Require Import rt.util.all.
Require Import rt.model.arrival.basic.task rt.model.arrival.basic.job rt.model.priority rt.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 is a safe response-time bound.
     This analysis can be 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.

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

    Hypothesis H_sequential_jobs: sequential_jobs sched.
    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.

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

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

    (* ... and assume that the schedule is a 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.

    (* 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.

    (* Assume that the response-time bounds are larger than task costs. *)
    Hypothesis H_response_time_bounds_ge_cost:
       hp_tsk R,
        (hp_tsk, R) \in hp_bounds R task_cost hp_tsk.

    (* Assume that no deadline is missed by any higher-priority task. *)
    Hypothesis H_interfering_tasks_miss_no_deadlines:
       hp_tsk R,
        (hp_tsk, R) \in hp_bounds R task_deadline hp_tsk.

    (* 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 tsk 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_job_parameters into PARAMS,
                 H_all_jobs_from_taskset into FROMTS,
                 H_valid_task_parameters into TASK_PARAMS,
                 H_constrained_deadlines into RESTR,
                 H_response_time_of_interfering_tasks_is_known into RESP,
                 H_interfering_tasks_miss_no_deadlines into NOMISS,
                 H_response_time_bounds_ge_cost into GE_COST.
          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:SCHED'; 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 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);
            try (by ins); last 2 first;
              [ by ins; apply GE_COST
              | by ins; apply NOMISS
              | by ins; apply TASK_PARAMS
              | by ins; apply RESTR
              | 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 subh1; last by rewrite [R](REC) // leq_addr.
          rewrite -addnBA // subnn addn0.
          move: (NOTCOMP) ⇒ /negP NOTCOMP'.
          rewrite neq_ltn in NOTCOMP.
          move: NOTCOMP ⇒ /orP [LT | BUG]; last first.
            exfalso; rewrite ltnNge in BUG; move: BUG ⇒ /negP BUG; apply BUG.
            by apply cumulative_service_le_job_cost.
          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.

        (* 1) Next, we prove that during the scheduling window of j, any job that is
              scheduled while j is backlogged comes from a different task.
              This follows from the fact that j is the first job not to complete
              by its response-time bound, so previous jobs of j's task must have
              completed by their periods and cannot be pending. *)

        Lemma bertogna_fp_interference_by_different_tasks :
           t j_other,
            job_arrival j t < job_arrival j + R
            arrives_in arr_seq j_other
            backlogged job_arrival job_cost sched j t
            scheduled sched j_other t
            job_task j_other != tsk.
          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_constrained_deadlines into CONSTR,
                 H_previous_jobs_of_tsk_completed into PREV,
                 H_response_time_no_larger_than_deadline into NOMISS.
          movet j_other /andP [LEt GEt] ARRother BACK SCHED.
          apply/eqP; red; intro SAMEtsk.
          move: SCHED ⇒ /existsP [cpu SCHED].
          assert (SCHED': scheduled sched j_other t).
            by apply/existsP; cpu.
          clear SCHED; rename SCHED' into SCHED.
          move: (SCHED) ⇒ PENDING.
          apply scheduled_implies_pending with (job_arrival0 := job_arrival)
                                               (job_cost0 := job_cost) in PENDING; try (by done).
          destruct (ltnP (job_arrival j_other) (job_arrival j)) as [BEFOREother | BEFOREj].
            move: (BEFOREother) ⇒ LT; rewrite -(ltn_add2r R) in LT.
            specialize (PREV j_other ARRother SAMEtsk BEFOREother).
            move: PENDING ⇒ /andP [_ /negP NOTCOMP]; apply NOTCOMP.
            apply completion_monotonic with (t0 := 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 CONSTR; rewrite -JOBtsk FROMTS.
            rewrite -SAMEtsk; apply SPO; try (by done); [ | by rewrite JOBtsk | by apply ltnW].
            by intro EQ; subst j_other; rewrite 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 CONSTR // -JOBtsk FROMTS].
            by red; intros EQtsk; subst j_other; rewrite /backlogged SCHED andbF in BACK.

        (* 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.

