Require Export prosa.analysis.facts.preemption.task.floating.
Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "_ + _" was already used in scope nat_scope.
[notation-overridden,parsing]
Notation "_ - _" was already used in scope nat_scope.
[notation-overridden,parsing]
Notation "_ <= _" was already used in scope nat_scope.
[notation-overridden,parsing]
Notation "_ < _" was already used in scope nat_scope.
[notation-overridden,parsing]
Notation "_ >= _" was already used in scope nat_scope.
[notation-overridden,parsing]
Notation "_ > _" was already used in scope nat_scope.
[notation-overridden,parsing]
Notation "_ <= _ <= _" was already used in scope
nat_scope. [notation-overridden,parsing]
Notation "_ < _ <= _" was already used in scope
nat_scope. [notation-overridden,parsing]
Notation "_ <= _ < _" was already used in scope
nat_scope. [notation-overridden,parsing]
Notation "_ < _ < _" was already used in scope
nat_scope. [notation-overridden,parsing]
Notation "_ * _" was already used in scope nat_scope.
[notation-overridden,parsing]
Require Export prosa.analysis.facts.preemption.rtc_threshold.job_preemptable.
Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Require Export prosa.model.task.preemption.floating_nonpreemptive.
Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]

(** * 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. *)
Section TaskRTCThresholdFloatingNonPreemptiveRegions.

  (** 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}.
  Context `{JobPreemptable Job}.

  (** We assume a task model with floating non-preemptive regions. *)
  #[local] Existing Instance floating_preemptive_rtc_threshold.

  (** Consider any arrival sequence. *)
  Variable arr_seq : arrival_sequence Job.

  (** 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 floating_preemptive_valid_task_run_to_completion_threshold:
    forall tsk, valid_task_run_to_completion_threshold arr_seq tsk.
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.
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
    intros; split.
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
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
    -
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
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 by rewrite /task_rtc_bounded_by_cost.
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
    -
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.
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_of_task tsk j
job_rtct j <= task_rtct tsk
      apply leq_trans with (job_cost j); rt_eauto.
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_of_task tsk j
job_cost j <= task_rtct tsk
      by move: TSK => /eqP <-; apply H_valid_job_cost.
  Qed.
  
End TaskRTCThresholdFloatingNonPreemptiveRegions.
Global Hint Resolve floating_preemptive_valid_task_run_to_completion_threshold : basic_rt_facts.