Library rt.restructuring.analysis.basic_facts.ideal_schedule


(* ----------------------------------[ coqtop ]---------------------------------

Welcome to Coq 8.10.1 (October 2019)

----------------------------------------------------------------------------- *)

From mathcomp Require Import all_ssreflect.
From rt.restructuring.model.processor Require Import ideal platform_properties.
Note: we do not re-export the basic definitions to avoid littering the global namespace with type class instances.
In this section we estlish the classes to which an ideal schedule belongs.
Section ScheduleClass.

  Local Transparent service_in 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 uniprocessor model.
  Lemma ideal_proc_model_is_a_uniprocessor_model:
    uniprocessor_model (processor_state Job).

(* ----------------------------------[ coqtop ]---------------------------------

1 subgoal (ID 267)
  
  Job : JobType
  H : JobArrival Job
  H0 : JobCost Job
  ============================
  uniprocessor_model (processor_state Job)

----------------------------------------------------------------------------- *)

  Proof.
    move⇒ j1 j2 sched t /existsP[[]/eqP E1] /existsP[[]/eqP E2].

(* ----------------------------------[ coqtop ]---------------------------------

1 subgoal (ID 427)
  
  Job : JobType
  H : JobArrival Job
  H0 : JobCost Job
  j1, j2 : Job
  sched : schedule (processor_state Job)
  t : instant
  E1 : sched t = Some j1
  E2 : sched t = Some j2
  ============================
  j1 = j2

----------------------------------------------------------------------------- *)

    by move: E1 E2=>->[].

(* ----------------------------------[ coqtop ]---------------------------------

No more subgoals.

----------------------------------------------------------------------------- *)

  Qed.

We observe that the ideal processor model falls into the category of ideal-progress models, i.e., a scheduled job always receives service.
  Lemma ideal_proc_model_ensures_ideal_progress:
    ideal_progress_proc_model (processor_state Job).

(* ----------------------------------[ coqtop ]---------------------------------

1 subgoal (ID 270)
  
  Job : JobType
  H : JobArrival Job
  H0 : JobCost Job
  ============================
  ideal_progress_proc_model (processor_state Job)

----------------------------------------------------------------------------- *)

  Proof.
    move⇒ j s /existsP[[]/eqP->]/=.

(* ----------------------------------[ coqtop ]---------------------------------

1 subgoal (ID 355)
  
  Job : JobType
  H : JobArrival Job
  H0 : JobCost Job
  j : Job
  s : processor_state Job
  ============================
  0 < (Some j == Some j)

----------------------------------------------------------------------------- *)

    by rewrite eqxx.

(* ----------------------------------[ coqtop ]---------------------------------

No more subgoals.

----------------------------------------------------------------------------- *)

  Qed.

The ideal processor model is a unit-service model.
  Lemma ideal_proc_model_provides_unit_service:
    unit_service_proc_model (processor_state Job).

(* ----------------------------------[ coqtop ]---------------------------------

1 subgoal (ID 273)
  
  Job : JobType
  H : JobArrival Job
  H0 : JobCost Job
  ============================
  unit_service_proc_model (processor_state Job)

----------------------------------------------------------------------------- *)

  Proof.
    move⇒ j s.

(* ----------------------------------[ coqtop ]---------------------------------

1 subgoal (ID 276)
  
  Job : JobType
  H : JobArrival Job
  H0 : JobCost Job
  j : Job
  s : processor_state Job
  ============================
  service_in j s <= 1

----------------------------------------------------------------------------- *)

    rewrite /service_in/=/nat_of_bool.

(* ----------------------------------[ coqtop ]---------------------------------

1 subgoal (ID 285)
  
  Job : JobType
  H : JobArrival Job
  H0 : JobCost Job
  j : Job
  s : processor_state Job
  ============================
  (if s == Some j then 1 else 0) <= 1

----------------------------------------------------------------------------- *)

    by case:ifP.

(* ----------------------------------[ coqtop ]---------------------------------

No more subgoals.

----------------------------------------------------------------------------- *)

  Qed.

  Lemma scheduled_in_def (j : Job) s :
    scheduled_in j s = (s == Some j).

(* ----------------------------------[ coqtop ]---------------------------------

1 subgoal (ID 287)
  
  Job : JobType
  H : JobArrival Job
  H0 : JobCost Job
  j : Job
  s : option_eqType Job
  ============================
  scheduled_in j s = (s == Some j)

----------------------------------------------------------------------------- *)

  Proof.
    rewrite /scheduled_in/scheduled_on/=.

(* ----------------------------------[ coqtop ]---------------------------------

1 subgoal (ID 302)
  
  Job : JobType
  H : JobArrival Job
  H0 : JobCost Job
  j : Job
  s : option_eqType Job
  ============================
  [exists c, s == Some j] = (s == Some j)

----------------------------------------------------------------------------- *)

    case: existsP=>[[_->]//|].

