Library prosa.implementation.refinements.arrival_curve

Require Export prosa.analysis.definitions.request_bound_function.
Require Export prosa.implementation.refinements.task.

Arrival Curve Refinements.

In this module, we provide a series of definitions related to arrival curves when working with generic tasks.

First, we define a helper function that yields the horizon of the arrival curve prefix of a task ...

Definition get_horizon_of_task (tsk : Task) : duration :=
horizon_of (get_arrival_curve_prefix tsk).

... another one that yields the time steps of the arrival curve, ...

Definition get_time_steps_of_task (tsk : Task) : seq duration :=
time_steps_of (get_arrival_curve_prefix tsk).

... a function that yields the same time steps, offset by δ, ...

Definition time_steps_with_offset tsk δ := [seq t + δ | t <- get_time_steps_of_task tsk ].

... and a generalization of the previous function that repeats the time steps for each given offset.

Definition repeat_steps_with_offset (tsk : Task) (offsets : seq nat): seq nat :=
flatten (map (time_steps_with_offset tsk) offsets).

For convenience, we provide a short form to indicate the request-bound function of a task.

Definition task_rbf (tsk : Task) (Δ : duration) :=
task_request_bound_function tsk Δ.

Further, we define a valid arrival bound as (1) a positive minimum inter-arrival time/period, or (2) a valid arrival curve prefix.

Definition valid_arrivals (tsk : Task) : bool :=
  match task_arrival tsk with
  | Periodic p ⇒ p ≥ 1
  | Sporadic m ⇒ m ≥ 1
  | ArrivalPrefix emax_vec ⇒ valid_arrival_curve_prefix_dec emax_vec
  end.

Next, we define some helper functions, that indicate whether the given task is periodic, ...

Definition is_periodic_arrivals (tsk : Task) : Prop :=
∃ p , task_arrival tsk = Periodic p.

... sporadic, ...

Definition is_sporadic_arrivals (tsk : Task) : Prop :=
∃ m , task_arrival tsk = Sporadic m.

... or bounded by an arrival curve.

Definition is_etamax_arrivals (tsk : Task) : Prop :=
∃ ac_prefix_vec , task_arrival tsk = ArrivalPrefix ac_prefix_vec.

Further, for convenience, we define the notion of task having a valid arrival curve.

Definition has_valid_arrival_curve_prefix (tsk: Task) :=
∃ ac_prefix_vec , get_arrival_curve_prefix tsk = ac_prefix_vec ∧
valid_arrival_curve_prefix ac_prefix_vec.

Finally, we define define the notion of task set with valid arrivals.

Definition task_set_with_valid_arrivals (ts : seq Task) :=
∀ tsk,
tsk \in ts → valid_arrivals tsk.

In this section, we prove some facts regarding the above definitions.

Section Facts.

Consider a task tsk.

Variable tsk : Task.

We show that a task is either periodic, sporadic, or bounded by an arrival-curve prefix.

  Lemma arrival_cases :
    is_periodic_arrivals tsk
    ∨ is_sporadic_arrivals tsk
    ∨ is_etamax_arrivals tsk.

End Facts.

In this fairly technical section, we prove a series of refinements aimed to be able to convert between a standard natural-number task arrival bound and an arrival bound that uses, instead of numbers, a generic type T.

Section Theory.

First, we prove a refinement for positive_horizon.

  Local Instance refine_positive_horizon :
    refines (prod_R Rnat (list_R (prod_R Rnat Rnat)) ==> bool_R)%rel
            positive_horizon positive_horizon_T.

Next, we prove a refinement for large_horizon.

  Local Instance refine_large_horizon :
    refines (prod_R Rnat (list_R (prod_R Rnat Rnat)) ==> bool_R)%rel
            large_horizon_dec large_horizon_T.

Next, we prove a refinement for no_inf_arrivals.

