Library prosa.analysis.facts.preemption.rtc_threshold.nonpreemptive


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

Welcome to Coq 8.11.2 (June 2020)

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


Require Export prosa.analysis.facts.preemption.job.nonpreemptive.

Furthermore, we assume the fully non-preemptive task model.

Task's Run to Completion Threshold

In this section, we prove that instantiation of function [task run to completion threshold] to the fully non-preemptive model indeed defines a valid run-to-completion threshold function.
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 `{JobArrival Job}.
  Context `{JobCost Job}.

Consider any arrival sequence with consistent arrivals.
Next, consider any ideal non-preemptive uniprocessor schedule of this arrival sequence ...
... where jobs do not execute before their arrival or after completion.
First we prove that if the cost of a job j is equal to 0, then [job_run_to_completion_threshold j = 0] ...
  Fact job_rtc_threshold_is_0:
     j,
      job_cost j = 0
      job_run_to_completion_threshold j = 0.

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

1 subgoal (ID 659)
  
  Task : TaskType
  H : TaskCost Task
  Job : JobType
  H0 : JobTask Job Task
  H1 : JobArrival Job
  H2 : JobCost Job
  arr_seq : arrival_sequence Job
  H_arrival_times_are_consistent : consistent_arrival_times arr_seq
  sched : schedule (ideal.processor_state Job)
  H_nonpreemptive_sched : nonpreemptive_schedule sched
  H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
  H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
  ============================
  forall j : Job, job_cost j = 0 -> job_run_to_completion_threshold j = 0

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


  Proof.
    intros.

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

1 subgoal (ID 661)
  
  Task : TaskType
  H : TaskCost Task
  Job : JobType
  H0 : JobTask Job Task
  H1 : JobArrival Job
  H2 : JobCost Job
  arr_seq : arrival_sequence Job
  H_arrival_times_are_consistent : consistent_arrival_times arr_seq
  sched : schedule (ideal.processor_state Job)
  H_nonpreemptive_sched : nonpreemptive_schedule sched
  H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
  H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
  j : Job
  H3 : job_cost j = 0
  ============================
  job_run_to_completion_threshold j = 0

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


    apply/eqP; rewrite eqn_leq; apply/andP; split; last by done.

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

1 subgoal (ID 746)
  
  Task : TaskType
  H : TaskCost Task
  Job : JobType
  H0 : JobTask Job Task
  H1 : JobArrival Job
  H2 : JobCost Job
  arr_seq : arrival_sequence Job
  H_arrival_times_are_consistent : consistent_arrival_times arr_seq
  sched : schedule (ideal.processor_state Job)
  H_nonpreemptive_sched : nonpreemptive_schedule sched
  H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
  H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
  j : Job
  H3 : job_cost j = 0
  ============================
  job_run_to_completion_threshold j <= 0

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


    unfold job_run_to_completion_threshold.

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

1 subgoal (ID 749)
  
  Task : TaskType
  H : TaskCost Task
  Job : JobType
  H0 : JobTask Job Task
  H1 : JobArrival Job
  H2 : JobCost Job
  arr_seq : arrival_sequence Job
  H_arrival_times_are_consistent : consistent_arrival_times arr_seq
  sched : schedule (ideal.processor_state Job)
  H_nonpreemptive_sched : nonpreemptive_schedule sched
  H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
  H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
  j : Job
  H3 : job_cost j = 0
  ============================
  job_cost j - (job_last_nonpreemptive_segment j - ε) <= 0

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


      by rewrite H3; compute.

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

No more subgoals.

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


  Qed.

... and ε otherwise.
  Fact job_rtc_threshold_is_ε:
     j,
      job_cost j > 0
      arrives_in arr_seq j
      job_run_to_completion_threshold j = ε.

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

1 subgoal (ID 671)
  
  Task : TaskType
  H : TaskCost Task
  Job : JobType
  H0 : JobTask Job Task
  H1 : JobArrival Job
  H2 : JobCost Job
  arr_seq : arrival_sequence Job
  H_arrival_times_are_consistent : consistent_arrival_times arr_seq
  sched : schedule (ideal.processor_state Job)
  H_nonpreemptive_sched : nonpreemptive_schedule sched
  H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
  H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
  ============================
  forall j : Job,
  0 < job_cost j ->
  arrives_in arr_seq j -> job_run_to_completion_threshold j = ε

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


  Proof.
    intros ? ARRj POSj; unfold ε in ×.

