nonpreemptive

Require Export prosa.analysis.facts.preemption.task.nonpreemptive.Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "_ + _" was already used in scope nat_scope.
[notation-overridden,parsing]
Notation "_ - _" was already used in scope nat_scope.
[notation-overridden,parsing]
Notation "_ <= _" was already used in scope nat_scope.
[notation-overridden,parsing]
Notation "_ < _" was already used in scope nat_scope.
[notation-overridden,parsing]
Notation "_ >= _" was already used in scope nat_scope.
[notation-overridden,parsing]
Notation "_ > _" was already used in scope nat_scope.
[notation-overridden,parsing]
Notation "_ <= _ <= _" was already used in scope
nat_scope. [notation-overridden,parsing]
Notation "_ < _ <= _" was already used in scope
nat_scope. [notation-overridden,parsing]
Notation "_ <= _ < _" was already used in scope
nat_scope. [notation-overridden,parsing]
Notation "_ < _ < _" was already used in scope
nat_scope. [notation-overridden,parsing]
Notation "_ * _" was already used in scope nat_scope.
[notation-overridden,parsing]
Require Export prosa.analysis.facts.preemption.rtc_threshold.nonpreemptive.Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Require Export prosa.analysis.facts.readiness.basic.Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Require Export prosa.analysis.definitions.tardiness.Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Require Export prosa.implementation.facts.ideal_uni.prio_aware.Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Require Export prosa.implementation.definitions.task.Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Require Export prosa.model.priority.edf.Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]
Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]

(** ** Fully-Nonpreemptive Earliest-Deadline-First Schedules  *)

(** In this section, we prove that the fully-nonpreemptive preemption policy
    under earliest-deadline-first schedules is valid, and that the scheduling policy is
    respected at each preemption point.

    Some lemmas in this file are not novel facts; they are used to uniform
    POET's certificates and minimize their verbosity. *)
Section Schedule.

  (** In this file, we adopt the Prosa standard implementation of jobs and tasks. *)
  Definition Task := [eqType of concrete_task].
  Definition Job := [eqType of concrete_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.

  (** ... assume basic readiness, ... *)
  Instance basic_ready_instance : JobReady Job (ideal.processor_state Job) :=
    basic.basic_ready_instance.

  (** ... and consider any fully-nonpreemptive, earliest-deadline-first schedule. *)
  #[local] Existing Instance fully_nonpreemptive_job_model.
  #[local] Existing Instance EDF.
  Definition sched := uni_schedule arr_seq.

  (** First, we show that only ready jobs execute. *)
  Lemma sched_jobs_must_be_ready_to_execute :
    jobs_must_be_ready_to_execute sched.arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
jobs_must_be_ready_to_execute sched
  Proof.arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
jobs_must_be_ready_to_execute sched
    move=> j t SCHED.arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
SCHED: scheduled_at sched j t
job_ready sched j t
    apply jobs_must_be_ready => //.arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
SCHED: scheduled_at sched j t
forall (t : instant) (s : seq Job) (j : Job),
choose_highest_prio_job t s = Some j -> j \in s
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
SCHED: scheduled_at sched j t
nonclairvoyant_readiness basic_ready_instance
    -arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
SCHED: scheduled_at sched j t
forall (t : instant) (s : seq Job) (j : Job),
choose_highest_prio_job t s = Some j -> j \in s
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
SCHED: scheduled_at sched j t
nonclairvoyant_readiness basic_ready_instance move=> tt ss jj IN.arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
SCHED: scheduled_at sched j t
tt: instant
ss: seq Job
jj: Job
IN: choose_highest_prio_job tt ss = Some jj
jj \in ss
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
SCHED: scheduled_at sched j t
nonclairvoyant_readiness basic_ready_instance
      rewrite  /choose_highest_prio_job in IN.arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
SCHED: scheduled_at sched j t
tt: instant
ss: seq Job
jj: Job
IN: supremum (hep_job_at tt) ss = Some jj
jj \in ss
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
SCHED: scheduled_at sched j t
nonclairvoyant_readiness basic_ready_instance
      by apply supremum_in in IN.arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
SCHED: scheduled_at sched j t
nonclairvoyant_readiness basic_ready_instance
    -arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
SCHED: scheduled_at sched j t
nonclairvoyant_readiness basic_ready_instance by apply basic_readiness_nonclairvoyance.
