Library prosa.analysis.facts.hyperperiod

Require Export prosa.analysis.definitions.hyperperiod.
Require Export prosa.analysis.facts.periodic.task_arrivals_size.
Require Export prosa.analysis.facts.sporadic.arrival_bound.
Require Export prosa.analysis.facts.model.workload.
Require Export prosa.analysis.facts.model.task_cost.
Require Export prosa.util.div_mod.
Require Export prosa.util.tactics.

In this file we prove some simple properties of hyperperiods of periodic tasks.

Section Hyperperiod.

Consider any type of periodic tasks, ...

Context {Task : TaskType} `{PeriodicModel Task}.

... any task set ts, ...

Variable ts : TaskSet Task.

... and any task tsk that belongs to this task set.

Variable tsk : Task.
Hypothesis H_tsk_in_ts : tsk \in ts.

A task set's hyperperiod is an integral multiple of each task's period in the task set.

  Lemma hyperperiod_int_mult_of_any_task :
    ∃ (k : nat ),
      hyperperiod ts = k × task_period tsk.

End Hyperperiod.

In this section we show a property of hyperperiod in context of task sets with valid periods.

Section ValidPeriodsImplyPositiveHP.

Consider any type of periodic tasks ...

Context {Task : TaskType} `{PeriodicModel Task}.

... and any task set ts ...

Variable ts : TaskSet Task.

... such that all tasks in ts have valid periods.

Hypothesis H_valid_periods : valid_periods ts.

We show that the hyperperiod of task set ts is positive.

Lemma valid_periods_imply_pos_hp :
hyperperiod ts > 0.

End ValidPeriodsImplyPositiveHP.

In this section we prove some lemmas about the hyperperiod in context of the periodic model.

Section PeriodicLemmas.

Consider any type of tasks, ...

  Context {Task : TaskType}.
  Context `{TaskOffset Task}.
  Context `{PeriodicModel Task}.

... any type of jobs, ...

  Context {Job : JobType}.
  Context `{JobTask Job Task}.
  Context `{JobArrival Job}.

... and a consistent arrival sequence with non-duplicate arrivals.

Variable arr_seq : arrival_sequence Job.
Hypothesis H_valid_arrival_sequence : valid_arrival_sequence arr_seq.

Consider a task set ts such that all tasks in ts have valid periods.

Variable ts : TaskSet Task.
Hypothesis H_valid_periods : valid_periods ts.

Let tsk be any periodic task in ts with a valid offset and period.

  Variable tsk : Task.
  Hypothesis H_task_in_ts : tsk \in ts.
  Hypothesis H_valid_offset : valid_offset arr_seq tsk.
  Hypothesis H_valid_period : valid_period tsk.
  Hypothesis H_periodic_task : respects_periodic_task_model arr_seq tsk.

Each task in ts continues releasing jobs indefinitely.

Hypothesis H_infinite_jobs : tasks_have_infinite_arrivals arr_seq ts.

Let O_max denote the maximum task offset in ts and let HP denote the hyperperiod of all tasks in ts.

Let O_max := max_task_offset ts.
Let HP := hyperperiod ts.

We show that the job corresponding to any job j1 in any other hyperperiod is a job of the same task as j1.

  Lemma corresponding_jobs_have_same_task :
    ∀ j1 j2,
      job_task (corresponding_job_in_hyperperiod ts arr_seq j1
               (starting_instant_of_corresponding_hyperperiod ts j2) (job_task j1)) = job_task j1.

We show that if a job j lies in the hyperperiod starting at instant t then j arrives in the interval [t, t + HP).

  Lemma all_jobs_arrive_within_hyperperiod :
    ∀ j t,
      j \in jobs_in_hyperperiod ts arr_seq t tsk →
      t ≤ job_arrival j < t + HP.

We show that the number of jobs in a hyperperiod starting at n1 × HP + O_max is the same as the number of jobs in a hyperperiod starting at n2 × HP + O_max given that n1 is less than or equal to n2.

  Lemma eq_size_hyp_lt :
    ∀ n1 n2,
      n1 ≤ n2 →
      size (jobs_in_hyperperiod ts arr_seq (n1 × HP + O_max) tsk)
      = size (jobs_in_hyperperiod ts arr_seq (n2 × HP + O_max) tsk).

We generalize the above lemma by lifting the condition on n1 and n2.

  Lemma eq_size_of_arrivals_in_hyperperiod :
    ∀ n1 n2,
      size (jobs_in_hyperperiod ts arr_seq (n1 × HP + O_max) tsk)
      = size (jobs_in_hyperperiod ts arr_seq (n2 × HP + O_max) tsk).

