Library rt.restructuring.analysis.basic_facts.service

From mathcomp Require Import ssrnat ssrbool fintype.
From rt.restructuring.behavior Require Export all.
From rt.restructuring.model.processor Require Export platform_properties.
From rt.util Require Import tactics step_function sum.

In this file, we establish basic facts about the service received by jobs.

Section Composition.

To begin with, we provide some simple but handy rewriting rules for service and service_during.

Consider any job type and any processor state.

  Context {Job: JobType}.
  Context {PState: Type}.
  Context `{ProcessorState Job PState}.

For any given schedule...

Variable sched: schedule PState.

...and any given job...

Variable j: Job.

...we establish a number of useful rewriting rules that decompose the service received during an interval into smaller intervals.

As a trivial base case, no job receives any service during an empty interval.

  Lemma service_during_geq:
    ∀ t1 t2,
      t1 ≥ t2 → service_during sched j t1 t2 = 0.

Equally trivially, no job has received service prior to time zero.

Corollary service0:
service sched j 0 = 0.

Trivially, an interval consiting of one time unit is equivalent to service_at.

  Lemma service_during_instant:
    ∀ t,
      service_during sched j t t .+1 = service_at sched j t.

Next, we observe that we can look at the service received during an interval t1, t3) as the sum of the service during [t1, t2) and [t2, t3) for any t2 \in [t1, t3]. (The "_cat" suffix denotes the concatenation of the two intervals.)

  Lemma service_during_cat:
    ∀ t1 t2 t3,
      t1 ≤ t2 ≤ t3 →
      (service_during sched j t1 t2 ) + (service_during sched j t2 t3 )
      = service_during sched j t1 t3.

Since service is just a special case of service_during, the same holds for service.

  Lemma service_cat:
    ∀ t1 t2,
      t1 ≤ t2 →
      (service sched j t1 ) + (service_during sched j t1 t2 )
      = service sched j t2.

As a special case, we observe that the service during an interval can be decomposed into the first instant and the rest of the interval.

  Lemma service_during_first_plus_later:
    ∀ t1 t2,
      t1 < t2 →
      (service_at sched j t1 ) + (service_during sched j t1 .+1 t2 )
      = service_during sched j t1 t2.

Symmetrically, we have the same for the end of the interval.

  Lemma service_during_last_plus_before:
    ∀ t1 t2,
      t1 ≤ t2 →
      (service_during sched j t1 t2 ) + (service_at sched j t2 )
      = service_during sched j t1 t2 .+1.

And hence also for service.

  Corollary service_last_plus_before:
    ∀ t,
      (service sched j t ) + (service_at sched j t )
      = service sched j t .+1.

Finally, we deconstruct the service received during an interval t1, t3) into the service at a midpoint t2 and the service in the intervals before and after.

  Lemma service_split_at_point:
    ∀ t1 t2 t3,
      t1 ≤ t2 < t3 →
      (service_during sched j t1 t2 ) + (service_at sched j t2 ) + (service_during sched j t2 .+1 t3 )
      = service_during sched j t1 t3.

End Composition.

Section UnitService.

As a common special case, we establish facts about schedules in which a job receives either 1 or 0 service units at all times.

Consider any job type and any processor state.

  Context {Job: JobType}.
  Context {PState: Type}.
  Context `{ProcessorState Job PState}.

Let's consider a unit-service model...

Hypothesis H_unit_service: unit_service_proc_model PState.

...and a given schedule.

Variable sched: schedule PState.

Let j be any job that is to be scheduled.

Variable j: Job.

First, we prove that the instantaneous service cannot be greater than 1, ...

Lemma service_at_most_one:
∀ t, service_at sched j t ≤ 1.

...which implies that the cumulative service received by job j in any interval of length delta is at most delta.

  Lemma cumulative_service_le_delta:
    ∀ t delta,
      service_during sched j t (t + delta) ≤ delta.

  Section ServiceIsAStepFunction.

We show that the service received by any job j is a step function.

Lemma service_is_a_step_function:
is_step_function (service sched j).

Next, consider any time t...

Variable t: instant.

...and let s0 be any value less than the service received by job j by time t.

Variable s0: duration.
Hypothesis H_less_than_s: s0 < service sched j t.

Then, we show that there exists an earlier time t0 where job j had s0 units of service.

    Corollary exists_intermediate_service:
      ∃ t0,
        t0 < t ∧
        service sched j t0 = s0.
  End ServiceIsAStepFunction.

End UnitService.

Section Monotonicity.

We establish a basic fact about the monotonicity of service.

Consider any job type and any processor model.

  Context {Job: JobType}.
  Context {PState: Type}.
  Context `{ProcessorState Job PState}.

Consider any given schedule...

Variable sched: schedule PState.

...and a given job that is to be scheduled.

Variable j: Job.

We observe that the amount of service received is monotonic by definition.

  Lemma service_monotonic:
    ∀ t1 t2,
      t1 ≤ t2 →
      service sched j t1 ≤ service sched j t2.

End Monotonicity.

Section RelationToScheduled.

Consider any job type and any processor model.

  Context {Job: JobType}.
  Context {PState: Type}.
  Context `{ProcessorState Job PState}.

Consider any given schedule...

Variable sched: schedule PState.

...and a given job that is to be scheduled.

Variable j: Job.

We observe that a job that isn't scheduled cannot possibly receive service.

  Lemma not_scheduled_implies_no_service:
    ∀ t,
      ~~ scheduled_at sched j t → service_at sched j t = 0.

Conversely, if a job receives service, then it must be scheduled.

