Library prosa.analysis.facts.busy_interval.ideal.priority_inversion_bounded

Require Export prosa.model.task.preemption.parameters.
Require Export prosa.analysis.facts.model.preemption.

Throughout this file, we assume ideal uni-processor schedules.
Require Import prosa.model.processor.ideal.
Require Export prosa.analysis.facts.busy_interval.ideal.hep_job_scheduled.

Priority inversion is bounded

In this file, we prove that any priority inversion that occurs in the model with bounded nonpreemptive segments is bounded.
Section PriorityInversionIsBounded.

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 `{Arrival : JobArrival Job}.
  Context `{Cost : JobCost Job}.

Consider any arrival sequence with consistent arrivals ...
  Variable arr_seq : arrival_sequence Job.
  Hypothesis H_arrival_times_are_consistent : consistent_arrival_times arr_seq.

... and any ideal uniprocessor schedule of this arrival sequence.
  Variable sched : schedule (ideal.processor_state Job).

Consider a JLFP policy that indicates a higher-or-equal priority relation, and assume that the relation is reflexive and transitive.
  Context {JLFP : JLFP_policy Job}.
  Hypothesis H_priority_is_reflexive: reflexive_priorities.
  Hypothesis H_priority_is_transitive: transitive_priorities.

In addition, we assume the existence of a function mapping a task to its maximal non-preemptive segment ...
  Context `{TaskMaxNonpreemptiveSegment Task}.

... and the existence of a function mapping a job and its progress to a boolean value saying whether this job is preemptable at its current point of execution.
  Context `{JobPreemptable Job}.

And assume that they define a valid preemption model with bounded nonpreemptive segments.
  Hypothesis H_valid_model_with_bounded_nonpreemptive_segments:
    valid_model_with_bounded_nonpreemptive_segments arr_seq sched.

Further, allow for any work-bearing notion of job readiness.
  Context `{@JobReady Job (ideal.processor_state Job) Cost Arrival}.
  Hypothesis H_job_ready : work_bearing_readiness arr_seq sched.

We assume that the schedule is valid and that all jobs come from the arrival sequence.
  Hypothesis H_sched_valid : valid_schedule sched arr_seq.
  Hypothesis H_jobs_come_from_arrival_sequence: jobs_come_from_arrival_sequence sched arr_seq.

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 scheduling policy at every preemption point.
  Hypothesis H_respects_policy :
    respects_JLFP_policy_at_preemption_point arr_seq sched JLFP.

Finally, we introduce the notion of the maximal length of (potential) priority inversion at a time instant t, which is defined as the maximum length of nonpreemptive segments among all jobs that arrived so far.
  Definition max_length_of_priority_inversion (j : Job) (t : instant) :=
    \max_(j_lp <- arrivals_before arr_seq t | (~~ hep_job j_lp j) && (job_cost j_lp > 0))
     (job_max_nonpreemptive_segment j_lp - ε).

Next we prove that a priority inversion of a job is bounded by function max_length_of_priority_inversion.
Note that any bound on function max_length_of_priority_inversion will also be a bound on the maximal priority inversion. This bound may be different for different scheduler and/or task models. Thus, we don't define such a bound in this module.
Consider any job j of tsk with positive job cost.
  Variable j : Job.
  Hypothesis H_j_arrives : arrives_in arr_seq j.
  Hypothesis H_job_cost_positive : job_cost_positive j.

Consider any busy interval prefix [t1, t2) of job j.
  Variable t1 t2 : instant.
  Hypothesis H_busy_interval_prefix:
    busy_interval_prefix arr_seq sched j t1 t2.

Processor Executes HEP jobs after Preemption Point

In this section, we prove that at any time instant after any preemption point (inside the busy interval), the processor is always busy scheduling a job with higher or equal priority.
  Section PreemptionTimeAndPriorityInversion.

