Library prosa.analysis.abstract.ideal.iw_instantiation

Throughout this file, we assume ideal uni-processor schedules.

JLFP instantiation of Interference and Interfering Workload for ideal uni-processor.

In this module we instantiate functions Interference and Interfering Workload for ideal uni-processor schedulers with an arbitrary JLFP-policy that satisfies the sequential-tasks hypothesis. We also prove equivalence of Interference and Interfering Workload to the more conventional notions of service or workload.
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 `{JobArrival Job}.
  Context `{JobCost Job}.

Consider any valid arrival sequence with consistent arrivals ...
... and any ideal uni-processor schedule of this arrival sequence ...
... where jobs do not execute before their arrival or after completion.
Consider a JLFP-policy that indicates a higher-or-equal priority relation, and assume that this relation is reflexive and transitive.
We also assume that the policy respects sequential tasks, meaning that later-arrived jobs of a task don't have higher priority than earlier-arrived jobs of the same task.
Let tsk be any task.
  Variable tsk : Task.

Interference and Interfering Workload

In the following, we introduce definitions of interference, interfering workload and a function that bounds cumulative interference.

Instantiation of Interference

Before we define the notion of interference, we need to recall the definition of priority inversion. We say that job j is incurring a priority inversion at time t if there exists a job with lower priority that executes at time t. In order to simplify things, we ignore the fact that according to this definition a job can incur priority inversion even before its release (or after completion). All such (potentially bad) cases do not cause problems, as each job is analyzed only within the corresponding busy interval where the priority inversion behaves in the expected way.
We say that job j incurs interference at time t iff it cannot execute due to a higher-or-equal-priority job being scheduled, or if it incurs a priority inversion.
  #[local,program] Instance ideal_jlfp_interference : Interference Job :=
    {
      interference (j : Job) (t : instant) :=
        priority_inversion arr_seq sched j t
        || another_hep_job_interference arr_seq sched j t
    }.

Instantiation of Interfering Workload

The interfering workload, in turn, is defined as the sum of the priority inversion predicate and interfering workload of jobs with higher or equal priority.
  #[local,program] Instance ideal_jlfp_interfering_workload : InterferingWorkload Job :=
    {
      interfering_workload (j : Job) (t : instant) :=
        priority_inversion arr_seq sched j t
        + other_hep_jobs_interfering_workload arr_seq j t
    }.

Equivalences

In this section we prove useful equivalences between the definitions obtained by instantiation of definitions from the Abstract RTA module (interference and interfering workload) and definitions corresponding to the conventional concepts.
Instantiated functions of interference and interfering workload usually do not have any useful lemmas about them. However, it is possible to prove their equivalence to the more conventional notions like service or workload. Next, we prove the equivalence between the instantiations and conventional notions as well as a few useful rewrite rules.
  Section Equivalences.

In the following subsection, we prove properties of the introduced functions under the assumption that the schedule is idle.
    Section IdleSchedule.

Consider a time instant t ...
      Variable t : instant.

... and assume that the schedule is idle at t.
      Hypothesis H_idle : ideal_is_idle sched t.

We prove that in this case: ...
... there is no interference, ...
      Lemma no_interference_when_idle :
         j, ~~ interference j t.
      Proof.
        movej; apply/negP.
        rewrite /interference /ideal_jlfp_interference ⇒ /orP [PI | HEPI].
        - have [j' schj'] : j' : Job, scheduled_at sched j' t
            by exact/priority_inversion_scheduled_at.
          exact: ideal_sched_implies_not_idle schj' H_idle.
        - by move: HEPI ⇒ /negPn; rewrite no_hep_job_interference_when_idle // is_idle_def //.
      Qed.

... as well as no interference for tsk.
      Lemma no_task_interference_when_idle :
         j, ~~ task_interference arr_seq sched j t.
      Proof.
        movej; rewrite /task_interference /cond_interference.
        by rewrite negb_and; apply/orP; right; apply no_interference_when_idle.
      Qed.

    End IdleSchedule.

Next, we prove properties of the introduced functions under the assumption that the scheduler is not idle.
    Section ScheduledJob.

Consider a job j of task tsk. In this subsection, job j is deemed to be the main job with respect to which the functions are computed.
      Variable j : Job.
      Hypothesis H_j_tsk : job_of_task tsk j.

Consider a time instant t.
      Variable t : instant.

