Library rt.restructuring.model.task.concept

From mathcomp Require Export ssrbool.
Require Export rt.restructuring.behavior.all.

Throughout the library we assume that tasks have decidable equality
Definition TaskType := eqType.

Definition of a generic type of parameter relating jobs to tasks
Class JobTask (Job : JobType) (Task : TaskType) := job_task : Job → Task.

Definition of a generic type of parameter for task deadlines
Class TaskDeadline (Task : TaskType) := task_deadline : Task → duration.

Definition of a generic type of parameter for task cost
Class TaskCost (Task : TaskType) := task_cost : Task → duration.

Section SameTask.

For any type of job associated with any type of tasks...
  Context {Job : JobType}.
  Context {Task : TaskType}.
  Context `{JobTask Job Task}.

... we say that two jobs j1 and j2 are from the same task iff job_task j1 is equal to job_task j2, ...
  Definition same_task (j1 j2 : Job) := job_task j1 == job_task j2.

... we also say that job j is a job of task tsk iff job_task j is equal to tsk.
  Definition job_of_task (tsk : Task) (j : Job) := job_task j == tsk.

End SameTask.

In this section, we introduce properties of a task.
Section PropertesOfTask.

Consider any type of tasks.
  Context {Task : TaskType}.
  Context `{TaskCost Task}.
  Context `{TaskDeadline Task}.

Next we intrdoduce attributes of a valid sporadic task.
  Section ValidTask.

Consider an arbitrary task.
    Variable tsk: Task.

The cost and deadline of the task must be positive.
    Definition task_cost_positive := task_cost tsk > 0.
    Definition task_deadline_positive := task_deadline tsk > 0.

The task cost cannot be larger than the deadline or the period.
    Definition task_cost_le_deadline := task_cost tsk ≤ task_deadline tsk.

  End ValidTask.

Job of task
  Section ValidJobOfTask.

Consider any type of jobs associated with the tasks ...
    Context {Job : JobType}.
    Context `{JobTask Job Task}.
    Context `{JobCost Job}.
    Context `{JobDeadline Job}.

Consider an arbitrary job.
    Variable j: Job.

The job cost cannot be larger than the task cost.
    Definition job_cost_le_task_cost :=
      job_cost j ≤ task_cost (job_task j).

  End ValidJobOfTask.

Jobs of task
  Section ValidJobsTask.

Consider any type of jobs associated with the tasks ...
    Context {Job : JobType}.
    Context `{JobTask Job Task}.
    Context `{JobCost Job}.

... and any arrival sequence.
    Variable arr_seq : arrival_sequence Job.

The cost of a job from the arrival sequence cannot be larger than the task cost.
    Definition cost_of_jobs_from_arrival_sequence_le_task_cost :=
      ∀ j,
        arrives_in arr_seq j →
        job_cost_le_task_cost j.

  End ValidJobsTask.

End PropertesOfTask.

In this section, we introduce properties of a task set.
Section PropertesOfTaskSet.

Consider any type of tasks ...
  Context {Task : TaskType}.
  Context `{TaskCost Task}.

... and any type of jobs associated with these tasks ...
  Context {Job : JobType}.
  Context `{JobTask Job Task}.
  Context `{JobCost Job}.

... and any arrival sequence.
  Variable arr_seq : arrival_sequence Job.

Consider an arbitrary task set.
  Variable ts : seq Task.

Tasks from the task set have a positive cost.
  Definition tasks_cost_positive :=
    ∀ tsk,
      tsk \in ts →
      task_cost_positive tsk.

All jobs in the arrival sequence come from the taskset.
  Definition all_jobs_from_taskset :=
    ∀ j,
      arrives_in arr_seq j →
      job_task j \in ts.

End PropertesOfTaskSet.