Library rt.restructuring.analysis.basic_facts.service

Require Export rt.util.all.
Require Export rt.restructuring.analysis.basic_facts.ideal_schedule.

From mathcomp Require Import ssrnat ssrbool fintype.

In this file, we establish basic facts about the service received by jobs.

Section Composition.
To begin with, we provide some simple but handy rewriting rules for service and service_during.
Consider any job type and any processor state.
  Context {Job: JobType}.
  Context {PState: Type}.
  Context `{ProcessorState Job PState}.

For any given schedule...
  Variable sched: schedule PState.

...and any given job...
  Variable j: Job.

...we establish a number of useful rewriting rules that decompose the service received during an interval into smaller intervals.
As a trivial base case, no job receives any service during an empty interval.
  Lemma service_during_geq:
    ∀ t1 t2,
      t1 ≥ t2 → service_during sched j t1 t2 = 0.
  Proof.
    move⇒ t1 t2 t1t2.
    rewrite /service_during big_geq //.
  Qed.

Equally trivially, no job has received service prior to time zero.
  Corollary service0:
    service sched j 0 = 0.
  Proof.
    rewrite /service service_during_geq //.
  Qed.

Trivially, an interval consiting of one time unit is equivalent to service_at.
  Lemma service_during_instant:
    ∀ t,
      service_during sched j t t.+1 = service_at sched j t.
  Proof.
    move ⇒ t.
     by rewrite /service_during big_nat_recr ?big_geq //.
  Qed.

Next, we observe that we can look at the service received during an interval t1, t3) as the sum of the service during [t1, t2) and [t2, t3) for any t2 \in [t1, t3]. (The "_cat" suffix denotes the concatenation of the two intervals.)
  Lemma service_during_cat:
    ∀ t1 t2 t3,
      t1 ≤ t2 ≤ t3 →
      (service_during sched j t1 t2) + (service_during sched j t2 t3)
      = service_during sched j t1 t3.
  Proof.
    move ⇒ t1 t2 t3 /andP [t1t2 t2t3].
      by rewrite /service_during -big_cat_nat /=.
  Qed.

Since service is just a special case of service_during, the same holds for service.
  Lemma service_cat:
    ∀ t1 t2,
      t1 ≤ t2 →
      (service sched j t1) + (service_during sched j t1 t2)
      = service sched j t2.
  Proof.
    move⇒ t1 t2 t1t2.
    rewrite /service service_during_cat //.
  Qed.

As a special case, we observe that the service during an interval can be decomposed into the first instant and the rest of the interval.
  Lemma service_during_first_plus_later:
    ∀ t1 t2,
      t1 < t2 →
      (service_at sched j t1) + (service_during sched j t1.+1 t2)
      = service_during sched j t1 t2.
  Proof.
    move ⇒ t1 t2 t1t2.
    have TIMES: t1 ≤ t1.+1 ≤ t2 by rewrite /(_ && _) ifT //.
    have SPLIT := service_during_cat t1 t1.+1 t2 TIMES.
      by rewrite -service_during_instant //.
  Qed.

Symmetrically, we have the same for the end of the interval.
  Lemma service_during_last_plus_before:
    ∀ t1 t2,
      t1 ≤ t2 →
      (service_during sched j t1 t2) + (service_at sched j t2)
      = service_during sched j t1 t2.+1.
  Proof.
    move⇒ t1 t2 t1t2.
    rewrite -(service_during_cat t1 t2 t2.+1); last by rewrite /(_ && _) ifT //.
    rewrite service_during_instant //.
  Qed.

And hence also for service.
  Corollary service_last_plus_before:
    ∀ t,
      (service sched j t) + (service_at sched j t)
      = service sched j t.+1.
  Proof.
    move⇒ t.
    rewrite /service. rewrite -service_during_last_plus_before //.
  Qed.

Finally, we deconstruct the service received during an interval t1, t3) into the service at a midpoint t2 and the service in the intervals before and after.
  Lemma service_split_at_point:
    ∀ t1 t2 t3,
      t1 ≤ t2 < t3 →
      (service_during sched j t1 t2) + (service_at sched j t2) + (service_during sched j t2.+1 t3)
      = service_during sched j t1 t3.
  Proof.
    move ⇒ t1 t2 t3 /andP [t1t2 t2t3].
    rewrite -addnA service_during_first_plus_later// service_during_cat// /(_ && _) ifT//.
      by exact: ltnW.
  Qed.