In the next subsection, we consider a case when a job j' from the same task (as job j) is scheduled.
      Section FromSameTask.

Consider a job j' that comes from task tsk and is scheduled at time instant t.
        Variable j' : Job.
        Hypothesis H_j'_tsk : job_of_task tsk j'.
        Hypothesis H_j'_sched : scheduled_at sched j' t.

Similarly, there is no task interference, since in order to incur the task interference, a job from a distinct task must be scheduled.
        Lemma task_interference_eq_false :
          ~~ task_interference arr_seq sched j t.
        Proof.
          apply/negP; move ⇒ /andP [+ _]; rewrite /nonself /task_scheduled_at.
          rewrite task_served_eq_task_scheduled //=; erewrite job_of_scheduled_task ⇒ //.
          move: H_j_tsk H_j'_tsk; rewrite /job_of_task ⇒ /eqP → /eqP →.
          by rewrite eq_refl.
        Qed.

      End FromSameTask.

In the next subsection, we consider a case when a job j' from a task other than j's task is scheduled.
      Section FromDifferentTask.

Consider a job j' that does _not comes from task tsk and is scheduled at time instant t.
        Variable j' : Job.
        Hypothesis H_j'_not_tsk : ~~ job_of_task tsk j'.
        Hypothesis H_j'_sched : scheduled_at sched j' t.

Hence, if we assume that j' has higher-or-equal priority, ...
        Hypothesis H_j'_hep : hep_job j' j.

Moreover, in this case, task tsk also incurs interference.
        Lemma sched_athep_implies_task_interference :
          task_interference arr_seq sched j t.
        Proof.
          apply/andP; split.
          - rewrite /nonself task_served_eq_task_scheduled ⇒ //.
            apply: job_of_other_task_scheduled ⇒ //.
            by move: H_j_tsk ⇒ /eqP → .
          - apply/orP; right.
            apply/hasP; j'.
            + by apply scheduled_at_implies_in_served_at ⇒ //.
            + rewrite another_hep_job_diff_task // same_task_sym.
              exact: (diff_task tsk).
        Qed.

      End FromDifferentTask.

    End ScheduledJob.

We prove that we can split cumulative interference into two parts: (1) cumulative priority inversion and (2) cumulative interference from jobs with higher or equal priority.
    Lemma cumulative_interference_split :
       j t1 t2,
        cumulative_interference j t1 t2
        = cumulative_priority_inversion arr_seq sched j t1 t2
          + cumulative_another_hep_job_interference arr_seq sched j t1 t2.
    Proof.
      rewrite /cumulative_interference /cumul_cond_interference /cond_interference /interference.
      movej t1 t2; rewrite -big_split //=.
      apply/eqP; rewrite eqn_leq; apply/andP; split; rewrite leq_sum//.
      { movet _; unfold ideal_jlfp_interference.
        by destruct (priority_inversion _ _ _ _), (another_hep_job_interference arr_seq sched j t).
      }
      { movet _; rewrite /ideal_jlfp_interference.
        destruct (ideal_proc_model_sched_case_analysis sched t) as [IDLE | [s SCHED]].
        - have → : priority_inversion arr_seq sched j t = false
            by exact/negbTE/idle_implies_no_priority_inversion.
          by rewrite add0n.
        - destruct (hep_job s j) eqn:PRIO.
          + by erewrite sched_hep_implies_no_priority_inversion ⇒ //; rewrite add0n.
          + erewrite !sched_lp_implies_priority_inversion ⇒ //; last by rewrite PRIO.
            rewrite orTb.
            by rewrite add1n ltnS //= leqn0 eqb0; erewrite no_ahep_interference_when_scheduled_lp; last by erewrite PRIO.
      }
    Qed.

Similarly, we prove that we can split cumulative interfering workload into two parts: (1) cumulative priority inversion and (2) cumulative interfering workload from jobs with higher or equal priority.
    Lemma cumulative_interfering_workload_split :
       j t1 t2,
        cumulative_interfering_workload j t1 t2 =
          cumulative_priority_inversion arr_seq sched j t1 t2
          + cumulative_other_hep_jobs_interfering_workload arr_seq j t1 t2.
    Proof.
      rewrite /cumulative_interfering_workload
              /cumulative_priority_inversion
              /cumulative_other_hep_jobs_interfering_workload.
      by movej t1 t2; rewrite -big_split //=.
    Qed.

