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Notation "_ + _" was already used in scope nat_scope. [notation-overridden,parsing,default]
Notation "_ - _" was already used in scope nat_scope. [notation-overridden,parsing,default]
Notation "_ <= _" was already used in scope nat_scope. [notation-overridden,parsing,default]
Notation "_ < _" was already used in scope nat_scope. [notation-overridden,parsing,default]
Notation "_ >= _" was already used in scope nat_scope. [notation-overridden,parsing,default]
Notation "_ > _" was already used in scope nat_scope. [notation-overridden,parsing,default]
Notation "_ <= _ <= _" was already used in scope nat_scope. [notation-overridden,parsing,default]
Notation "_ < _ <= _" was already used in scope nat_scope. [notation-overridden,parsing,default]
Notation "_ <= _ < _" was already used in scope nat_scope. [notation-overridden,parsing,default]
Notation "_ < _ < _" was already used in scope nat_scope. [notation-overridden,parsing,default]
Notation "_ * _" was already used in scope nat_scope. [notation-overridden,parsing,default]
Require Export prosa.analysis.facts.behavior.deadlines. (** In this file, we establish invariants about schedules in which two allocations have been swapped, as for instance it is done in the classic EDF optimality proof. *) Section SwappedFacts. (** For any given type of jobs... *) Context {Job : JobType}. (** ... any given type of processor states: *) Context {PState: ProcessorState Job}. (** ...consider any given reference schedule. *) Variable sched: schedule PState. (** Suppose we are given two specific times [t1] and [t2]. *) Variable t1 t2: instant. (** In the following, let [sched'] denote the schedule in which the allocations at [t1] and [t2] have been swapped. *) Let sched' := swapped sched t1 t2. (** First, we note that the trivial case where t1 == t2 is not interesting because then the two schedules are identical. *)
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState

t1 = t2 -> forall t : instant, sched t = sched' t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState

t1 = t2 -> forall t : instant, sched t = sched' t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
t: instant

sched t = swapped sched t1 t1 t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
t: instant

sched t = (if t1 == t then sched t1 else if t1 == t then sched t1 else sched t)
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
t: instant

t1 == t -> sched t = sched t1
by move /eqP ->. Qed. (** In this trivial case, the amount of service received hence is obviously always identical. *)
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState

t1 = t2 -> forall (t : instant) (j : Job), service sched j t = service sched' j t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState

t1 = t2 -> forall (t : instant) (j : Job), service sched j t = service sched' j t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
SAME: t1 = t2
t: instant
j: Job

service sched j t = service sched' j t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
SAME: t1 = t2
t: instant
j: Job

\sum_(0 <= t < t) service_in j (sched t) = \sum_(0 <= t < t) service_in j (sched' t)
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
SAME: t1 = t2
t: instant
j: Job
t': nat
RANGE: 0 <= t' < t

service_in j (sched t') = service_in j (sched' t')
by rewrite trivial_swap. Qed. (** In any case, the two schedules do not differ at non-swapped times. *)
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState

forall t : instant, t <> t1 -> t <> t2 -> sched t = sched' t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState

forall t : instant, t <> t1 -> t <> t2 -> sched t = sched' t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
t: instant
NOT_t1: t <> t1
NOT_t2: t <> t2

sched t = sched' t
by rewrite /sched' /swapped !rest_of_schedule_invariant //. Qed. (** By definition, if a job is scheduled at t2 in the original schedule, then it is found at t1 in the new schedule. *)
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState

forall j : Job, scheduled_at sched' j t1 = scheduled_at sched j t2
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState

forall j : Job, scheduled_at sched' j t1 = scheduled_at sched j t2
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job

scheduled_at sched' j t1 = scheduled_at sched j t2
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job

scheduled_in j (if t2 == t1 then sched t1 else if t1 == t1 then sched t2 else sched t1) = scheduled_in j (sched t2)
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
EQ: t2 = t1

scheduled_in j (sched t1) = scheduled_in j (sched t2)
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
NE: t2 != t1
scheduled_in j (if t1 == t1 then sched t2 else sched t1) = scheduled_in j (sched t2)
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
EQ: t2 = t1

scheduled_in j (sched t1) = scheduled_in j (sched t2)
by rewrite EQ.
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
NE: t2 != t1

scheduled_in j (if t1 == t1 then sched t2 else sched t1) = scheduled_in j (sched t2)
by rewrite ifT. Qed. (** Similarly, a job scheduled at t1 in the original schedule is scheduled at t2 after the swap. *)
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState

forall j : Job, scheduled_at sched' j t2 = scheduled_at sched j t1
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState

forall j : Job, scheduled_at sched' j t2 = scheduled_at sched j t1
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job

scheduled_at sched' j t2 = scheduled_at sched j t1
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job

scheduled_in j (if t2 == t2 then sched t1 else if t1 == t2 then sched t2 else sched t2) = scheduled_in j (sched t1)
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
EQ: t2 = t1

scheduled_in j (if t2 == t2 then sched t1 else if t1 == t2 then sched t2 else sched t2) = scheduled_in j (sched t1)
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
NE: t2 != t1
scheduled_in j (if t2 == t2 then sched t1 else if t1 == t2 then sched t2 else sched t2) = scheduled_in j (sched t1)
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
EQ: t2 = t1

scheduled_in j (if t2 == t2 then sched t1 else if t1 == t2 then sched t2 else sched t2) = scheduled_in j (sched t1)
by rewrite ifT.
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
NE: t2 != t1

scheduled_in j (if t2 == t2 then sched t1 else if t1 == t2 then sched t2 else sched t2) = scheduled_in j (sched t1)
by rewrite ifT. Qed. (** If a job is scheduled at any time not involved in the swap, then it remains scheduled at that time in the new schedule. *)
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState

forall (j : Job) (t : Datatypes_nat__canonical__eqtype_Equality), t1 != t -> t2 != t -> scheduled_at sched' j t = scheduled_at sched j t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState

forall (j : Job) (t : Datatypes_nat__canonical__eqtype_Equality), t1 != t -> t2 != t -> scheduled_at sched' j t = scheduled_at sched j t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: Datatypes_nat__canonical__eqtype_Equality
NOT_t1: t1 <> t
NOT_t2: t2 <> t

scheduled_at sched' j t = scheduled_at sched j t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: Datatypes_nat__canonical__eqtype_Equality
NOT_t1: t1 <> t
NOT_t2: t2 <> t