  Local Instance refine_no_inf_arrivals :
    refines (prod_R Rnat (list_R (prod_R Rnat Rnat)) ==> bool_R)%rel
            no_inf_arrivals no_inf_arrivals_T.

Next, we prove a refinement for specified_bursts.

  Local Instance refine_specified_bursts :
    refines (prod_R Rnat (list_R (prod_R Rnat Rnat)) ==> bool_R)%rel
            specified_bursts specified_bursts_T.

Next, we prove a refinement for the arrival curve prefix validity.

  Local Instance refine_valid_arrival_curve_prefix :
    refines
      (prod_R Rnat (list_R (prod_R Rnat Rnat)) ==> bool_R)%rel
      valid_arrival_curve_prefix_dec
      valid_extrapolated_arrival_curve_T.

Next, we prove a refinement for the arrival curve validity.

  Global Instance refine_valid_arrivals :
    ∀ tsk,
      refines (bool_R)%rel
              (valid_arrivals (taskT_to_task tsk))
              (valid_arrivals_T tsk) | 0.

Next, we prove a refinement for repeat_steps_with_offset.

  Global Instance refine_repeat_steps_with_offset :
    refines (Rtask ==> list_R Rnat ==> list_R Rnat)%rel
            repeat_steps_with_offset repeat_steps_with_offset_T.

Next, we prove a refinement for get_horizon_of_task.

Global Instance refine_get_horizon_of_task :
refines (Rtask ==> Rnat)%rel get_horizon_of_task get_horizon_of_task_T.

Next, we prove a refinement for the extrapolated arrival curve.

  Local Instance refine_extrapolated_arrival_curve :
    ∀ arrival_curve_prefix arrival_curve_prefixT δ δ',
      refines Rnat δ δ' →
      Rtask_ab (ArrivalPrefix arrival_curve_prefix) (ArrivalPrefix_T arrival_curve_prefixT) →
      refines Rnat (extrapolated_arrival_curve arrival_curve_prefix δ) (extrapolated_arrival_curve_T arrival_curve_prefixT δ').

Next, we prove a refinement for the arrival bound definition.

Local Instance refine_ConcreteMaxArrivals :
refines ( Rtask ==> Rnat ==> Rnat )%rel ConcreteMaxArrivals ConcreteMaxArrivals_T.

Next, we prove a refinement for the arrival bound definition applied to the task conversion function.

  Global Instance refine_ConcreteMaxArrivals' :
    ∀ tsk,
      refines (Rnat ==> Rnat)%rel (ConcreteMaxArrivals (taskT_to_task tsk))
              (ConcreteMaxArrivals_T tsk) | 0.

Next, we prove a refinement for get_arrival_curve_prefix.

  Global Instance refine_get_arrival_curve_prefix :
    refines (Rtask ==> prod_R Rnat (list_R (prod_R Rnat Rnat)))%rel
            get_arrival_curve_prefix get_extrapolated_arrival_curve_T.

Next, we prove a refinement for get_arrival_curve_prefix applied to lists.

  Global Instance refine_get_arrival_curve_prefix' :
    ∀ tsk,
      refines
        (prod_R Rnat (list_R (prod_R Rnat Rnat)))%rel
        (get_arrival_curve_prefix (taskT_to_task tsk))
        (get_extrapolated_arrival_curve_T tsk) | 0.

Next, we prove a refinement for sorted when applied to leq_steps.

  Global Instance refine_sorted_leq_steps :
    ∀ tsk,
      refines (bool_R)%rel
              (sorted leq_steps (steps_of (get_arrival_curve_prefix (taskT_to_task tsk))))
              (sorted leq_steps_T (get_extrapolated_arrival_curve_T tsk ).2) | 0.

Lastly, we prove a refinement for the task request-bound function.

Global Instance refine_task_rbf :
refines ( Rtask ==> Rnat ==> Rnat )%rel task_rbf task_rbf_T.

End Theory.