        (* 2) 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 (3). *)

        Lemma bertogna_fp_all_cpus_are_busy:
            job_arrival j t < job_arrival j + R
            backlogged job_arrival job_cost sched j t
            count (other_scheduled_task t) ts = num_cpus.
          rename H_valid_task_parameters into PARAMS,
                 H_all_jobs_from_taskset into FROMTS,
                 H_job_of_tsk into JOBtsk,
                 H_sporadic_tasks into SPO,
                 H_valid_job_parameters into JOBPARAMS,
                 H_constrained_deadlines into RESTR,
                 H_hp_bounds_has_interfering_tasks into HAS,
                 H_interfering_tasks_miss_no_deadlines into NOMISS,
                 H_response_time_of_interfering_tasks_is_known into PREV.
          unfold sporadic_task_model, is_response_time_bound_of_task in ×.
          movet /andP [LEt LTt] BACK.
          apply platform_fp_cpus_busy_with_interfering_tasks with (task_cost0 := task_cost)
          (task_period0 := task_period) (task_deadline0 := task_deadline) (job_task0 := job_task)
          (arr_seq0 := arr_seq) (ts0 := ts) (tsk0 := tsk) (higher_eq_priority0 := higher_eq_priority)
            in BACK; try (by done); first by apply PARAMS.
            apply leq_trans with (n := job_arrival j + R); first by done.
            rewrite leq_add2l.
            by apply leq_trans with (n := task_deadline tsk); last by apply RESTR.
            intros j_other tsk_other ARRother JOBother INTERF.
            feed (HAS tsk_other); first by rewrite -JOBother FROMTS.
            move: (HAS INTERF) ⇒ [R' IN].
            apply completion_monotonic with (t0 := job_arrival j_other + R'); try (by done);
              last by apply PREV with (hp_tsk := tsk_other).
              rewrite leq_add2l.
              apply leq_trans with (n := task_deadline tsk_other); first by apply NOMISS.
              by apply RESTR; rewrite -JOBother FROMTS.
            ins; apply completion_monotonic with (t0 := job_arrival j0 + R); try (by done);
              last by apply H_previous_jobs_of_tsk_completed.
            rewrite leq_add2l.
            by apply leq_trans with (n := task_deadline tsk); last by apply RESTR.

        (* 3) 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 (4). *)

        Lemma bertogna_fp_interference_on_all_cpus :
          \sum_(tsk_k <- hp_tasks) x tsk_k = X × num_cpus.
          have DIFFTASK := bertogna_fp_interference_by_different_tasks.
          rename H_work_conserving into WORK, H_respects_FP_policy into FP,
                 H_jobs_come_from_arrival_sequence into SEQ,
                 H_all_jobs_from_taskset into FROMTS, H_job_of_tsk into JOBtsk.
          unfold sporadic_task_model in ×.
          unfold x, X, total_interference, task_interference.
          rewrite -big_mkcond -exchange_big big_distrl /= mul1n.
          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 by rewrite big1 //; ins; rewrite big1.
          rewrite big_mkcond /=.
          rewrite exchange_big /=.
          apply eq_trans with (y := \sum_(cpu < num_cpus) 1); last by simpl_sum_const.
          apply eq_bigr; intros cpu _.
          move: (WORK j t H_j_arrives BACK cpu) ⇒ [j_other /eqP SCHED]; unfold scheduled_on in ×.
          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 (SEQ 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 DIFFTASK with (t := t); try (by done); first by auto.
          by apply/existsP; cpu; apply/eqP.

        (* Before stating the next lemma, let (num_tasks_exceeding delta) be the
           number of interfering tasks whose interference x is larger than delta. *)

        Let num_tasks_exceeding delta := count (fun ix i delta) (hp_tasks).

        (* 4) Now we prove that, for any delta, if (num_task_exceeding delta > 0), then the
              cumulative interference caused by the complementary set of interfering tasks fills
              the remaining, not-completely-full (num_cpus - num_tasks_exceeding delta)
              processors. *)

        Lemma bertogna_fp_interference_in_non_full_processors :
            0 < num_tasks_exceeding delta < num_cpus
            \sum_(i <- hp_tasks | x i < delta) x i delta × (num_cpus - num_tasks_exceeding delta).
          have INV := bertogna_fp_all_cpus_are_busy.
          rename H_all_jobs_from_taskset into FROMTS, H_jobs_come_from_arrival_sequence into FROMSEQ,
                 H_valid_task_parameters into PARAMS, H_job_of_tsk into JOBtsk,
                 H_sporadic_tasks into SPO, H_previous_jobs_of_tsk_completed into BEFOREok,
                 H_response_time_no_larger_than_deadline into NOMISS,
                 H_constrained_deadlines into CONSTR, H_sequential_jobs into SEQ,
                 H_respects_FP_policy into FP, H_hp_bounds_has_interfering_tasks into HASHP,
                 H_interfering_tasks_miss_no_deadlines into NOMISSHP.
          unfold sporadic_task_model in ×.
          movedelta /andP [HAS LT].
          rewrite -has_count in HAS.