scheduled_in j (sched' t) = scheduled_in j (sched t)
by rewrite swap_other_times_invariant //; apply: not_eq_sym. Qed. (** To make case analysis more convenient, we summarize the preceding three lemmas as a disjunction. *)
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState

forall (j : Job) (t : instant), scheduled_at sched' j t -> scheduled_at sched' j t = scheduled_at sched j t \/ t = t1 /\ scheduled_at sched' j t = scheduled_at sched j t2 \/ t = t2 /\ scheduled_at sched' j t = scheduled_at sched j t1
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState

forall (j : Job) (t : instant), scheduled_at sched' j t -> scheduled_at sched' j t = scheduled_at sched j t \/ t = t1 /\ scheduled_at sched' j t = scheduled_at sched j t2 \/ t = t2 /\ scheduled_at sched' j t = scheduled_at sched j t1
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched' j t

scheduled_at sched' j t = scheduled_at sched j t \/ t = t1 /\ scheduled_at sched' j t = scheduled_at sched j t2 \/ t = t2 /\ scheduled_at sched' j t = scheduled_at sched j t1
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched' j t
T1: (t1 == t) = true

scheduled_at sched' j t = scheduled_at sched j t \/ t = t1 /\ scheduled_at sched' j t = scheduled_at sched j t2 \/ t = t2 /\ scheduled_at sched' j t = scheduled_at sched j t1
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched' j t
T1: (t1 == t) = false
scheduled_at sched' j t = scheduled_at sched j t \/ t = t1 /\ scheduled_at sched' j t = scheduled_at sched j t2 \/ t = t2 /\ scheduled_at sched' j t = scheduled_at sched j t1
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched' j t
T1: (t1 == t) = true

scheduled_at sched' j t = scheduled_at sched j t \/ t = t1 /\ scheduled_at sched' j t = scheduled_at sched j t2 \/ t = t2 /\ scheduled_at sched' j t = scheduled_at sched j t1
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched' j t
T1: (t1 == t) = true

t = t1 /\ scheduled_at sched' j t = scheduled_at sched j t2 \/ t = t2 /\ scheduled_at sched' j t = scheduled_at sched j t1
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched' j t
T1: (t1 == t) = true

t = t1 /\ scheduled_at sched' j t = scheduled_at sched j t2
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched' j t
T1: t1 = t

t = t1 /\ scheduled_at sched' j t = scheduled_at sched j t2
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched' j t
T1: t1 = t

scheduled_at sched' j t = scheduled_at sched j t2
by rewrite -swap_job_scheduled_t1 T1.
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched' j t
T1: (t1 == t) = false

scheduled_at sched' j t = scheduled_at sched j t \/ t = t1 /\ scheduled_at sched' j t = scheduled_at sched j t2 \/ t = t2 /\ scheduled_at sched' j t = scheduled_at sched j t1
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched' j t
T1: (t1 == t) = false
T2: (t2 == t) = true

scheduled_at sched' j t = scheduled_at sched j t \/ t = t1 /\ scheduled_at sched' j t = scheduled_at sched j t2 \/ t = t2 /\ scheduled_at sched' j t = scheduled_at sched j t1
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched' j t
T1: (t1 == t) = false
T2: (t2 == t) = false
scheduled_at sched' j t = scheduled_at sched j t \/ t = t1 /\ scheduled_at sched' j t = scheduled_at sched j t2 \/ t = t2 /\ scheduled_at sched' j t = scheduled_at sched j t1
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched' j t
T1: (t1 == t) = false
T2: (t2 == t) = true

scheduled_at sched' j t = scheduled_at sched j t \/ t = t1 /\ scheduled_at sched' j t = scheduled_at sched j t2 \/ t = t2 /\ scheduled_at sched' j t = scheduled_at sched j t1
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched' j t
T1: (t1 == t) = false
T2: (t2 == t) = true

t = t1 /\ scheduled_at sched' j t = scheduled_at sched j t2 \/ t = t2 /\ scheduled_at sched' j t = scheduled_at sched j t1
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched' j t
T1: (t1 == t) = false
T2: (t2 == t) = true

t = t2 /\ scheduled_at sched' j t = scheduled_at sched j t1
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched' j t
T1: (t1 == t) = false
T2: t2 = t

t = t2 /\ scheduled_at sched' j t = scheduled_at sched j t1
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched' j t
T1: (t1 == t) = false
T2: t2 = t

scheduled_at sched' j t = scheduled_at sched j t1
by rewrite -swap_job_scheduled_t2 T2.
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched' j t
T1: (t1 == t) = false
T2: (t2 == t) = false

scheduled_at sched' j t = scheduled_at sched j t \/ t = t1 /\ scheduled_at sched' j t = scheduled_at sched j t2 \/ t = t2 /\ scheduled_at sched' j t = scheduled_at sched j t1
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched' j t
T1: (t1 == t) = false
T2: (t2 == t) = false

scheduled_at sched' j t = scheduled_at sched j t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched' j t
T1: t1 != t
T2: t2 != t

scheduled_at sched' j t = scheduled_at sched j t
by rewrite -swap_job_scheduled_other_times. Qed. (** From this, we can easily conclude that no jobs have appeared out of thin air: if a job scheduled at some time in the new schedule, then it was also scheduled at some time in the original schedule. *)
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState

forall (j : Job) (t : instant), scheduled_at sched' j t -> exists t' : instant, scheduled_at sched j t'
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState

forall (j : Job) (t : instant), scheduled_at sched' j t -> exists t' : instant, scheduled_at sched j t'
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched' j t

exists t' : instant, scheduled_at sched j t'
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched' j t

scheduled_at sched' j t -> exists t' : instant, scheduled_at sched j t'
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched' j t

scheduled_at sched j t -> exists t' : instant, scheduled_at sched j t'
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched' j t
scheduled_at sched j t2 -> exists t' : instant, scheduled_at sched j t'
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched' j t
scheduled_at sched j t1 -> exists t' : instant, scheduled_at sched j t'
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched' j t

scheduled_at sched j t -> exists t' : instant, scheduled_at sched j t'
by exists t.
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched' j t

scheduled_at sched j t2 -> exists t' : instant, scheduled_at sched j t'
by exists t2.
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched' j t