  Lemma service_at_implies_scheduled_at:
    ∀ t,
      service_at sched j t > 0 → scheduled_at sched j t.

Thus, if the cumulative amount of service changes, then it must be scheduled, too.

  Lemma service_delta_implies_scheduled:
    ∀ t,
      service sched j t < service sched j t .+1 → scheduled_at sched j t.

We observe that a job receives cumulative service during some interval iff it receives services at some specific time in the interval.

  Lemma service_during_service_at:
    ∀ t1 t2,
      service_during sched j t1 t2 > 0
      ↔
      ∃ t,
        t1 ≤ t < t2 ∧
        service_at sched j t > 0.

Thus, any job that receives some service during an interval must be scheduled at some point during the interval...

  Corollary cumulative_service_implies_scheduled:
    ∀ t1 t2,
      service_during sched j t1 t2 > 0 →
      ∃ t,
        t1 ≤ t < t2 ∧
        scheduled_at sched j t.

...which implies that any job with positive cumulative service must have been scheduled at some point.

  Corollary positive_service_implies_scheduled_before:
    ∀ t,
      service sched j t > 0 → ∃ t', (t' < t ∧ scheduled_at sched j t').

  Section GuaranteedService.

If we can assume that a scheduled job always receives service, we can further prove the converse.

Assume j always receives some positive service.

Hypothesis H_scheduled_implies_serviced: ideal_progress_proc_model PState.

In other words, not being scheduled is equivalent to receiving zero service.

    Lemma no_service_not_scheduled:
      ∀ t,
        ~~ scheduled_at sched j t ↔ service_at sched j t = 0.

Then, if a job does not receive any service during an interval, it is not scheduled.

    Lemma no_service_during_implies_not_scheduled:
      ∀ t1 t2,
        service_during sched j t1 t2 = 0 →
        ∀ t,
          t1 ≤ t < t2 → ~~ scheduled_at sched j t.

If a job is scheduled at some point in an interval, it receivees positive cumulative service during the interval...

    Lemma scheduled_implies_cumulative_service:
      ∀ t1 t2,
        (∃ t,
            t1 ≤ t < t2 ∧
            scheduled_at sched j t ) →
        service_during sched j t1 t2 > 0.

...which again applies to total service, too.

    Corollary scheduled_implies_nonzero_service:
      ∀ t,
        (∃ t',
            t' < t ∧
            scheduled_at sched j t') →
        service sched j t > 0.

  End GuaranteedService.

  Section AfterArrival.

Futhermore, if we know that jobs are not released early, then we can narrow the interval during which they must have been scheduled.

Context `{JobArrival Job}.

Assume that jobs must arrive to execute.

Hypothesis H_jobs_must_arrive:
jobs_must_arrive_to_execute sched.

We prove that any job with positive cumulative service at time t must have been scheduled some time since its arrival and before time t.

    Lemma positive_service_implies_scheduled_since_arrival:
      ∀ t,
        service sched j t > 0 →
        ∃ t', (job_arrival j ≤ t' < t ∧ scheduled_at sched j t').

    Lemma not_scheduled_before_arrival:
      ∀ t, t < job_arrival j → ~~ scheduled_at sched j t.

We show that job j does not receive service at any time t prior to its arrival.

    Lemma service_before_job_arrival_zero:
      ∀ t,
        t < job_arrival j →
        service_at sched j t = 0.

Note that the same property applies to the cumulative service.

    Lemma cumulative_service_before_job_arrival_zero :
      ∀ t1 t2 : instant,
        t2 ≤ job_arrival j →
        service_during sched j t1 t2 = 0.

Hence, one can ignore the service received by a job before its arrival time...

    Lemma ignore_service_before_arrival:
      ∀ t1 t2,
        t1 ≤ job_arrival j →
        t2 ≥ job_arrival j →
        service_during sched j t1 t2 = service_during sched j (job_arrival j) t2.

... which we can also state in terms of cumulative service.

    Corollary no_service_before_arrival:
      ∀ t,
        t ≤ job_arrival j → service sched j t = 0.

  End AfterArrival.

  Section TimesWithSameService.

In this section, we prove some lemmas about time instants with same service.

Consider any time instants t1 and t2...

Variable t1 t2: instant.

...where t1 is no later than t2...

Hypothesis H_t1_le_t2: t1 ≤ t2.

...and where job j has received the same amount of service.

Hypothesis H_same_service: service sched j t1 = service sched j t2.

First, we observe that this means that the job receives no service during t1, t2)...

Lemma constant_service_implies_no_service_during:
service_during sched j t1 t2 = 0.

...which of course implies that it does not receive service at any point, either.

    Lemma constant_service_implies_not_scheduled:
      ∀ t,
        t1 ≤ t < t2 → service_at sched j t = 0.

We show that job j receives service at some point t < t1 iff j receives service at some point t' < t2.

    Lemma same_service_implies_serviced_at_earlier_times:
      [∃ t: 'I_t1 , service_at sched j t > 0] =
      [∃ t': 'I_t2 , service_at sched j t' > 0].

Then, under the assumption that scheduled jobs receives service, we can translate this into a claim about scheduled_at.

Assume j always receives some positive service.

Hypothesis H_scheduled_implies_serviced: ideal_progress_proc_model PState.

We show that job j is scheduled at some point t < t1 iff j is scheduled at some point t' < t2.

    Lemma same_service_implies_scheduled_at_earlier_times:
      [∃ t: 'I_t1 , scheduled_at sched j t ] =
      [∃ t': 'I_t2 , scheduled_at sched j t'].
  End TimesWithSameService.

End RelationToScheduled.