Library prosa.classic.model.arrival.basic.arrival_bounds
Require Import prosa.classic.util.all.
Require Import prosa.classic.model.arrival.basic.arrival_sequence prosa.classic.model.arrival.basic.task prosa.classic.model.arrival.basic.job prosa.classic.model.arrival.basic.task_arrival
prosa.classic.model.priority.
From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq path div.
Module ArrivalBounds.
Import ArrivalSequence SporadicTaskset TaskArrival Priority.
Section Lemmas.
Context {Task: eqType}.
Variable task_period: Task → time.
Context {Job: eqType}.
Variable job_arrival: Job → time.
Variable job_cost: Job → time.
Variable job_task: Job → Task.
Variable arr_seq: arrival_sequence Job.
Hypothesis H_arrival_times_are_consistent: arrival_times_are_consistent job_arrival arr_seq.
Hypothesis H_arrival_sequence_is_a_set: arrival_sequence_is_a_set arr_seq.
Section BoundOnSporadicArrivals.
Hypothesis H_sporadic_tasks: sporadic_task_model task_period job_arrival job_task arr_seq.
Variable t1 t2: time.
Variable tsk: Task.
Hypothesis H_period_gt_zero: task_period tsk > 0.
Let arriving_jobs := arrivals_of_task_between job_task arr_seq tsk t1 t2.
Let num_arrivals := num_arrivals_of_task job_task arr_seq tsk t1 t2.
Section NoJobs.
Hypothesis H_no_jobs: num_arrivals = 0.
Lemma sporadic_arrival_bound_no_jobs:
num_arrivals ≤ div_ceil (t2 - t1) (task_period tsk).
End NoJobs.
Section OneJob.
Lemma sporadic_arrival_bound_more_than_one_point:
num_arrivals > 0 →
t1 < t2.
Hypothesis H_no_jobs: num_arrivals = 1.
Lemma sporadic_arrival_bound_one_job:
num_arrivals ≤ div_ceil (t2 - t1) (task_period tsk).
End OneJob.
Section AtLeastTwoJobs.
Hypothesis H_at_least_two_jobs: num_arrivals ≥ 2.
Section DerivingContradiction.
Hypothesis H_many_arrivals: div_ceil (t2 - t1) (task_period tsk) < num_arrivals.
Let by_arrival_time j j' := job_arrival j ≤ job_arrival j'.
Let sorted_jobs := sort by_arrival_time arriving_jobs.
Variable elem: Job.
Let nth_job := nth elem sorted_jobs.
Let j_first := nth_job 0.
Let j_last := nth_job (num_arrivals.-1).
Let a_first := job_arrival j_first.
Let a_last := job_arrival j_last.
Corollary sporadic_arrival_bound_properties_of_nth:
∀ idx,
idx < num_arrivals →
t1 ≤ job_arrival (nth_job idx) < t2 ∧
job_task (nth_job idx) = tsk ∧
arrives_in arr_seq (nth_job idx).
Corollary sporadic_arrival_bound_distance_between_first_and_last:
a_last ≥ a_first + (num_arrivals-1) × task_period tsk.
Lemma sporadic_arrival_bound_last_job_too_far:
a_first + t2 - t1 ≤ a_last.
Lemma sporadic_arrival_bound_last_arrives_too_late:
a_last ≥ t2.
Lemma sporadic_arrival_bound_case_3_contradiction: False.
End DerivingContradiction.
Lemma sporadic_task_arrival_bound_at_least_two_jobs:
num_arrivals ≤ div_ceil (t2 - t1) (task_period tsk).
End AtLeastTwoJobs.
Theorem sporadic_task_arrival_bound:
num_arrivals ≤ div_ceil (t2 - t1) (task_period tsk).
End BoundOnSporadicArrivals.
End Lemmas.
End ArrivalBounds.
Require Import prosa.classic.model.arrival.basic.arrival_sequence prosa.classic.model.arrival.basic.task prosa.classic.model.arrival.basic.job prosa.classic.model.arrival.basic.task_arrival
prosa.classic.model.priority.
From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq path div.
Module ArrivalBounds.
Import ArrivalSequence SporadicTaskset TaskArrival Priority.
Section Lemmas.
Context {Task: eqType}.
Variable task_period: Task → time.
Context {Job: eqType}.
Variable job_arrival: Job → time.
Variable job_cost: Job → time.
Variable job_task: Job → Task.
Variable arr_seq: arrival_sequence Job.
Hypothesis H_arrival_times_are_consistent: arrival_times_are_consistent job_arrival arr_seq.
Hypothesis H_arrival_sequence_is_a_set: arrival_sequence_is_a_set arr_seq.
Section BoundOnSporadicArrivals.
Hypothesis H_sporadic_tasks: sporadic_task_model task_period job_arrival job_task arr_seq.
Variable t1 t2: time.
Variable tsk: Task.
Hypothesis H_period_gt_zero: task_period tsk > 0.
Let arriving_jobs := arrivals_of_task_between job_task arr_seq tsk t1 t2.
Let num_arrivals := num_arrivals_of_task job_task arr_seq tsk t1 t2.
Section NoJobs.
Hypothesis H_no_jobs: num_arrivals = 0.
Lemma sporadic_arrival_bound_no_jobs:
num_arrivals ≤ div_ceil (t2 - t1) (task_period tsk).
End NoJobs.
Section OneJob.
Lemma sporadic_arrival_bound_more_than_one_point:
num_arrivals > 0 →
t1 < t2.
Hypothesis H_no_jobs: num_arrivals = 1.
Lemma sporadic_arrival_bound_one_job:
num_arrivals ≤ div_ceil (t2 - t1) (task_period tsk).
End OneJob.
Section AtLeastTwoJobs.
Hypothesis H_at_least_two_jobs: num_arrivals ≥ 2.
Section DerivingContradiction.
Hypothesis H_many_arrivals: div_ceil (t2 - t1) (task_period tsk) < num_arrivals.
Let by_arrival_time j j' := job_arrival j ≤ job_arrival j'.
Let sorted_jobs := sort by_arrival_time arriving_jobs.
Variable elem: Job.
Let nth_job := nth elem sorted_jobs.
Let j_first := nth_job 0.
Let j_last := nth_job (num_arrivals.-1).
Let a_first := job_arrival j_first.
Let a_last := job_arrival j_last.
Corollary sporadic_arrival_bound_properties_of_nth:
∀ idx,
idx < num_arrivals →
t1 ≤ job_arrival (nth_job idx) < t2 ∧
job_task (nth_job idx) = tsk ∧
arrives_in arr_seq (nth_job idx).
Corollary sporadic_arrival_bound_distance_between_first_and_last:
a_last ≥ a_first + (num_arrivals-1) × task_period tsk.
Lemma sporadic_arrival_bound_last_job_too_far:
a_first + t2 - t1 ≤ a_last.
Lemma sporadic_arrival_bound_last_arrives_too_late:
a_last ≥ t2.
Lemma sporadic_arrival_bound_case_3_contradiction: False.
End DerivingContradiction.
Lemma sporadic_task_arrival_bound_at_least_two_jobs:
num_arrivals ≤ div_ceil (t2 - t1) (task_period tsk).
End AtLeastTwoJobs.
Theorem sporadic_task_arrival_bound:
num_arrivals ≤ div_ceil (t2 - t1) (task_period tsk).
End BoundOnSporadicArrivals.
End Lemmas.
End ArrivalBounds.