scheduled_at sched j t1 -> exists t' : instant, scheduled_at sched j t'
by exists t1. Qed. (** Mirroring swap_job_scheduled_cases above, we also state a disjunction for case analysis under the premise that a job is scheduled in the original schedule. *)
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState

forall (j : Job) (t : instant), scheduled_at sched j t -> scheduled_at sched' j t = scheduled_at sched j t \/ t = t1 /\ scheduled_at sched' j t2 = scheduled_at sched j t \/ t = t2 /\ scheduled_at sched' j t1 = scheduled_at sched j t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState

forall (j : Job) (t : instant), scheduled_at sched j t -> scheduled_at sched' j t = scheduled_at sched j t \/ t = t1 /\ scheduled_at sched' j t2 = scheduled_at sched j t \/ t = t2 /\ scheduled_at sched' j t1 = scheduled_at sched j t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched j t

scheduled_at sched' j t = scheduled_at sched j t \/ t = t1 /\ scheduled_at sched' j t2 = scheduled_at sched j t \/ t = t2 /\ scheduled_at sched' j t1 = scheduled_at sched j t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched j t
T1: (t1 == t) = true

scheduled_at sched' j t = scheduled_at sched j t \/ t = t1 /\ scheduled_at sched' j t2 = scheduled_at sched j t \/ t = t2 /\ scheduled_at sched' j t1 = scheduled_at sched j t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched j t
T1: (t1 == t) = false
scheduled_at sched' j t = scheduled_at sched j t \/ t = t1 /\ scheduled_at sched' j t2 = scheduled_at sched j t \/ t = t2 /\ scheduled_at sched' j t1 = scheduled_at sched j t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched j t
T1: (t1 == t) = true

scheduled_at sched' j t = scheduled_at sched j t \/ t = t1 /\ scheduled_at sched' j t2 = scheduled_at sched j t \/ t = t2 /\ scheduled_at sched' j t1 = scheduled_at sched j t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched j t
T1: (t1 == t) = true

t = t1 /\ scheduled_at sched' j t2 = scheduled_at sched j t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched j t
T1: t1 = t

t = t1 /\ scheduled_at sched' j t2 = scheduled_at sched j t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched j t
T1: t1 = t

scheduled_at sched' j t2 = scheduled_at sched j t
by rewrite swap_job_scheduled_t2 T1.
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched j t
T1: (t1 == t) = false

scheduled_at sched' j t = scheduled_at sched j t \/ t = t1 /\ scheduled_at sched' j t2 = scheduled_at sched j t \/ t = t2 /\ scheduled_at sched' j t1 = scheduled_at sched j t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched j t
T1: (t1 == t) = false
T2: (t2 == t) = true

scheduled_at sched' j t = scheduled_at sched j t \/ t = t1 /\ scheduled_at sched' j t2 = scheduled_at sched j t \/ t = t2 /\ scheduled_at sched' j t1 = scheduled_at sched j t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched j t
T1: (t1 == t) = false
T2: (t2 == t) = false
scheduled_at sched' j t = scheduled_at sched j t \/ t = t1 /\ scheduled_at sched' j t2 = scheduled_at sched j t \/ t = t2 /\ scheduled_at sched' j t1 = scheduled_at sched j t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched j t
T1: (t1 == t) = false
T2: (t2 == t) = true

scheduled_at sched' j t = scheduled_at sched j t \/ t = t1 /\ scheduled_at sched' j t2 = scheduled_at sched j t \/ t = t2 /\ scheduled_at sched' j t1 = scheduled_at sched j t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched j t
T1: (t1 == t) = false
T2: (t2 == t) = true

t = t2 /\ scheduled_at sched' j t1 = scheduled_at sched j t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched j t
T1: (t1 == t) = false
T2: t2 = t

t = t2 /\ scheduled_at sched' j t1 = scheduled_at sched j t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched j t
T1: (t1 == t) = false
T2: t2 = t

scheduled_at sched' j t1 = scheduled_at sched j t
by rewrite swap_job_scheduled_t1 T2.
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched j t
T1: (t1 == t) = false
T2: (t2 == t) = false

scheduled_at sched' j t = scheduled_at sched j t \/ t = t1 /\ scheduled_at sched' j t2 = scheduled_at sched j t \/ t = t2 /\ scheduled_at sched' j t1 = scheduled_at sched j t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched j t
T1: (t1 == t) = false
T2: (t2 == t) = false

scheduled_at sched' j t = scheduled_at sched j t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched j t
T1: t1 != t
T2: t2 != t

scheduled_at sched' j t = scheduled_at sched j t
by rewrite swap_job_scheduled_other_times. Qed. (** Thus, we can conclude that no jobs are lost: if a job is scheduled at some point in the original schedule, then it is also scheduled at some point in the new schedule. *)
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState

forall (j : Job) (t : instant), scheduled_at sched j t -> exists t' : instant, scheduled_at sched' j t'
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState

forall (j : Job) (t : instant), scheduled_at sched j t -> exists t' : instant, scheduled_at sched' j t'
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched j t

exists t' : instant, scheduled_at sched' j t'
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched j t

scheduled_at sched j t -> exists t' : instant, scheduled_at sched' j t'
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched j t

scheduled_at sched' j t -> exists t' : instant, scheduled_at sched' j t'
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched j t
scheduled_at sched' j t2 -> exists t' : instant, scheduled_at sched' j t'
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched j t
scheduled_at sched' j t1 -> exists t' : instant, scheduled_at sched' j t'
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched j t

scheduled_at sched' j t -> exists t' : instant, scheduled_at sched' j t'
by exists t.
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched j t

scheduled_at sched' j t2 -> exists t' : instant, scheduled_at sched' j t'
by exists t2.
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
j: Job
t: instant
SCHED: scheduled_at sched j t

scheduled_at sched' j t1 -> exists t' : instant, scheduled_at sched' j t'
by exists t1. Qed. (** In the following, we lift the above basic invariants to statements about invariants about the cumulative amount of service received. *) (** To avoid dealing with symmetric cases, assume in the following that t1 and t2 are ordered. *) Hypothesis H_well_ordered: t1 <= t2. (** As another trivial invariant, we observe that nothing has changed before time t1. *)
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
H_well_ordered: t1 <= t2

forall t : nat, t < t1 -> sched t = sched' t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
H_well_ordered: t1 <= t2

forall t : nat, t < t1 -> sched t = sched' t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
H_well_ordered: t1 <= t2
t: nat
t_lt_t1: t < t1

