Library prosa.analysis.facts.preemption.rtc_threshold.floating
(* ----------------------------------[ coqtop ]---------------------------------
Welcome to Coq 8.13.0 (January 2021)
----------------------------------------------------------------------------- *)
Require Export prosa.analysis.facts.preemption.task.floating.
Require Export prosa.analysis.facts.preemption.rtc_threshold.job_preemptable.
Furthermore, we assume the task model with floating non-preemptive regions.
Require Import prosa.model.preemption.limited_preemptive.
Require Import prosa.model.task.preemption.floating_nonpreemptive.
Require Import prosa.model.task.preemption.floating_nonpreemptive.
Task's Run to Completion Threshold
In this section, we instantiate function [task run to completion threshold] for the model with floating non-preemptive regions.
Consider any type of tasks ...
... and any type of jobs associated with these tasks.
Context {Job : JobType}.
Context `{JobTask Job Task}.
Context `{JobCost Job}.
Context `{JobPreemptable Job}.
Context `{JobTask Job Task}.
Context `{JobCost Job}.
Context `{JobPreemptable Job}.
Consider any arrival sequence.
Assume that a job cost cannot be larger than a task cost.
Then, we prove that [task_rtct] function
defines a valid task's run to completion threshold.
Lemma floating_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
H2 : JobPreemptable 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
----------------------------------------------------------------------------- *)
Proof.
intros; split.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 36)
Task : TaskType
H : TaskCost Task
Job : JobType
H0 : JobTask Job Task
H1 : JobCost Job
H2 : JobPreemptable 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.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 37)
Task : TaskType
H : TaskCost Task
Job : JobType
H0 : JobTask Job Task
H1 : JobCost Job
H2 : JobPreemptable 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
H2 : JobPreemptable 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.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 52)
Task : TaskType
H : TaskCost Task
Job : JobType
H0 : JobTask Job Task
H1 : JobCost Job
H2 : JobPreemptable 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
----------------------------------------------------------------------------- *)
by rewrite-TSK; apply H_valid_job_cost.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
End TaskRTCThresholdFloatingNonPreemptiveRegions.
Global Hint Resolve floating_preemptive_valid_task_run_to_completion_threshold : basic_facts.
∀ 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
H2 : JobPreemptable 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
----------------------------------------------------------------------------- *)
Proof.
intros; split.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 36)
Task : TaskType
H : TaskCost Task
Job : JobType
H0 : JobTask Job Task
H1 : JobCost Job
H2 : JobPreemptable 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.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 37)
Task : TaskType
H : TaskCost Task
Job : JobType
H0 : JobTask Job Task
H1 : JobCost Job
H2 : JobPreemptable 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
H2 : JobPreemptable 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.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 52)
Task : TaskType
H : TaskCost Task
Job : JobType
H0 : JobTask Job Task
H1 : JobCost Job
H2 : JobPreemptable 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
----------------------------------------------------------------------------- *)
by rewrite-TSK; apply H_valid_job_cost.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
End TaskRTCThresholdFloatingNonPreemptiveRegions.
Global Hint Resolve floating_preemptive_valid_task_run_to_completion_threshold : basic_facts.