Let j be any job of task tsk. Then the cumulative task interference received by job j is bounded to the sum of the cumulative priority inversion of job j and the cumulative interference incurred by job j due to higher-or-equal priority jobs from other tasks.
    Lemma cumulative_task_interference_split :
       j t1 t2,
        arrives_in arr_seq j
        job_of_task tsk j
        ~~ completed_by sched j t2
        cumul_task_interference arr_seq sched j t1 t2
         cumulative_priority_inversion arr_seq sched j t1 t2
          + cumulative_another_task_hep_job_interference arr_seq sched j t1 t2.
    Proof.
      movej t1 R ARR TSK NCOMPL.
      rewrite /cumul_task_interference /cumul_cond_interference.
      rewrite -big_split //= big_seq_cond [leqRHS]big_seq_cond.
      apply leq_sum; movet /andP [IN _].
      rewrite /cond_interference /nonself /interference /ideal_jlfp_interference.
      have [IDLE|[s SCHEDs]] := ideal_proc_model_sched_case_analysis sched t.
      { move: (IDLE) ⇒ IIDLE; erewrite <-is_idle_def in IDLE ⇒ //.
        have ->: priority_inversion arr_seq sched j t = false
          by apply/eqP; rewrite eqbF_neg; apply: no_priority_inversion_when_idle ⇒ //.
        have ->: another_hep_job_interference arr_seq sched j t = false
          by apply/eqP; rewrite eqbF_neg; apply no_hep_job_interference_when_idle.
        have ->: another_task_hep_job_interference arr_seq sched j t = false
          by apply/eqP; rewrite eqbF_neg; apply/negP; rewrite_neg @no_hep_task_interference_when_idle.
        by rewrite andbF.
      }
      { rewrite (interference_ahep_def _ _ _ _ _ _ _ _ _ _ _ SCHEDs) //.
        rewrite (interference_athep_def _ _ _ _ _ _ _ _ _ _ _ SCHEDs) ⇒ //.
        rewrite (priority_inversion_equiv_sched_lower_priority _ _ _ _ _ _ _ _ _ SCHEDs) ⇒ //.
        rewrite task_served_eq_task_scheduled // (job_of_scheduled_task _ _ _ _ _ _ _ _ _ SCHEDs) ⇒ //.
        rewrite /another_hep_job /another_task_hep_job.
        have [EQj|NEQj] := eqVneq s j.
        { by subst; rewrite /job_of_task eq_refl H_priority_is_reflexive. }
        have [/eqP EQt|NEQt] := eqVneq (job_task s) (job_task j).
        { apply/eqP; move: (EQt) ⇒ /eqP <-.
          by rewrite /job_of_task eq_refl //= andbF addn0 eq_sym eqb0; apply/negPLPs. }
        { by rewrite /job_of_task NEQt //= andbT; case: hep_job. }
      }
    Qed.

In this section, we prove that the (abstract) cumulative interfering workload is equivalent to the conventional workload, i.e., the one defined with concrete schedule parameters.
Let [t1,t2) be any time interval.
      Variables t1 t2 : instant.

Consider any job j of tsk.
      Variable j : Job.
      Hypothesis H_j_arrives : arrives_in arr_seq j.
      Hypothesis H_job_of_tsk : job_of_task tsk j.

Then for any job j, the cumulative interfering workload is equal to the conventional workload.
      Lemma cumulative_iw_hep_eq_workload_of_ohep :
        cumulative_other_hep_jobs_interfering_workload arr_seq j t1 t2
        = workload_of_other_hep_jobs arr_seq j t1 t2.
      Proof.
        rewrite /cumulative_other_hep_jobs_interfering_workload /workload_of_other_hep_jobs.
        case NEQ: (t1 < t2); last first.
        { move: NEQ ⇒ /negP /negP; rewrite -leqNgt; moveNEQ.
          rewrite big_geq//.
          rewrite /arrivals_between /arrival_sequence.arrivals_between big_geq//.
          by rewrite /workload_of_jobs big_nil.
        }
        { unfold other_hep_jobs_interfering_workload, workload_of_jobs.
          interval_to_duration t1 t2 k.
          elim: k ⇒ [|k IHk].
          - rewrite !addn0.
            rewrite big_geq//.
            rewrite /arrivals_between /arrival_sequence.arrivals_between big_geq//.
            by rewrite /workload_of_jobs big_nil.
          - rewrite addnS big_nat_recr //=; last by rewrite leq_addr.
            rewrite IHk.
            rewrite /arrivals_between /arrival_sequence.arrivals_between big_nat_recr //=.
            + by rewrite big_cat.
            + by rewrite leq_addr.
        }
      Qed.