End Composition.

Section UnitService.
As a common special case, we establish facts about schedules in which a job receives either 1 or 0 service units at all times.
Consider any job type and any processor state.
  Context {Job: JobType}.
  Context {PState: Type}.
  Context `{ProcessorState Job PState}.

Let's consider a unit-service model...
  Hypothesis H_unit_service: unit_service_proc_model PState.

...and a given schedule.
  Variable sched: schedule PState.

Let j be any job that is to be scheduled.
  Variable j: Job.

First, we prove that the instantaneous service cannot be greater than 1, ...
  Lemma service_at_most_one:
    ∀ t, service_at sched j t ≤ 1.
  Proof.
      by move⇒ t; rewrite /service_at.
  Qed.

...which implies that the cumulative service received by job j in any interval of length delta is at most delta.
  Lemma cumulative_service_le_delta:
    ∀ t delta,
      service_during sched j t (t + delta) ≤ delta.
  Proof.
    unfold service_during; intros t delta.
    apply leq_trans with (n := \sum_(t ≤ t0 < t + delta) 1);
      last by rewrite sum_of_ones.
    by apply: leq_sum ⇒ t' _; apply: service_at_most_one.
  Qed.

  Section ServiceIsAStepFunction.

We show that the service received by any job j is a step function.
    Lemma service_is_a_step_function:
      is_step_function (service sched j).
    Proof.
      rewrite /is_step_function ⇒ t.
      rewrite addn1 -service_last_plus_before leq_add2l.
      apply service_at_most_one.
    Qed.

Next, consider any time t...
    Variable t: instant.

...and let s0 be any value less than the service received by job j by time t.
    Variable s0: duration.
    Hypothesis H_less_than_s: s0 < service sched j t.

Then, we show that there exists an earlier time t0 where job j had s0 units of service.
    Corollary exists_intermediate_service:
      ∃ t0,
        t0 < t ∧
        service sched j t0 = s0.
    Proof.
      feed (exists_intermediate_point (service sched j));
        [by apply service_is_a_step_function | intros EX].
      feed (EX 0 t); first by done.
      feed (EX s0);
        first by rewrite /service /service_during big_geq //.
        by move: EX ⇒ /= [x_mid EX]; ∃ x_mid.
    Qed.
  End ServiceIsAStepFunction.

End UnitService.

Section Monotonicity.
We establish a basic fact about the monotonicity of service.
Consider any job type and any processor model.
  Context {Job: JobType}.
  Context {PState: Type}.
  Context `{ProcessorState Job PState}.

Consider any given schedule...
  Variable sched: schedule PState.

...and a given job that is to be scheduled.
  Variable j: Job.

We observe that the amount of service received is monotonic by definition.
  Lemma service_monotonic:
    ∀ t1 t2,
      t1 ≤ t2 →
      service sched j t1 ≤ service sched j t2.
  Proof.
    move⇒ t1 t2 let1t2.
      by rewrite -(service_cat sched j t1 t2 let1t2); apply: leq_addr.
  Qed.

End Monotonicity.

Section RelationToScheduled.
Consider any job type and any processor model.
  Context {Job: JobType}.
  Context {PState: Type}.
  Context `{ProcessorState Job PState}.

Consider any given schedule...
  Variable sched: schedule PState.

...and a given job that is to be scheduled.
  Variable j: Job.

We observe that a job that isn't scheduled cannot possibly receive service.
  Lemma not_scheduled_implies_no_service:
    ∀ t,
      ~~ scheduled_at sched j t → service_at sched j t = 0.
  Proof.
    rewrite /service_at /scheduled_at.
    move⇒ t NOT_SCHED.
    rewrite service_implies_scheduled //.
  Qed.

Conversely, if a job receives service, then it must be scheduled.
  Lemma service_at_implies_scheduled_at:
    ∀ t,
      service_at sched j t > 0 → scheduled_at sched j t.
  Proof.
    move⇒ t.
    destruct (scheduled_at sched j t) eqn:SCHEDULED; first trivial.
    rewrite not_scheduled_implies_no_service // negbT //.
  Qed.

Thus, if the cumulative amount of service changes, then it must be scheduled, too.
  Lemma service_delta_implies_scheduled:
    ∀ t,
      service sched j t < service sched j t.+1 → scheduled_at sched j t.
  Proof.
    move ⇒ t.
    rewrite -service_last_plus_before -{1}(addn0 (service sched j t)) ltn_add2l.
    apply: service_at_implies_scheduled_at.
  Qed.

