Require Import prosa.classic.util.all.
Require Import prosa.classic.model.arrival.basic.task prosa.classic.model.arrival.basic.job prosa.classic.model.arrival.basic.task_arrival
Require Import
Require Import
Require Import
From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq fintype bigop div path.

Module ResponseTimeAnalysisFP.

  Export Job SporadicTaskset ScheduleOfSporadicTask Workload Interference
         InterferenceBoundFP Platform Schedulability ResponseTime
         Priority TaskArrival WorkloadBound ConstrainedDeadlines.

  (* In this section, we prove that any fixed point in Bertogna and
     Cirinei's RTA for FP scheduling modified to consider (potentially)
     parallel jobs yields a safe response-time bound. This is an extension
     of the analysis found in Chapter 18.2 of Baruah et al.'s book
     Multiprocessor Scheduling for Real-time Systems. *)

  Section ResponseTimeBound.

    Context {sporadic_task: eqType}.
    Variable task_cost: sporadic_task time.
    Variable task_period: sporadic_task time.
    Variable task_deadline: sporadic_task time.

    Context {Job: eqType}.
    Variable job_arrival: Job time.
    Variable job_cost: Job time.
    Variable job_deadline: Job time.
    Variable job_task: Job sporadic_task.

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

    (* ... in which jobs arrive sporadically and have valid parameters. *)
    Hypothesis H_sporadic_tasks:
      sporadic_task_model task_period job_arrival job_task arr_seq.
    Hypothesis H_valid_job_parameters:
        arrives_in arr_seq j
        valid_sporadic_job task_cost task_deadline job_cost job_deadline job_task j.

    (* Assume that we have a task set where all tasks have valid
       parameters and constrained deadlines, ... *)

    Variable ts: taskset_of sporadic_task.
    Hypothesis H_valid_task_parameters:
      valid_sporadic_taskset task_cost task_period task_deadline ts.
    Hypothesis H_constrained_deadlines:
       tsk, tsk \in ts task_deadline tsk task_period tsk.

    (* ... and that all jobs in the arrival sequence come from the task set. *)
    Hypothesis H_all_jobs_from_taskset:
       j, arrives_in arr_seq j job_task j \in ts.

    (* Next, consider any schedule such that...*)
    Variable num_cpus: nat.
    Variable sched: schedule Job num_cpus.
    Hypothesis H_jobs_come_from_arrival_sequence:
      jobs_come_from_arrival_sequence sched arr_seq.

    (* do not execute before their arrival times nor longer
       than their execution costs. *)

    Hypothesis H_jobs_must_arrive_to_execute:
      jobs_must_arrive_to_execute job_arrival sched.
    Hypothesis H_completed_jobs_dont_execute:
      completed_jobs_dont_execute job_cost sched.

    (* Consider a given FP policy, ... *)
    Variable higher_eq_priority: FP_policy sporadic_task.

    (* ... and assume that the schedule is an APA work-conserving
       schedule that respects this policy. *)

    Hypothesis H_work_conserving: work_conserving job_arrival job_cost arr_seq sched.
    Hypothesis H_respects_FP_policy:
      respects_FP_policy job_arrival job_cost job_task arr_seq sched higher_eq_priority.

    (* Assume that there exists at least one processor. *)
    Hypothesis H_at_least_one_cpu: num_cpus > 0.

    (* Let's define some local names to avoid passing many parameters. *)
    Let no_deadline_is_missed_by_tsk (tsk: sporadic_task) :=
      task_misses_no_deadline job_arrival job_cost job_deadline job_task arr_seq sched tsk.
    Let response_time_bounded_by (tsk: sporadic_task) :=
      is_response_time_bound_of_task job_arrival job_cost job_task arr_seq sched tsk.

    (* Next, we consider the response-time recurrence.
       Let tsk be a task in ts that is to be analyzed. *)

    Variable tsk: sporadic_task.
    Hypothesis task_in_ts: tsk \in ts.