sched t = sched' t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
H_well_ordered: t1 <= t2
t: nat
t_lt_t1: t < t1
t_lt_t2: t < t2

sched t = sched' t
by apply swap_other_times_invariant => /eqP /eqP EQ; subst; [move: t_lt_t1|move: t_lt_t2]; rewrite ltnn. Qed. (** Similarly, nothing has changed after time t2. *)
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
H_well_ordered: t1 <= t2

forall t : nat, t2 < t -> sched t = sched' t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
H_well_ordered: t1 <= t2

forall t : nat, t2 < t -> sched t = sched' t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
H_well_ordered: t1 <= t2
t: nat
t2_lt_t: t2 < t

sched t = sched' t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
H_well_ordered: t1 <= t2
t: nat
t2_lt_t: t2 < t
t1_lt_t: t1 < t

sched t = sched' t
by apply swap_other_times_invariant => /eqP /eqP EQ; subst; [move: t1_lt_t|move: t2_lt_t]; rewrite ltnn. Qed. (** Thus, we observe that, before t1, the two schedules are identical with regard to the service received by any job because they are identical. *)
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
H_well_ordered: t1 <= t2

forall t : nat, t <= t1 -> forall j : Job, service sched j t = service sched' j t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
H_well_ordered: t1 <= t2

forall t : nat, t <= t1 -> forall j : Job, service sched j t = service sched' j t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
H_well_ordered: t1 <= t2
t: nat
le_tt1: t <= t1
j: Job

service sched j t = service sched' j t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
H_well_ordered: t1 <= t2
t: nat
le_tt1: t <= t1
j: Job

service_during sched j 0 t = service_during sched' j 0 t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
H_well_ordered: t1 <= t2
t: nat
le_tt1: t <= t1
j: Job

t <= t2
by apply leq_trans with (n := t1) => //; apply: ltnW. Qed. (** Likewise, we observe that, *after* t2, the swapped schedule again does not differ with regard to the service received by any job. *)
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
H_well_ordered: t1 <= t2

forall t : nat, t2 < t -> forall j : Job, service sched j t = service sched' j t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
H_well_ordered: t1 <= t2

forall t : nat, t2 < t -> forall j : Job, service sched j t = service sched' j t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
H_well_ordered: t1 <= t2
t: nat
t2t: t2 < t
j: Job

service sched j t = service sched' j t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
H_well_ordered: t1 <= t2
t: nat
t2t: t2 < t
j: Job

t1 <= t2 -> service sched j t = service sched' j t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
H_well_ordered: t1 <= t2
t: nat
t2t: t2 < t
j: Job
t1_lt_t2: t1 < t2

service sched j t = service sched' j t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
H_well_ordered: t1 <= t2
t: nat
t2t: t2 < t
j: Job
t1_lt_t2: t1 < t2
TIME: 0 <= t1 < t

service sched j t = service sched' j t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
H_well_ordered: t1 <= t2
t: nat
t2t: t2 < t
j: Job
t1_lt_t2: t1 < t2
TIME: 0 <= t1 < t

service_during sched j 0 t = service_during sched j 0 t + service_in j (if t1 == t1 then sched t2 else sched t1) - service_in j (sched t1) + service_in j (if t2 == t2 then sched t1 else if t1 == t2 then sched t2 else sched t2) - service_in j (if t1 == t2 then sched t2 else sched t2)
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
H_well_ordered: t1 <= t2
t: nat
t2t: t2 < t
j: Job
t1_lt_t2: t1 < t2
TIME: 0 <= t1 < t

service_during sched j 0 t = service_during sched j 0 t + service_in j (sched t2) - service_in j (sched t1) + service_in j (sched t1) - service_in j (sched t2)
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
H_well_ordered: t1 <= t2
t: nat
t2t: t2 < t
j: Job
t1_lt_t2: t1 < t2
TIME: 0 <= t1 < t

service_in j (sched t1) <= service_during sched j 0 t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
H_well_ordered: t1 <= t2
t: nat
t2t: t2 < t
j: Job
t1_lt_t2: t1 < t2
TIME: 0 <= t1 < t
service_in_t1: service_in j (sched t1) <= service_during sched j 0 t
service_during sched j 0 t = service_during sched j 0 t + service_in j (sched t2) - service_in j (sched t1) + service_in j (sched t1) - service_in j (sched t2)
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
H_well_ordered: t1 <= t2
t: nat
t2t: t2 < t
j: Job
t1_lt_t2: t1 < t2
TIME: 0 <= t1 < t

service_in j (sched t1) <= service_during sched j 0 t
by rewrite -(service_split_at_point _ _ _ t1 _)// addnAC leq_addl.
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
H_well_ordered: t1 <= t2
t: nat
t2t: t2 < t
j: Job
t1_lt_t2: t1 < t2
TIME: 0 <= t1 < t
service_in_t1: service_in j (sched t1) <= service_during sched j 0 t

service_during sched j 0 t = service_during sched j 0 t + service_in j (sched t2) - service_in j (sched t1) + service_in j (sched t1) - service_in j (sched t2)
by rewrite subnK ?(leq_trans service_in_t1) ?leq_addr// addnK. Qed. (** Finally, we note that, trivially, jobs that are not involved in the swap receive invariant service. *)
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
H_well_ordered: t1 <= t2

forall (t : instant) (j : Job), ~~ scheduled_in j (sched t1) -> ~~ scheduled_in j (sched t2) -> service sched j t = service sched' j t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
H_well_ordered: t1 <= t2

forall (t : instant) (j : Job), ~~ scheduled_in j (sched t1) -> ~~ scheduled_in j (sched t2) -> service sched j t = service sched' j t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
H_well_ordered: t1 <= t2
t: instant
j: Job
NOT1: ~~ scheduled_in j (sched t1)
NOT2: ~~ scheduled_in j (sched t2)

service sched j t = service sched' j t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
H_well_ordered: t1 <= t2
t: instant
j: Job
NOT1: ~~ scheduled_in j (sched t1)
NOT2: ~~ scheduled_in j (sched t2)

t1 <= t2 -> service sched j t = service sched' j t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
H_well_ordered: t1 <= t2
t: instant
j: Job
NOT1: ~~ scheduled_in j (sched t1)
NOT2: ~~ scheduled_in j (sched t2)
t1_lt_t2: t1 < t2

service sched j t = service sched' j t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
H_well_ordered: t1 <= t2
t: instant
j: Job
NOT1: ~~ scheduled_in j (sched t1)
NOT2: ~~ scheduled_in j (sched t2)
t1_lt_t2: t1 < t2

service_during sched j 0 t = service_during sched' j 0 t
Job: JobType
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
sched':= swapped sched t1 t2: schedule PState
H_well_ordered: t1 <= t2
t: instant
j: Job
NOT1: ~~ scheduled_in j (sched t1)
NOT2: ~~ scheduled_in j (sched t2)
t1_lt_t2: t1 < t2