  Qed.

  (** Next, we remark that such a schedule is valid. *)
  Remark sched_valid :
    valid_schedule sched arr_seq.arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
valid_schedule sched arr_seq
  Proof.arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
valid_schedule sched arr_seq
    split.arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
jobs_come_from_arrival_sequence sched arr_seq
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
jobs_must_be_ready_to_execute sched
    -arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
jobs_come_from_arrival_sequence sched arr_seq
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
jobs_must_be_ready_to_execute sched apply np_schedule_jobs_from_arrival_sequence.arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
forall (t : instant) (s : seq Job) (j : Job),
choose_highest_prio_job t s = Some j -> j \in s
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
jobs_must_be_ready_to_execute sched
      by move=> t ???; eapply (supremum_in (hep_job_at t)).arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
jobs_must_be_ready_to_execute sched
    -arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
jobs_must_be_ready_to_execute sched by apply sched_jobs_must_be_ready_to_execute.
  Qed.

  (** Next, we show that an unfinished job scheduled at time [t] is
      always scheduled at time [t+1] as well. Note that the validity of
      this fact also depends on which readiness model is used. *)
  Lemma sched_nonpreemptive_next :
    forall j t,
      scheduled_at sched j t ->
      ~~ completed_by sched j t.+1 ->
      scheduled_at sched j t.+1.arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
forall (j : Job) (t : instant),
scheduled_at sched j t ->
~~ completed_by sched j t.+1 ->
scheduled_at sched j t.+1
  Proof.arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
forall (j : Job) (t : instant),
scheduled_at sched j t ->
~~ completed_by sched j t.+1 ->
scheduled_at sched j t.+1
    move=> j t SCHED NCOMP.arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
SCHED: scheduled_at sched j t
NCOMP: ~~ completed_by sched j t.+1
scheduled_at sched j t.+1
    have PENDING: job_ready sched j t by apply sched_jobs_must_be_ready_to_execute.arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
SCHED: scheduled_at sched j t
NCOMP: ~~ completed_by sched j t.+1
PENDING: job_ready sched j t
scheduled_at sched j t.+1
    unfold job_ready, basic_ready_instance, basic.basic_ready_instance in *.arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
SCHED: scheduled_at sched j t
NCOMP: ~~ completed_by sched j t.+1
PENDING: pending sched j t
scheduled_at sched j t.+1
    set chp:= choose_highest_prio_job.arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
SCHED: scheduled_at sched j t
NCOMP: ~~ completed_by sched j t.+1
PENDING: pending sched j t
chp:= choose_highest_prio_job: instant ->
seq [eqType of concrete_job] ->
option [eqType of concrete_job]
scheduled_at sched j t.+1
    have SERV_ZERO: 0 < service sched j t.+1.arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
SCHED: scheduled_at sched j t
NCOMP: ~~ completed_by sched j t.+1
PENDING: pending sched j t
chp:= choose_highest_prio_job: instant ->
seq [eqType of concrete_job] ->
option [eqType of concrete_job]
0 < service sched j t.+1
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
SCHED: scheduled_at sched j t
NCOMP: ~~ completed_by sched j t.+1
PENDING: pending sched j t
chp:= choose_highest_prio_job: instant ->
seq [eqType of concrete_job] ->
option [eqType of concrete_job]
SERV_ZERO: 0 < service sched j t.+1
scheduled_at sched j t.+1
    {arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