    (* Let is_hp_task denote whether a task is a higher-priority task
       (with respect to tsk). *)

    Let is_hp_task := higher_priority_task higher_eq_priority tsk.

    (* Assume a response-time bound is known... *)
    Let task_with_response_time := (sporadic_task × time)%type.
    Variable hp_bounds: seq task_with_response_time.
    Hypothesis H_response_time_of_interfering_tasks_is_known:
       hp_tsk R,
        (hp_tsk, R) \in hp_bounds
        response_time_bounded_by hp_tsk R.

    (* ... for every higher-priority task. *)
    Hypothesis H_hp_bounds_has_interfering_tasks:
        hp_tsk \in ts
        is_hp_task hp_tsk
           R, (hp_tsk, R) \in hp_bounds.

    (* Let R be the fixed point of Bertogna and Cirinei's recurrence, ...*)
    Variable R: time.
    Hypothesis H_response_time_recurrence_holds :
      R = task_cost tsk +
            (total_interference_bound_fp task_cost task_period hp_bounds R)

    (* ... and assume that R is no larger than the deadline of tsk.*)
    Hypothesis H_response_time_no_larger_than_deadline:
      R task_deadline tsk.

    (* In order to prove that R is a response-time bound, we first provide some lemmas. *)
    Section Lemmas.

      (* Consider any job j of tsk. *)
      Variable j: Job.
      Hypothesis H_j_arrives: arrives_in arr_seq j.
      Hypothesis H_job_of_tsk: job_task j = tsk.

      (* Assume that job j is the first job of tsk not to complete by the response time bound. *)
      Hypothesis H_j_not_completed: ~~ completed job_cost sched j (job_arrival j + R).
      Hypothesis H_previous_jobs_of_tsk_completed :
          arrives_in arr_seq j0
          job_task j0 = tsk
          job_arrival j0 < job_arrival j
          completed job_cost sched j0 (job_arrival j0 + R).

      (* Let's call x the interference incurred by job j due to tsk_other, ...*)
      Let x (tsk_other: sporadic_task) :=
        task_interference job_arrival job_cost job_task sched j
                          tsk_other (job_arrival j) (job_arrival j + R).

      (* ...and X the total interference incurred by job j due to any task. *)
      Let X := total_interference job_arrival job_cost sched j (job_arrival j) (job_arrival j + R).

      (* Recall Bertogna and Cirinei's workload bound. *)
      Let workload_bound (tsk_other: sporadic_task) (R_other: time) :=
        W task_cost task_period tsk_other R_other R.

      (* Let hp_tasks denote the set of higher-priority tasks. *)
      Let hp_tasks := [seq tsk_other <- ts | is_hp_task tsk_other].

      (* Now we establish results about the higher-priority tasks. *)
      Section LemmasAboutHPTasks.

        (* Let (tsk_other, R_other) be any pair of higher-priority task and
           response-time bound computed in previous iterations. *)

        Variable tsk_other: sporadic_task.
        Variable R_other: time.
        Hypothesis H_response_time_of_tsk_other: (tsk_other, R_other) \in hp_bounds.

        (* Since tsk_other cannot interfere more than it executes, we show that
           the interference caused by tsk_other is no larger than workload bound W. *)