          set some_interference_A := fun t
            has (fun tsk_kbacklogged job_arrival job_cost sched j t &&
                              (x tsk_k delta) &&
                              task_is_scheduled job_task sched tsk_k t) hp_tasks.
          set total_interference_B := fun t
              backlogged job_arrival job_cost sched j t ×
              count (fun tsk_k(x tsk_k < delta) &&
                    task_is_scheduled job_task sched tsk_k t) hp_tasks.

          apply leq_trans with ((\sum_(job_arrival j t < job_arrival j + R)
                                some_interference_A t) × (num_cpus - num_tasks_exceeding delta)).
            rewrite leq_mul2r; apply/orP; right.
            move: HAS ⇒ /hasP HAS; destruct HAS as [tsk_a INa LEa].
            apply leq_trans with (n := x tsk_a); first by apply LEa.
            unfold x, task_interference, some_interference_A.
            apply leq_sum_nat; movet /andP [GEt LTt] _.
            destruct (backlogged job_arrival job_cost sched j t) eqn:BACK;
              last by rewrite (eq_bigr (fun x ⇒ 0)); [by simpl_sum_const | by ins].
            destruct ([ cpu, task_scheduled_on job_task sched tsk_a cpu t]) eqn:SCHED;
              last first.
              apply negbT in SCHED; rewrite negb_exists in SCHED; move: SCHED ⇒ /forallP ALL.
              rewrite (eq_bigr (fun x ⇒ 0)); first by simpl_sum_const.
              by intros cpu _; specialize (ALL cpu); apply negbTE in ALL; rewrite ALL.
            move: SCHED ⇒ /existsP [cpu SCHED].
            apply leq_trans with (n := 1); last first.
              rewrite lt0b; apply/hasP; tsk_a; first by done.
              by rewrite LEa 2!andTb; apply/existsP; cpu.
            rewrite (bigD1 cpu) /= // SCHED.
            rewrite (eq_bigr (fun x ⇒ 0)); first by simpl_sum_const; rewrite leq_b1.
            intros cpu' DIFF.
            apply/eqP; rewrite eqb0; apply/negP.
            intros SCHED'.
            move: DIFF ⇒ /negP DIFF; apply DIFF; apply/eqP.
            unfold task_scheduled_on in ×.
            destruct (sched cpu t) as [j1|] eqn:SCHED1; last by done.
            destruct (sched cpu' t) as [j2|] eqn:SCHED2; last by done.
            move: SCHED SCHED' ⇒ /eqP JOB /eqP JOB'.
            subst tsk_a; symmetry in JOB'.
            have ARR1: arrives_in arr_seq j1.
              by apply (FROMSEQ j1 t); apply/existsP; cpu; apply/eqP.
            have ARR2: arrives_in arr_seq j2.
              by apply (FROMSEQ j2 t); apply/existsP; cpu'; apply/eqP.
            assert (PENDING1: pending job_arrival job_cost sched j1 t).
              apply scheduled_implies_pending; try by done.
              by apply/existsP; cpu; apply/eqP.
            assert (PENDING2: pending job_arrival job_cost sched j2 t).
              apply scheduled_implies_pending; try by done.
              by apply/existsP; cpu'; apply/eqP.
            assert (BUG: j1 = j2).
              destruct (job_task j1 == tsk) eqn:SAMEtsk.
                move: SAMEtsk ⇒ /eqP SAMEtsk.
                move: (PENDING1) ⇒ SAMEjob.
                apply platform_fp_no_multiple_jobs_of_tsk with (task_cost0 := task_cost)
                  (arr_seq0 := arr_seq) (task_period0 := task_period) (task_deadline0 := task_deadline)
                  (job_task0 := job_task) (tsk0 := tsk) (j0 := j) in SAMEjob; try (by done);
                  [ | by apply PARAMS | |]; last 2 first.
                    apply (leq_trans LTt); rewrite leq_add2l.
                    by apply leq_trans with (n := task_deadline tsk); last by apply CONSTR.
                    intros j0 ARR0 JOB0 LT0.
                    apply completion_monotonic with (t0 := job_arrival j0 + R); try (by done);
                      last by apply BEFOREok.
                    rewrite leq_add2l.
                    by apply leq_trans with (n := task_deadline tsk); last by apply CONSTR.
                move: BACK ⇒ /andP [_ /negP NOTSCHED]; exfalso; apply NOTSCHED.
                by rewrite -SAMEjob; apply/existsP; cpu; apply/eqP.
                assert (INTERF: is_hp_task (job_task j1)).
                  apply/andP; split; last by rewrite SAMEtsk.
                  rewrite -JOBtsk; apply FP with (t := t); try (by done).
                  by apply/existsP; cpu; apply/eqP.
                apply platform_fp_no_multiple_jobs_of_interfering_tasks with
                  (job_arrival0 := job_arrival) (arr_seq0 := arr_seq) (task_period0 := task_period)
                  (tsk0 := tsk) (higher_eq_priority0 := higher_eq_priority)
                  (job_cost0 := job_cost) (job_task0 := job_task) (sched0 := sched) (t0 := t);
                  rewrite ?JOBtsk ?SAMEtsk //.
                  intros j0 tsk0 ARR0 JOB0 INTERF0.
                  feed (HASHP tsk0); first by rewrite -JOB0 FROMTS.
                  move: (HASHP INTERF0) ⇒ [R0 IN0].
                  apply completion_monotonic with (t0 := job_arrival j0 + R0); try (by done);
                    last by eapply H_response_time_of_interfering_tasks_is_known; first by apply IN0.
                  rewrite leq_add2l.
                  by apply leq_trans with (n := task_deadline tsk0);
                    [by apply NOMISSHP | by apply CONSTR; rewrite -JOB0 FROMTS].
            by subst j2; apply SEQ with (j := j1) (t := t).