SCHED: scheduled_at sched j t
NCOMP: ~~ completed_by sched j t.+1
PENDING: pending sched j t
chp:= choose_highest_prio_job: instant ->
seq [eqType of concrete_job] ->
option [eqType of concrete_job]
0 < service sched j t.+1 apply scheduled_implies_nonzero_service => //; last by simpl; exists t; split.arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
SCHED: scheduled_at sched j t
NCOMP: ~~ completed_by sched j t.+1
PENDING: pending sched j t
chp:= choose_highest_prio_job: instant ->
seq [eqType of concrete_job] ->
option [eqType of concrete_job]
ideal_progress_proc_model (ideal.processor_state Job)
      by apply ideal_proc_model_ensures_ideal_progress. }arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
SCHED: scheduled_at sched j t
NCOMP: ~~ completed_by sched j t.+1
PENDING: pending sched j t
chp:= choose_highest_prio_job: instant ->
seq [eqType of concrete_job] ->
option [eqType of concrete_job]
SERV_ZERO: 0 < service sched j t.+1
scheduled_at sched j t.+1
    have SERV_COST: service sched j t.+1 < job_cost j by apply less_service_than_cost_is_incomplete.arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
SCHED: scheduled_at sched j t
NCOMP: ~~ completed_by sched j t.+1
PENDING: pending sched j t
chp:= choose_highest_prio_job: instant ->
seq [eqType of concrete_job] ->
option [eqType of concrete_job]
SERV_ZERO: 0 < service sched j t.+1
SERV_COST: service sched j t.+1 < job_cost j
scheduled_at sched j t.+1
    move: SCHED NCOMP.arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
PENDING: pending sched j t
chp:= choose_highest_prio_job: instant ->
seq [eqType of concrete_job] ->
option [eqType of concrete_job]
SERV_ZERO: 0 < service sched j t.+1
SERV_COST: service sched j t.+1 < job_cost j
scheduled_at sched j t ->
~~ completed_by sched j t.+1 ->
scheduled_at sched j t.+1
    rewrite !scheduled_at_def /sched /uni_schedule
            /pmc_uni_schedule /generic_schedule => /eqP SCHED NCOMP.arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
PENDING: pending sched j t
chp:= choose_highest_prio_job: instant ->
seq [eqType of concrete_job] ->
option [eqType of concrete_job]
SERV_ZERO: 0 < service sched j t.+1
SERV_COST: service sched j t.+1 < job_cost j
SCHED: schedule_up_to
  (allocation_at arr_seq
     choose_highest_prio_job) None t t =
Some j
NCOMP: ~~
completed_by
  (fun t : instant =>
   schedule_up_to
     (allocation_at arr_seq
        choose_highest_prio_job) None t t) j
  t.+1
schedule_up_to
  (allocation_at arr_seq choose_highest_prio_job) None
  t.+1 t.+1 == Some j
    rewrite schedule_up_to_def {1}/allocation_at ifT; first by rewrite SCHED.arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
PENDING: pending sched j t
chp:= choose_highest_prio_job: instant ->
seq [eqType of concrete_job] ->
option [eqType of concrete_job]
SERV_ZERO: 0 < service sched j t.+1
SERV_COST: service sched j t.+1 < job_cost j
SCHED: schedule_up_to
  (allocation_at arr_seq
     choose_highest_prio_job) None t t =
Some j
NCOMP: ~~
completed_by
  (fun t : instant =>
   schedule_up_to
     (allocation_at arr_seq
        choose_highest_prio_job) None t t) j
  t.+1
prev_job_nonpreemptive
  (schedule_up_to
     (allocation_at arr_seq choose_highest_prio_job)
     None t) t.+1
    rewrite /prev_job_nonpreemptive SCHED /job_preemptable /fully_nonpreemptive_job_model.arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
PENDING: pending sched j t
chp:= choose_highest_prio_job: instant ->
seq [eqType of concrete_job] ->
option [eqType of concrete_job]
SERV_ZERO: 0 < service sched j t.+1
SERV_COST: service sched j t.+1 < job_cost j
SCHED: schedule_up_to
  (allocation_at arr_seq