We observe that a job receives cumulative service during some interval iff it receives services at some specific time in the interval.
  Lemma service_during_service_at:
    ∀ t1 t2,
      service_during sched j t1 t2 > 0
      ↔
      ∃ t,
        t1 ≤ t < t2 ∧
        service_at sched j t > 0.
  Proof.
    split.
    {
      move⇒ NONZERO.
      case (boolP([∃ t: 'I_t2,
                      (t ≥ t1) && (service_at sched j t > 0)])) ⇒ [EX | ALL].
      - move: EX ⇒ /existsP [x /andP [GE SERV]].
        ∃ x; split ⇒ //.
        apply /andP; by split.
      - rewrite negb_exists in ALL; move: ALL ⇒ /forallP ALL.
        rewrite /service_during big_nat_cond in NONZERO.
        rewrite big1 ?ltn0 // in NONZERO ⇒ i.
        rewrite andbT; move ⇒ /andP [GT LT].
        specialize (ALL (Ordinal LT)); simpl in ALL.
        rewrite GT andTb -eqn0Ngt in ALL.
        apply /eqP ⇒ //.
    }
    {
      move⇒ [t [TT SERVICE]].
      case (boolP (0 < service_during sched j t1 t2)) ⇒ // NZ.
      exfalso.
      rewrite -eqn0Ngt in NZ. move/eqP: NZ.
      rewrite big_nat_eq0 ⇒ IS_ZERO.
      have NO_SERVICE := IS_ZERO t TT.
      apply lt0n_neq0 in SERVICE.
        by move/neqP in SERVICE; contradiction.
    }
  Qed.

Thus, any job that receives some service during an interval must be scheduled at some point during the interval...
  Corollary cumulative_service_implies_scheduled:
    ∀ t1 t2,
      service_during sched j t1 t2 > 0 →
      ∃ t,
        t1 ≤ t < t2 ∧
        scheduled_at sched j t.
  Proof.
    move⇒ t1 t2.
    rewrite service_during_service_at.
    move⇒ [t' [TIMES SERVICED]].
    ∃ t'; split ⇒ //.
    by apply: service_at_implies_scheduled_at.
 Qed.

...which implies that any job with positive cumulative service must have been scheduled at some point.
  Corollary positive_service_implies_scheduled_before:
    ∀ t,
      service sched j t > 0 → ∃ t', (t' < t ∧ scheduled_at sched j t').
  Proof.
    move⇒ t2.
    rewrite /service ⇒ NONZERO.
    have EX_SCHED := cumulative_service_implies_scheduled 0 t2 NONZERO.
    by move: EX_SCHED ⇒ [t [TIMES SCHED_AT]]; ∃ t; split.
  Qed.

  Section GuaranteedService.
If we can assume that a scheduled job always receives service, we can further prove the converse.
Assume j always receives some positive service.
    Hypothesis H_scheduled_implies_serviced: ideal_progress_proc_model PState.

In other words, not being scheduled is equivalent to receiving zero service.
    Lemma no_service_not_scheduled:
      ∀ t,
        ~~ scheduled_at sched j t ↔ service_at sched j t = 0.
    Proof.
      move⇒ t. rewrite /scheduled_at /service_at.
      split ⇒ [NOT_SCHED | NO_SERVICE].
      - by rewrite service_implies_scheduled //.
      - apply (contra (H_scheduled_implies_serviced j (sched t))).
        by rewrite -eqn0Ngt; apply /eqP ⇒ //.
    Qed.

Then, if a job does not receive any service during an interval, it is not scheduled.
    Lemma no_service_during_implies_not_scheduled:
      ∀ t1 t2,
        service_during sched j t1 t2 = 0 →
        ∀ t,
          t1 ≤ t < t2 → ~~ scheduled_at sched j t.
    Proof.
      move⇒ t1 t2 ZERO_SUM t /andP [GT_t1 LT_t2].
      rewrite no_service_not_scheduled.
      move: ZERO_SUM.
      rewrite /service_during big_nat_eq0 ⇒ IS_ZERO.
      by apply (IS_ZERO t); apply /andP; split ⇒ //.
    Qed.