    End InstantiatedWorkloadEquivalence.

In this section, we prove that the (abstract) cumulative interference of jobs with higher or equal priority is equal to total service of jobs with higher or equal priority.
Consider any job j of tsk.
      Variable j : Job.
      Hypothesis H_j_arrives : arrives_in arr_seq j.
      Hypothesis H_job_of_tsk : job_of_task tsk j.

We consider an arbitrary time interval [t1, t) that starts with a quiet time.
      Variable t1 t : instant.
      Hypothesis H_quiet_time : busy_interval.quiet_time arr_seq sched j t1.

As follows from lemma cumulative_pred_served_eq_service, the (abstract) instantiated function of interference is equal to the total service of any subset of jobs with higher or equal priority.
The above is in particular true for the jobs other than j with higher or equal priority...
      Lemma cumulative_i_ohep_eq_service_of_ohep :
        cumulative_another_hep_job_interference arr_seq sched j t1 t
        = service_of_other_hep_jobs arr_seq sched j t1 t.
      Proof. by apply: cumulative_pred_served_eq_service ⇒ // ⇒ ? /andP[]. Qed.

...and for jobs from other tasks than j with higher or equal priority.
In this section we prove that the abstract definition of busy interval is equivalent to the conventional, concrete definition of busy interval for JLFP scheduling.
    Section BusyIntervalEquivalence.

In order to avoid confusion, we denote the notion of a quiet time in the classical sense as quiet_time_cl, and the notion of quiet time in the abstract sense as quiet_time_ab.
Same for the two notions of a busy interval prefix ...
... and the two notions of a busy interval.
Consider any job j of tsk.
      Variable j : Job.
      Hypothesis H_j_arrives : arrives_in arr_seq j.
      Hypothesis H_job_cost_positive : job_cost_positive j.

To show the equivalence of the notions of busy intervals, we first show that the notions of quiet time are also equivalent.
First, we show that the classical notion of quiet time implies the abstract notion of quiet time.
      Lemma quiet_time_cl_implies_quiet_time_ab :
         t, quiet_time_cl j t quiet_time_ab j t.
      Proof.
        clear H_JLFP_respects_sequential_tasks.
        have zero_is_quiet_time: j, quiet_time_cl j 0.
        { by movejhp ARR HP AB; move: AB; rewrite /arrived_before ltn0. }
        movet QT; apply/andP; split; last first.
        { rewrite negb_and negbK; apply/orP.
          by case ARR: (arrived_before j t); [right | left]; [apply QT | ]. }
        { erewrite cumulative_interference_split, cumulative_interfering_workload_split; rewrite eqn_add2l.
          rewrite cumulative_i_ohep_eq_service_of_ohep//.
          rewrite //= cumulative_iw_hep_eq_workload_of_ohep eq_sym; apply/eqP.
          apply all_jobs_have_completed_equiv_workload_eq_service ⇒ //.
          movej0 IN HEP; apply QT.
          - by apply in_arrivals_implies_arrived in IN.
          - by move: HEP ⇒ /andP [H6 H7].
          - by apply in_arrivals_implies_arrived_between in IN.
        }
      Qed.