     choose_highest_prio_job) None t t =
Some j
NCOMP: ~~
completed_by
  (fun t : instant =>
   schedule_up_to
     (allocation_at arr_seq
        choose_highest_prio_job) None t t) j
  t.+1
job_ready
  (schedule_up_to
     (allocation_at arr_seq choose_highest_prio_job)
     None t) j t.+1 &&
~~
((service
    (schedule_up_to
       (allocation_at arr_seq choose_highest_prio_job)
       None t) j t.+1 == 0)
 || (service
       (schedule_up_to
          (allocation_at arr_seq
             choose_highest_prio_job) None t) j t.+1 ==
     job_cost j))
    move: SERV_COST SERV_ZERO; rewrite /uni_schedule /pmc_uni_schedule /generic_schedule.arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
PENDING: pending sched j t
chp:= choose_highest_prio_job: instant ->
seq [eqType of concrete_job] ->
option [eqType of concrete_job]
SCHED: schedule_up_to
  (allocation_at arr_seq
     choose_highest_prio_job) None t t =
Some j
NCOMP: ~~
completed_by
  (fun t : instant =>
   schedule_up_to
     (allocation_at arr_seq
        choose_highest_prio_job) None t t) j
  t.+1
service sched j t.+1 < job_cost j ->
0 < service sched j t.+1 ->
job_ready
  (schedule_up_to
     (allocation_at arr_seq choose_highest_prio_job)
     None t) j t.+1 &&
~~
((service
    (schedule_up_to
       (allocation_at arr_seq choose_highest_prio_job)
       None t) j t.+1 == 0)
 || (service
       (schedule_up_to
          (allocation_at arr_seq
             choose_highest_prio_job) None t) j t.+1 ==
     job_cost j))
    replace (_ (_ (_ _ _) _ t) j _) with (service sched j t.+1); last first.arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
PENDING: pending sched j t
chp:= choose_highest_prio_job: instant ->
seq [eqType of concrete_job] ->
option [eqType of concrete_job]
SCHED: schedule_up_to
  (allocation_at arr_seq
     choose_highest_prio_job) None t t =
Some j
NCOMP: ~~
completed_by
  (fun t : instant =>
   schedule_up_to
     (allocation_at arr_seq
        choose_highest_prio_job) None t t) j
  t.+1
service sched j t.+1 =
service
  (schedule_up_to
     (allocation_at arr_seq choose_highest_prio_job)
     None t) j t.+1
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
PENDING: pending sched j t
chp:= choose_highest_prio_job: instant ->
seq [eqType of concrete_job] ->
option [eqType of concrete_job]
SCHED: schedule_up_to
  (allocation_at arr_seq
     choose_highest_prio_job) None t t =
Some j
NCOMP: ~~
completed_by
  (fun t : instant =>
   schedule_up_to
     (allocation_at arr_seq
        choose_highest_prio_job) None t t) j
  t.+1
service sched j t.+1 < job_cost j ->
0 < service sched j t.+1 ->
job_ready
  (schedule_up_to
     (allocation_at arr_seq choose_highest_prio_job)
     None t) j t.+1 &&
~~
((service sched j t.+1 == 0)
 || (service sched j t.+1 == job_cost j))
    {arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
PENDING: pending sched j t
chp:= choose_highest_prio_job: instant ->
seq [eqType of concrete_job] ->
option [eqType of concrete_job]
SCHED: schedule_up_to
  (allocation_at arr_seq
     choose_highest_prio_job) None t t =
Some j
NCOMP: ~~
completed_by
  (fun t : instant =>
   schedule_up_to
     (allocation_at arr_seq
        choose_highest_prio_job) None t t) j
  t.+1
service sched j t.+1 =
service
  (schedule_up_to
     (allocation_at arr_seq choose_highest_prio_job)
     None t) j t.+1 rewrite /uni_schedule /pmc_uni_schedule /generic_schedule /service
              /service_during /service_at.arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
PENDING: pending sched j t
chp:= choose_highest_prio_job: instant ->
seq [eqType of concrete_job] ->
option [eqType of concrete_job]
SCHED: schedule_up_to