If a job is scheduled at some point in an interval, it receivees positive cumulative service during the interval...
    Lemma scheduled_implies_cumulative_service:
      ∀ t1 t2,
        (∃ t,
            t1 ≤ t < t2 ∧
            scheduled_at sched j t) →
        service_during sched j t1 t2 > 0.
    Proof.
      move⇒ t1 t2 [t' [TIMES SCHED]].
      rewrite service_during_service_at.
      ∃ t'; split ⇒ //.
      move: SCHED. rewrite /scheduled_at /service_at.
        by apply (H_scheduled_implies_serviced j (sched t')).
    Qed.

...which again applies to total service, too.
    Corollary scheduled_implies_nonzero_service:
      ∀ t,
        (∃ t',
            t' < t ∧
            scheduled_at sched j t') →
        service sched j t > 0.
    Proof.
      move⇒ t [t' [TT SCHED]].
      rewrite /service. apply scheduled_implies_cumulative_service.
      ∃ t'; split ⇒ //.
    Qed.

  End GuaranteedService.

  Section AfterArrival.
Futhermore, if we know that jobs are not released early, then we can narrow the interval during which they must have been scheduled.

    Context `{JobArrival Job}.

Assume that jobs must arrive to execute.
    Hypothesis H_jobs_must_arrive:
      jobs_must_arrive_to_execute sched.

We prove that any job with positive cumulative service at time t must have been scheduled some time since its arrival and before time t.
    Lemma positive_service_implies_scheduled_since_arrival:
      ∀ t,
        service sched j t > 0 →
        ∃ t', (job_arrival j ≤ t' < t ∧ scheduled_at sched j t').
    Proof.
      move⇒ t SERVICE.
      have EX_SCHED := positive_service_implies_scheduled_before t SERVICE.
      inversion EX_SCHED as [t'' [TIMES SCHED_AT]].
      ∃ t''; split; last by assumption.
      rewrite /(_ && _) ifT //.
      move: H_jobs_must_arrive. rewrite /jobs_must_arrive_to_execute /has_arrived ⇒ ARR.
        by apply: ARR; exact.
    Qed.

    Lemma not_scheduled_before_arrival:
      ∀ t, t < job_arrival j → ~~ scheduled_at sched j t.
    Proof.
      move⇒ t EARLY.
      apply: (contra (H_jobs_must_arrive j t)).
      rewrite /has_arrived -ltnNge //.
   Qed.

We show that job j does not receive service at any time t prior to its arrival.
    Lemma service_before_job_arrival_zero:
      ∀ t,
        t < job_arrival j →
        service_at sched j t = 0.
    Proof.
      move⇒ t NOT_ARR.
      rewrite not_scheduled_implies_no_service // not_scheduled_before_arrival //.
    Qed.

Note that the same property applies to the cumulative service.
    Lemma cumulative_service_before_job_arrival_zero :
      ∀ t1 t2 : instant,
        t2 ≤ job_arrival j →
        service_during sched j t1 t2 = 0.
    Proof.
      move⇒ t1 t2 EARLY.
      rewrite /service_during.
      have ZERO_SUM: \sum_(t1 ≤ t < t2) service_at sched j t = \sum_(t1 ≤ t < t2) 0;
        last by rewrite ZERO_SUM sum0.
      rewrite big_nat_cond [in RHS]big_nat_cond.
      apply: eq_bigr ⇒ i /andP [TIMES _]. move: TIMES ⇒ /andP [le_t1_i lt_i_t2].
      apply (service_before_job_arrival_zero i).
        by apply leq_trans with (n := t2); auto.
    Qed.

Hence, one can ignore the service received by a job before its arrival time...
    Lemma ignore_service_before_arrival:
      ∀ t1 t2,
        t1 ≤ job_arrival j →
        t2 ≥ job_arrival j →
        service_during sched j t1 t2 = service_during sched j (job_arrival j) t2.
    Proof.
      move⇒ t1 t2 le_t1 le_t2.
      rewrite -(service_during_cat sched j t1 (job_arrival j) t2).
      rewrite cumulative_service_before_job_arrival_zero //.
        by apply/andP; split; exact.
    Qed.

... which we can also state in terms of cumulative service.
    Corollary no_service_before_arrival:
      ∀ t,
        t ≤ job_arrival j → service sched j t = 0.
    Proof.
      move⇒ t EARLY.
      rewrite /service cumulative_service_before_job_arrival_zero //.
    Qed.

  End AfterArrival.

  Section TimesWithSameService.
In this section, we prove some lemmas about time instants with same service.
Consider any time instants t1 and t2...
    Variable t1 t2: instant.