And vice versa, the abstract notion of quiet time implies the classical notion of quiet time.
      Lemma quiet_time_ab_implies_quiet_time_cl :
         t, quiet_time_ab j t quiet_time_cl j t.
      Proof.
        clear H_JLFP_respects_sequential_tasks.
        have zero_is_quiet_time: j, quiet_time_cl j 0.
        { by movejhp ARR HP AB; move: AB; rewrite /arrived_before ltn0. }
        movet /andP [T0 T1] jhp ARR HP ARB.
        eapply all_jobs_have_completed_equiv_workload_eq_service with
          (P := fun jhphep_job jhp j) (t1 := 0) (t2 := t) ⇒ //.
        erewrite service_of_jobs_case_on_pred with (P2 := fun j'j' != j).
        erewrite workload_of_jobs_case_on_pred with (P' := fun j'j' != j) ⇒ //.
        replace ((fun j0 : Jobhep_job j0 j && (j0 != j))) with (another_hep_job^~j); last by rewrite /another_hep_job.
        rewrite -/(service_of_other_hep_jobs arr_seq _ j 0 t) -cumulative_i_ohep_eq_service_of_ohep; eauto;
          last by move ⇒ ? _ _ ; unfold arrived_before; lia.
        rewrite -/(workload_of_other_hep_jobs _ j 0 t) -cumulative_iw_hep_eq_workload_of_ohep; eauto.
        move: T1; rewrite negb_and ⇒ /orP [NA | /negPn COMP].
        { have PRED__eq: {in arrivals_between arr_seq 0 t, (fun j__copy : Jobhep_job j__copy j && ~~ (j__copy != j)) =1 pred0}.
          { movej__copy IN; apply negbTE.
            rewrite negb_and; apply/orP; right; apply/negPn/eqPEQ; subst j__copy.
            move: NA ⇒ /negP CONTR; apply: CONTR.
            by apply in_arrivals_implies_arrived_between in IN. }
          erewrite service_of_jobs_equiv_pred with (P2 := pred0) ⇒ [|//].
          erewrite workload_of_jobs_equiv_pred with (P' := pred0) ⇒ [|//].
          move: T0; erewrite cumulative_interference_split, cumulative_interfering_workload_split; rewrite eqn_add2l ⇒ /eqP EQ.
          rewrite EQ; clear EQ; apply/eqP; rewrite eqn_add2l.
          by erewrite workload_of_jobs_pred0, service_of_jobs_pred0.
        }
        { have PRED__eq: {in arrivals_between arr_seq 0 t, (fun j0 : Jobhep_job j0 j && ~~ (j0 != j)) =1 eq_op j}.
          { movej__copy IN.
            replace (~~ (j__copy != j)) with (j__copy == j); last by case: (j__copy == j).
            rewrite eq_sym; destruct (j == j__copy) eqn:EQ; last by rewrite Bool.andb_false_r.
            by move: EQ ⇒ /eqP EQ; rewrite Bool.andb_true_r; apply/eqP; rewrite eqb_id; subst. }
          erewrite service_of_jobs_equiv_pred with (P2 := eq_op j) ⇒ [|//].
          erewrite workload_of_jobs_equiv_pred with (P' := eq_op j) ⇒ [|//].
          move: T0; erewrite cumulative_interference_split, cumulative_interfering_workload_split; rewrite eqn_add2l ⇒ /eqP EQ.
          rewrite EQ; clear EQ; apply/eqP; rewrite eqn_add2l.
          apply/eqP; eapply all_jobs_have_completed_equiv_workload_eq_service with
            (P := eq_op j) (t1 := 0) (t2 := t) ⇒ //.
          by movej__copy _ /eqP EQ; subst j__copy.
        }
      Qed.

The equivalence trivially follows from the lemmas above.
      Corollary instantiated_quiet_time_equivalent_quiet_time :
         t,
          quiet_time_cl j t quiet_time_ab j t.
      Proof.
        clear H_JLFP_respects_sequential_tasks.
        move ⇒ ?; split.
        - by apply quiet_time_cl_implies_quiet_time_ab.
        - by apply quiet_time_ab_implies_quiet_time_cl.
      Qed.

Based on that, we prove that the concept of busy interval prefix obtained by instantiating the abstract definition of busy interval prefix coincides with the conventional definition of busy interval prefix.
      Lemma instantiated_busy_interval_prefix_equivalent_busy_interval_prefix :
         t1 t2, busy_interval_prefix_cl j t1 t2 busy_interval_prefix_ab j t1 t2.
      Proof.
        movet1 t2; split.
        { move ⇒ [NEQ [QTt1 [NQT REL]]].
          split⇒ [//|]; split.
          - by apply instantiated_quiet_time_equivalent_quiet_time in QTt1.
          - by movet NE QT; eapply NQT; eauto 2; apply instantiated_quiet_time_equivalent_quiet_time.
        }
        { move ⇒ [/andP [NEQ1 NEQ2] [QTt1 NQT]].
          split; [ | split; [ |split] ].
          - by apply leq_ltn_trans with (job_arrival j).
          - by eapply instantiated_quiet_time_equivalent_quiet_time.
          - by movet NEQ QT; eapply NQT; eauto 2; eapply instantiated_quiet_time_equivalent_quiet_time in QT.
          - by apply/andP.
        }
      Qed.