  (allocation_at arr_seq
     choose_highest_prio_job) None t t =
Some j
NCOMP: ~~
completed_by
  (fun t : instant =>
   schedule_up_to
     (allocation_at arr_seq
        choose_highest_prio_job) None t t) j
  t.+1
\sum_(0 <= t < t.+1) service_in j (sched t) =
\sum_(0 <= t0 < t.+1)
   service_in j
     (schedule_up_to
        (allocation_at arr_seq choose_highest_prio_job)
        None t t0)
      apply eq_big_nat; intros.arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
PENDING: pending sched j t
chp:= choose_highest_prio_job: instant ->
seq [eqType of concrete_job] ->
option [eqType of concrete_job]
SCHED: schedule_up_to
  (allocation_at arr_seq
     choose_highest_prio_job) None t t =
Some j
NCOMP: ~~
completed_by
  (fun t : instant =>
   schedule_up_to
     (allocation_at arr_seq
        choose_highest_prio_job) None t t) j
  t.+1
i: nat
H: 0 <= i < t.+1
service_in j (sched i) =
service_in j
  (schedule_up_to
     (allocation_at arr_seq choose_highest_prio_job)
     None t i)
      replace (_ (_ _ _) _ t i) with (schedule_up_to (allocation_at arr_seq chp) None i i) => //.arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
PENDING: pending sched j t
chp:= choose_highest_prio_job: instant ->
seq [eqType of concrete_job] ->
option [eqType of concrete_job]
SCHED: schedule_up_to
  (allocation_at arr_seq
     choose_highest_prio_job) None t t =
Some j
NCOMP: ~~
completed_by
  (fun t : instant =>
   schedule_up_to
     (allocation_at arr_seq
        choose_highest_prio_job) None t t) j
  t.+1
i: nat
H: 0 <= i < t.+1
schedule_up_to (allocation_at arr_seq chp) None i i =
schedule_up_to
  (allocation_at arr_seq choose_highest_prio_job) None
  t i
      by apply schedule_up_to_prefix_inclusion. }arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
PENDING: pending sched j t
chp:= choose_highest_prio_job: instant ->
seq [eqType of concrete_job] ->
option [eqType of concrete_job]
SCHED: schedule_up_to
  (allocation_at arr_seq
     choose_highest_prio_job) None t t =
Some j
NCOMP: ~~
completed_by
  (fun t : instant =>
   schedule_up_to
     (allocation_at arr_seq
        choose_highest_prio_job) None t t) j
  t.+1
service sched j t.+1 < job_cost j ->
0 < service sched j t.+1 ->
job_ready
  (schedule_up_to
     (allocation_at arr_seq choose_highest_prio_job)
     None t) j t.+1 &&
~~
((service sched j t.+1 == 0)
 || (service sched j t.+1 == job_cost j))
    move=> SERV_COST SERV_ZERO; apply /andP.arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
PENDING: pending sched j t
chp:= choose_highest_prio_job: instant ->
seq [eqType of concrete_job] ->
option [eqType of concrete_job]
SCHED: schedule_up_to
  (allocation_at arr_seq
     choose_highest_prio_job) None t t =
Some j
NCOMP: ~~
completed_by
  (fun t : instant =>
   schedule_up_to
     (allocation_at arr_seq
        choose_highest_prio_job) None t t) j
  t.+1
SERV_COST: service sched j t.+1 < job_cost j
SERV_ZERO: 0 < service sched j t.+1
job_ready
  (schedule_up_to
     (allocation_at arr_seq choose_highest_prio_job)
     None t) j t.+1 /\
~~
((service sched j t.+1 == 0)
 || (service sched j t.+1 == job_cost j))
    split; [|by apply /norP;split;[rewrite -lt0n|move: SERV_COST;rewrite ltn_neqAle=> /andP [??]]].arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
PENDING: pending sched j t
chp:= choose_highest_prio_job: instant ->
seq [eqType of concrete_job] ->
option [eqType of concrete_job]
SCHED: schedule_up_to
  (allocation_at arr_seq
     choose_highest_prio_job) None t t =
Some j
NCOMP: ~~
completed_by
  (fun t : instant =>
   schedule_up_to
     (allocation_at arr_seq
        choose_highest_prio_job) None t t) j
  t.+1