Consider any two jobs j1 and j2 that stem from the arrival sequence arr_seq such that j1 is of task tsk.

  Variable j1 : Job.
  Variable j2 : Job.
  Hypothesis H_j1_from_arr_seq : arrives_in arr_seq j1.
  Hypothesis H_j2_from_arr_seq : arrives_in arr_seq j2.
  Hypothesis H_j1_task : job_task j1 = tsk.

Assume that both j1 and j2 arrive after O_max.

Hypothesis H_j1_arr_after_O_max : O_max ≤ job_arrival j1.
Hypothesis H_j2_arr_after_O_max : O_max ≤ job_arrival j2.

We show that any job j that arrives in task arrivals in the same hyperperiod as j2 also arrives in task arrivals up to job_arrival j2 + HP.

  Lemma job_in_hp_arrives_in_task_arrivals_up_to :
    ∀ j,
      j \in jobs_in_hyperperiod ts arr_seq ((job_arrival j2 - O_max ) %/ HP × HP + O_max) tsk →
      j \in task_arrivals_up_to arr_seq tsk (job_arrival j2 + HP).

We show that job j1 arrives in its own hyperperiod.

Lemma job_in_own_hp :
j1 \in jobs_in_hyperperiod ts arr_seq ((job_arrival j1 - O_max ) %/ HP × HP + O_max) tsk.

We show that the corresponding_job_in_hyperperiod of j1 in j2's hyperperiod arrives in task arrivals up to job_arrival j2 + HP.

  Lemma corr_job_in_task_arrivals_up_to :
    corresponding_job_in_hyperperiod ts arr_seq j1 (starting_instant_of_corresponding_hyperperiod ts j2) tsk \in
      task_arrivals_up_to arr_seq tsk (job_arrival j2 + HP).

Finally, we show that the corresponding_job_in_hyperperiod of j1 in j2's hyperperiod arrives in the arrival sequence arr_seq.

Lemma corresponding_job_arrives :
arrives_in arr_seq (corresponding_job_in_hyperperiod ts arr_seq j1 (starting_instant_of_corresponding_hyperperiod ts j2) tsk).

End PeriodicLemmas.

Jobs in Adjacent Hyperperiods

Section AdjacentHyperperiodJobs.

Consider periodic tasks, their jobs, ...

Context {Task : TaskType} `{PeriodicModel Task}.
Context {Job : JobType} `{JobTask Job Task} `{JobArrival Job}.

... and a valid arrival sequence of such jobs.

Variable arr_seq : arrival_sequence Job.
Hypothesis H_valid_arrival_sequence : valid_arrival_sequence arr_seq.

Consider a given task set ...

Variable ts : TaskSet Task.

... and any valid periodic task in this set.

  Variable tsk : Task.
  Hypothesis H_task_in_ts : tsk \in ts.
  Hypothesis H_valid_period : valid_period tsk.
  Hypothesis H_periodic_task : respects_periodic_task_model arr_seq tsk.

Trivially, the hyperperiod can be expressed in terms of the task's period.

Lemma hyperperiod_as_job_count :
hyperperiod ts = jobs_per_hyperperiod ts tsk × task_period tsk.

By definition, the task stays the same going forwards ...

  Lemma next_hyperperiod_job_task :
    ∀ j,
      job_task (next_hyperperiod_job ts arr_seq j) = job_task j.

... and also backwards.

  Lemma prev_hyperperiod_job_task :
    ∀ j,
      job_task (prev_hyperperiod_job ts arr_seq j) = job_task j.

The list of task-specific arrivals is a singleton list.

  Local Fact prev_next_hyperperiod_job_selection :
    ∀ j j' t,
      arrives_in arr_seq j' →
      job_task j = tsk →
      job_task j' = tsk →
      job_arrival j' = t →
      [seq j'' <- arr_seq t | same_task j'' j ] = [::j'].

We can identify the next job based on its arrival time.

  Lemma next_hyperperiod_job_from_arrival :
    ∀ j j',
      arrives_in arr_seq j' →
      job_task j = tsk →
      job_task j' = tsk →
      job_arrival j' = job_arrival j + hyperperiod ts →
      next_hyperperiod_job ts arr_seq j = j'.

We can also identify the previous job based on its arrival time.

  Lemma prev_hyperperiod_job_from_arrival :
    ∀ j j',
      arrives_in arr_seq j' →
      job_task j = tsk →
      job_task j' = tsk →
      job_arrival j' = job_arrival j - hyperperiod ts →
      prev_hyperperiod_job ts arr_seq j = j'.

Forward Correspondence

Section ForwardCorrespondence.

Suppose tsk releases jobs indefinitely.