(t1 == t2) = false
by apply ltn_eqF. Qed. End SwappedFacts. (** In the following section, we observe that, if the original schedule satisfies certain properties, then so does the new scheduled obtained by swapping the allocations at times [t1] and [t2]. *) Section SwappedScheduleProperties. (** For any given type of jobs... *) Context {Job : JobType} `{JobCost Job} `{JobDeadline Job} `{JobArrival Job}. (** ... any given type of processor states: *) Context {PState: ProcessorState Job}. (** ...consider any given reference schedule. *) Variable sched: schedule PState. (** Suppose we are given two specific times [t1] and [t2]... *) Variable t1 t2: instant. (** ...such that [t1] is no later than [t2]. *) Hypothesis H_order: t1 <= t2. (** We let [sched'] denote the schedule in which the allocations at [t1] and [t2] have been swapped. *) Let sched' := swapped sched t1 t2. (** First, we observe that if jobs never accumulate more service than required, then that's still the case after the swap. *)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
H_order: t1 <= t2
sched':= swapped sched t1 t2: schedule PState

(forall (j : Job) (t : instant), service sched j t <= job_cost j) -> forall (j : Job) (t : instant), service sched' j t <= job_cost j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
H_order: t1 <= t2
sched':= swapped sched t1 t2: schedule PState

(forall (j : Job) (t : instant), service sched j t <= job_cost j) -> forall (j : Job) (t : instant), service sched' j t <= job_cost j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
H_order: t1 <= t2
sched':= swapped sched t1 t2: schedule PState
COMP: forall (j : Job) (t : instant), service sched j t <= job_cost j

forall (j : Job) (t : instant), service sched' j t <= job_cost j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
H_order: t1 <= t2
sched':= swapped sched t1 t2: schedule PState
COMP: forall (j : Job) (t : instant), service sched j t <= job_cost j
j: Job
t: instant

service sched' j t <= job_cost j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
H_order: t1 <= t2
sched':= swapped sched t1 t2: schedule PState
COMP: forall (j : Job) (t : instant), service sched j t <= job_cost j
j: Job
t: instant

t2 < t -> service sched' j t <= job_cost j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
H_order: t1 <= t2
sched':= swapped sched t1 t2: schedule PState
COMP: forall (j : Job) (t : instant), service sched j t <= job_cost j
j: Job
t: instant
(forall t : instant, t2 < t -> service sched' j t <= job_cost j) -> service sched' j t <= job_cost j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
H_order: t1 <= t2
sched':= swapped sched t1 t2: schedule PState
COMP: forall (j : Job) (t : instant), service sched j t <= job_cost j
j: Job
t: instant

t2 < t -> service sched' j t <= job_cost j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
H_order: t1 <= t2
sched':= swapped sched t1 t2: schedule PState
COMP: forall (j : Job) (t : instant), service sched j t <= job_cost j
j: Job
t: instant
LATE: t2 < t

service sched' j t <= job_cost j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
H_order: t1 <= t2
sched':= swapped sched t1 t2: schedule PState
COMP: forall (j : Job) (t : instant), service sched j t <= job_cost j
j: Job
t: instant
LATE: t2 < t

t1 <= t2 -> service sched' j t <= job_cost j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
H_order: t1 <= t2
sched':= swapped sched t1 t2: schedule PState
COMP: forall (j : Job) (t : instant), service sched j t <= job_cost j
j: Job
t: instant
LATE: t2 < t
EQ: t1 = t2

service sched' j t <= job_cost j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
H_order: t1 <= t2
sched':= swapped sched t1 t2: schedule PState
COMP: forall (j : Job) (t : instant), service sched j t <= job_cost j
j: Job
t: instant
LATE: t2 < t
LT: t1 < t2
service sched' j t <= job_cost j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
H_order: t1 <= t2
sched':= swapped sched t1 t2: schedule PState
COMP: forall (j : Job) (t : instant), service sched j t <= job_cost j
j: Job
t: instant
LATE: t2 < t
EQ: t1 = t2

service sched' j t <= job_cost j
by rewrite -(trivial_swap_service_invariant sched t1 t2) //.
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
H_order: t1 <= t2
sched':= swapped sched t1 t2: schedule PState
COMP: forall (j : Job) (t : instant), service sched j t <= job_cost j
j: Job
t: instant
LATE: t2 < t
LT: t1 < t2

service sched' j t <= job_cost j
by rewrite -(service_after_swap_invariant sched t1 t2) //.
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
H_order: t1 <= t2
sched':= swapped sched t1 t2: schedule PState
COMP: forall (j : Job) (t : instant), service sched j t <= job_cost j
j: Job
t: instant

(forall t : instant, t2 < t -> service sched' j t <= job_cost j) -> service sched' j t <= job_cost j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
H_order: t1 <= t2
sched':= swapped sched t1 t2: schedule PState
COMP: forall (j : Job) (t : instant), service sched j t <= job_cost j
j: Job
t: instant

(forall t : instant, t2 < t -> service sched' j t <= job_cost j) -> service sched' j t <= job_cost j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
H_order: t1 <= t2
sched':= swapped sched t1 t2: schedule PState
COMP: forall (j : Job) (t : instant), service sched j t <= job_cost j
j: Job
t: instant
ARGUMENT: forall t : instant, t2 < t -> service sched' j t <= job_cost j

service sched' j t <= job_cost j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
H_order: t1 <= t2
sched':= swapped sched t1 t2: schedule PState
COMP: forall (j : Job) (t : instant), service sched j t <= job_cost j
j: Job
t: instant
ARGUMENT: forall t : instant, t2 < t -> service sched' j t <= job_cost j

