limited.v

Require Import rt.util.all.
Require Import rt.restructuring.behavior.all.
Require Import rt.restructuring.analysis.basic_facts.all.
Require Import rt.restructuring.analysis.definitions.job_properties.
Require Import rt.restructuring.model.task.concept.
Require Import rt.restructuring.model.aggregate.workload.
Require Import rt.restructuring.model.processor.ideal.
Require Import rt.restructuring.model.readiness.basic.
Require Import rt.restructuring.model.arrival.arrival_curves.
Require Import rt.restructuring.model.schedule.limited_preemptive.
Require Import rt.restructuring.model.task.preemption.limited_preemptive.
Require Import rt.restructuring.model.schedule.work_conserving.
Require Import rt.restructuring.model.priority.classes.
Require Import rt.restructuring.analysis.facts.edf.
Require Import rt.restructuring.model.schedule.priority_driven.
Require Import rt.restructuring.analysis.arrival.workload_bound.
Require Import rt.restructuring.analysis.arrival.rbf.
Require Import rt.restructuring.analysis.edf.rta.nonpr_reg.response_time_bound.
Require Export rt.restructuring.analysis.basic_facts.preemption.job.limited.
Require Export rt.restructuring.analysis.basic_facts.preemption.task.limited.
Require Export rt.restructuring.analysis.basic_facts.preemption.rtc_threshold.limited.

From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq path fintype bigop.

(** * RTA for EDF-schedulers with Fixed Premption Points *)
(** In this module we prove the RTA theorem for EDF-schedulers with fixed preemption points. *)
Section RTAforFixedPreemptionPointsModelwithArrivalCurves.
  
  (** Consider any type of tasks ... *)
  Context {Task : TaskType}.
  Context `{TaskCost Task}.
  Context `{TaskDeadline Task}.

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

  (** For clarity, let's denote the relative deadline of a task as D. *)
  Let D tsk := task_deadline tsk.
   
  (** Consider the EDF policy that indicates a higher-or-equal priority relation. *)
  Let EDF := EDF Job.

  (** Consider any arrival sequence with consistent, non-duplicate arrivals. *)
  Variable arr_seq : arrival_sequence Job.
  Hypothesis H_arrival_times_are_consistent : consistent_arrival_times arr_seq.
  Hypothesis H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq.

  (** Consider an arbitrary task set ts, ... *)
  Variable ts : list Task.

  (** ... assume that all jobs come from this task set, ... *)
  Hypothesis H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts.

  (** ... and the cost of a job cannot be larger than the task cost. *)
  Hypothesis H_job_cost_le_task_cost:
    cost_of_jobs_from_arrival_sequence_le_task_cost arr_seq.
  
  (** Next, we assume we have the model with fixed preemption points.
     I.e., each task is divided into a number of nonpreemptive segments 
     by inserting staticaly predefined preemption points. *)
  Context `{JobPreemptionPoints Job}.
  Context `{TaskPreemptionPoints Task}.
  Hypothesis H_valid_model_with_fixed_preemption_points:
    valid_fixed_preemption_points_model arr_seq ts.

  (** Let max_arrivals be a family of valid arrival curves, i.e., for
     any task tsk in ts [max_arrival tsk] is (1) an arrival bound of
     tsk, and (2) it is a monotonic function that equals 0 for the
     empty interval delta = 0. *)
  Context `{MaxArrivals Task}.
  Hypothesis H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals.
  Hypothesis H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts.

  (** Let tsk be any task in ts that is to be analyzed. *)
  Variable tsk : Task.
  Hypothesis H_tsk_in_ts : tsk \in ts.

  (** Next, consider any ideal uniprocessor schedule with limited
      preemptionsof this arrival sequence ...  *)
  Variable sched : schedule (ideal.processor_state Job).
  Hypothesis H_jobs_come_from_arrival_sequence:
    jobs_come_from_arrival_sequence sched arr_seq.
  Hypothesis H_schedule_with_limited_preemptions:
    valid_schedule_with_limited_preemptions arr_seq sched.
  
  (** ... where jobs do not execute before their arrival or after completion. *)
  Hypothesis H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched.
  Hypothesis H_completed_jobs_dont_execute : completed_jobs_dont_execute sched.

  (** Assume we have sequential tasks, i.e, jobs from the 
      same task execute in the order of their arrival. *)
  Hypothesis H_sequential_tasks : sequential_tasks sched.

  (** Next, we assume that the schedule is a work-conserving schedule... *)
  Hypothesis H_work_conserving : work_conserving arr_seq sched.
  
