Library prosa.classic.model.schedule.global.schedulability
Require Import prosa.classic.util.all.
Require Import prosa.classic.model.arrival.basic.job prosa.classic.model.arrival.basic.task.
Require Import prosa.classic.model.schedule.global.basic.schedule.
From mathcomp Require Import ssreflect eqtype ssrbool ssrnat seq bigop.
(* Definitions of deadline miss. *)
Module Schedulability.
Import Schedule SporadicTaskset Job.
Section SchedulableDefs.
Context {sporadic_task: eqType}.
Context {Job: eqType}.
Variable job_arrival: Job → time.
Variable job_cost: Job → time.
Variable job_deadline: Job → time.
Variable job_task: Job → sporadic_task.
(* Consider any job arrival sequence... *)
Variable arr_seq: arrival_sequence Job.
(* ...and any multiprocessor schedule of these jobs. *)
Context {num_cpus: nat}.
Variable sched: schedule Job num_cpus.
Section ScheduleOfJobs.
(* Let j be any job. *)
Variable j: Job.
(* We say that job j misses no deadline in sched if it completed by its absolute deadline. *)
Definition job_misses_no_deadline :=
completed job_cost sched j (job_arrival j + job_deadline j).
End ScheduleOfJobs.
Section ScheduleOfTasks.
(* Consider any task tsk. *)
Variable tsk: sporadic_task.
(* Task tsk doesn't miss its deadline iff all of its jobs don't miss their deadline. *)
Definition task_misses_no_deadline :=
∀ j,
arrives_in arr_seq j →
job_task j = tsk →
job_misses_no_deadline j.
(* Task tsk doesn't miss its deadline before time t' iff all of its jobs don't miss
their deadline by that time. *)
Definition task_misses_no_deadline_before (t': time) :=
∀ j,
arrives_in arr_seq j →
job_task j = tsk →
job_arrival j + job_deadline j < t' →
job_misses_no_deadline j.
End ScheduleOfTasks.
End SchedulableDefs.
Section BasicLemmas.
Context {sporadic_task: eqType}.
Variable task_cost: sporadic_task → time.
Variable task_period: sporadic_task → time.
Variable task_deadline: sporadic_task → time.
Context {Job: eqType}.
Variable job_arrival: Job → time.
Variable job_cost: Job → time.
Variable job_deadline: Job → time.
Variable job_task: Job → sporadic_task.
(* Consider any job arrival sequence... *)
Variable arr_seq: arrival_sequence Job.
(* ...and any schedule of these jobs... *)
Context {num_cpus : nat}.
Variable sched: schedule Job num_cpus.
(* ... where jobs dont execute after completion. *)
Hypothesis H_completed_jobs_dont_execute:
completed_jobs_dont_execute job_cost sched.
Section SpecificJob.
(* Then, for any job j ...*)
Variable j: Job.
Hypothesis H_j_arrives: arrives_in arr_seq j.
(* ...that doesn't miss a deadline in this schedule, ... *)
Hypothesis no_deadline_miss:
job_misses_no_deadline job_arrival job_cost job_deadline sched j.
(* the service received by j at any time t' after its deadline is 0. *)
Lemma service_after_job_deadline_zero :
∀ t',
t' ≥ job_arrival j + job_deadline j →
service_at sched j t' = 0.
Proof.
intros t' LE.
rename no_deadline_miss into RT,
H_completed_jobs_dont_execute into EXEC.
unfold job_misses_no_deadline, completed, completed_jobs_dont_execute in ×.
apply/eqP; rewrite -leqn0.
eapply completion_monotonic in RT; eauto 2.
apply completed_implies_not_scheduled in RT; eauto 2.
by move: RT; rewrite not_scheduled_no_service; move ⇒ /eqP RT; rewrite RT.
Qed.
(* The same applies for the cumulative service of job j. *)
Lemma cumulative_service_after_job_deadline_zero :
∀ t' t'',
t' ≥ job_arrival j + job_deadline j →
\sum_(t' ≤ t < t'') service_at sched j t = 0.