~~ (t2 < t) -> service sched' j t <= job_cost j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
H_order: t1 <= t2
sched':= swapped sched t1 t2: schedule PState
COMP: forall (j : Job) (t : instant), service sched j t <= job_cost j
j: Job
t: instant
ARGUMENT: forall t : instant, t2 < t -> service sched' j t <= job_cost j
EARLY: t <= t2

service sched' j t <= job_cost j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
H_order: t1 <= t2
sched':= swapped sched t1 t2: schedule PState
COMP: forall (j : Job) (t : instant), service sched j t <= job_cost j
j: Job
t: instant
ARGUMENT: forall t : instant, t2 < t -> service sched' j t <= job_cost j
EARLY: t <= t2

service sched' j t <= service sched' j t2.+1
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
H_order: t1 <= t2
sched':= swapped sched t1 t2: schedule PState
COMP: forall (j : Job) (t : instant), service sched j t <= job_cost j
j: Job
t: instant
ARGUMENT: forall t : instant, t2 < t -> service sched' j t <= job_cost j
EARLY: t <= t2
service sched' j t2.+1 <= job_cost j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
H_order: t1 <= t2
sched':= swapped sched t1 t2: schedule PState
COMP: forall (j : Job) (t : instant), service sched j t <= job_cost j
j: Job
t: instant
ARGUMENT: forall t : instant, t2 < t -> service sched' j t <= job_cost j
EARLY: t <= t2

service sched' j t <= service sched' j t2.+1
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
H_order: t1 <= t2
sched':= swapped sched t1 t2: schedule PState
COMP: forall (j : Job) (t : instant), service sched j t <= job_cost j
j: Job
t: instant
ARGUMENT: forall t : instant, t2 < t -> service sched' j t <= job_cost j
EARLY: t <= t2

t <= t2.+1
by apply leq_trans with (n := t2) => //.
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
H_order: t1 <= t2
sched':= swapped sched t1 t2: schedule PState
COMP: forall (j : Job) (t : instant), service sched j t <= job_cost j
j: Job
t: instant
ARGUMENT: forall t : instant, t2 < t -> service sched' j t <= job_cost j
EARLY: t <= t2

service sched' j t2.+1 <= job_cost j
by apply (ARGUMENT t2.+1). } Qed. (** From the above service bound, we conclude that, if if completed jobs don't execute in the original schedule, then that's still the case after the swap, assuming an ideal unit-service model (i.e., scheduled jobs receive exactly one unit of service). *)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
H_order: t1 <= t2
sched':= swapped sched t1 t2: schedule PState

unit_service_proc_model PState -> ideal_progress_proc_model PState -> completed_jobs_dont_execute sched -> completed_jobs_dont_execute sched'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
H_order: t1 <= t2
sched':= swapped sched t1 t2: schedule PState

unit_service_proc_model PState -> ideal_progress_proc_model PState -> completed_jobs_dont_execute sched -> completed_jobs_dont_execute sched'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
H_order: t1 <= t2
sched':= swapped sched t1 t2: schedule PState
UNIT: unit_service_proc_model PState
IDEAL: ideal_progress_proc_model PState
COMP: completed_jobs_dont_execute sched

completed_jobs_dont_execute sched'
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
H_order: t1 <= t2
sched':= swapped sched t1 t2: schedule PState
UNIT: unit_service_proc_model PState
IDEAL: ideal_progress_proc_model PState
COMP: completed_jobs_dont_execute sched

forall (j : Job) (t : instant), service sched' j t <= job_cost j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
H_order: t1 <= t2
sched':= swapped sched t1 t2: schedule PState
UNIT: unit_service_proc_model PState
IDEAL: ideal_progress_proc_model PState
COMP: completed_jobs_dont_execute sched

forall (j : Job) (t : instant), service sched j t <= job_cost j
by move=> j; apply service_at_most_cost. Qed. (** Suppose all jobs in the original schedule come from some arrival sequence... *) Variable arr_seq: arrival_sequence Job. Hypothesis H_from_arr_seq: jobs_come_from_arrival_sequence sched arr_seq. (** ...then that's still the case in the new schedule. *)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
H_order: t1 <= t2
sched':= swapped sched t1 t2: schedule PState
arr_seq: arrival_sequence Job
H_from_arr_seq: jobs_come_from_arrival_sequence sched arr_seq

jobs_come_from_arrival_sequence sched' arr_seq
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
H_order: t1 <= t2
sched':= swapped sched t1 t2: schedule PState
arr_seq: arrival_sequence Job
H_from_arr_seq: jobs_come_from_arrival_sequence sched arr_seq

jobs_come_from_arrival_sequence sched' arr_seq
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
H_order: t1 <= t2
sched':= swapped sched t1 t2: schedule PState
arr_seq: arrival_sequence Job
H_from_arr_seq: jobs_come_from_arrival_sequence sched arr_seq

jobs_come_from_arrival_sequence sched arr_seq -> jobs_come_from_arrival_sequence sched' arr_seq
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
H_order: t1 <= t2
sched':= swapped sched t1 t2: schedule PState
arr_seq: arrival_sequence Job
H_from_arr_seq: jobs_come_from_arrival_sequence sched arr_seq
ARR_sched: forall (j : Job) (t : instant), scheduled_at sched j t -> arrives_in arr_seq j
j: Job
t: instant
SCHED: scheduled_at sched' j t

arrives_in arr_seq j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
H1: JobArrival Job
PState: ProcessorState Job
sched: schedule PState
t1, t2: instant
H_order: t1 <= t2
sched':= swapped sched t1 t2: schedule PState
arr_seq: arrival_sequence Job
H_from_arr_seq: jobs_come_from_arrival_sequence sched arr_seq
ARR_sched: forall (j : Job) (t : instant), scheduled_at sched j t -> arrives_in arr_seq j
j: Job
t: instant
SCHED: scheduled_at sched' j t
t': instant
SCHED': scheduled_at sched j t'