          apply leq_trans with (\sum_(job_arrival j t < job_arrival j + R)
                                     total_interference_B t).
            rewrite big_distrl /=.
            apply leq_sum_nat; movet LEt _.
            unfold some_interference_A, total_interference_B.
            destruct (backlogged job_arrival job_cost sched j t) eqn:BACK;
              [rewrite mul1n /= | by rewrite has_pred0 //].

            destruct (has (fun tsk_k : sporadic_task(delta x tsk_k) &&
                       task_is_scheduled job_task sched tsk_k t) hp_tasks) eqn:HAS';
              last by done.
            rewrite mul1n; move: HAS ⇒ /hasP [tsk_k INk LEk].
            unfold num_tasks_exceeding.
            apply leq_trans with (n := num_cpus -
                         count (fun i(x i delta) &&
                            task_is_scheduled job_task sched i t) hp_tasks).
              apply leq_sub2l.
              rewrite -2!sum1_count big_mkcond /=.
              rewrite [\sum_(_ <- _ | _ _)_]big_mkcond /=.
              apply leq_sum; intros i _.
              by destruct (task_is_scheduled job_task sched i t);
                [by rewrite andbT | by rewrite andbF].
            rewrite -count_filter -[count _ hp_tasks]count_filter.
            eapply leq_trans with (n := count (predC (fun tskdelta x tsk)) _);
              last by apply eq_leq, eq_in_count; red; ins; rewrite ltnNge.
            rewrite leq_subLR count_predC size_filter.
            apply leq_trans with (n := count (other_scheduled_task t) ts);
              [by rewrite INV | by rewrite count_filter].
            unfold x at 2, total_interference_B.
            rewrite exchange_big /=; apply leq_sum; intros t _.
            destruct (backlogged job_arrival job_cost sched j t) eqn:BACK; last by ins.
            rewrite mul1n -sum1_count.
            rewrite big_mkcond [\sum_(i <- hp_tasks | _ < _) _]big_mkcond /=.
            apply leq_sum_seq; movetsk_k IN _.
            destruct (x tsk_k < delta); [rewrite andTb | by rewrite andFb].
            destruct (task_is_scheduled job_task sched tsk_k t) eqn:SCHED; last by done.
            move: SCHED ⇒ /existsP [cpu SCHED].
            by rewrite (bigD1 cpu) /= // SCHED.

        (* 5) Based on lemma (4), we prove that, for any interval delta, if the sum of per-task
              interference exceeds (delta * num_cpus), the same applies for the
              sum of the minimum of the interference and delta. *)

        Lemma bertogna_fp_minimum_exceeds_interference :
            \sum_(tsk_k <- hp_tasks) x tsk_k delta × num_cpus
               \sum_(tsk_k <- hp_tasks) minn (x tsk_k) delta
               delta × num_cpus.
          intros delta SUMLESS.
          set more_interf := fun tsk_kx tsk_k delta.
          rewrite [\sum_(_ <- _) minn _ _](bigID more_interf) /=.
          unfold more_interf, minn.
          rewrite [\sum_(_ <- _ | delta _)_](eq_bigr (fun idelta));
            last by intros i COND; rewrite leqNgt in COND; destruct (delta > x i).
          rewrite [\sum_(_ <- _ | ~~_)_](eq_big (fun ix i < delta)
                                                (fun ix i));
            [| by red; ins; rewrite ltnNge
             | by intros i COND; rewrite -ltnNge in COND; rewrite COND].

