Library prosa.classic.model.schedule.global.basic.schedule
Require Import prosa.classic.util.all
prosa.classic.model.arrival.basic.job prosa.classic.model.arrival.basic.task prosa.classic.model.arrival.basic.arrival_sequence.
From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq fintype bigop.
Module Schedule.
Export ArrivalSequence.
Definition processor (num_cpus: nat) := 'I_num_cpus.
Section ScheduleDef.
Variable Job: eqType.
Variable num_cpus: nat.
Definition schedule :=
processor num_cpus → time → option Job.
End ScheduleDef.
Section ScheduledJobs.
Context {Job: eqType}.
Variable job_arrival: Job → time.
Context {arr_seq: arrival_sequence Job}.
Variable job_cost: Job → time.
Context {num_cpus: nat}.
Variable sched: schedule Job num_cpus.
Variable j: Job.
Definition scheduled_on (cpu: processor num_cpus) (t: time) :=
sched cpu t == Some j.
Definition scheduled (t: time) :=
[∃ cpu, scheduled_on cpu t].
Definition is_idle (cpu: processor num_cpus) (t: time) :=
sched cpu t = None.
Definition service_at (t: time) :=
\sum_(cpu < num_cpus | scheduled_on cpu t) 1.
Definition service (t': time) := \sum_(0 ≤ t < t') service_at t.
Definition service_during (t1 t2: time) := \sum_(t1 ≤ t < t2) service_at t.
Definition completed (t: time) := service t ≥ job_cost j.
Definition pending (t: time) := has_arrived job_arrival j t && ~~completed t.
Definition backlogged (t: time) := pending t && ~~scheduled t.
Definition carried_in (t1: time) := arrived_before job_arrival j t1 && ~~ completed t1.
Definition carried_out (t1 t2: time) := arrived_before job_arrival j t2 && ~~ completed t2.
Definition jobs_scheduled_at (t: time) :=
\cat_(cpu < num_cpus) make_sequence (sched cpu t).
Definition jobs_scheduled_between (t1 t2: time) :=
undup (\cat_(t1 ≤ t < t2) jobs_scheduled_at t).
End ScheduledJobs.
Section ValidSchedules.
Context {Job: eqType}.
Variable job_arrival: Job → time.
Variable job_cost: Job → time.
Context {num_cpus: nat}.
Variable sched: schedule Job num_cpus.
Definition sequential_jobs :=
∀ j t cpu1 cpu2,
sched cpu1 t = Some j → sched cpu2 t = Some j → cpu1 = cpu2.
Definition jobs_must_arrive_to_execute :=
∀ j t,
scheduled sched j t →
has_arrived job_arrival j t.
Definition completed_jobs_dont_execute :=
∀ j t, service sched j t ≤ job_cost j.
Definition jobs_come_from_arrival_sequence (arr_seq: arrival_sequence Job) :=
∀ j t, scheduled sched j t → arrives_in arr_seq j.
End ValidSchedules.
Section JobLemmas.
Context {Job: eqType}.
Variable job_arrival: Job → time.
Variable job_cost: Job → time.
Context {num_cpus: nat}.
Variable sched: schedule Job num_cpus.
Variable j: Job.
Section Basic.
Lemma not_scheduled_no_service :
∀ t,
~~ scheduled sched j t = (service_at sched j t == 0).
Lemma cumulative_service_implies_service :
∀ t1 t2,
service_during sched j t1 t2 != 0 →
∃ t,
t1 ≤ t < t2 ∧
service_at sched j t != 0.
Lemma service_implies_cumulative_service:
∀ t t1 t2,
t1 ≤ t < t2 →
service_at sched j t != 0 →
service_during sched j t1 t2 != 0.
End Basic.
Section SequentialJobs.
Hypothesis H_sequential_jobs: sequential_jobs sched.
Lemma service_at_most_one :
∀ t, service_at sched j t ≤ 1.
Lemma cumulative_service_le_delta :
∀ t delta, service_during sched j t (t + delta) ≤ delta.
End SequentialJobs.
Section Completion.
Hypothesis H_completed_jobs:
completed_jobs_dont_execute job_cost sched.
Lemma completion_monotonic :
∀ t t',
t ≤ t' →
completed job_cost sched j t →
completed job_cost sched j t'.