  (** ... and the schedule respects the policy defined by the
      job_preemptable function (i.e., jobs have bounded nonpreemptive
      segments). *)
  Hypothesis H_respects_policy : respects_policy_at_preemption_point arr_seq sched.
  
  (** Let's define some local names for clarity. *)
  Let response_time_bounded_by := task_response_time_bound arr_seq sched.
  Let task_rbf_changes_at A := task_rbf_changes_at tsk A.
  Let bound_on_total_hep_workload_changes_at :=
    bound_on_total_hep_workload_changes_at ts tsk.
  
  (** We introduce the abbreviation "rbf" for the task request bound function,
     which is defined as [task_cost(T) × max_arrivals(T,Δ)] for a task T. *)
  Let rbf := task_request_bound_function.

  (** Next, we introduce task_rbf as an abbreviation
     for the task request bound function of task tsk. *)
  Let task_rbf := rbf tsk.

  (** Using the sum of individual request bound functions, we define the request bound 
     function of all tasks (total request bound function). *)
  Let total_rbf := total_request_bound_function ts.

  (** We define a bound for the priority inversion caused by jobs with lower priority. *)
  Let blocking_bound :=
    \max_(tsk_other <- ts | (tsk_other != tsk) && (D tsk_other > D tsk))
     (task_max_nonpreemptive_segment tsk_other - ε).
  
  (** Next, we define an upper bound on interfering workload received from jobs 
     of other tasks with higher-than-or-equal priority. *)
  Let bound_on_total_hep_workload A Δ :=
    \sum_(tsk_o <- ts | tsk_o != tsk)
     rbf tsk_o (minn ((A + ε) + D tsk - D tsk_o) Δ).
  
  (** Let L be any positive fixed point of the busy interval recurrence. *)
  Variable L : duration.
  Hypothesis H_L_positive : L > 0.
  Hypothesis H_fixed_point : L = total_rbf L.

  (** To reduce the time complexity of the analysis, recall the notion of search space. *)
  Let is_in_search_space A :=
    (A < L) && (task_rbf_changes_at A || bound_on_total_hep_workload_changes_at A). 
  
  (** Consider any value R, and assume that for any given arrival offset A in the search space,
     there is a solution of the response-time bound recurrence which is bounded by R. *)
  Variable R : duration.
  Hypothesis H_R_is_maximum:
    forall (A : duration),
      is_in_search_space A -> 
      exists (F : duration),
        A + F = blocking_bound
                + (task_rbf (A + ε) - (task_last_nonpr_segment tsk - ε)) 
                + bound_on_total_hep_workload A (A + F) /\
        F + (task_last_nonpr_segment tsk - ε) <= R.
  
  (** Now, we can leverage the results for the abstract model with bounded nonpreemptive segments
      to establish a response-time bound for the more concrete model of fixed preemption points.  *)
  Theorem uniprocessor_response_time_bound_edf_with_fixed_preemption_points:
    response_time_bounded_by tsk R.  
  Proof. 
    move: (H_valid_model_with_fixed_preemption_points) => [MLP [BEG [END [INCR [HYP1 [HYP2 HYP3]]]]]].
    move: (MLP) => [BEGj [ENDj _]].
    case: (posnP (task_cost tsk)) => [ZERO|POSt].
    { intros j ARR TSK.
      move: (H_job_cost_le_task_cost _ ARR) => POSt.
      move: POSt; rewrite /job_cost_le_task_cost TSK ZERO leqn0; move => /eqP Z.
        by rewrite /job_response_time_bound /completed_by Z.
    }
    eapply uniprocessor_response_time_bound_edf_with_bounded_nonpreemptive_segments with (L0 := L).
    all: eauto 2 with basic_facts.
    { rewrite subKn; first by done.
      rewrite /task_last_nonpr_segment  -(leq_add2r 1) subn1 !addn1 prednK; last first.
      {  rewrite /last0 -nth_last.
        apply HYP3; try by done. 
        rewrite -(ltn_add2r 1) !addn1 prednK //.
        move: (number_of_preemption_points_in_task_at_least_two
                 _ _ H_valid_model_with_fixed_preemption_points _ H_tsk_in_ts POSt) => Fact2.
        move: (Fact2) => Fact3.
          by rewrite size_of_seq_of_distances // addn1 ltnS // in Fact2.
      }        
      { apply leq_trans with (task_max_nonpreemptive_segment tsk).
        - by apply last_of_seq_le_max_of_seq. 
        - rewrite -END; last by done.
          apply ltnW; rewrite ltnS; try done.
            by apply max_distance_in_seq_le_last_element_of_seq; eauto 2. 
      }
    }
  Qed.
      
End RTAforFixedPreemptionPointsModelwithArrivalCurves.