        Lemma bertogna_fp_workload_bounds_interference :
          x tsk_other workload_bound tsk_other R_other.
          unfold response_time_bounded_by, is_response_time_bound_of_task,
                 completed, completed_jobs_dont_execute, valid_sporadic_job in ×.
          rename H_valid_task_parameters into TASK_PARAMS,
                 H_all_jobs_from_taskset into FROMTS,
                 H_response_time_of_interfering_tasks_is_known into RESP.
          unfold x, workload_bound.
          destruct ([ t: 'I_(job_arrival j + R),
                       task_is_scheduled job_task sched tsk_other t]) eqn: SCHED;
            last first.
            apply negbT in SCHED; rewrite negb_exists in SCHED.
            move: SCHED ⇒ /forallP SCHED.
            apply leq_trans with (n := 0); last by done.
            apply leq_trans with (n := \sum_(job_arrival j t < job_arrival j + R) 0);
              last by rewrite big1.
            apply leq_sum_nat; movei /andP [_ LTi] _.
            specialize (SCHED (Ordinal LTi)).
            rewrite negb_exists in SCHED; move: SCHED ⇒ /forallP SCHED.
            rewrite big1 //; intros cpu _.
            specialize (SCHED cpu); apply negbTE in SCHED.
            by rewrite SCHED andbF.
          move: SCHED ⇒ /existsP [t /existsP [cpu SCHED]].
          unfold task_scheduled_on in SCHED.
          destruct (sched cpu t) as [j0 |] eqn:SCHED0; last by done.
          assert (INts: tsk_other \in ts).
            move: SCHED ⇒ /eqP <-; apply FROMTS, (H_jobs_come_from_arrival_sequence j0 t).
            by apply/existsP; cpu; apply/eqP.
          apply leq_trans with (n := workload job_task sched tsk_other
                                              (job_arrival j) (job_arrival j + R));
            first by apply task_interference_le_workload.
          by ( try ( apply workload_bounded_by_W with (task_deadline0 := task_deadline) (arr_seq0 := arr_seq)
              (job_arrival0 := job_arrival) (job_cost0 := job_cost) (job_deadline0 := job_deadline) ) ||
          apply workload_bounded_by_W with (task_deadline := task_deadline) (arr_seq := arr_seq)
              (job_arrival := job_arrival) (job_cost := job_cost) (job_deadline := job_deadline));
            try (by ins);
              [ by ins; apply TASK_PARAMS
              | by ins; apply RESP with (hp_tsk := tsk_other)].

      End LemmasAboutHPTasks.

      (* Next we prove some lemmas that help to derive a contradiction.*)
      Section DerivingContradiction.

        (* 0) Since job j did not complete by its response time bound, it follows that
              the total interference X >= R - e_k + 1. *)

        Lemma bertogna_fp_too_much_interference : X R - task_cost tsk + 1.
          rename H_completed_jobs_dont_execute into COMP,
                 H_valid_job_parameters into PARAMS,
                 H_response_time_recurrence_holds into REC,
                 H_job_of_tsk into JOBtsk,
                 H_j_not_completed into NOTCOMP.
          unfold completed, valid_sporadic_job in ×.
          unfold X, total_interference; rewrite addn1.
          rewrite -(ltn_add2r (task_cost tsk)).
          rewrite addnBAC; last by rewrite REC leq_addr.
          rewrite -addnBA // subnn addn0.
          move: (NOTCOMP) ⇒ /negP NOTCOMP'.
          rewrite -ltnNge in NOTCOMP.
          apply leq_ltn_trans with (n := (\sum_(job_arrival j t < job_arrival j + R)
                                       backlogged job_arrival job_cost sched j t) +
                                     service sched j (job_arrival j + R)); last first.
            rewrite -addn1 -addnA leq_add2l addn1.
            apply leq_trans with (n := job_cost j); first by done.
            by specialize (PARAMS j H_j_arrives); des; rewrite -JOBtsk.
          unfold service; rewrite service_before_arrival_eq_service_during //.
          rewrite -big_split /=.
          apply leq_trans with (n := \sum_(job_arrival j i < job_arrival j + R) 1);
            first by rewrite big_const_nat iter_addn mul1n addn0 addKn.
          rewrite big_nat_cond [\sum_(_ _ < _ | true) _]big_nat_cond.
          apply leq_sum; movei /andP [/andP [GEi LTi] _].
          destruct (backlogged job_arrival job_cost sched j i) eqn:BACK;
            first by rewrite -addn1 addnC; apply leq_add.
          apply negbT in BACK.
          rewrite add0n lt0n -not_scheduled_no_service negbK.
          rewrite /backlogged negb_and negbK in BACK.
          move: BACK ⇒ /orP [/negP NOTPENDING | SCHED]; last by done.
          exfalso; apply NOTPENDING; unfold pending; apply/andP; split; first by done.
          apply/negP; red; intro BUG; apply NOTCOMP'.
          by apply completion_monotonic with (t := i); try (by done); apply ltnW.

