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# Library prosa.model.task.sequentiality

Require Export prosa.model.task.concept.
Require Export prosa.model.task.arrivals.
In this module, we give a precise definition of the common notion of "sequential tasks", which is commonly understood to mean that the jobs of a sequential task execute in sequence and in a non-overlapping fashion.
Section PropertyOfSequentiality.
Consider any type of jobs stemming from any type of tasks ...
Context {Job : JobType}.
Context {Task : TaskType}.
Context `{JobTask Job Task}.
... with arrival times and costs ...
Context `{JobArrival Job}.
Context `{JobCost Job}.
... and any kind of processor model.
Context {PState : ProcessorState Job}.
Consider any arrival sequence ...
Variable arr_seq : arrival_sequence Job.
... and any schedule of this arrival sequence.
Variable sched : schedule PState.
We say that the tasks execute sequentially if each task's jobs are executed in arrival order and in a non-overlapping fashion.
Definition sequential_tasks :=
∀ (j1 j2 : Job) (t : instant),
arrives_in arr_seq j1 →
arrives_in arr_seq j2 →
same_task j1 j2 →
job_arrival j1 < job_arrival j2 →
scheduled_at sched j2 t →
completed_by sched j1 t.
Given a job j and a time instant t, we say that all prior jobs are completed if any job j_tsk of task job_task j that arrives before time job_arrival j is completed by time t.
Definition prior_jobs_complete (j : Job) (t : instant) :=
all
(fun j_tsk ⇒ completed_by sched j_tsk t)
(task_arrivals_before arr_seq (job_task j) (job_arrival j)).
End PropertyOfSequentiality.