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# Library prosa.analysis.facts.blocking_bound.elf

Require Export prosa.analysis.definitions.blocking_bound.elf.
Require Export prosa.analysis.facts.busy_interval.pi.
Require Export prosa.model.task.absolute_deadline.
Require Export prosa.analysis.facts.model.arrival_curves.
Require Import prosa.analysis.facts.priority.classes.
Require Export prosa.analysis.facts.priority.elf.

# Lower-Priority Non-Preemptive Segment is Bounded

In this file, we prove that, under the ELF scheduling policy, the length of the maximum non-preemptive segment of a lower-priority job (w.r.t. a job of a task tsk) is bounded by blocking_bound.
Section MaxNPSegmentIsBounded.
Consider any type of tasks, each characterized by a WCET task_cost, an arrival curve max_arrivals, a relative deadline task_deadline, a bound on the task's longest non-preemptive segment task_max_nonpreemptive_segment and relative priority points, ...
Context {Task : TaskType}.
Context `{TaskCost Task}.
Context `{MaxArrivals Task}.
Context `{TaskMaxNonpreemptiveSegment Task}.
Context `{PriorityPoint Task}.
... and any type of jobs associated with these tasks, where each job has a task job_task, a cost job_cost, and an arrival time job_arrival.
Context {Job : JobType}.
Context `{JobTask Job Task}.
Context `{JobCost Job}.
Context `{JobArrival Job}.
Consider any kind of processor state model.
Context `{PState : ProcessorState Job}.
Consider any valid arrival sequence ...
Variable arr_seq : arrival_sequence Job.
Hypothesis H_valid_arrival_sequence : valid_arrival_sequence arr_seq.
... and any schedule of this arrival sequence.
Variable sched : schedule PState.
We further require that a job's cost cannot exceed its task's stated WCET ...
Hypothesis H_valid_job_cost : arrivals_have_valid_job_costs arr_seq.
... and assume that jobs have bounded non-preemptive segments.
Context `{JobPreemptable Job}.
Hypothesis H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched.
Consider an arbitrary task set ts, ...
Variable ts : seq Task.
... assume that all jobs come from the task set, ...
Hypothesis H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts.
... and let tsk be any task in ts that is to be analyzed.
Variable tsk : Task.
Hypothesis H_tsk_in_ts : tsk \in ts.
Let max_arrivals be a family of arrival curves.
Hypothesis H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts.
Consider any job j of tsk.
Variable j : Job.
Hypothesis H_job_of_tsk : job_of_task tsk j.
Consider an FP policy that indicates a higher-or-equal priority relation, and assume that the relation is reflexive, transitive and total.
Context (FP : FP_policy Task).
Hypothesis H_reflexive_priorities : reflexive_task_priorities FP.
Hypothesis H_transitive_priorities : transitive_task_priorities FP.
Hypothesis H_total_priorities : total_task_priorities FP.
Assume ELF scheduling.
Context {JLFP : JLFP_policy Job}.
Hypothesis H_policy_is_ELF : policy_is_ELF FP JLFP.
Then, the maximum length of a nonpreemptive segment among all lower-priority jobs (w.r.t. the given job j) arrived so far is bounded by blocking_bound.
Lemma nonpreemptive_segments_bounded_by_blocking :
∀ t1 t2,
busy_interval_prefix arr_seq sched j t1 t2 →
max_lp_nonpreemptive_segment arr_seq j t1 ≤ blocking_bound ts tsk (job_arrival j - t1).
Proof.
move⇒ t1 t2 BUSY; rewrite /max_lp_nonpreemptive_segment /blocking_bound //.
apply: leq_trans; first by apply: max_np_job_segment_bounded_by_max_np_task_segment.
apply /bigmax_leq_seqP ⇒ j' JINB NOTHEP.
have ARR': arrives_in arr_seq j'
by apply: in_arrivals_implies_arrived; exact: JINB.
apply leq_bigmax_cond_seq with (x := (job_task j')) (F := fun tsk ⇒ task_max_nonpreemptive_segment tsk - 1);
first by apply H_all_jobs_from_taskset.
apply in_arrivals_implies_arrived_between in JINB ⇒ //.
move: JINB ⇒ /andP [_ TJ'].
have ->: (max_arrivals (job_task j') ε > 0) && (task_cost (job_task j') > 0).
{ apply /andP; split.
{ apply: non_pathological_max_arrivals; last exact: ARR'; try done.
by rewrite /job_of_task. }
{ move: NOTHEP ⇒ /andP [_ NZ].
move: (H_valid_job_cost j' ARR'); rewrite /valid_job_cost.
by lia. } }
{ rewrite ?andbT.
move: NOTHEP ⇒ /andP [NOTHEP _].
move: H_job_of_tsk; rewrite /job_of_task ⇒ /eqP <-.
case: (boolP (hep_task (job_task j') (job_task j))) ⇒ HEP'
; rewrite /hp_task HEP' //=
; last by have ->: hep_task (job_task j) (job_task j')
by apply: ELF_policy_not_hep_task_priority_order.
rewrite andbF orFb /ep_task; repeat (apply/andP; split) ⇒ //.
have EP : ep_task (job_task j') (job_task j)
by rewrite /ep_task; apply/andP; split.
have: (job_priority_point j ≤ job_priority_point j')%R
by apply: ELF_policy_not_hep_priority_point_order.
rewrite /job_priority_point/jpp_from_tpp ⇒ PP.
have LT: ((job_arrival j)%:R + task_priority_point (job_task j)
< (t1)%:R + task_priority_point (job_task j'))%R by lia.
by rewrite natrB
; [|move: BUSY; rewrite /busy_interval_prefix]
; lia. }
Qed.
End MaxNPSegmentIsBounded.