Lemma completed_implies_not_scheduled :
∀ t,
completed job_cost sched j t →
~~ scheduled sched j t.
Lemma cumulative_service_le_job_cost :
∀ t t',
service_during sched j t t' ≤ job_cost j.
End Completion.
Section Arrival.
Hypothesis H_jobs_must_arrive:
jobs_must_arrive_to_execute job_arrival sched.
Lemma service_before_job_arrival_zero :
∀ t,
t < job_arrival j →
service_at sched j t = 0.
Lemma cumulative_service_before_job_arrival_zero :
∀ t1 t2,
t2 ≤ job_arrival j →
\sum_(t1 ≤ i < t2) service_at sched j i = 0.
Lemma service_before_arrival_eq_service_during :
∀ t0 t,
t0 ≤ job_arrival j →
\sum_(t0 ≤ t < job_arrival j + t) service_at sched j t =
\sum_(job_arrival j ≤ t < job_arrival j + t) service_at sched j t.
End Arrival.
Section Pending.
Hypothesis H_jobs_must_arrive:
jobs_must_arrive_to_execute job_arrival sched.
Hypothesis H_completed_jobs:
completed_jobs_dont_execute job_cost sched.
Lemma scheduled_implies_pending:
∀ t,
scheduled sched j t →
pending job_arrival job_cost sched j t.
End Pending.
End JobLemmas.
Section ScheduledJobsLemmas.
Context {Job: eqType}.
Context {num_cpus: nat}.
Variable sched: schedule Job num_cpus.
Section Membership.
Lemma mem_scheduled_jobs_eq_scheduled :
∀ j t,
j \in jobs_scheduled_at sched t = scheduled sched j t.
End Membership.
Section Uniqueness.
Hypothesis H_sequential_jobs : sequential_jobs sched.
Lemma scheduled_jobs_uniq :
∀ t,
uniq (jobs_scheduled_at sched t).
End Uniqueness.
Section NumberOfJobs.
Lemma num_scheduled_jobs_le_num_cpus :
∀ t,
size (jobs_scheduled_at sched t) ≤ num_cpus.
End NumberOfJobs.
End ScheduledJobsLemmas.
End Schedule.
Module ScheduleOfSporadicTask.
Import SporadicTask Job.
Export Schedule.
Section ScheduledJobs.
Context {sporadic_task: eqType}.
Context {Job: eqType}.
Variable job_task: Job → sporadic_task.
Context {num_cpus: nat}.
Variable sched: schedule Job num_cpus.
Variable tsk: sporadic_task.
Definition task_scheduled_on (cpu: processor num_cpus) (t: time) :=
if (sched cpu t) is Some j then
(job_task j == tsk)
else false.
Definition task_is_scheduled (t: time) :=
[∃ cpu, task_scheduled_on cpu t].
Definition jobs_of_task_scheduled_between (t1 t2: time) :=
filter (fun j ⇒ job_task j == tsk)
(jobs_scheduled_between sched t1 t2).
End ScheduledJobs.
Section ScheduleProperties.
Context {sporadic_task: eqType}.
Context {Job: eqType}.
Variable job_cost: Job → time.
Variable job_task: Job → sporadic_task.
Context {num_cpus: nat}.
Variable sched: schedule Job num_cpus.
Definition jobs_of_same_task_dont_execute_in_parallel :=
∀ j j' t,
job_task j = job_task j' →
scheduled sched j t →
scheduled sched j' t →
j = j'.
End ScheduleProperties.
Section BasicLemmas.
Context {sporadic_task: eqType}.
Variable task_cost: sporadic_task → time.
Variable task_deadline: sporadic_task → time.
Context {Job: eqType}.
Variable job_cost: Job → time.
Variable job_deadline: Job → time.
Variable job_task: Job → sporadic_task.
Context {num_cpus: nat}.
Variable sched: schedule Job num_cpus.
Hypothesis jobs_dont_execute_after_completion :
completed_jobs_dont_execute job_cost sched.
Variable tsk: sporadic_task.
Variable j: Job.
Hypothesis H_job_of_task: job_task j = tsk.
Hypothesis valid_job:
valid_sporadic_job task_cost task_deadline job_cost job_deadline job_task j.
Lemma cumulative_service_le_task_cost :
∀ t t',
service_during sched j t t' ≤ task_cost tsk.
End BasicLemmas.
