Library prosa.analysis.facts.model.ideal.schedule

Note: we do not re-export the basic definitions to avoid littering the global namespace with type class instances.
In this section we establish the classes to which an ideal schedule belongs.
Section ScheduleClass.

We assume ideal uni-processor schedules.
  #[local] Existing Instance ideal.processor_state.

  Local Transparent scheduled_in scheduled_on.
Consider any job type and the ideal processor model.
  Context {Job: JobType}.
  Context `{JobArrival Job}.
  Context `{JobCost Job}.

We note that the ideal processor model is indeed a uni-processor model.
By definition, service_in is the sum of the service received in total across all cores. In the ideal uniprocessor model, however, there is only one "core," which is expressed by using unit as the type of cores. The type unit only has a single member tt, which serves as a placeholder. Consequently, the definition of service_in simplifies to a single term of the sum, the service on the one core, which we note with the following lemma that relates service_in to service_on.
  Lemma service_in_service_on (j : Job) s :
    service_in j s = service_on j s tt.

Furthermore, since the ideal uniprocessor state is represented by the option Job type, service_in further simplifies to a simple equality comparison, which we note next.
  Lemma service_in_def (j : Job) (s : processor_state Job) :
    service_in j s = (s == Some j).

We observe that the ideal processor model falls into the category of ideal-progress models, i.e., a scheduled job always receives service.
The ideal processor model is a unit-service model.
Next we prove a lemma which helps us to do a case analysis on the state of an ideal schedule.
We prove that if a job j is scheduled at a time instant t, then the scheduler is not idle at t.
On a similar note, if a scheduler is idle at a time instant t, then no job can receive service at t.

Incremental Service in Ideal Schedule

In the following section we prove a few facts about service in ideal schedule.
(* Note that these lemmas can be generalized to an arbitrary scheduler. *)
Section IncrementalService.

Consider any job type, ...
  Context {Job : JobType}.
  Context `{JobArrival Job}.
  Context `{JobCost Job}.

... any arrival sequence, ...
  Variable arr_seq : arrival_sequence Job.

... and any ideal uni-processor schedule of this arrival sequence.
As a base case, we prove that if a job j receives service in some time interval [t1,t2), then there exists a time instant t ∈ [t1,t2) such that j is scheduled at time t and t is the first instant where j receives service.
  Lemma positive_service_during:
     j t1 t2,
      0 < service_during sched j t1 t2
       t : nat, t1 t < t2 scheduled_at sched j t service_during sched j t1 t = 0.

Furthermore, we observe that, if a job receives some positive amount of service during an interval [t1, t2), then the interval can't be empty and hence t1 < t2.
  Lemma service_during_ge :
     j t1 t2 k,
      service_during sched j t1 t2 > k
      t1 < t2.

Next, we prove that if in some time interval [t1,t2) a job j receives k units of service, then there exists a time instant t ∈ [t1,t2) such that j is scheduled at time t and service of job j within interval [t1,t) is equal to k.


We add the above lemmas into a "Hint Database" basic_rt_facts, so Coq will be able to apply them automatically.
Global Hint Resolve ideal_proc_model_is_a_uniprocessor_model
     ideal_proc_model_provides_unit_service : basic_rt_facts.

We also provide tactics for case analysis on ideal processor state.
The first tactic generates two sub-goals: one with idle processor and the other with processor executing a job named JobName.
Ltac ideal_proc_model_sched_case_analysis sched t JobName :=
  let Idle := fresh "Idle" in
  let Sched := fresh "Sched_" JobName in
  destruct (ideal_proc_model_sched_case_analysis sched t) as [Idle | [JobName Sched]].

The second tactic is similar to the first, but it additionally generates two equalities: sched t = None and sched t = Some j.
Ltac ideal_proc_model_sched_case_analysis_eq sched t JobName :=
  let Idle := fresh "Idle" in
  let IdleEq := fresh "Eq" Idle in
  let Sched := fresh "Sched_" JobName in
  let SchedEq := fresh "Eq" Sched in
  destruct (ideal_proc_model_sched_case_analysis sched t) as [Idle | [JobName Sched]];
  [move: (Idle) ⇒ /eqP IdleEq; rewrite ?IdleEq
  | move: (Sched); simpl; move ⇒ /eqP SchedEq; rewrite ?SchedEq].