First, recall from file busy_interval/ideal/hep_job_scheduled we already know that the processor at any preemptive point is always busy scheduling a job with higher or equal priority.
We show that at any time instant after a preemption point the processor is always busy with a job with higher or equal priority.
    Lemma not_quiet_implies_exists_scheduled_hp_job_after_preemption_point:
      ∀ tp t,
        preemption_time sched tp →
        t1 ≤ tp < t2 →
        tp ≤ t < t2 →
        ∃ j_hp,
          arrived_between j_hp t1 t.+1 ∧
          hep_job j_hp j ∧
          scheduled_at sched j_hp t.
    Proof.
      move ⇒ tp t PRPOINT /andP [GEtp LTtp] /andP [LEtp LTt].
      ideal_proc_model_sched_case_analysis_eq sched t jhp.
      { by eapply instant_t_is_not_idle in Idle; rt_eauto;
        [ | apply/andP; split; first apply leq_trans with tp]. }
      ∃ jhp.
      have HP: hep_job jhp j.
      { intros.
         move:(H_valid_model_with_bounded_nonpreemptive_segments) ⇒ [PREE ?].
        specialize (scheduling_of_any_segment_starts_with_preemption_time arr_seq sched H_sched_valid PREE jhp t Sched_jhp) ⇒ SOAS.
        move: SOAS ⇒ [prt [/andP [_ LE] [PR SCH]]].
        case E:(t1 ≤ prt).
        - move: E ⇒ /eqP /eqP E; rewrite subn_eq0 in E.
          edestruct not_quiet_implies_exists_scheduled_hp_job_at_preemption_point as [jlp [_ [HEP SCHEDjhp]]]; eauto 2.
          { by apply /andP; split; last by apply leq_ltn_trans with t. }
          enough (EQ : jhp = jlp); first by subst.
          apply: (ideal_proc_model_is_a_uniprocessor_model _ _ _ prt); eauto;
            by apply SCH; apply/andP; split.
        - move: E ⇒ /eqP /neqP E; rewrite -lt0n subn_gt0 in E.
          apply negbNE; apply/negP; intros LP; rename jhp into jlp.
          edestruct not_quiet_implies_exists_scheduled_hp_job_at_preemption_point
            as [jhp [_ [HEP SCHEDjhp]]]; try apply PRPOINT; rt_eauto; first by apply/andP; split.
          move: LP ⇒ /negP LP; apply: LP.
          enough (EQ : jhp = jlp); first by subst.
          apply: (ideal_proc_model_is_a_uniprocessor_model jhp _ _ tp); eauto.
            by apply SCH; apply/andP; split; first apply leq_trans with t1; auto.
      }
      repeat split; try done.
      move: (H_busy_interval_prefix) ⇒ [SL [QUIET [NOTQUIET EXj]]]; move: (Sched_jhp) ⇒ PENDING.
      eapply scheduled_implies_pending in PENDING; rt_eauto.
      apply/andP; split; last by apply leq_ltn_trans with (n := t); first by move: PENDING ⇒ /andP [ARR _].
      apply contraT; rewrite -ltnNge; intro LT; exfalso.
      feed (QUIET jhp); first by eapply H_jobs_come_from_arrival_sequence, Sched_jhp.
      specialize (QUIET HP LT).
      have COMP: completed_by sched jhp t.
      { apply: completion_monotonic QUIET; exact: leq_trans LEtp. }
      apply completed_implies_not_scheduled in COMP; rt_eauto.
      by move : COMP ⇒ /negP COMP; apply : COMP.
    Qed.

Now, suppose there exists some constant K that bounds the distance to a preemption time from the beginning of the busy interval.
    Variable K : duration.
    Hypothesis H_preemption_time_exists :
      ∃ pr_t, preemption_time sched pr_t ∧ t1 ≤ pr_t ≤ t1 + K.

Then we prove that the processor is always busy with a job with higher-or-equal priority after time instant t1 + K.
    Lemma not_quiet_implies_exists_scheduled_hp_job:
      ∀ t,
        t1 + K ≤ t < t2 →
        ∃ j_hp,
          arrived_between j_hp t1 t.+1 ∧
          hep_job j_hp j ∧
          scheduled_at sched j_hp t.
    Proof.
      move ⇒ t /andP [GE LT].
      move: H_preemption_time_exists ⇒ [prt [PR /andP [GEprt LEprt]]].
      apply not_quiet_implies_exists_scheduled_hp_job_after_preemption_point with (tp := prt); eauto 2.
      - apply/andP; split; first by done.
        apply leq_ltn_trans with (t1 + K); first by done.
        by apply leq_ltn_trans with t.
      - apply/andP; split; last by done.
        by apply leq_trans with (t1 + K).
    Qed.

  End PreemptionTimeAndPriorityInversion.

Preemption Time Exists

In this section we prove that the function max_length_of_priority_inversion indeed upper-bounds the priority inversion length.
  Section PreemptionTimeExists.

