Library prosa.model.task.arrival.periodic_as_sporadic

Require Export prosa.model.task.arrival.periodic.

Treating Periodic Tasks as Sporadic Tasks

Since the periodic-arrivals assumption is stricter than the sporadic-arrivals assumption (i.e., any periodic arrival sequence is also a valid sporadic arrivals sequence), it is sometimes convenient to apply analyses for sporadic tasks to periodic tasks. We therefore provide an automatic "conversion" of periodic tasks to sporadic tasks, i.e., we tell Coq that it may use a periodic task's task_period parameter also as the task's minimum inter-arrival time task_min_inter_arrival_time parameter.

Section PeriodicTasksAsSporadicTasks.

Any type of periodic tasks ...
  Context {Task : TaskType} `{PeriodicModel Task}.

... and their corresponding jobs from a consistent arrival sequence with non-duplicate arrivals ...
  Context {Job : JobType} `{JobTask Job Task} `{JobArrival Job}.

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

... may be interpreted as a type of sporadic tasks by using each task's period as its minimum inter-arrival time ...
  Global Instance periodic_as_sporadic : SporadicModel Task :=
    {
      task_min_inter_arrival_time := task_period
    }.

... such that the model interpretation is valid, both w.r.t. the minimum inter-arrival time parameter ...
  Remark valid_period_is_valid_inter_arrival_time:
    ∀ tsk, valid_period tsk → valid_task_min_inter_arrival_time tsk.

... and the separation of job arrivals.
  Remark periodic_task_respects_sporadic_task_model:
    ∀ tsk, valid_period tsk →
           respects_periodic_task_model arr_seq tsk →
           respects_sporadic_task_model arr_seq tsk.

For convenience, we state these obvious correspondences also at the level of entire task sets.
First, we show that all tasks in a task set with valid periods also have valid min inter-arrival times.
  Remark valid_periods_are_valid_inter_arrival_times :
    ∀ ts, valid_periods ts → valid_taskset_inter_arrival_times ts.

Second, we show that each task in a periodic task set respects the sporadic task model.
  Remark periodic_task_sets_respect_sporadic_task_model :
    ∀ ts,
      valid_periods ts →
      taskset_respects_periodic_task_model arr_seq ts →
      taskset_respects_sporadic_task_model ts arr_seq.

End PeriodicTasksAsSporadicTasks.

We add the lemmas into the "Hint Database" basic_rt_facts so that they become available for proof automation.
Global Hint Resolve
  periodic_task_respects_sporadic_task_model
  valid_period_is_valid_inter_arrival_time
  valid_periods_are_valid_inter_arrival_times
  periodic_task_sets_respect_sporadic_task_model
  : basic_rt_facts.