...where t1 is no later than t2...
    Hypothesis H_t1_le_t2: t1 ≤ t2.

...and where job j has received the same amount of service.
    Hypothesis H_same_service: service sched j t1 = service sched j t2.

First, we observe that this means that the job receives no service during t1, t2)...
    Lemma constant_service_implies_no_service_during:
      service_during sched j t1 t2 = 0.
    Proof.
      move: H_same_service.
      rewrite -(service_cat sched j t1 t2) // -[service sched j t1 in LHS]addn0 ⇒ /eqP.
      by rewrite eqn_add2l ⇒ /eqP //.
    Qed.

...which of course implies that it does not receive service at any point, either.
    Lemma constant_service_implies_not_scheduled:
      ∀ t,
        t1 ≤ t < t2 → service_at sched j t = 0.
    Proof.
      move⇒ t /andP [GE_t1 LT_t2].
      move: constant_service_implies_no_service_during.
      rewrite /service_during big_nat_eq0 ⇒ IS_ZERO.
      apply IS_ZERO. apply /andP; split ⇒ //.
    Qed.

We show that job j receives service at some point t < t1 iff j receives service at some point t' < t2.
    Lemma same_service_implies_serviced_at_earlier_times:
      [∃ t: 'I_t1, service_at sched j t > 0] =
      [∃ t': 'I_t2, service_at sched j t' > 0].
    Proof.
      apply /idP/idP ⇒ /existsP [t SERVICED].
      {
        have LE: t < t2
          by apply: (leq_trans _ H_t1_le_t2) ⇒ //.
        by apply /existsP; ∃ (Ordinal LE).
      }
      {
        case (boolP (t < t1)) ⇒ LE.
        - by apply /existsP; ∃ (Ordinal LE).
        - exfalso.
          have TIMES: t1 ≤ t < t2
            by apply /andP; split ⇒ //; rewrite leqNgt //.
          have NO_SERVICE := constant_service_implies_not_scheduled t TIMES.
          by rewrite NO_SERVICE in SERVICED.
      }
    Qed.

Then, under the assumption that scheduled jobs receives service, we can translate this into a claim about scheduled_at.
Assume j always receives some positive service.
    Hypothesis H_scheduled_implies_serviced: ideal_progress_proc_model PState.

We show that job j is scheduled at some point t < t1 iff j is scheduled at some point t' < t2.
    Lemma same_service_implies_scheduled_at_earlier_times:
      [∃ t: 'I_t1, scheduled_at sched j t] =
      [∃ t': 'I_t2, scheduled_at sched j t'].
    Proof.
      have CONV: ∀ B, [∃ b: 'I_B, scheduled_at sched j b]
                           = [∃ b: 'I_B, service_at sched j b > 0].
      {
        move⇒ B. apply/idP/idP ⇒ /existsP [b P]; apply /existsP; ∃ b.
        - by move: P; rewrite /scheduled_at /service_at;
            apply (H_scheduled_implies_serviced j (sched b)).
        - by apply service_at_implies_scheduled_at.
      }
      rewrite !CONV.
      apply same_service_implies_serviced_at_earlier_times.
    Qed.
  End TimesWithSameService.

End RelationToScheduled.

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

Incremental Service in Ideal Schedule

In the following section we prove a few facts about service in ideal schedeule.
(* Note that these lemmas can be generalized to an arbitrary scheduler. *)
Section IncrementalService.

Consider any job type, ...
  Context {Job : JobType}.
  Context `{JobArrival Job}.
  Context `{JobCost Job}.

... any arrival sequence, ...
  Variable arr_seq : arrival_sequence Job.