Hypothesis H_infinite_jobs : infinite_jobs arr_seq tsk.

Consider any job of the task under analysis.

    Variable j : Job.
    Hypothesis H_arrives : arrives_in arr_seq j.
    Hypothesis H_tsk : job_task j = tsk.

We establish some basic facts about the "matching" job in the next hyperperiod in one go, which we then expose as individual facts.

    Local Fact next_hyperperiod_job_properties :
      let
        j' := next_hyperperiod_job ts arr_seq j
      in
      [/\ arrives_in arr_seq j'
       , job_task j' = tsk
       , job_arrival j' = job_arrival j + hyperperiod ts
       & job_index arr_seq j' = job_index arr_seq j + jobs_per_hyperperiod ts tsk ].

The successor belongs to the arrival sequence, ...

Fact next_hyperperiod_job_arrives :
arrives_in arr_seq (next_hyperperiod_job ts arr_seq j).

... has an arrival time exactly one hyperperiod apart, ...

    Fact next_hyperperiod_job_arrival :
      job_arrival (next_hyperperiod_job ts arr_seq j)
      = job_arrival j + hyperperiod ts.

... and the obvious job index.

    Fact next_hyperperiod_job_index :
      job_index arr_seq (next_hyperperiod_job ts arr_seq j)
      = job_index arr_seq j + jobs_per_hyperperiod ts tsk.

The "prev" and "next" operations cancel out.

    Fact prev_next_hyperperiod_job :
      prev_hyperperiod_job ts arr_seq (next_hyperperiod_job ts arr_seq j)
      = j.

  End ForwardCorrespondence.

Backward Correspondence

Section BackwardCorrespondence.

Consider a job of the task with at least one hyperperiod's worth of earlier releases.

    Variable j : Job.
    Hypothesis H_arrives : arrives_in arr_seq j.
    Hypothesis H_tsk : job_task j = tsk.
    Hypothesis H_index : jobs_per_hyperperiod ts tsk ≤ job_index arr_seq j.

We establish the properties of the matching job in the previous hyperperiod together, then expose them individually.

    Local Fact prev_hyperperiod_job_properties :
      let j' := prev_hyperperiod_job ts arr_seq j in
      [/\ arrives_in arr_seq j'
       , job_task j' = tsk
       , job_arrival j' + hyperperiod ts = job_arrival j
       & job_index arr_seq j' = job_index arr_seq j - jobs_per_hyperperiod ts tsk ].

The predecessor belongs to the arrival sequence, ...

Fact prev_hyperperiod_job_arrives :
arrives_in arr_seq (prev_hyperperiod_job ts arr_seq j).

... arrives exactly one hyperperiod earlier, ...

    Lemma prev_hyperperiod_job_arrival :
      job_arrival (prev_hyperperiod_job ts arr_seq j)
      = job_arrival j - hyperperiod ts.

... and has the preceding hyperperiod's job index.

    Fact prev_hyperperiod_job_index :
      job_index arr_seq (prev_hyperperiod_job ts arr_seq j)
      = job_index arr_seq j - jobs_per_hyperperiod ts tsk.

The "next" and "prev" operations cancel out.

    Fact next_prev_hyperperiod_job :
      next_hyperperiod_job ts arr_seq (prev_hyperperiod_job ts arr_seq j) = j.

  End BackwardCorrespondence.

End AdjacentHyperperiodJobs.

Workload in a Hyperperiod

The task parameters bound the work released in any hyperperiod.

Section HyperperiodWorkloadBound.

Consider periodic tasks with worst-case execution costs ...

Context {Task : TaskType} `{PeriodicModel Task} `{TaskCost Task}.

... and their jobs.

Context {Job : JobType} `{JobTask Job Task} `{JobArrival Job} `{JobCost Job}.

Consider a given set of such periodic tasks ...

Variable ts : TaskSet Task.
Hypothesis H_valid_periods : valid_periods ts.

... and a corresponding valid arrival sequence.

  Variable arr_seq : arrival_sequence Job.
  Hypothesis H_valid_arrival_sequence : valid_arrival_sequence arr_seq.
  Hypothesis H_periodic_arrivals : taskset_respects_periodic_task_model arr_seq ts.
  Hypothesis H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts.

If task WCETs bound the execution requirements of arriving jobs, ...

Hypothesis H_valid_job_costs : arrivals_have_valid_job_costs arr_seq.

... then the total workload in any hyperperiod-sized interval is upper-bounded by the hyperperiod workload.

  Lemma workload_in_hyperperiod_bounded :
    ∀ start,
      total_workload_between arr_seq start (start + hyperperiod ts)
      ≤ hyperperiod_workload ts.

End HyperperiodWorkloadBound.