Proof.
ins; apply/eqP; rewrite -leqn0.
rewrite big_nat_cond; rewrite → eq_bigr with (F2 := fun i ⇒ 0);
first by rewrite big_const_seq iter_addn mul0n addn0 leqnn.
intro i; rewrite andbT; move ⇒ /andP [LE _].
by rewrite service_after_job_deadline_zero;
[by ins | by apply leq_trans with (n := t')].
Qed.
End SpecificJob.
Section AllJobs.
(* Consider any task tsk ...*)
Variable tsk: sporadic_task.
(* ... that doesn't miss any deadline. *)
Hypothesis no_deadline_misses:
task_misses_no_deadline job_arrival job_cost job_deadline job_task arr_seq sched tsk.
(* Then, for any valid job j of this task, ...*)
Variable j: Job.
Hypothesis H_j_arrives: arrives_in arr_seq j.
Hypothesis H_job_of_task: job_task j = tsk.
Hypothesis H_valid_job:
valid_sporadic_job task_cost task_deadline job_cost job_deadline job_task j.
(* the service received by job j at any time t' after the deadline is 0. *)
Lemma service_after_task_deadline_zero :
∀ t',
t' ≥ job_arrival j + task_deadline tsk →
service_at sched j t' = 0.
Proof.
rename H_valid_job into PARAMS; unfold valid_sporadic_job in *; des; intros t'.
rewrite -H_job_of_task -PARAMS1.
by apply service_after_job_deadline_zero, no_deadline_misses.
Qed.
(* The same applies for the cumulative service of job j. *)
Lemma cumulative_service_after_task_deadline_zero :
∀ t' t'',
t' ≥ job_arrival j + task_deadline tsk →
\sum_(t' ≤ t < t'') service_at sched j t = 0.
Proof.
rename H_valid_job into PARAMS; unfold valid_sporadic_job in *; des; intros t' t''.
rewrite -H_job_of_task -PARAMS1.
by apply cumulative_service_after_job_deadline_zero, no_deadline_misses.
Qed.
End AllJobs.
End BasicLemmas.
End Schedulability.
Require Import prosa.classic.model.arrival.basic.job prosa.classic.model.arrival.basic.task.
Require Import prosa.classic.model.schedule.global.basic.schedule.
From mathcomp Require Import ssreflect eqtype ssrbool ssrnat seq bigop.
(* Definitions of deadline miss. *)
Module Schedulability.
Import Schedule SporadicTaskset Job.
Section SchedulableDefs.
Context {sporadic_task: eqType}.
Context {Job: eqType}.
Variable job_arrival: Job → time.
Variable job_cost: Job → time.
Variable job_deadline: Job → time.
Variable job_task: Job → sporadic_task.
(* Consider any job arrival sequence... *)
Variable arr_seq: arrival_sequence Job.
(* ...and any multiprocessor schedule of these jobs. *)
Context {num_cpus: nat}.
Variable sched: schedule Job num_cpus.
Section ScheduleOfJobs.
(* Let j be any job. *)
Variable j: Job.
(* We say that job j misses no deadline in sched if it completed by its absolute deadline. *)
Definition job_misses_no_deadline :=
completed job_cost sched j (job_arrival j + job_deadline j).
End ScheduleOfJobs.
Section ScheduleOfTasks.
(* Consider any task tsk. *)
Variable tsk: sporadic_task.
(* Task tsk doesn't miss its deadline iff all of its jobs don't miss their deadline. *)
Definition task_misses_no_deadline :=
∀ j,
arrives_in arr_seq j →
job_task j = tsk →
job_misses_no_deadline j.
(* Task tsk doesn't miss its deadline before time t' iff all of its jobs don't miss
their deadline by that time. *)
Definition task_misses_no_deadline_before (t': time) :=
∀ j,
arrives_in arr_seq j →
job_task j = tsk →
job_arrival j + job_deadline j < t' →
job_misses_no_deadline j.
End ScheduleOfTasks.
End SchedulableDefs.
Section BasicLemmas.
Context {sporadic_task: eqType}.
