Library prosa.implementation.definitions.arrival_bound

Require Export prosa.implementation.definitions.extrapolated_arrival_curve.

Implementation of a Task's Arrival Bound

In this file, we define a reference implementation of the notion of a task's arrival bound.

Note that its use is entirely optional: clients of Prosa may choose to use this type or implement their own notion of arrival bounds.

A task's arrival bound is an inductive type comprised of three types of arrival patterns: (a) periodic, characterized by a period between consequent activation of a task, (b) sporadic, characterized by a minimum inter-arrival time, or (c) arrival-curve prefix, characterized by a finite prefix of an arrival curve.

Inductive task_arrivals_bound :=
| Periodic : nat → task_arrivals_bound
| Sporadic : nat → task_arrivals_bound
| ArrivalPrefix : ArrivalCurvePrefix → task_arrivals_bound.

To make it compatible with ssreflect, we define a decidable equality for arrival bounds.

Definition task_arrivals_bound_eqdef (tb1 tb2 : task_arrivals_bound) :=
  match tb1, tb2 with
  | Periodic p1, Periodic p2 ⇒ p1 == p2
  | Sporadic s1, Sporadic s2 ⇒ s1 == s2
  | ArrivalPrefix s1, ArrivalPrefix s2 ⇒ s1 == s2
  | _, _ ⇒ false
  end.

Next, we prove that task_eqdef is indeed an equality, ...

Lemma eqn_task_arrivals_bound : Equality.axiom task_arrivals_bound_eqdef.
Proof.
  intros x y.
  destruct (task_arrivals_bound_eqdef x y) eqn:EQ.
  { apply ReflectT; destruct x, y.
    all: try by move: EQ ⇒ /eqP EQ; subst.
  }
  { apply ReflectF; destruct x, y.
    all: try by move: EQ ⇒ /eqP EQ.
    all: by move: EQ ⇒ /neqP; intros NEQ1 NEQ2; apply NEQ1; inversion NEQ2.
  }
Qed.

..., which allows instantiating the canonical structure for [eqType of task_arrivals_bound].

Canonical task_arrivals_bound_eqMixin := EqMixin eqn_task_arrivals_bound.
Canonical task_arrivals_bound_eqType := Eval hnf in EqType task_arrivals_bound task_arrivals_bound_eqMixin.