(* ----------------------------------[ coqtop ]---------------------------------

1 subgoal (ID 325)
  
  Job : JobType
  H : JobArrival Job
  H0 : JobCost Job
  j : Job
  s : option_eqType Job
  ============================
  ~ (exists _ : unit_finType, s == Some j) -> false = (s == Some j)

----------------------------------------------------------------------------- *)

    case: (s == Some j)=>//=[[]].

(* ----------------------------------[ coqtop ]---------------------------------

1 subgoal (ID 385)
  
  Job : JobType
  H : JobArrival Job
  H0 : JobCost Job
  j : Job
  s : option_eqType Job
  ============================
  exists _ : unit, true

----------------------------------------------------------------------------- *)

      by ∃.

(* ----------------------------------[ coqtop ]---------------------------------

No more subgoals.

----------------------------------------------------------------------------- *)

  Qed.

  Lemma scheduled_at_def sched (j : Job) t :
    scheduled_at sched j t = (sched t == Some j).

(* ----------------------------------[ coqtop ]---------------------------------

1 subgoal (ID 303)
  
  Job : JobType
  H : JobArrival Job
  H0 : JobCost Job
  sched : schedule (option_eqType Job)
  j : Job
  t : instant
  ============================
  scheduled_at sched j t = (sched t == Some j)

----------------------------------------------------------------------------- *)

  Proof.
      by rewrite /scheduled_at scheduled_in_def.

(* ----------------------------------[ coqtop ]---------------------------------

No more subgoals.

----------------------------------------------------------------------------- *)

  Qed.

  Lemma service_in_is_scheduled_in (j : Job) s :
    service_in j s = scheduled_in j s.

(* ----------------------------------[ coqtop ]---------------------------------

1 subgoal (ID 317)
  
  Job : JobType
  H : JobArrival Job
  H0 : JobCost Job
  j : Job
  s : option Job
  ============================
  service_in j s = scheduled_in j s

----------------------------------------------------------------------------- *)

  Proof.
    by rewrite /service_in scheduled_in_def/=.

(* ----------------------------------[ coqtop ]---------------------------------

No more subgoals.

----------------------------------------------------------------------------- *)

  Qed.

  Lemma service_at_is_scheduled_at sched (j : Job) t :
    service_at sched j t = scheduled_at sched j t.

(* ----------------------------------[ coqtop ]---------------------------------

1 subgoal (ID 333)
  
  Job : JobType
  H : JobArrival Job
  H0 : JobCost Job
  sched : schedule (option Job)
  j : Job
  t : instant
  ============================
  service_at sched j t = scheduled_at sched j t

----------------------------------------------------------------------------- *)

  Proof.
      by rewrite /service_at service_in_is_scheduled_in.

(* ----------------------------------[ coqtop ]---------------------------------

No more subgoals.

----------------------------------------------------------------------------- *)

  Qed.

Next we prove a lemma which helps us to do a case analysis on the state of an ideal schedule.
  Lemma ideal_proc_model_sched_case_analysis:
    ∀ (sched : schedule (ideal.processor_state Job)) (t : instant),
      is_idle sched t ∨ ∃ j, scheduled_at sched j t.
(* ----------------------------------[ coqtop ]---------------------------------

1 subgoal (ID 341)
  
  Job : JobType
  H : JobArrival Job
  H0 : JobCost Job
  ============================
  forall (sched : schedule (processor_state Job)) (t : instant),
  is_idle sched t \/ (exists j : Job, scheduled_at sched j t)

----------------------------------------------------------------------------- *)

  Proof.
    intros.

(* ----------------------------------[ coqtop ]---------------------------------

1 subgoal (ID 343)
  
  Job : JobType
  H : JobArrival Job
  H0 : JobCost Job
  sched : schedule (processor_state Job)
  t : instant
  ============================
  is_idle sched t \/ (exists j : Job, scheduled_at sched j t)

----------------------------------------------------------------------------- *)

    unfold is_idle; simpl; destruct (sched t) eqn:EQ.

(* ----------------------------------[ coqtop ]---------------------------------

2 subgoals (ID 360)
  
  Job : JobType
  H : JobArrival Job
  H0 : JobCost Job
  sched : schedule (processor_state Job)
  t : instant
  s : Job
  EQ : sched t = Some s
  ============================
  Some s == None \/ (exists j : Job, scheduled_at sched j t)

subgoal 2 (ID 361) is:
 None == None \/ (exists j : Job, scheduled_at sched j t)

----------------------------------------------------------------------------- *)

    - by right; ∃ s; auto; rewrite scheduled_at_def EQ.

(* ----------------------------------[ coqtop ]---------------------------------

1 subgoal (ID 361)
  
  Job : JobType
  H : JobArrival Job
  H0 : JobCost Job
  sched : schedule (processor_state Job)
  t : instant
  EQ : sched t = None
  ============================
  None == None \/ (exists j : Job, scheduled_at sched j t)

----------------------------------------------------------------------------- *)

    - by left; auto.

(* ----------------------------------[ coqtop ]---------------------------------

No more subgoals.

----------------------------------------------------------------------------- *)

  Qed.

End ScheduleClass.

Automation

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

We also provide tactics for case analysis on ideal processor state.
The first tactic generates two subgoals: 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].