End ScheduleOfSporadicTask.
prosa.classic.model.arrival.basic.job prosa.classic.model.arrival.basic.task prosa.classic.model.arrival.basic.arrival_sequence.
From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq fintype bigop.
Module Schedule.
Export ArrivalSequence.
Definition processor (num_cpus: nat) := 'I_num_cpus.
Section ScheduleDef.
Variable Job: eqType.
Variable num_cpus: nat.
Definition schedule :=
processor num_cpus → time → option Job.
End ScheduleDef.
Section ScheduledJobs.
Context {Job: eqType}.
Variable job_arrival: Job → time.
Context {arr_seq: arrival_sequence Job}.
Variable job_cost: Job → time.
Context {num_cpus: nat}.
Variable sched: schedule Job num_cpus.
Variable j: Job.
Definition scheduled_on (cpu: processor num_cpus) (t: time) :=
sched cpu t == Some j.
Definition scheduled (t: time) :=
[∃ cpu, scheduled_on cpu t].
Definition is_idle (cpu: processor num_cpus) (t: time) :=
sched cpu t = None.
Definition service_at (t: time) :=
\sum_(cpu < num_cpus | scheduled_on cpu t) 1.
Definition service (t': time) := \sum_(0 ≤ t < t') service_at t.
Definition service_during (t1 t2: time) := \sum_(t1 ≤ t < t2) service_at t.
Definition completed (t: time) := service t ≥ job_cost j.
Definition pending (t: time) := has_arrived job_arrival j t && ~~completed t.
Definition backlogged (t: time) := pending t && ~~scheduled t.
Definition carried_in (t1: time) := arrived_before job_arrival j t1 && ~~ completed t1.
Definition carried_out (t1 t2: time) := arrived_before job_arrival j t2 && ~~ completed t2.
Definition jobs_scheduled_at (t: time) :=
\cat_(cpu < num_cpus) make_sequence (sched cpu t).
Definition jobs_scheduled_between (t1 t2: time) :=
undup (\cat_(t1 ≤ t < t2) jobs_scheduled_at t).
End ScheduledJobs.
Section ValidSchedules.
Context {Job: eqType}.
Variable job_arrival: Job → time.
Variable job_cost: Job → time.
Context {num_cpus: nat}.
Variable sched: schedule Job num_cpus.
Definition sequential_jobs :=
∀ j t cpu1 cpu2,
sched cpu1 t = Some j → sched cpu2 t = Some j → cpu1 = cpu2.
Definition jobs_must_arrive_to_execute :=
∀ j t,
scheduled sched j t →
has_arrived job_arrival j t.
Definition completed_jobs_dont_execute :=
∀ j t, service sched j t ≤ job_cost j.
Definition jobs_come_from_arrival_sequence (arr_seq: arrival_sequence Job) :=
∀ j t, scheduled sched j t → arrives_in arr_seq j.
End ValidSchedules.
Section JobLemmas.
Context {Job: eqType}.
Variable job_arrival: Job → time.
Variable job_cost: Job → time.
Context {num_cpus: nat}.
Variable sched: schedule Job num_cpus.
Variable j: Job.
Section Basic.
Lemma not_scheduled_no_service :
∀ t,
~~ scheduled sched j t = (service_at sched j t == 0).
Lemma cumulative_service_implies_service :
∀ t1 t2,
service_during sched j t1 t2 != 0 →
∃ t,
t1 ≤ t < t2 ∧
service_at sched j t != 0.
Lemma service_implies_cumulative_service:
∀ t t1 t2,
t1 ≤ t < t2 →
service_at sched j t != 0 →
service_during sched j t1 t2 != 0.
End Basic.
Section SequentialJobs.
Hypothesis H_sequential_jobs: sequential_jobs sched.
Lemma service_at_most_one :
∀ t, service_at sched j t ≤ 1.
Lemma cumulative_service_le_delta :
∀ t delta, service_during sched j t (t + delta) ≤ delta.
End SequentialJobs.
Section Completion.
Hypothesis H_completed_jobs:
completed_jobs_dont_execute job_cost sched.
Lemma completion_monotonic :
∀ t t',
t ≤ t' →
completed job_cost sched j t →
completed job_cost sched j t'.
Lemma completed_implies_not_scheduled :
∀ t,
completed job_cost sched j t →
~~ scheduled sched j t.