arrives_in arr_seq j
by apply (ARR_sched j t' SCHED'). Qed. End SwappedScheduleProperties. (** In the next section, we consider a special case of swaps: namely, when the job moved earlier has an earlier deadline than the job being moved to a later allocation, assuming that the former has not yet missed its deadline, which is the core transformation of the classic EDF optimality proof. *) Section EDFSwap. (** For any given type of jobs with costs and deadlines... *) Context {Job : JobType} `{JobCost Job} `{JobDeadline Job}. (** ... any given type of processor states... *) Context {PState: ProcessorState Job}. (** ...consider a given reference schedule... *) Variable sched: schedule PState. (** ...in which complete jobs don't execute... *) Hypothesis H_completed_jobs: completed_jobs_dont_execute sched. (** ...and scheduled jobs always receive service. *) Hypothesis H_scheduled_implies_serviced: ideal_progress_proc_model PState. (** Suppose we are given two specific times [t1] and [t2]... *) Variable t1 t2: instant. (** ...that are ordered. *) Hypothesis H_well_ordered: t1 <= t2. (** Further, assume that, if there are jobs scheduled at times t1 and t2, then they either have the same deadline or violate EDF, ... *) Hypothesis H_not_EDF: forall j1 j2, scheduled_at sched j1 t1 -> scheduled_at sched j2 t2 -> job_deadline j1 >= job_deadline j2. (** ...and that we don't move idle times or deadline misses earlier, i.e., if t1 is not an idle time, then neither is t2 and whatever job is scheduled at time t2 has not yet missed its deadline. *) Hypothesis H_no_idle_time_at_t2: forall j1, scheduled_at sched j1 t1 -> exists j2, scheduled_at sched j2 t2 /\ job_deadline j2 > t2. (** Consider the schedule obtained from swapping the allocations at times t1 and t2. *) Let sched' := swapped sched t1 t2. (** The key property of this transformation is that, for any job that meets its deadline in the original schedule, we have not introduced any deadline misses, which we establish by considering a number of different cases. *) Section NoNewDeadlineMissesCases. (** Consider any job... *) Variable j: Job. (** ... that meets its deadline in the original schedule. *) Hypothesis H_deadline_met: job_meets_deadline sched j. (** First we observe that jobs that are not involved in the swap still meet their deadlines. *)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
PState: ProcessorState Job
sched: schedule PState
H_completed_jobs: completed_jobs_dont_execute sched
H_scheduled_implies_serviced: ideal_progress_proc_model PState
t1, t2: instant
H_well_ordered: t1 <= t2
H_not_EDF: forall j1 j2 : Job, scheduled_at sched j1 t1 -> scheduled_at sched j2 t2 -> job_deadline j2 <= job_deadline j1
H_no_idle_time_at_t2: forall j1 : Job, scheduled_at sched j1 t1 -> exists j2 : Job, scheduled_at sched j2 t2 /\ t2 < job_deadline j2
sched':= swapped sched t1 t2: schedule PState
j: Job
H_deadline_met: job_meets_deadline sched j

~~ scheduled_at sched j t1 -> ~~ scheduled_at sched j t2 -> job_meets_deadline sched' j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
PState: ProcessorState Job
sched: schedule PState
H_completed_jobs: completed_jobs_dont_execute sched
H_scheduled_implies_serviced: ideal_progress_proc_model PState
t1, t2: instant
H_well_ordered: t1 <= t2
H_not_EDF: forall j1 j2 : Job, scheduled_at sched j1 t1 -> scheduled_at sched j2 t2 -> job_deadline j2 <= job_deadline j1
H_no_idle_time_at_t2: forall j1 : Job, scheduled_at sched j1 t1 -> exists j2 : Job, scheduled_at sched j2 t2 /\ t2 < job_deadline j2
sched':= swapped sched t1 t2: schedule PState
j: Job
H_deadline_met: job_meets_deadline sched j

~~ scheduled_at sched j t1 -> ~~ scheduled_at sched j t2 -> job_meets_deadline sched' j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
PState: ProcessorState Job
sched: schedule PState
H_completed_jobs: completed_jobs_dont_execute sched
H_scheduled_implies_serviced: ideal_progress_proc_model PState
t1, t2: instant
H_well_ordered: t1 <= t2
H_not_EDF: forall j1 j2 : Job, scheduled_at sched j1 t1 -> scheduled_at sched j2 t2 -> job_deadline j2 <= job_deadline j1
H_no_idle_time_at_t2: forall j1 : Job, scheduled_at sched j1 t1 -> exists j2 : Job, scheduled_at sched j2 t2 /\ t2 < job_deadline j2
sched':= swapped sched t1 t2: schedule PState
j: Job
H_deadline_met: job_meets_deadline sched j
NOT_t1: ~~ scheduled_in j (sched t1)
NOT_t2: ~~ scheduled_in j (sched t2)

job_meets_deadline sched' j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
PState: ProcessorState Job
sched: schedule PState
H_completed_jobs: completed_jobs_dont_execute sched
H_scheduled_implies_serviced: ideal_progress_proc_model PState
t1, t2: instant
H_well_ordered: t1 <= t2
H_not_EDF: forall j1 j2 : Job, scheduled_at sched j1 t1 -> scheduled_at sched j2 t2 -> job_deadline j2 <= job_deadline j1
H_no_idle_time_at_t2: forall j1 : Job, scheduled_at sched j1 t1 -> exists j2 : Job, scheduled_at sched j2 t2 /\ t2 < job_deadline j2
sched':= swapped sched t1 t2: schedule PState
j: Job
H_deadline_met: job_meets_deadline sched j
NOT_t1: ~~ scheduled_in j (sched t1)
NOT_t2: ~~ scheduled_in j (sched t2)

service sched j (job_deadline j) = service sched' j (job_deadline j)
by apply: service_of_others_invariant. Qed. (** Next, we observe that a swap is unproblematic for the job scheduled at time t2. *)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
PState: ProcessorState Job
sched: schedule PState
H_completed_jobs: completed_jobs_dont_execute sched
H_scheduled_implies_serviced: ideal_progress_proc_model PState
t1, t2: instant
H_well_ordered: t1 <= t2
H_not_EDF: forall j1 j2 : Job, scheduled_at sched j1 t1 -> scheduled_at sched j2 t2 -> job_deadline j2 <= job_deadline j1
H_no_idle_time_at_t2: forall j1 : Job, scheduled_at sched j1 t1 -> exists j2 : Job, scheduled_at sched j2 t2 /\ t2 < job_deadline j2
sched':= swapped sched t1 t2: schedule PState
j: Job
H_deadline_met: job_meets_deadline sched j