First, we prove that, if a job with higher-or-equal priority is scheduled at a quiet time t+1, then this is the first time when this job is scheduled.
    Lemma hp_job_not_scheduled_before_quiet_time:
      ∀ jhp t,
        quiet_time arr_seq sched j t.+1 →
        scheduled_at sched jhp t.+1 →
        hep_job jhp j →
        ~~ scheduled_at sched jhp t.
    Proof.
      intros jhp t QT SCHED1 HP.
      apply/negP; intros SCHED2.
      specialize (QT jhp).
      feed_n 3 QT; eauto.
      - have MATE: jobs_must_arrive_to_execute sched by rt_eauto.
        have HA: has_arrived jhp t by apply MATE.
        by done.
      apply completed_implies_not_scheduled in QT; rt_eauto.
      by move: QT ⇒ /negP NSCHED; apply: NSCHED.
    Qed.

Also, we show that lower-priority jobs that are scheduled inside the busy-interval prefix [t1,t2) must arrive before that interval.
    Lemma low_priority_job_arrives_before_busy_interval_prefix:
      ∀ jlp t,
        t1 ≤ t < t2 →
        scheduled_at sched jlp t →
        ~~ hep_job jlp j →
        job_arrival jlp < t1.
    Proof.
      move ⇒ jlp t /andP [GE LT] SCHED LP.
      move: (H_busy_interval_prefix) ⇒ [NEM [QT [NQT HPJ]]].
      apply negbNE; apply/negP; intros ARR; rewrite -leqNgt in ARR.
      move:(H_valid_model_with_bounded_nonpreemptive_segments) ⇒ [PREE ?].
      specialize (scheduling_of_any_segment_starts_with_preemption_time arr_seq sched H_sched_valid PREE jlp t SCHED) ⇒ SCH.
      move: SCH ⇒ [pt [/andP [NEQ1 NEQ2] [PT FA]]].
      have NEQ: t1 ≤ pt < t2.
      { apply/andP; split.
        - by apply leq_trans with (job_arrival jlp).
        - by apply leq_ltn_trans with t. }
      edestruct not_quiet_implies_exists_scheduled_hp_job_at_preemption_point as [jhp [_ [HEP SCHEDjhp]]]; rt_eauto.
      feed (FA pt); first (by apply/andP; split).
      move: LP ⇒ /negP LP; apply: LP.
      by have ->: jlp = jhp by eapply ideal_proc_model_is_a_uniprocessor_model; eauto.
    Qed.

Moreover, we show that lower-priority jobs that are scheduled inside the busy-interval prefix [t1,t2) must be scheduled before that interval.
    Lemma low_priority_job_scheduled_before_busy_interval_prefix:
      ∀ jlp t,
        t1 ≤ t < t2 →
        scheduled_at sched jlp t →
        ~~ hep_job jlp j →
        ∃ t', t' < t1 ∧ scheduled_at sched jlp t'.
    Proof.
      move ⇒ jlp t NEQ SCHED LP; move: (NEQ) ⇒ /andP [GE LT].
      have ARR := low_priority_job_arrives_before_busy_interval_prefix _ _ NEQ SCHED LP.
      ∃ t1.-1; split.
      { by rewrite prednK; last apply leq_ltn_trans with (job_arrival jlp). }
      move: (H_busy_interval_prefix) ⇒ [NEM [QT [NQT HPJ]]].
      move:(H_valid_model_with_bounded_nonpreemptive_segments) ⇒ [PREE ?].
      specialize (scheduling_of_any_segment_starts_with_preemption_time arr_seq sched H_sched_valid PREE jlp t SCHED) ⇒ SCHEDST.
      move: SCHEDST ⇒ [pt [NEQpt [PT SCHEDc]]].
      have LT2: pt < t1.
      { rewrite ltnNge; apply/negP; intros CONTR.
        edestruct not_quiet_implies_exists_scheduled_hp_job_at_preemption_point
          as [jhp [_ [HEP SCHEDjhp]]]; try apply PT; rt_eauto; first lia.
        specialize (SCHEDc pt).
        feed SCHEDc; first by apply/andP; split; last move: NEQpt ⇒ /andP [_ T].
        move: LP ⇒ /negP LP; apply: LP.
        by have ->: jlp = jhp by eapply ideal_proc_model_is_a_uniprocessor_model; eauto.
      }
      apply SCHEDc; apply/andP; split.
      - by rewrite -add1n in LT2; apply leq_subRL_impl in LT2; rewrite subn1 in LT2.
      - by apply leq_trans with t1; first apply leq_pred.
    Qed.

