# Library prosa.analysis.facts.model.scheduled

Require Export prosa.model.schedule.scheduled.
Require Export prosa.analysis.definitions.service.
Require Export prosa.model.processor.platform_properties.
Require Export prosa.analysis.facts.behavior.arrivals.
Require Export prosa.util.tactics.

# Correctness of the Scheduled Job(s)

In this file, we establish the correctness of the computable definition of the (set of) scheduled job(s) and a useful case-analysis lemma.
Section ScheduledJobs.
Consider any type of jobs with arrival times.
Context {Job : JobType}.
Context `{JobArrival Job}.
Next, consider any kind of processor state model, ...
Context {PState : ProcessorState Job}.
... any valid arrival sequence, ...
Variable arr_seq : arrival_sequence Job.
Hypothesis H_valid_arrivals : valid_arrival_sequence arr_seq.
... and any schedule.
Variable sched : schedule PState.
Now we establish the validity conditions under which these definitions yield the expected results: all jobs must come from the arrival sequence and do not execute before their arrival.
NB: We do not use valid_schedule here to avoid introducing a dependency on a readiness mode.
Hypothesis H_jobs_come_from_arrival_sequence :
jobs_come_from_arrival_sequence sched arr_seq.
Hypothesis H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched.

## The Set of Jobs Scheduled at a Given Time

Under these assumptions, scheduled_jobs_at is correct: a job is included in the sequence if and only if it scheduled.
Lemma scheduled_jobs_at_iff :
∀ j t,
j \in scheduled_jobs_at arr_seq sched t = scheduled_at sched j t.
Conversely, if no jobs are in the sequence, then no job is scheduled.
Lemma scheduled_jobs_at_nil :
∀ t,
nilp (scheduled_jobs_at arr_seq sched t)
↔ (∀ j, ~~ scheduled_at sched j t).
We restate the previous claim for convenience.
Corollary not_scheduled_when_idle :
∀ j t,
is_idle arr_seq sched t → ~~ scheduled_at sched j t.

## The Job Scheduled on an Ideal Progress Processor

In this section, we prove a simple fact about the relation between scheduled_at and served_at.
Section IdealProgress.
Assume a scheduled job always receives some positive service.
Hypothesis H_ideal_progress_model : ideal_progress_proc_model PState.
We prove that if a job j is scheduled at time t, then j is in the set of jobs that are served at time t.
Lemma scheduled_at_implies_in_served_at :
∀ j t,
scheduled_at sched j t →
j \in served_jobs_at arr_seq sched t.
End IdealProgress.

## The Job Scheduled on a Uniprocessor

For convenience, we note similar rewriting rules for uniprocessors.
Section Uniprocessors.
Suppose we are looking at a uniprocessor.
Hypothesis H_uni : uniprocessor_model PState.
Then clearly there is at most one job scheduled at any time t.
Corollary scheduled_jobs_at_seq1 :
∀ t,
size (scheduled_jobs_at arr_seq sched t) ≤ 1.
For convenience, we restate the fact that there is at most one job as a case analysis.
Corollary scheduled_jobs_at_uni_cases :
∀ t,
scheduled_jobs_at arr_seq sched t == [::]
∨ ∃ j, scheduled_jobs_at arr_seq sched t == [:: j].
We note the obvious relationship between scheduled_jobs_at and scheduled_job_at ...
Lemma scheduled_jobs_at_uni :
∀ j t,
(scheduled_jobs_at arr_seq sched t == [::j])
= (scheduled_job_at arr_seq sched t == Some j).
... and observe that scheduled_job_at t behaves as expected: it yields some job j if and only if j is scheduled at time t.
Corollary scheduled_job_at_scheduled_at :
∀ j t,
(scheduled_job_at arr_seq sched t == Some j) = scheduled_at sched j t.
Similarly to scheduled_job_at_scheduled_at, we relate scheduled_jobs_at to scheduled_at.
Corollary scheduled_jobs_at_scheduled_at :
∀ j t,
(scheduled_jobs_at arr_seq sched t == [:: j])
= scheduled_at sched j t.
Conversely, if scheduled_job_at t returns None, then no job is scheduled at time t.
Corollary scheduled_job_at_none :
∀ t,
scheduled_job_at arr_seq sched t = None
↔ (∀ j, ~~ scheduled_at sched j t).
For convenience, we state a similar observation also for the is_idle wrapper, both for the idle case ...
Corollary is_idle_iff :
∀ t,
is_idle arr_seq sched t = (scheduled_job_at arr_seq sched t == None).
... and the non-idle case.
Corollary is_nonidle_iff :
∀ t,
~~ is_idle arr_seq sched t ↔ ∃ j, scheduled_at sched j t.
End Uniprocessors.

## Case Analysis: a Scheduled Job Exists or no Job is Scheduled

Last but not least, we note a case analysis that results from the above considerations: at any point in time, either some job is scheduled, or it holds that all jobs are not scheduled.
Lemma scheduled_at_dec :
∀ t,
{∃ j, scheduled_at sched j t} + {∀ j, ~~ scheduled_at sched j t}.
For ease of porting, we restate the above case analysis in a form closer to what was used in earlier versions of Prosa.
Corollary scheduled_at_cases :
∀ t,
is_idle arr_seq sched t ∨ ∃ j, scheduled_at sched j t.
End ScheduledJobs.