SERV_COST: service sched j t.+1 < job_cost j
SERV_ZERO: 0 < service sched j t.+1
job_ready
  (schedule_up_to
     (allocation_at arr_seq choose_highest_prio_job)
     None t) j t.+1
    {arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
PENDING: pending sched j t
chp:= choose_highest_prio_job: instant ->
seq [eqType of concrete_job] ->
option [eqType of concrete_job]
SCHED: schedule_up_to
  (allocation_at arr_seq
     choose_highest_prio_job) None t t =
Some j
NCOMP: ~~
completed_by
  (fun t : instant =>
   schedule_up_to
     (allocation_at arr_seq
        choose_highest_prio_job) None t t) j
  t.+1
SERV_COST: service sched j t.+1 < job_cost j
SERV_ZERO: 0 < service sched j t.+1
job_ready
  (schedule_up_to
     (allocation_at arr_seq choose_highest_prio_job)
     None t) j t.+1 unfold job_ready, basic_ready_instance, basic.basic_ready_instance.arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
PENDING: pending sched j t
chp:= choose_highest_prio_job: instant ->
seq [eqType of concrete_job] ->
option [eqType of concrete_job]
SCHED: schedule_up_to
  (allocation_at arr_seq
     choose_highest_prio_job) None t t =
Some j
NCOMP: ~~
completed_by
  (fun t : instant =>
   schedule_up_to
     (allocation_at arr_seq
        choose_highest_prio_job) None t t) j
  t.+1
SERV_COST: service sched j t.+1 < job_cost j
SERV_ZERO: 0 < service sched j t.+1
pending
  (schedule_up_to
     (allocation_at arr_seq choose_highest_prio_job)
     None t) j t.+1
      apply/andP; split; first by move: PENDING => /andP [ARR _]; unfold has_arrived in *; lia.arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
PENDING: pending sched j t
chp:= choose_highest_prio_job: instant ->
seq [eqType of concrete_job] ->
option [eqType of concrete_job]
SCHED: schedule_up_to
  (allocation_at arr_seq
     choose_highest_prio_job) None t t =
Some j
NCOMP: ~~
completed_by
  (fun t : instant =>
   schedule_up_to
     (allocation_at arr_seq
        choose_highest_prio_job) None t t) j
  t.+1
SERV_COST: service sched j t.+1 < job_cost j
SERV_ZERO: 0 < service sched j t.+1
~~
completed_by
  (schedule_up_to
     (allocation_at arr_seq choose_highest_prio_job)
     None t) j t.+1
      unfold completed_by in *; rewrite <- ltnNge in *.arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
PENDING: pending sched j t
chp:= choose_highest_prio_job: instant ->
seq [eqType of concrete_job] ->
option [eqType of concrete_job]
SCHED: schedule_up_to
  (allocation_at arr_seq
     choose_highest_prio_job) None t t =
Some j
NCOMP: service
  (fun t : instant =>
   schedule_up_to
     (allocation_at arr_seq
        choose_highest_prio_job) None t t) j
  t.+1 < job_cost j
SERV_COST: service sched j t.+1 < job_cost j
SERV_ZERO: 0 < service sched j t.+1
service
  (schedule_up_to
     (allocation_at arr_seq choose_highest_prio_job)
     None t) j t.+1 < job_cost j
      replace (service (_ (_ _ _) _ t) j t.+1) with
          (service (fun t => schedule_up_to (allocation_at arr_seq chp) None t t) j t.+1) => //.arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
PENDING: pending sched j t
chp:= choose_highest_prio_job: instant ->
seq [eqType of concrete_job] ->
option [eqType of concrete_job]
SCHED: schedule_up_to
  (allocation_at arr_seq
     choose_highest_prio_job) None t t =
Some j
NCOMP: service
  (fun t : instant =>
   schedule_up_to
     (allocation_at arr_seq
        choose_highest_prio_job) None t t) j
  t.+1 < job_cost j
SERV_COST: service sched j t.+1 < job_cost j
SERV_ZERO: 0 < service sched j t.+1
service
  (fun t : instant =>
   schedule_up_to (allocation_at arr_seq chp) None t t)
  j t.+1 =
service
  (schedule_up_to
     (allocation_at arr_seq choose_highest_prio_job)
