Library prosa.analysis.facts.preemption.rtc_threshold.preemptive


(* ----------------------------------[ coqtop ]---------------------------------

Welcome to Coq 8.13.0 (January 2021)

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Furthermore, we assume the fully preemptive task model.
Require Import prosa.model.preemption.fully_preemptive.
Require Import prosa.model.task.preemption.fully_preemptive.
Require Import prosa.analysis.facts.preemption.rtc_threshold.job_preemptable.

Task's Run to Completion Threshold

In this section, we prove that instantiation of function [task run to completion threshold] to the fully preemptive model indeed defines a valid run-to-completion threshold function.
Section TaskRTCThresholdFullyPreemptiveModel.

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}.

Next, consider any arrival sequence ...
  Variable arr_seq : arrival_sequence Job.

... and assume that a job cost cannot be larger than a task cost.
  Hypothesis H_valid_job_cost:
    arrivals_have_valid_job_costs arr_seq.

Then, we prove that [task_rtct] function defines a valid task's run to completion threshold.
  Lemma fully_preemptive_valid_task_run_to_completion_threshold:
    ∀ tsk, valid_task_run_to_completion_threshold arr_seq tsk.

(* ----------------------------------[ coqtop ]---------------------------------

1 subgoal (ID 33)
  
  Task : TaskType
  H : TaskCost Task
  Job : JobType
  H0 : JobTask Job Task
  H1 : JobCost Job
  arr_seq : arrival_sequence Job
  H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
  ============================
  forall tsk : Task, valid_task_run_to_completion_threshold arr_seq tsk

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  Proof.
    intros; split.

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2 subgoals (ID 36)
  
  Task : TaskType
  H : TaskCost Task
  Job : JobType
  H0 : JobTask Job Task
  H1 : JobCost Job
  arr_seq : arrival_sequence Job
  H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
  tsk : Task
  ============================
  task_rtc_bounded_by_cost tsk

subgoal 2 (ID 37) is:
 job_respects_task_rtc arr_seq tsk

----------------------------------------------------------------------------- *)

    - by rewrite /task_rtc_bounded_by_cost.

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1 subgoal (ID 37)
  
  Task : TaskType
  H : TaskCost Task
  Job : JobType
  H0 : JobTask Job Task
  H1 : JobCost Job
  arr_seq : arrival_sequence Job
  H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
  tsk : Task
  ============================
  job_respects_task_rtc arr_seq tsk

----------------------------------------------------------------------------- *)

    - intros j ARR TSK.

(* ----------------------------------[ coqtop ]---------------------------------

1 subgoal (ID 48)
  
  Task : TaskType
  H : TaskCost Task
  Job : JobType
  H0 : JobTask Job Task
  H1 : JobCost Job
  arr_seq : arrival_sequence Job
  H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
  tsk : Task
  j : Job
  ARR : arrives_in arr_seq j
  TSK : job_task j = tsk
  ============================
  job_rtct j <= task_rtct tsk

----------------------------------------------------------------------------- *)

      apply leq_trans with (job_cost j); eauto 2 with basic_facts.

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1 subgoal (ID 52)
  
  Task : TaskType
  H : TaskCost Task
  Job : JobType
  H0 : JobTask Job Task
  H1 : JobCost Job
  arr_seq : arrival_sequence Job
  H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
  tsk : Task
  j : Job
  ARR : arrives_in arr_seq j
  TSK : job_task j = tsk
  ============================
  job_cost j <= task_rtct tsk

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        by rewrite-TSK; apply H_valid_job_cost.

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No more subgoals.

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  Qed.

End TaskRTCThresholdFullyPreemptiveModel.
Global Hint Resolve fully_preemptive_valid_task_run_to_completion_threshold : basic_facts.