Thus, there must be a preemption time in the interval t1, t1 + max_priority_inversion t1. That is, if a job with higher-or-equal priority is scheduled at time instant t1, then t1 is a preemption time. Otherwise, if a job with lower priority is scheduled at time t1, then this job also should be scheduled before the beginning of the busy interval. So, the next preemption time will be no more than max_priority_inversion t1 time units later.
We proceed by doing a case analysis.
    Section CaseAnalysis.

(1) Case when the schedule is idle at time t1.
      Section Case1.

Assume that the schedule is idle at time t1.
        Hypothesis H_is_idle : is_idle sched t1.

Then time instant t1 is a preemption time.
        Lemma preemption_time_exists_case1:
          ∃ pr_t,
            preemption_time sched pr_t ∧
            t1 ≤ pr_t ≤ t1 + max_length_of_priority_inversion j t1.
        Proof.
          set (service := service sched).
          move: (H_valid_model_with_bounded_nonpreemptive_segments) ⇒ CORR.
          move: (H_busy_interval_prefix) ⇒ [NEM [QT1 [NQT HPJ]]].
          ∃ t1; split.
          - by rewrite /preemption_time; move: H_is_idle ⇒ /eqP →.
          - by apply/andP; split; last rewrite leq_addr.
        Qed.

      End Case1.

(2) Case when a job with higher-or-equal priority is scheduled at time t1.
      Section Case2.

Assume that a job jhp with higher-or-equal priority is scheduled at time t1.
        Variable jhp : Job.
        Hypothesis H_jhp_is_scheduled : scheduled_at sched jhp t1.
        Hypothesis H_jhp_hep_priority : hep_job jhp j.

Then time instant t1 is a preemption time.
        Lemma preemption_time_exists_case2:
          ∃ pr_t,
            preemption_time sched pr_t ∧
            t1 ≤ pr_t ≤ t1 + max_length_of_priority_inversion j t1.
        Proof.
          set (service := service sched).
          move : (H_valid_model_with_bounded_nonpreemptive_segments) ⇒ [VALID BOUNDED].
          move: (H_valid_model_with_bounded_nonpreemptive_segments) ⇒ CORR.
          move: (H_busy_interval_prefix) ⇒ [NEM [QT1 [NQT HPJ]]].
          ∃ t1; split; last first.
          apply/andP; split; [by done | by rewrite leq_addr].
          destruct t1.
          - by eapply zero_is_pt; eauto 2.
          - apply: first_moment_is_pt H_jhp_is_scheduled; eauto 2.
            by eapply hp_job_not_scheduled_before_quiet_time; try by done.
        Qed.

      End Case2.

(3) Case when a job with lower priority is scheduled at time t1.
      Section Case3.

Assume that a job jhp with lower priority is scheduled at time t1.
        Variable jlp : Job.
        Hypothesis H_jlp_is_scheduled : scheduled_at sched jlp t1.
        Hypothesis H_jlp_low_priority : ~~ hep_job jlp j.

To prove the lemma in this case we need a few auxiliary facts about the first preemption point of job jlp.
        Section FirstPreemptionPointOfjlp.

Let's denote the progress of job jlp at time t1 as progr_t1.
          Let progr_t1 := service sched jlp t1.

Consider the first preemption point of job jlp after progr_t1.
          Variable fpt : instant.
          Hypothesis H_fpt_is_preemptio_point : job_preemptable jlp (progr_t1 + fpt).
          Hypothesis H_fpt_is_first_preemption_point:
            ∀ ρ,
              progr_t1 ≤ ρ ≤ progr_t1 + (job_max_nonpreemptive_segment jlp - ε) →
              job_preemptable jlp ρ →
              service sched jlp t1 + fpt ≤ ρ.

For correctness, we also assume that fpt does not exceed the length of the maximum non-preemptive segment.
          Hypothesis H_progr_le_max_nonp_segment:
            fpt ≤ job_max_nonpreemptive_segment jlp - ε.

First we show that fpt is indeed the first preemption point after progr_t1.
          Lemma no_intermediate_preemption_point:
            ∀ ρ,
              progr_t1 ≤ ρ < progr_t1 + fpt →
              ~~ job_preemptable jlp ρ.
          Proof.
            move ⇒ prog /andP [GE LT].
            apply/negP; intros PPJ.
            move: H_fpt_is_first_preemption_point ⇒ K; specialize (K prog).
            feed_n 2 K; first (apply/andP; split); try done.
            { apply leq_trans with (service sched jlp t1 + fpt).
              + by apply ltnW.
              + by rewrite leq_add2l; apply H_progr_le_max_nonp_segment.
            }
            by move: K; rewrite leqNgt; move ⇒ /negP NLT; apply: NLT.
          Qed.