          (* Case 1: num_tasks_exceeding = 0 *)
          destruct (~~ has (fun idelta x i) hp_tasks) eqn:HASa.
            rewrite [\sum_(_ <- _ | _ _) _]big_hasC; last by apply HASa.
            rewrite big_seq_cond; move: HASa ⇒ /hasPn HASa.
            rewrite add0n (eq_bigl (fun i(i \in hp_tasks) && true));
              last by red; intros tsk_k; destruct (tsk_k \in hp_tasks) eqn:INk;
                [by rewrite andTb ltnNge; apply HASa | by rewrite andFb].
            by rewrite -big_seq_cond.
          } apply negbFE in HASa.

          (* Case 2: num_tasks_exceeding >= num_cpus *)
          destruct (num_tasks_exceeding delta num_cpus) eqn:CARD.
            apply leq_trans with (delta × num_tasks_exceeding delta);
              first by rewrite leq_mul2l; apply/orP; right.
            unfold num_tasks_exceeding; rewrite -sum1_count big_distrr /=.
            rewrite -[\sum_(_ <- _ | _) _]addn0.
            by apply leq_add; [by apply leq_sum; ins; rewrite muln1|by ins].
          } apply negbT in CARD; rewrite -ltnNge in CARD.

          (* Case 3: num_tasks_exceeding < num_cpus *)
          rewrite big_const_seq iter_addn addn0; fold num_tasks_exceeding.
          apply leq_trans with (n := delta × num_tasks_exceeding delta +
                                     delta × (num_cpus - num_tasks_exceeding delta));
            first by rewrite -mulnDr subnKC //; apply ltnW.
          rewrite leq_add2l; apply bertogna_fp_interference_in_non_full_processors.
          by apply/andP; split; first by rewrite -has_count.

        (* 6) Next, using lemmas (0), (3) and (5) 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)
            minn (x tsk_k) (R - task_cost tsk + 1) >
          total_interference_bound_fp task_cost task_period tsk hp_bounds R.
          have EXCEEDS := bertogna_fp_minimum_exceeds_interference.
          have ALLBUSY := bertogna_fp_interference_on_all_cpus.
          have TOOMUCH := bertogna_fp_too_much_interference.
          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) minn (x tsk_k) (R - task_cost tsk + 1));
            last first.
            rewrite (eq_bigr (fun iminn (x (fst i)) (R - task_cost tsk + 1)));
              last by ins; destruct i.
            have MAP := @big_map _ 0 addn _ _ (fun xfst x) hp_bounds (fun xtrue) (fun yminn (x y) (R - task_cost tsk + 1)).
            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 - _]subh1; last by rewrite REC; apply leq_addr.
          rewrite leq_divRL; last by apply H_at_least_one_cpu.
          apply EXCEEDS.
          apply leq_trans with (n := X × num_cpus); last by rewrite ALLBUSY.
          by rewrite leq_mul2r; apply/orP; right; apply TOOMUCH.

        (* 7) 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
            (minn (x tsk_k) (R - task_cost tsk + 1) >
              minn (workload_bound tsk_k R_k) (R - task_cost tsk + 1)).
          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
                               (minn (x tsk_k) (R - task_cost tsk + 1) >
                                minn (workload_bound tsk_k R_k)(R - task_cost tsk + 1)))
              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 iminn (x (fst i)) (R - task_cost tsk + 1)));
                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 WORKLOAD := bertogna_fp_workload_bounds_interference.
      have EX := bertogna_fp_exists_task_that_exceeds_bound.
      rename H_response_time_bounds_ge_cost into GE_COST,
             H_interfering_tasks_miss_no_deadlines into NOMISS,
             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]]].
      unfold minn at 1 in LTmin.
      specialize (WORKLOAD j tsk_k R_k HPk).
      destruct (W task_cost task_period tsk_k R_k R < R - task_cost tsk + 1); rewrite leq_min in LTmin;
        last by move: LTmin ⇒ /andP [_ BUG]; rewrite ltnn in BUG.
      move: LTmin ⇒ /andP [BUG _]; des.
      apply leq_trans with (p := W task_cost task_period tsk_k R_k R) in BUG; last by done.
      by rewrite ltnn in BUG.

  End ResponseTimeBound.

End ResponseTimeAnalysisFP.