     None t) j t.+1
      rewrite /service /service_during //=; apply eq_big_nat.arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
PENDING: pending sched j t
chp:= choose_highest_prio_job: instant ->
seq [eqType of concrete_job] ->
option [eqType of concrete_job]
SCHED: schedule_up_to
  (allocation_at arr_seq
     choose_highest_prio_job) None t t =
Some j
NCOMP: service
  (fun t : instant =>
   schedule_up_to
     (allocation_at arr_seq
        choose_highest_prio_job) None t t) j
  t.+1 < job_cost j
SERV_COST: service sched j t.+1 < job_cost j
SERV_ZERO: 0 < service sched j t.+1
forall i : nat,
0 <= i < t.+1 ->
service_at
  (fun t : instant =>
   schedule_up_to (allocation_at arr_seq chp) None t t)
  j i =
service_at
  (schedule_up_to
     (allocation_at arr_seq choose_highest_prio_job)
     None t) j i
      move=> t' /andP[GEQ0 LEQt]; rewrite /service_at.arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
PENDING: pending sched j t
chp:= choose_highest_prio_job: instant ->
seq [eqType of concrete_job] ->
option [eqType of concrete_job]
SCHED: schedule_up_to
  (allocation_at arr_seq
     choose_highest_prio_job) None t t =
Some j
NCOMP: service
  (fun t : instant =>
   schedule_up_to
     (allocation_at arr_seq
        choose_highest_prio_job) None t t) j
  t.+1 < job_cost j
SERV_COST: service sched j t.+1 < job_cost j
SERV_ZERO: 0 < service sched j t.+1
t': nat
GEQ0: 0 <= t'
LEQt: t' < t.+1
service_in j
  (schedule_up_to (allocation_at arr_seq chp) None t'
     t') =
service_in j
  (schedule_up_to
     (allocation_at arr_seq choose_highest_prio_job)
     None t t')
      replace (_ (_ _ _) _ t' t') with (schedule_up_to (allocation_at arr_seq chp) None t t') => //.arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
PENDING: pending sched j t
chp:= choose_highest_prio_job: instant ->
seq [eqType of concrete_job] ->
option [eqType of concrete_job]
SCHED: schedule_up_to
  (allocation_at arr_seq
     choose_highest_prio_job) None t t =
Some j
NCOMP: service
  (fun t : instant =>
   schedule_up_to
     (allocation_at arr_seq
        choose_highest_prio_job) None t t) j
  t.+1 < job_cost j
SERV_COST: service sched j t.+1 < job_cost j
SERV_ZERO: 0 < service sched j t.+1
t': nat
GEQ0: 0 <= t'
LEQt: t' < t.+1
schedule_up_to (allocation_at arr_seq chp) None t t' =
schedule_up_to (allocation_at arr_seq chp) None t' t'
      by symmetry; apply schedule_up_to_prefix_inclusion; lia. }
  Qed.

  (** Using the lemma above, we show that the schedule is nonpreemptive. *)
  Lemma sched_nonpreemptive :
    nonpreemptive_schedule (uni_schedule arr_seq).arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
nonpreemptive_schedule (uni_schedule arr_seq)
  Proof.arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
nonpreemptive_schedule (uni_schedule arr_seq)
    rewrite /nonpreemptive_schedule.arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
forall (j : Job) (t t' : instant),
t <= t' ->
scheduled_at (uni_schedule arr_seq) j t ->
~~ completed_by (uni_schedule arr_seq) j t' ->
scheduled_at (uni_schedule arr_seq) j t'
    induction t'; first by move=> t'; have->: t = 0 by lia.arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
t': nat
IHt': t <= t' ->
scheduled_at (uni_schedule arr_seq) j t ->
~~ completed_by (uni_schedule arr_seq) j t' ->
scheduled_at (uni_schedule arr_seq) j t'
t <= t'.+1 ->
scheduled_at (uni_schedule arr_seq) j t ->
~~ completed_by (uni_schedule arr_seq) j t'.+1 ->
scheduled_at (uni_schedule arr_seq) j t'.+1
    move=> LEQ SCHED NCOMP.arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
t': nat
IHt': t <= t' ->
scheduled_at (uni_schedule arr_seq) j t ->