Lemma cumulative_service_le_job_cost :
∀ t t',
service_during sched j t t' ≤ job_cost j.
End Completion.
Section Arrival.
Hypothesis H_jobs_must_arrive:
jobs_must_arrive_to_execute job_arrival sched.
Lemma service_before_job_arrival_zero :
∀ t,
t < job_arrival j →
service_at sched j t = 0.
Lemma cumulative_service_before_job_arrival_zero :
∀ t1 t2,
t2 ≤ job_arrival j →
\sum_(t1 ≤ i < t2) service_at sched j i = 0.
Lemma service_before_arrival_eq_service_during :
∀ t0 t,
t0 ≤ job_arrival j →
\sum_(t0 ≤ t < job_arrival j + t) service_at sched j t =
\sum_(job_arrival j ≤ t < job_arrival j + t) service_at sched j t.
End Arrival.
Section Pending.
Hypothesis H_jobs_must_arrive:
jobs_must_arrive_to_execute job_arrival sched.
Hypothesis H_completed_jobs:
completed_jobs_dont_execute job_cost sched.
Lemma scheduled_implies_pending:
∀ t,
scheduled sched j t →
pending job_arrival job_cost sched j t.
End Pending.
End JobLemmas.
Section ScheduledJobsLemmas.
Context {Job: eqType}.
Context {num_cpus: nat}.
Variable sched: schedule Job num_cpus.
Section Membership.
Lemma mem_scheduled_jobs_eq_scheduled :
∀ j t,
j \in jobs_scheduled_at sched t = scheduled sched j t.
End Membership.
Section Uniqueness.
Hypothesis H_sequential_jobs : sequential_jobs sched.
Lemma scheduled_jobs_uniq :
∀ t,
uniq (jobs_scheduled_at sched t).
End Uniqueness.
Section NumberOfJobs.
Lemma num_scheduled_jobs_le_num_cpus :
∀ t,
size (jobs_scheduled_at sched t) ≤ num_cpus.
End NumberOfJobs.
End ScheduledJobsLemmas.
End Schedule.
Module ScheduleOfSporadicTask.
Import SporadicTask Job.
Export Schedule.
Section ScheduledJobs.
Context {sporadic_task: eqType}.
Context {Job: eqType}.
Variable job_task: Job → sporadic_task.
Context {num_cpus: nat}.
Variable sched: schedule Job num_cpus.
Variable tsk: sporadic_task.
Definition task_scheduled_on (cpu: processor num_cpus) (t: time) :=
if (sched cpu t) is Some j then
(job_task j == tsk)
else false.
Definition task_is_scheduled (t: time) :=
[∃ cpu, task_scheduled_on cpu t].
Definition jobs_of_task_scheduled_between (t1 t2: time) :=
filter (fun j ⇒ job_task j == tsk)
(jobs_scheduled_between sched t1 t2).
End ScheduledJobs.
Section ScheduleProperties.
Context {sporadic_task: eqType}.
Context {Job: eqType}.
Variable job_cost: Job → time.
Variable job_task: Job → sporadic_task.
Context {num_cpus: nat}.
Variable sched: schedule Job num_cpus.
Definition jobs_of_same_task_dont_execute_in_parallel :=
∀ j j' t,
job_task j = job_task j' →
scheduled sched j t →
scheduled sched j' t →
j = j'.
End ScheduleProperties.
Section BasicLemmas.
Context {sporadic_task: eqType}.
Variable task_cost: sporadic_task → time.
Variable task_deadline: sporadic_task → time.
Context {Job: eqType}.
Variable job_cost: Job → time.
Variable job_deadline: Job → time.
Variable job_task: Job → sporadic_task.
Context {num_cpus: nat}.
Variable sched: schedule Job num_cpus.
Hypothesis jobs_dont_execute_after_completion :
completed_jobs_dont_execute job_cost sched.
Variable tsk: sporadic_task.
Variable j: Job.
Hypothesis H_job_of_task: job_task j = tsk.
Hypothesis valid_job:
valid_sporadic_job task_cost task_deadline job_cost job_deadline job_task j.
Lemma cumulative_service_le_task_cost :
∀ t t',
service_during sched j t t' ≤ task_cost tsk.
End BasicLemmas.
End ScheduleOfSporadicTask.