Thanks to the fact that the scheduler respects the notion of preemption points we show that jlp is continuously scheduled in time interval [t1, t1 + fpt).
          Lemma continuously_scheduled_between_preemption_points:
            ∀ t',
              t1 ≤ t' < t1 + fpt →
              scheduled_at sched jlp t'.
          Proof.
            move: (H_valid_model_with_bounded_nonpreemptive_segments) ⇒ CORR.
            move: (H_jlp_is_scheduled) ⇒ ARRs; apply H_jobs_come_from_arrival_sequence in ARRs.
            move ⇒ t' /andP [GE LT].
            have Fact: ∃ Δ, t' = t1 + Δ.
            { by ∃ (t' - t1); apply/eqP; rewrite eq_sym; apply/eqP; rewrite subnKC. }
            move: Fact ⇒ [Δ EQ]; subst t'.
            have NPPJ := @no_intermediate_preemption_point (@service _ _ sched jlp (t1 + Δ)).
            apply proj1 in CORR; specialize (CORR jlp ARRs).
            move: CORR ⇒ [_ [_ [T _] ]].
            apply T; apply: NPPJ; apply/andP; split.
            { by apply service_monotonic; rewrite leq_addr. }
            rewrite /service -(service_during_cat _ _ _ t1).
            { rewrite ltn_add2l; rewrite ltn_add2l in LT.
              apply leq_ltn_trans with Δ; last by done.
              rewrite -{2}(sum_of_ones t1 Δ).
              rewrite leq_sum //; intros t _.
              apply service_at_most_one.
              by apply ideal_proc_model_provides_unit_service.
            }
            { by apply/andP; split; [done | rewrite leq_addr]. }
          Qed.

Thus, job jlp reaches its preemption point at time instant t1 + fpt, which implies that time instant t1 + fpt is a preemption time.
          Lemma first_preemption_time:
            preemption_time sched (t1 + fpt).
          Proof.
            rewrite /preemption_time.
            move: (H_valid_model_with_bounded_nonpreemptive_segments) ⇒ CORR.
            ideal_proc_model_sched_case_analysis_eq sched (t1 + fpt) s'; try done.
            clear EqSched_s'; move: (Sched_s'); rewrite scheduled_at_def;
              move ⇒ /eqP EqSched_s'; rewrite EqSched_s'.
            destruct (jlp == s') eqn: EQ.
            - move: EQ ⇒ /eqP EQ; subst s'.
              rewrite /service -(service_during_cat _ _ _ t1); last first.
              { by apply/andP; split; last rewrite leq_addr. }
              have ->: service_during sched jlp t1 (t1 + fpt) = fpt.
              { rewrite -{2}(sum_of_ones t1 fpt) /service_during.
                apply/eqP; rewrite eqn_leq //; apply/andP; split.
                + rewrite leq_sum //; intros t _.
                  apply service_at_most_one.
                    by apply ideal_proc_model_provides_unit_service.
                + rewrite big_nat_cond [in X in _ ≤ X]big_nat_cond.
                  rewrite leq_sum //.
                  move ⇒ x /andP [HYP _].
                  rewrite service_at_def lt0b -scheduled_at_def.
                    by apply continuously_scheduled_between_preemption_points.
              } by done.
            - case: (posnP fpt) ⇒ [ZERO|POS].
              { subst fpt.
                exfalso; move: EQ ⇒ /negP EQ; apply: EQ.
                move: H_jlp_is_scheduled; rewrite scheduled_at_def; move ⇒ /eqP SCHED2.
                rewrite addn0 in EqSched_s'; rewrite EqSched_s' in SCHED2.
                  by inversion SCHED2.
              }
              { have EX: ∃ sm, sm.+1 = fpt.
                { ∃ fpt.-1. lia. }
                destruct EX as [sm EQ2]. rewrite -EQ2.
                rewrite addnS.
                move: ((proj1 CORR) s' (H_jobs_come_from_arrival_sequence _ _ Sched_s')) ⇒ T.
                apply T; clear T. apply /negP; intros CONTR.
                move: EQ ⇒ /negP EQ; apply: EQ.
                move: (continuously_scheduled_between_preemption_points (t1 + sm)) ⇒ SCHEDs0.
                feed SCHEDs0; first by apply/andP; split; [rewrite leq_addr | rewrite -EQ2 addnS].
                apply/eqP; eapply ideal_proc_model_is_a_uniprocessor_model; eauto 2.
                  by rewrite -addnS EQ2.
              }
          Qed.