Similarly, we prove that the concept of busy interval obtained by instantiating the abstract definition of busy interval coincides with the conventional definition of busy interval.
      Lemma instantiated_busy_interval_equivalent_busy_interval :
         t1 t2, busy_interval_cl j t1 t2 busy_interval_ab j t1 t2.
      Proof.
        movet1 t2; split.
        { move ⇒ [PREF QTt2]; split.
          - by apply instantiated_busy_interval_prefix_equivalent_busy_interval_prefix.
          - by eapply instantiated_quiet_time_equivalent_quiet_time in QTt2.
        }
        { move ⇒ [PREF QTt2]; split.
          - by apply instantiated_busy_interval_prefix_equivalent_busy_interval_prefix.
          - by eapply instantiated_quiet_time_equivalent_quiet_time.
        }
      Qed.

For the sake of proof automation, we note the frequently needed special case of an abstract busy window implying the existence of a classic quiet time.
      Fact abstract_busy_interval_classic_quiet_time :
         t1 t2,
          busy_interval_ab j t1 t2 quiet_time_cl j t1.
      Proof.
        by move⇒ ? ? /instantiated_busy_interval_equivalent_busy_interval [[_ [? _]] _].
      Qed.

Also for automation, we note a similar fact about classic busy-window prefixes.
      Fact abstract_busy_interval_classic_busy_interval_prefix :
         t1 t2,
          busy_interval_ab j t1 t2 busy_interval_prefix_cl j t1 t2.
      Proof. by move⇒ ? ? /instantiated_busy_interval_equivalent_busy_interval [+ _]. Qed.

    End BusyIntervalEquivalence.

  End Equivalences.

In this section we prove some properties about the interference and interfering workload as defined in this file.
  Section I_IW_correctness.

Consider work-bearing readiness.
Assume that the schedule is valid and work-conserving.
    Hypothesis H_sched_valid : valid_schedule sched arr_seq.

Note that we differentiate between abstract and classical notions of work-conserving schedule.
We assume that the schedule is a work-conserving schedule in the classical sense, and later prove that the hypothesis about abstract work-conservation also holds.
    Hypothesis H_work_conserving : work_conserving_cl.

Assume the scheduling policy under consideration is reflexive.
In this section, we prove the correctness of interference inside the busy interval, i.e., we prove that if interference for a job is false then the job is scheduled and vice versa. This property is referred to as abstract work conservation.

    Section Abstract_Work_Conservation.

Consider a job j that is in the arrival sequence and has a positive job cost.
      Variable j : Job.
      Hypothesis H_arrives : arrives_in arr_seq j.
      Hypothesis H_job_cost_positive : 0 < job_cost j.

Let the busy interval of the job be [t1,t2).
      Variable t1 t2 : instant.
      Hypothesis H_busy_interval_prefix : busy_interval_prefix_ab j t1 t2.

Consider a time t inside the busy interval of the job.
      Variable t : instant.
      Hypothesis H_t_in_busy_interval : t1 t < t2.

First, we prove that if interference is false at a time t then the job is scheduled.
      Lemma not_interference_implies_scheduled :
        ¬ interference j t receives_service_at sched j t.
      Proof.
        move ⇒ /negP HYP; move : HYP.
        rewrite negb_or /another_hep_job_interference.
        move ⇒ /andP [HYP1 HYP2].
        have [Idle|[jo Sched_jo]] := ideal_proc_model_sched_case_analysis sched t.
        { exfalso; clear HYP1 HYP2.
          eapply instantiated_busy_interval_prefix_equivalent_busy_interval_prefix in H_busy_interval_prefix ⇒ //.
          apply: not_quiet_implies_not_idle ⇒ //.
          by rewrite is_idle_def.
        }
        { have PINV := negP HYP1.
          have: ~~ another_hep_job jo j by apply: contraNN HYP2; erewrite <-interference_ahep_def ⇒ //.
          rewrite negb_and ⇒ /orP [NHEP | EQ].
          - rewrite/receives_service_at; move_neq_up ZS; move: ZS; rewrite leqn0 ⇒ /eqP ZS.
            apply no_service_not_scheduled in ZS ⇒ //; apply: PINV.
            exact/uni_priority_inversion_P.
          - apply negbNE in EQ; move: EQ ⇒ /eqP EQ; subst jo.
            by rewrite /receives_service_at service_at_is_scheduled_at Sched_jo.
        }
      Qed.