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

As a base case, we prove that if a job j receives service in some time interval t1,t2), then there ∃ a time instant t ∈ [t1,t2) such that j is scheduled at time t and t is the first instant where j receives service.
  Lemma positive_service_during:
    ∀ j t1 t2,
      0 < service_during sched j t1 t2 →
      ∃ t : nat, t1 ≤ t < t2 ∧ scheduled_at sched j t ∧ service_during sched j t1 t = 0.
  Proof.
    intros j t1 t2 SERV.
    have LE: t1 ≤ t2.
    { rewrite leqNgt; apply/negP; intros CONTR.
        by apply ltnW in CONTR; move: SERV; rewrite /service_during big_geq.
    }
    destruct (scheduled_at sched j t1) eqn:SCHED.
    { ∃ t1; repeat split; try done.
      - apply/andP; split; first by done.
        rewrite ltnNge; apply/negP; intros CONTR.
          by move: SERV; rewrite/service_during big_geq.
      - by rewrite /service_during big_geq.
    }
    { apply negbT in SCHED.
      move: SERV; rewrite /service /service_during; move ⇒ /sum_seq_gt0P [t [IN SCHEDt]].
      rewrite lt0b in SCHEDt.
      rewrite mem_iota subnKC in IN; last by done.
      move: IN ⇒ /andP [IN1 IN2].
      move: (exists_first_intermediate_point
               ((fun t ⇒ scheduled_at sched j t)) t1 t IN1 SCHED) ⇒ A.
      feed A; first by rewrite scheduled_at_def/=.
      move: A ⇒ [x [/andP [T1 T4] [T2 T3]]].
      ∃ x; repeat split; try done.
      - apply/andP; split; first by apply ltnW.
          by apply leq_ltn_trans with t.
      - apply/eqP; rewrite big_nat_cond big1 //.
        move ⇒ y /andP [T5 _].
          by apply/eqP; rewrite eqb0; specialize (T2 y); rewrite scheduled_at_def/= in T2; apply T2.
    }
  Qed.

Next, we prove that if in some time interval t1,t2) a job j receives k units of service, then there ∃ a time instant t ∈ [t1,t2) such that j is scheduled at time t and service of job j within interval [t1,t) is equal to k.
  Lemma incremental_service_during:
    ∀ j t1 t2 k,
      service_during sched j t1 t2 > k →
      ∃ t, t1 ≤ t < t2 ∧ scheduled_at sched j t ∧ service_during sched j t1 t = k.
  Proof.
    intros j t1 t2 k SERV.
    have LE: t1 ≤ t2.
    { rewrite leqNgt; apply/negP; intros CONTR.
        by apply ltnW in CONTR; move: SERV; rewrite /service_during big_geq.
    }
    induction k; first by apply positive_service_during in SERV.
    feed IHk; first by apply ltn_trans with k.+1.
    move: IHk ⇒ [t [/andP [NEQ1 NEQ2] [SCHEDt SERVk]]].
    have SERVk1: service_during sched j t1 t.+1 = k.+1.
    { rewrite -(service_during_cat _ _ _ t); last by apply/andP; split.
      rewrite SERVk -[X in _ = X]addn1; apply/eqP; rewrite eqn_add2l.
        by rewrite /service_during big_nat1 /service_at eqb1 -scheduled_at_def/=.
    }
    move: SERV; rewrite -(service_during_cat _ _ _ t.+1); last first.
    { by apply/andP; split; first apply leq_trans with t. }
    rewrite SERVk1 -addn1 leq_add2l; move ⇒ SERV.
    destruct (scheduled_at sched j t.+1) eqn:SCHED.
    - ∃ t.+1; repeat split; try done. apply/andP; split.
      + apply leq_trans with t; by done.
      + rewrite ltnNge; apply/negP; intros CONTR.
          by move: SERV; rewrite /service_during big_geq.
    - apply negbT in SCHED.
       move: SERV; rewrite /service /service_during; move ⇒ /sum_seq_gt0P [x [INx SCHEDx]].
       rewrite lt0b in SCHEDx.
       rewrite mem_iota subnKC in INx; last by done.
       move: INx ⇒ /andP [INx1 INx2].
       move: (exists_first_intermediate_point _ _ _ INx1 SCHED) ⇒ A.
       feed A; first by rewrite scheduled_at_def/=.
       move: A ⇒ [y [/andP [T1 T4] [T2 T3]]].
       ∃ y; repeat split; try done.
       + apply/andP; split.
         apply leq_trans with t; first by done.
         apply ltnW, ltn_trans with t.+1; by done.
           by apply leq_ltn_trans with x.
       + rewrite (@big_cat_nat _ _ _ t.+1) //=; [ | by apply leq_trans with t | by apply ltn_trans with t.+1].
         unfold service_during in SERVk1; rewrite SERVk1; apply/eqP.
         rewrite -{2}[k.+1]addn0 eqn_add2l.
         rewrite big_nat_cond big1 //; move ⇒ z /andP [H5 _].
           by apply/eqP; rewrite eqb0; specialize (T2 z); rewrite scheduled_at_def/= in T2; apply T2.
  Qed.

End IncrementalService.