        (* Let's define a predicate to identify the other tasks that are scheduled. *)
        Let other_scheduled_task (t: time) (tsk_other: sporadic_task) :=
          task_is_scheduled job_task sched tsk_other t &&
          is_hp_task tsk_other.

        (* 1) Now we prove that, at all times that j is backlogged, the number
              of tasks other than tsk that are scheduled is exactly the number
              of processors in the system. This is required to prove lemma (2). *)

        Lemma bertogna_fp_all_cpus_are_busy:
          \sum_(tsk_k <- hp_tasks) x tsk_k = X × num_cpus.
          rename H_all_jobs_from_taskset into FROMTS, H_valid_task_parameters into PARAMS,
                 H_job_of_tsk into JOBtsk, H_sporadic_tasks into SPO,
                 H_work_conserving into WORK, H_jobs_come_from_arrival_sequence into FROMarr,
                 H_constrained_deadlines into RESTR,
                 H_respects_FP_policy into FP,
                 H_previous_jobs_of_tsk_completed into BEFOREok,
                 H_response_time_no_larger_than_deadline into NOMISS.
          unfold sporadic_task_model, respects_FP_policy,
                 respects_JLDP_policy, FP_to_JLDP in ×.
          unfold x, X, total_interference, task_interference.
          rewrite -big_mkcond -exchange_big big_distrl /=.
          rewrite [\sum_(_ _ < _ | backlogged _ _ _ _ _) _]big_mkcond.
          apply eq_big_nat; movet /andP [GEt LTt].
          destruct (backlogged job_arrival job_cost sched j t) eqn:BACK; last first.
            rewrite (eq_bigr (fun i ⇒ 0));
              first by rewrite big_const_seq iter_addn mul0n addn0.
            by intros i _; rewrite (eq_bigr (fun i ⇒ 0));
              first by rewrite big_const_seq iter_addn mul0n addn0.
          rewrite big_mkcond mul1n /=.
          rewrite exchange_big /=.
          apply eq_trans with (y := \sum_(cpu < num_cpus) 1);
            last by rewrite big_const_ord iter_addn mul1n addn0.
          apply eq_bigr; intros cpu _.
          specialize (WORK j t H_j_arrives BACK cpu); des.
          move: WORK ⇒ /eqP SCHED.
          rewrite (bigD1_seq (job_task j_other)) /=; last by rewrite filter_uniq; destruct ts.
            rewrite (eq_bigr (fun i ⇒ 0));
              last by intros i DIFF; rewrite /task_scheduled_on SCHED;apply/eqP;rewrite eqb0 eq_sym.
            rewrite big_const_seq iter_addn mul0n 2!addn0; apply/eqP; rewrite eqb1.
            by unfold task_scheduled_on; rewrite SCHED.
          have ARRother: arrives_in arr_seq j_other.
            by apply (FROMarr j_other t); apply/existsP; cpu; apply/eqP.
          rewrite mem_filter; apply/andP; split; last by apply FROMTS.
          apply/andP; split.
            rewrite -JOBtsk; apply FP with (t := t); try (by done).
            by apply/existsP; cpu; apply/eqP.
            apply/eqP; intro SAMEtsk.
            assert (SCHED': scheduled sched j_other t).
              unfold scheduled, scheduled_on.
              by apply/existsP; cpu; rewrite SCHED.
            } clear SCHED; rename SCHED' into SCHED.
            move: (SCHED) ⇒ PENDING.
            try ( apply scheduled_implies_pending with (job_arrival0 := job_arrival)
                                                 (job_cost0 := job_cost) in PENDING; try (by done) ) ||
            apply scheduled_implies_pending with (job_arrival := job_arrival)
                                                 (job_cost := job_cost) in PENDING; try (by done).
            destruct (ltnP (job_arrival j_other) (job_arrival j)) as [BEFOREother | BEFOREj].
              specialize (BEFOREok j_other ARRother SAMEtsk BEFOREother).
              move: PENDING ⇒ /andP [_ /negP NOTCOMP]; apply NOTCOMP.
              try ( apply completion_monotonic with (t0 := job_arrival j_other + R); try (by done) ) ||
              apply completion_monotonic with (t := job_arrival j_other + R); try (by done).
              apply leq_trans with (n := job_arrival j); last by done.
              apply leq_trans with (n := job_arrival j_other + task_deadline tsk);
              first by rewrite leq_add2l; apply NOMISS.
              apply leq_trans with (n := job_arrival j_other + task_period tsk);
                first by rewrite leq_add2l; apply RESTR; rewrite -JOBtsk FROMTS.
              rewrite -SAMEtsk; apply SPO; try (by done); [ | by rewrite JOBtsk | by apply ltnW].
              by red; intro EQ; rewrite EQ ltnn in BEFOREother.
              move: PENDING ⇒ /andP [ARRIVED _].
              exploit (SPO j j_other); try (by done); [ | by rewrite SAMEtsk | ]; last first.
                apply/negP; rewrite -ltnNge.
                apply leq_ltn_trans with (n := t); first by done.
                apply leq_trans with (n := job_arrival j + R); first by done.
                by rewrite leq_add2l; apply leq_trans with (n := task_deadline tsk);
                [by apply NOMISS | by rewrite JOBtsk RESTR // -JOBtsk FROMTS].
              by red; intros EQjob; rewrite EQjob /backlogged SCHED andbF in BACK.