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

1 subgoal (ID 676)
  
  Task : TaskType
  H : TaskCost Task
  Job : JobType
  H0 : JobTask Job Task
  H1 : JobArrival Job
  H2 : JobCost Job
  arr_seq : arrival_sequence Job
  H_arrival_times_are_consistent : consistent_arrival_times arr_seq
  sched : schedule (ideal.processor_state Job)
  H_nonpreemptive_sched : nonpreemptive_schedule sched
  H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
  H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
  j : Job
  ARRj : 0 < job_cost j
  POSj : arrives_in arr_seq j
  ============================
  job_run_to_completion_threshold j = 1

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


    unfold job_run_to_completion_threshold.

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

1 subgoal (ID 678)
  
  Task : TaskType
  H : TaskCost Task
  Job : JobType
  H0 : JobTask Job Task
  H1 : JobArrival Job
  H2 : JobCost Job
  arr_seq : arrival_sequence Job
  H_arrival_times_are_consistent : consistent_arrival_times arr_seq
  sched : schedule (ideal.processor_state Job)
  H_nonpreemptive_sched : nonpreemptive_schedule sched
  H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
  H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
  j : Job
  ARRj : 0 < job_cost j
  POSj : arrives_in arr_seq j
  ============================
  job_cost j - (job_last_nonpreemptive_segment j - ε) = 1

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


    rewrite job_last_nps_is_job_cost.

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

1 subgoal (ID 685)
  
  Task : TaskType
  H : TaskCost Task
  Job : JobType
  H0 : JobTask Job Task
  H1 : JobArrival Job
  H2 : JobCost Job
  arr_seq : arrival_sequence Job
  H_arrival_times_are_consistent : consistent_arrival_times arr_seq
  sched : schedule (ideal.processor_state Job)
  H_nonpreemptive_sched : nonpreemptive_schedule sched
  H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
  H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
  j : Job
  ARRj : 0 < job_cost j
  POSj : arrives_in arr_seq j
  ============================
  job_cost j - (job_cost j - ε) = 1

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


      by rewrite subKn.

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

No more subgoals.

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


  Qed.

Consider a task with a positive cost.
  Variable tsk : Task.
  Hypothesis H_positive_cost : 0 < task_cost tsk.

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

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

1 subgoal (ID 682)
  
  Task : TaskType
  H : TaskCost Task
  Job : JobType
  H0 : JobTask Job Task
  H1 : JobArrival Job
  H2 : JobCost Job
  arr_seq : arrival_sequence Job
  H_arrival_times_are_consistent : consistent_arrival_times arr_seq
  sched : schedule (ideal.processor_state Job)
  H_nonpreemptive_sched : nonpreemptive_schedule sched
  H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
  H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
  tsk : Task
  H_positive_cost : 0 < task_cost tsk
  ============================
  valid_task_run_to_completion_threshold arr_seq tsk

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


  Proof.
    intros; split.

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

2 subgoals (ID 684)
  
  Task : TaskType
  H : TaskCost Task
  Job : JobType
  H0 : JobTask Job Task
  H1 : JobArrival Job
  H2 : JobCost Job
  arr_seq : arrival_sequence Job
  H_arrival_times_are_consistent : consistent_arrival_times arr_seq
  sched : schedule (ideal.processor_state Job)
  H_nonpreemptive_sched : nonpreemptive_schedule sched
  H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
  H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
  tsk : Task
  H_positive_cost : 0 < task_cost tsk
  ============================
  task_rtc_bounded_by_cost tsk

subgoal 2 (ID 685) is:
 job_respects_task_rtc arr_seq tsk

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


    - by unfold task_rtc_bounded_by_cost.

