequential_tasks JLFP.

Consider a consistent arrival sequence that does not contain duplicates.

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.

Consider any valid ideal uniprocessor schedule defined over this arrival sequence.

Variable sched : schedule (ideal.processor_state Job).

Hypothesis H_valid_schedule : valid_schedule sched arr_seq.

Hypothesis H_valid_schedule : valid_schedule sched arr_seq.

Consider a set of tasks ts that contains all the jobs in the
arrival sequence.

Consider a task tsk.

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.

Consider the ideal JLFP definitions of interference and
interfering workload.

#[local] Instance ideal_jlfp_interference : Interference Job :=

ideal_jlfp_interference arr_seq sched.

#[local] Instance ideal_jlfp_interfering_workload : InterferingWorkload Job :=

ideal_jlfp_interfering_workload arr_seq sched.

ideal_jlfp_interference arr_seq sched.

#[local] Instance ideal_jlfp_interfering_workload : InterferingWorkload Job :=

ideal_jlfp_interfering_workload arr_seq sched.

Consider the busy interval for j is given by

`[t1,t2)`

.
Let us denote the relative arrival time by A.

Consider any arbitrary time Δ inside the busy interval.

First, we prove that if the priority inversion is bounded then,
the cumulative priority inversion is also bounded.

Consider the priority inversion in any given interval
is bounded by a constant.

Variable priority_inversion_bound : duration → duration.

Hypothesis H_priority_inversion_is_bounded :

priority_inversion_is_bounded_by arr_seq sched tsk priority_inversion_bound.

Hypothesis H_priority_inversion_is_bounded :

priority_inversion_is_bounded_by arr_seq sched tsk priority_inversion_bound.

Then, the cumulative priority inversion in any interval
is also bounded.

Lemma cumulative_priority_inversion_is_bounded:

cumulative_priority_inversion arr_seq sched j t1 (t1 + Δ)

≤ priority_inversion_bound (job_arrival j - t1).

Proof.

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

- rewrite [X in _ ≤ X](@big_cat_nat _ _ _ (t1 + Δ)) //=; try by lia.

by rewrite /priority_inversion leq_addr.

- apply: H_priority_inversion_is_bounded ⇒ //.

by move: H_busy_interval; rewrite -instantiated_busy_interval_equivalent_busy_interval // ⇒ -[PREF _].

Qed.

End PIBound.

cumulative_priority_inversion arr_seq sched j t1 (t1 + Δ)

≤ priority_inversion_bound (job_arrival j - t1).

Proof.

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

- rewrite [X in _ ≤ X](@big_cat_nat _ _ _ (t1 + Δ)) //=; try by lia.

by rewrite /priority_inversion leq_addr.

- apply: H_priority_inversion_is_bounded ⇒ //.

by move: H_busy_interval; rewrite -instantiated_busy_interval_equivalent_busy_interval // ⇒ -[PREF _].

Qed.

End PIBound.

Next, we prove that the cumulative interference from higher priority
jobs of other tasks in an interval is bounded by the total service
received by the higher priority jobs of those tasks.

Lemma cumulative_interference_is_bounded_by_total_service:

cumulative_another_task_hep_job_interference arr_seq sched j t1 (t1 + Δ)

≤ service_of_jobs sched (fun jo ⇒ another_task_hep_job jo j) jobs t1 (t1 + Δ).

Proof.

move: (H_busy_interval) ⇒ [[/andP [JINBI JINBI2] [QT _]] _].

rewrite cumulative_i_thep_eq_service_of_othep ⇒ //.

have /eqP TSK := H_job_of_tsk.

by rewrite instantiated_quiet_time_equivalent_quiet_time.

Qed.

End BusyIntervalInequalities.

cumulative_another_task_hep_job_interference arr_seq sched j t1 (t1 + Δ)

≤ service_of_jobs sched (fun jo ⇒ another_task_hep_job jo j) jobs t1 (t1 + Δ).

Proof.

move: (H_busy_interval) ⇒ [[/andP [JINBI JINBI2] [QT _]] _].

rewrite cumulative_i_thep_eq_service_of_othep ⇒ //.

have /eqP TSK := H_job_of_tsk.

by rewrite instantiated_quiet_time_equivalent_quiet_time.

Qed.

End BusyIntervalInequalities.