        (* 2) Next, using lemmas (0) and (1) we prove that the reduction-based
              interference bound is not enough to cover the sum of the minima over all tasks
              (artifact of the proof by contradiction). *)

        Lemma bertogna_fp_sum_exceeds_total_interference:
          \sum_((tsk_k, R_k) <- hp_bounds)
           x tsk_k > total_interference_bound_fp task_cost task_period hp_bounds R.
          have TOOMUCH := bertogna_fp_too_much_interference.
          have ALLBUSY := bertogna_fp_all_cpus_are_busy.
          rename H_hp_bounds_has_interfering_tasks into HAS,
                 H_response_time_recurrence_holds into REC.
          apply leq_trans with (n := \sum_(tsk_k <- hp_tasks) x tsk_k);
              last first.
            rewrite (eq_bigr (fun ix (fst i))); last by ins; destruct i.
            have MAP := @big_map _ 0 addn _ _ (fun xfst x) hp_bounds (fun xtrue) (fun y ⇒ (x y)).
            rewrite -MAP.
            apply leq_sum_sub_uniq; first by apply filter_uniq; destruct ts.
            red; movetsk0 IN0.
            rewrite mem_filter in IN0; move: IN0 ⇒ /andP [INTERF0 IN0].
            feed (HAS tsk0); first by done.
            move: (HAS INTERF0) ⇒ [R0 IN].
            by (tsk0, R0).
          apply ltn_div_trunc with (d := num_cpus);
            first by apply H_at_least_one_cpu.
          rewrite -(ltn_add2l (task_cost tsk)) -REC.
          rewrite -addn1 -leq_subLR.
          rewrite -[R + 1 - _]addnBAC; last by rewrite REC; apply leq_addr.
          rewrite leq_divRL; last by apply H_at_least_one_cpu.
          rewrite ALLBUSY.
          by rewrite leq_mul2r; apply/orP; right; apply TOOMUCH.