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

1 subgoal (ID 685)
  
  Task : TaskType
  H : TaskCost Task
  Job : JobType
  H0 : JobTask Job Task
  H1 : JobArrival Job
  H2 : JobCost Job
  arr_seq : arrival_sequence Job
  H_arrival_times_are_consistent : consistent_arrival_times arr_seq
  sched : schedule (ideal.processor_state Job)
  H_nonpreemptive_sched : nonpreemptive_schedule sched
  H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
  H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
  tsk : Task
  H_positive_cost : 0 < task_cost tsk
  ============================
  job_respects_task_rtc arr_seq tsk

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


    - intros j ARR TSK.

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

1 subgoal (ID 691)
  
  Task : TaskType
  H : TaskCost Task
  Job : JobType
  H0 : JobTask Job Task
  H1 : JobArrival Job
  H2 : JobCost Job
  arr_seq : arrival_sequence Job
  H_arrival_times_are_consistent : consistent_arrival_times arr_seq
  sched : schedule (ideal.processor_state Job)
  H_nonpreemptive_sched : nonpreemptive_schedule sched
  H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
  H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
  tsk : Task
  H_positive_cost : 0 < task_cost tsk
  j : Job
  ARR : arrives_in arr_seq j
  TSK : job_task j = tsk
  ============================
  job_run_to_completion_threshold j <= task_run_to_completion_threshold tsk

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


      rewrite -TSK /task_run_to_completion_threshold /fully_nonpreemptive.

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

1 subgoal (ID 701)
  
  Task : TaskType
  H : TaskCost Task
  Job : JobType
  H0 : JobTask Job Task
  H1 : JobArrival Job
  H2 : JobCost Job
  arr_seq : arrival_sequence Job
  H_arrival_times_are_consistent : consistent_arrival_times arr_seq
  sched : schedule (ideal.processor_state Job)
  H_nonpreemptive_sched : nonpreemptive_schedule sched
  H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
  H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
  tsk : Task
  H_positive_cost : 0 < task_cost tsk
  j : Job
  ARR : arrives_in arr_seq j
  TSK : job_task j = tsk
  ============================
  job_run_to_completion_threshold j <= ε

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


      edestruct (posnP (job_cost j)) as [ZERO|POS].

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

2 subgoals (ID 726)
  
  Task : TaskType
  H : TaskCost Task
  Job : JobType
  H0 : JobTask Job Task
  H1 : JobArrival Job
  H2 : JobCost Job
  arr_seq : arrival_sequence Job
  H_arrival_times_are_consistent : consistent_arrival_times arr_seq
  sched : schedule (ideal.processor_state Job)
  H_nonpreemptive_sched : nonpreemptive_schedule sched
  H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
  H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
  tsk : Task
  H_positive_cost : 0 < task_cost tsk
  j : Job
  ARR : arrives_in arr_seq j
  TSK : job_task j = tsk
  ZERO : job_cost j = 0
  ============================
  job_run_to_completion_threshold j <= ε

subgoal 2 (ID 727) is:
 job_run_to_completion_threshold j <= ε

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


      + by rewrite job_rtc_threshold_is_0.

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

1 subgoal (ID 727)
  
  Task : TaskType
  H : TaskCost Task
  Job : JobType
  H0 : JobTask Job Task
  H1 : JobArrival Job
  H2 : JobCost Job
  arr_seq : arrival_sequence Job
  H_arrival_times_are_consistent : consistent_arrival_times arr_seq
  sched : schedule (ideal.processor_state Job)
  H_nonpreemptive_sched : nonpreemptive_schedule sched
  H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
  H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
  tsk : Task
  H_positive_cost : 0 < task_cost tsk
  j : Job
  ARR : arrives_in arr_seq j
  TSK : job_task j = tsk
  POS : 0 < job_cost j
  ============================
  job_run_to_completion_threshold j <= ε

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


      + by erewrite job_rtc_threshold_is_ε; eauto 2.
(* ----------------------------------[ coqtop ]---------------------------------

No more subgoals.

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


  Qed.

End TaskRTCThresholdFullyNonPreemptive.
Hint Resolve fully_nonpreemptive_valid_task_run_to_completion_threshold : basic_facts.