Variable task_cost: sporadic_task → time.
Variable task_period: sporadic_task → time.
Variable task_deadline: sporadic_task → time.
Context {Job: eqType}.
Variable job_arrival: Job → time.
Variable job_cost: Job → time.
Variable job_deadline: Job → time.
Variable job_task: Job → sporadic_task.
(* Consider any job arrival sequence... *)
Variable arr_seq: arrival_sequence Job.
(* ...and any schedule of these jobs... *)
Context {num_cpus : nat}.
Variable sched: schedule Job num_cpus.
(* ... where jobs dont execute after completion. *)
Hypothesis H_completed_jobs_dont_execute:
completed_jobs_dont_execute job_cost sched.
Section SpecificJob.
(* Then, for any job j ...*)
Variable j: Job.
Hypothesis H_j_arrives: arrives_in arr_seq j.
(* ...that doesn't miss a deadline in this schedule, ... *)
Hypothesis no_deadline_miss:
job_misses_no_deadline job_arrival job_cost job_deadline sched j.
(* the service received by j at any time t' after its deadline is 0. *)
Lemma service_after_job_deadline_zero :
∀ t',
t' ≥ job_arrival j + job_deadline j →
service_at sched j t' = 0.
Proof.
intros t' LE.
rename no_deadline_miss into RT,
H_completed_jobs_dont_execute into EXEC.
unfold job_misses_no_deadline, completed, completed_jobs_dont_execute in ×.
apply/eqP; rewrite -leqn0.
eapply completion_monotonic in RT; eauto 2.
apply completed_implies_not_scheduled in RT; eauto 2.
by move: RT; rewrite not_scheduled_no_service; move ⇒ /eqP RT; rewrite RT.
Qed.
(* The same applies for the cumulative service of job j. *)
Lemma cumulative_service_after_job_deadline_zero :
∀ t' t'',
t' ≥ job_arrival j + job_deadline j →
\sum_(t' ≤ t < t'') service_at sched j t = 0.
Proof.
ins; apply/eqP; rewrite -leqn0.
rewrite big_nat_cond; rewrite → eq_bigr with (F2 := fun i ⇒ 0);
first by rewrite big_const_seq iter_addn mul0n addn0 leqnn.
intro i; rewrite andbT; move ⇒ /andP [LE _].
by rewrite service_after_job_deadline_zero;
[by ins | by apply leq_trans with (n := t')].
Qed.
End SpecificJob.
Section AllJobs.
(* Consider any task tsk ...*)
Variable tsk: sporadic_task.
(* ... that doesn't miss any deadline. *)
Hypothesis no_deadline_misses:
task_misses_no_deadline job_arrival job_cost job_deadline job_task arr_seq sched tsk.
(* Then, for any valid job j of this task, ...*)
Variable j: Job.
Hypothesis H_j_arrives: arrives_in arr_seq j.
Hypothesis H_job_of_task: job_task j = tsk.
Hypothesis H_valid_job:
valid_sporadic_job task_cost task_deadline job_cost job_deadline job_task j.
(* the service received by job j at any time t' after the deadline is 0. *)
Lemma service_after_task_deadline_zero :
∀ t',
t' ≥ job_arrival j + task_deadline tsk →
service_at sched j t' = 0.
Proof.
rename H_valid_job into PARAMS; unfold valid_sporadic_job in *; des; intros t'.
rewrite -H_job_of_task -PARAMS1.
by apply service_after_job_deadline_zero, no_deadline_misses.
Qed.
(* The same applies for the cumulative service of job j. *)
Lemma cumulative_service_after_task_deadline_zero :
∀ t' t'',
t' ≥ job_arrival j + task_deadline tsk →
\sum_(t' ≤ t < t'') service_at sched j t = 0.
Proof.
rename H_valid_job into PARAMS; unfold valid_sporadic_job in *; des; intros t' t''.
rewrite -H_job_of_task -PARAMS1.
by apply cumulative_service_after_job_deadline_zero, no_deadline_misses.
Qed.
End AllJobs.
End BasicLemmas.
End Schedulability.