Require Import rt.restructuring.model.processor.ideal.
Require Import rt.restructuring.model.readiness.basic.

Require Import rt.restructuring.analysis.edf.rta.nonpr_reg.response_time_bound.
Require Export rt.restructuring.analysis.basic_facts.preemption.task.preemptive.
Require Export rt.restructuring.analysis.basic_facts.preemption.rtc_threshold.preemptive.

(** Assume we have a fully preemptive model. *)
Require Import rt.restructuring.model.task.preemption.fully_preemptive.

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

(** * RTA for Fully Preemptive EDF Model *)
(** In this section we prove the RTA theorem for the fully preemptive EDF model *)
Section RTAforFullyPreemptiveEDFModelwithArrivalCurves.

  (** 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.

  (** 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 of the arrival sequence ... *)
  Variable sched : schedule (ideal.processor_state Job).
  Hypothesis H_jobs_come_from_arrival_sequence:
    jobs_come_from_arrival_sequence sched arr_seq.

  (** ... 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.

  (** 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 = task_rbf (A + ε) + bound_on_total_hep_workload A (A + F) /\
        F <= 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 fully preemptive scheduling. *)
  Theorem uniprocessor_response_time_bound_fully_preemptive_edf:
    response_time_bounded_by tsk R.
  Proof.
    have BLOCK: blocking_bound  ts tsk = 0.
    { by rewrite /blocking_bound /parameters.task_max_nonpreemptive_segment
                 /fully_preemptive.fully_preemptive_model subnn big1_eq. } 
    eapply uniprocessor_response_time_bound_edf_with_bounded_nonpreemptive_segments with (L0 := L) .
    all: eauto 2 with basic_facts.
    - move => A /andP [LT NEQ].
      specialize (H_R_is_maximum A); feed H_R_is_maximum.
      { by apply/andP; split. }
      move: H_R_is_maximum => [F [FIX BOUND]].
      exists F; split.
      + by rewrite BLOCK add0n subnn subn0. 
      + by rewrite subnn addn0. 
  Qed.

End RTAforFullyPreemptiveEDFModelwithArrivalCurves.