~~ completed_by (uni_schedule arr_seq) j t' ->
scheduled_at (uni_schedule arr_seq) j t'
LEQ: t <= t'.+1
SCHED: scheduled_at (uni_schedule arr_seq) j t
NCOMP: ~~ completed_by (uni_schedule arr_seq) j t'.+1
scheduled_at (uni_schedule arr_seq) j t'.+1
    destruct (ltngtP t t'.+1) as [LT | _ | EQ] => //; last by rewrite -EQ.arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
t': nat
IHt': t <= t' ->
scheduled_at (uni_schedule arr_seq) j t ->
~~ completed_by (uni_schedule arr_seq) j t' ->
scheduled_at (uni_schedule arr_seq) j t'
LEQ: true
SCHED: scheduled_at (uni_schedule arr_seq) j t
NCOMP: ~~ completed_by (uni_schedule arr_seq) j t'.+1
LT: t < t'.+1
scheduled_at (uni_schedule arr_seq) j t'.+1
    feed_n 3 IHt'=> //.arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
t': nat
IHt': ~~ completed_by (uni_schedule arr_seq) j t' ->
scheduled_at (uni_schedule arr_seq) j t'
LEQ: true
SCHED: scheduled_at (uni_schedule arr_seq) j t
NCOMP: ~~ completed_by (uni_schedule arr_seq) j t'.+1
LT: t < t'.+1
~~ completed_by (uni_schedule arr_seq) j t'
arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
t': nat
IHt': scheduled_at (uni_schedule arr_seq) j t'
LEQ: true
SCHED: scheduled_at (uni_schedule arr_seq) j t
NCOMP: ~~ completed_by (uni_schedule arr_seq) j t'.+1
LT: t < t'.+1
scheduled_at (uni_schedule arr_seq) j t'.+1
    {arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
t': nat
IHt': ~~ completed_by (uni_schedule arr_seq) j t' ->
scheduled_at (uni_schedule arr_seq) j t'
LEQ: true
SCHED: scheduled_at (uni_schedule arr_seq) j t
NCOMP: ~~ completed_by (uni_schedule arr_seq) j t'.+1
LT: t < t'.+1
~~ completed_by (uni_schedule arr_seq) j t' specialize (completion_monotonic sched j t' t'.+1) => MONO.arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
t': nat
IHt': ~~ completed_by (uni_schedule arr_seq) j t' ->
scheduled_at (uni_schedule arr_seq) j t'
LEQ: true
SCHED: scheduled_at (uni_schedule arr_seq) j t
NCOMP: ~~ completed_by (uni_schedule arr_seq) j t'.+1
LT: t < t'.+1
MONO: t' <= t'.+1 ->
completed_by sched j t' ->
completed_by sched j t'.+1
~~ completed_by (uni_schedule arr_seq) j t'
      feed MONO; first by lia.arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
t': nat
IHt': ~~ completed_by (uni_schedule arr_seq) j t' ->
scheduled_at (uni_schedule arr_seq) j t'
LEQ: true
SCHED: scheduled_at (uni_schedule arr_seq) j t
NCOMP: ~~ completed_by (uni_schedule arr_seq) j t'.+1
LT: t < t'.+1
MONO: completed_by sched j t' ->
completed_by sched j t'.+1
~~ completed_by (uni_schedule arr_seq) j t'
      by apply contra in MONO. }arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
j: Job
t: instant
t': nat
IHt': scheduled_at (uni_schedule arr_seq) j t'
LEQ: true
SCHED: scheduled_at (uni_schedule arr_seq) j t
NCOMP: ~~ completed_by (uni_schedule arr_seq) j t'.+1
LT: t < t'.+1
scheduled_at (uni_schedule arr_seq) j t'.+1
    by apply sched_nonpreemptive_next.
  Qed.

  (** Finally, we show that the earliest-deadline-first policy is respected
      at each preemption point. *)
  Lemma respects_policy_at_preemption_point_edf_np :
    respects_JLFP_policy_at_preemption_point arr_seq sched (EDF Job).arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
respects_JLFP_policy_at_preemption_point arr_seq sched
  (EDF Job)
  Proof.arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
respects_JLFP_policy_at_preemption_point arr_seq sched
  (EDF Job)
    apply schedule_respects_policy => //; auto with basic_rt_facts.arr_seq: arrival_sequence Job
H_arrival_times_are_consistent: consistent_arrival_times arr_seq
nonclairvoyant_readiness basic_ready_instance
    by apply basic_readiness_nonclairvoyance.
  Qed.

End Schedule.