Conversely, if the job is scheduled at t then interference is false.
      Lemma scheduled_implies_no_interference :
        receives_service_at sched j t ¬ interference j t.
      Proof.
        moveRSERV /orP[PINV | INT].
        - rewrite (sched_hep_implies_no_priority_inversion _ _ _ _ _ _ _ _ j) in PINV ⇒ //.
          by rewrite /receives_service_at service_at_is_scheduled_at lt0b in RSERV.
        - rewrite /receives_service_at service_at_is_scheduled_at lt0b in RSERV.
          by move: INT; rewrite_neg @no_ahep_interference_when_scheduled.
      Qed.

    End Abstract_Work_Conservation.

Using the above two lemmas, we can prove that abstract work conservation always holds for these instantiations of I and IW.
    Corollary instantiated_i_and_w_are_coherent_with_schedule :
      work_conserving_ab.
    Proof.
      movej t1 t2 t ARR POS BUSY NEQ; split.
      - exact: (not_interference_implies_scheduled j ARR POS).
      - exact: scheduled_implies_no_interference.
    Qed.

Next, in order to prove that these definitions of I and IW are consistent with sequential tasks, we need to assume that the policy under consideration respects sequential tasks.
We prove that these definitions of I and IW are consistent with sequential tasks.
    Lemma instantiated_interference_and_workload_consistent_with_sequential_tasks :
      interference_and_workload_consistent_with_sequential_tasks arr_seq sched tsk.
    Proof.
      movej t1 t2 ARR /eqP TSK POS BUSY.
      eapply instantiated_busy_interval_equivalent_busy_interval in BUSY ⇒ //.
      eapply all_jobs_have_completed_equiv_workload_eq_service ⇒ //.
      moves INs /eqP TSKs.
      move: (INs) ⇒ NEQ.
      eapply in_arrivals_implies_arrived_between in NEQ ⇒ //.
      move: NEQ ⇒ /andP [_ JAs].
      move: (BUSY) ⇒ [[ _ [QT [_ /andP [JAj _]]] _]].
      apply QT ⇒ //; first exact: in_arrivals_implies_arrived.
      apply H_policy_respects_sequential_tasks; first by rewrite TSK TSKs.
      by apply leq_trans with t1; [lia |].
    Qed.

Since interfering and interfering workload are sufficient to define the busy window, next, we reason about the bound on the length of the busy window.
    Section BusyWindowBound.

Consider an arrival curve.
      Context `{MaxArrivals Task}.

Consider a set of tasks that respects the arrival curve.
      Variable ts : list Task.
      Hypothesis H_taskset_respects_max_arrivals : taskset_respects_max_arrivals arr_seq ts.

Assume that all jobs come from this task set.
Consider a constant L such that...
      Variable L : duration.
... L is greater than 0, and...
      Hypothesis H_L_positive : L > 0.

L is the fixed point of the following equation.
      Hypothesis H_fixed_point : L = total_request_bound_function ts L.

Assume all jobs have a valid job cost.
Then, we prove that L is a bound on the length of the busy window.
      Lemma instantiated_busy_intervals_are_bounded:
        busy_intervals_are_bounded_by arr_seq sched tsk L.
      Proof.
        movej ARR TSK POS.
        edestruct busy_interval_from_total_workload_bound
          with (Δ := L) as [t1 [t2 [T1 [T2 GGG]]]] ⇒ //.
        { by movet; rewrite {2}H_fixed_point; apply total_workload_le_total_rbf. }
         t1, t2; split⇒ [//|]; split⇒ [//|].
        by apply instantiated_busy_interval_equivalent_busy_interval.
      Qed.

    End BusyWindowBound.

  End I_IW_correctness.

End JLFPInstantiation.

To preserve modularity and hide the implementation details of a technical definition presented in this file, we make the definition opaque. This way, we ensure that the system will treat each of these definitions as a single entity.
We add some facts into the "Hint Database" basic_rt_facts, so Coq will be able to apply them automatically where needed.
Global Hint Resolve
       abstract_busy_interval_classic_quiet_time
       abstract_busy_interval_classic_busy_interval_prefix
  : basic_rt_facts.