        (* 3) After concluding that the sum of the minima exceeds (R - e_i + 1),
              we prove that there exists a tuple (tsk_k, R_k) that satisfies
              min (x_k, R - e_i + 1) > min (W_k, R - e_i + 1).
              This implies that x_k > W_k, which is of course a contradiction,
              since W_k is a valid task interference bound. *)

        Lemma bertogna_fp_exists_task_that_exceeds_bound :
           tsk_k R_k,
            (tsk_k, R_k) \in hp_bounds
            x tsk_k > workload_bound tsk_k R_k.
          have SUM := bertogna_fp_sum_exceeds_total_interference.
          rename H_hp_bounds_has_interfering_tasks into HASHP.
          assert (HAS: has (fun tup : task_with_response_time
                             let (tsk_k, R_k) := tup in
                               x tsk_k > workload_bound tsk_k R_k)
            apply/negP; unfold not; intro NOTHAS.
            move: NOTHAS ⇒ /negP /hasPn ALL.
            rewrite -[_ < _]negbK in SUM.
            move: SUM ⇒ /negP SUM; apply SUM; rewrite -leqNgt.
            rewrite (eq_bigr (fun ix (fst i))); last by ins; destruct i.
            unfold total_interference_bound_fp.
            rewrite big_seq_cond.
            rewrite [\sum_(_ <- _ | true)_]big_seq_cond.
            apply leq_sum.
            intros p; rewrite andbT; intros IN.
            by specialize (ALL p IN); destruct p; rewrite leqNgt.
          move: HAS ⇒ /hasP HAS; destruct HAS as [[tsk_k R_k] HPk MINk]; tsk_k, R_k.
          by repeat split.

      End DerivingContradiction.

    End Lemmas.

    (* Using the lemmas above, we prove that R bounds the response time of task tsk. *)
    Theorem bertogna_cirinei_response_time_bound_fp :
      response_time_bounded_by tsk R.
      have EX := bertogna_fp_exists_task_that_exceeds_bound.
      have BOUND := bertogna_fp_workload_bounds_interference.
      rename H_response_time_recurrence_holds into REC,
             H_response_time_of_interfering_tasks_is_known into RESP,
             H_hp_bounds_has_interfering_tasks into HAS,
             H_response_time_no_larger_than_deadline into LEdl.
      intros j ARRj JOBtsk.

      (* First, rewrite the claim in terms of the *absolute* response-time bound (arrival + R) *)
      remember (job_arrival j + R) as ctime.

      (* Now, we apply strong induction on the absolute response-time bound. *)
      generalize dependent j.
      induction ctime as [ctime IH] using strong_ind.

      intros j ARRj JOBtsk EQc; subst ctime.

      (* First, let's simplify the induction hypothesis. *)
      assert (BEFOREok: j0,
                          arrives_in arr_seq j0
                          job_task j0 = tsk
                          job_arrival j0 < job_arrival j
                          service sched j0 (job_arrival j0 + R) job_cost j0).
        by ins; apply IH; try (by done); rewrite ltn_add2r.
      } clear IH.

      unfold response_time_bounded_by, is_response_time_bound_of_task,
             completed, completed_jobs_dont_execute, valid_sporadic_job in ×.

      (* Now we start the proof. Assume by contradiction that job j
         is not complete at time (job_arrival j + R). *)

      destruct (completed job_cost sched j (job_arrival j + R)) eqn:NOTCOMP;
        first by done.
      apply negbT in NOTCOMP; exfalso.

      (* We derive a contradiction using the previous lemmas. *)
      specialize (EX j ARRj JOBtsk NOTCOMP BEFOREok).
      destruct EX as [tsk_k [R_k [HPk LTmin]]].
      specialize (BOUND j tsk_k R_k HPk).
      by apply (leq_ltn_trans BOUND) in LTmin; rewrite ltnn in LTmin.

  End ResponseTimeBound.

End ResponseTimeAnalysisFP.