And since fpt ≤ max_length_of_priority_inversion j t1, t1 ≤ t1 + fpt ≤ t1 + max_length_of_priority_inversion j t1.
          Lemma preemption_time_le_max_len_of_priority_inversion:
            t1 ≤ t1 + fpt ≤ t1 + max_length_of_priority_inversion j t1.
          Proof.
            move: (H_jlp_is_scheduled) ⇒ ARRs; apply H_jobs_come_from_arrival_sequence in ARRs.
            apply/andP; split; first by rewrite leq_addr.
            rewrite leq_add2l.
            unfold max_length_of_priority_inversion.
            rewrite (big_rem jlp) //=.
            { rewrite H_jlp_low_priority //=.
              have NZ: service sched jlp t1 < job_cost jlp
                by apply: service_lt_cost; rt_eauto.
              rewrite ifT; last by lia.
              apply leq_trans with (job_max_nonpreemptive_segment jlp - ε).
              - by apply H_progr_le_max_nonp_segment.
              - by rewrite leq_maxl.
            }
            eapply arrived_between_implies_in_arrivals; eauto 2.
            apply/andP; split; first by done.
            eapply low_priority_job_arrives_before_busy_interval_prefix with t1; eauto 2.
            by move: (H_busy_interval_prefix) ⇒ [NEM [QT1 [NQT HPJ]]]; apply/andP; split.
          Qed.

        End FirstPreemptionPointOfjlp.

Next, we combine the above facts to conclude the lemma.
        Lemma preemption_time_exists_case3:
          ∃ pr_t,
            preemption_time sched pr_t ∧
            t1 ≤ pr_t ≤ t1 + max_length_of_priority_inversion j t1.
        Proof.
          set (service := service sched).
          have EX: ∃ pt,
              ((service jlp t1) ≤ pt ≤ (service jlp t1) + (job_max_nonpreemptive_segment jlp - 1)) && job_preemptable jlp pt.
          { move: (H_jlp_is_scheduled) ⇒ ARRs; apply H_jobs_come_from_arrival_sequence in ARRs.
            move: (proj2 (H_valid_model_with_bounded_nonpreemptive_segments) jlp ARRs) =>[_ EXPP].
            destruct H_sched_valid as [A B].
            specialize (EXPP (service jlp t1)).
            feed EXPP.
            { apply/andP; split; first by done.
              apply service_at_most_cost; rt_eauto.
            }
            move: EXPP ⇒ [pt [NEQ PP]].
            ∃ pt; apply/andP; split; by done.
          }
          move: (ex_minnP EX) ⇒ [sm_pt /andP [NEQ PP] MIN]; clear EX.
          have Fact: ∃ Δ, sm_pt = service jlp t1 + Δ.
          { ∃ (sm_pt - service jlp t1).
            apply/eqP; rewrite eq_sym; apply/eqP; rewrite subnKC //.
            by move: NEQ ⇒ /andP [T _]. }
          move: Fact ⇒ [Δ EQ]; subst sm_pt; rename Δ into sm_pt.
          ∃ (t1 + sm_pt); split.
          { apply first_preemption_time; rewrite /service.service; try done.
            + by intros; apply MIN; apply/andP; split.
            + by rewrite /ε; lia.
          }
          by apply preemption_time_le_max_len_of_priority_inversion;
            [ done | by rewrite /ε; lia].
        Qed.

      End Case3.

    End CaseAnalysis.

By doing the case analysis, we show that indeed there is a preemption time in the time interval [t1, t1 + max_length_of_priority_inversion j t1].
    Lemma preemption_time_exists:
      ∃ pr_t,
        preemption_time sched pr_t ∧
        t1 ≤ pr_t ≤ t1 + max_length_of_priority_inversion j t1.
    Proof.
      set (service := service sched).
      move: (H_valid_model_with_bounded_nonpreemptive_segments) ⇒ CORR.
      move: (H_busy_interval_prefix) ⇒ [NEM [QT1 [NQT HPJ]]].
      ideal_proc_model_sched_case_analysis sched t1 s.
      - by apply preemption_time_exists_case1.
      - destruct (hep_job s j) eqn:PRIO.
        + by eapply preemption_time_exists_case2; eauto.
        + eapply preemption_time_exists_case3 with s; eauto.
          by rewrite -eqbF_neg; apply /eqP.
    Qed.

  End PreemptionTimeExists.

End PriorityInversionIsBounded.