scheduled_at sched j t2 -> job_meets_deadline sched' j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
PState: ProcessorState Job
sched: schedule PState
H_completed_jobs: completed_jobs_dont_execute sched
H_scheduled_implies_serviced: ideal_progress_proc_model PState
t1, t2: instant
H_well_ordered: t1 <= t2
H_not_EDF: forall j1 j2 : Job, scheduled_at sched j1 t1 -> scheduled_at sched j2 t2 -> job_deadline j2 <= job_deadline j1
H_no_idle_time_at_t2: forall j1 : Job, scheduled_at sched j1 t1 -> exists j2 : Job, scheduled_at sched j2 t2 /\ t2 < job_deadline j2
sched':= swapped sched t1 t2: schedule PState
j: Job
H_deadline_met: job_meets_deadline sched j

scheduled_at sched j t2 -> job_meets_deadline sched' j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
PState: ProcessorState Job
sched: schedule PState
H_completed_jobs: completed_jobs_dont_execute sched
H_scheduled_implies_serviced: ideal_progress_proc_model PState
t1, t2: instant
H_well_ordered: t1 <= t2
H_not_EDF: forall j1 j2 : Job, scheduled_at sched j1 t1 -> scheduled_at sched j2 t2 -> job_deadline j2 <= job_deadline j1
H_no_idle_time_at_t2: forall j1 : Job, scheduled_at sched j1 t1 -> exists j2 : Job, scheduled_at sched j2 t2 /\ t2 < job_deadline j2
sched':= swapped sched t1 t2: schedule PState
j: Job
H_deadline_met: job_meets_deadline sched j
AT_t2: scheduled_at sched j t2

job_meets_deadline sched' j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
PState: ProcessorState Job
sched: schedule PState
H_completed_jobs: completed_jobs_dont_execute sched
H_scheduled_implies_serviced: ideal_progress_proc_model PState
t1, t2: instant
H_well_ordered: t1 <= t2
H_not_EDF: forall j1 j2 : Job, scheduled_at sched j1 t1 -> scheduled_at sched j2 t2 -> job_deadline j2 <= job_deadline j1
H_no_idle_time_at_t2: forall j1 : Job, scheduled_at sched j1 t1 -> exists j2 : Job, scheduled_at sched j2 t2 /\ t2 < job_deadline j2
sched':= swapped sched t1 t2: schedule PState
j: Job
H_deadline_met: job_meets_deadline sched j
AT_t2: scheduled_at sched j t2

service sched j (job_deadline j) = service sched' j (job_deadline j)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
PState: ProcessorState Job
sched: schedule PState
H_completed_jobs: completed_jobs_dont_execute sched
H_scheduled_implies_serviced: ideal_progress_proc_model PState
t1, t2: instant
H_well_ordered: t1 <= t2
H_not_EDF: forall j1 j2 : Job, scheduled_at sched j1 t1 -> scheduled_at sched j2 t2 -> job_deadline j2 <= job_deadline j1
H_no_idle_time_at_t2: forall j1 : Job, scheduled_at sched j1 t1 -> exists j2 : Job, scheduled_at sched j2 t2 /\ t2 < job_deadline j2
sched':= swapped sched t1 t2: schedule PState
j: Job
H_deadline_met: job_meets_deadline sched j
AT_t2: scheduled_at sched j t2

t2 < job_deadline j
by apply scheduled_at_implies_later_deadline with (sched := sched) => //. Qed. (** Finally, we observe is also unproblematic for the job that is moved to a later allocation. *)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
PState: ProcessorState Job
sched: schedule PState
H_completed_jobs: completed_jobs_dont_execute sched
H_scheduled_implies_serviced: ideal_progress_proc_model PState
t1, t2: instant
H_well_ordered: t1 <= t2
H_not_EDF: forall j1 j2 : Job, scheduled_at sched j1 t1 -> scheduled_at sched j2 t2 -> job_deadline j2 <= job_deadline j1
H_no_idle_time_at_t2: forall j1 : Job, scheduled_at sched j1 t1 -> exists j2 : Job, scheduled_at sched j2 t2 /\ t2 < job_deadline j2
sched':= swapped sched t1 t2: schedule PState
j: Job
H_deadline_met: job_meets_deadline sched j

scheduled_at sched j t1 -> job_meets_deadline sched' j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
PState: ProcessorState Job
sched: schedule PState
H_completed_jobs: completed_jobs_dont_execute sched
H_scheduled_implies_serviced: ideal_progress_proc_model PState
t1, t2: instant
H_well_ordered: t1 <= t2
H_not_EDF: forall j1 j2 : Job, scheduled_at sched j1 t1 -> scheduled_at sched j2 t2 -> job_deadline j2 <= job_deadline j1
H_no_idle_time_at_t2: forall j1 : Job, scheduled_at sched j1 t1 -> exists j2 : Job, scheduled_at sched j2 t2 /\ t2 < job_deadline j2
sched':= swapped sched t1 t2: schedule PState
j: Job
H_deadline_met: job_meets_deadline sched j

scheduled_at sched j t1 -> job_meets_deadline sched' j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
PState: ProcessorState Job
sched: schedule PState
H_completed_jobs: completed_jobs_dont_execute sched
H_scheduled_implies_serviced: ideal_progress_proc_model PState
t1, t2: instant
H_well_ordered: t1 <= t2
H_not_EDF: forall j1 j2 : Job, scheduled_at sched j1 t1 -> scheduled_at sched j2 t2 -> job_deadline j2 <= job_deadline j1
H_no_idle_time_at_t2: forall j1 : Job, scheduled_at sched j1 t1 -> exists j2 : Job, scheduled_at sched j2 t2 /\ t2 < job_deadline j2
sched':= swapped sched t1 t2: schedule PState
j: Job
H_deadline_met: job_meets_deadline sched j
AT_t1: scheduled_at sched j t1

job_meets_deadline sched' j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
PState: ProcessorState Job
sched: schedule PState
H_completed_jobs: completed_jobs_dont_execute sched
H_scheduled_implies_serviced: ideal_progress_proc_model PState
t1, t2: instant
H_well_ordered: t1 <= t2
H_not_EDF: forall j1 j2 : Job, scheduled_at sched j1 t1 -> scheduled_at sched j2 t2 -> job_deadline j2 <= job_deadline j1
H_no_idle_time_at_t2: forall j1 : Job, scheduled_at sched j1 t1 -> exists j2 : Job, scheduled_at sched j2 t2 /\ t2 < job_deadline j2
sched':= swapped sched t1 t2: schedule PState
j: Job
H_deadline_met: job_meets_deadline sched j
AT_t1: scheduled_at sched j t1
j2: Job
AT_t2: scheduled_at sched j2 t2
DL2: t2 < job_deadline j2

job_meets_deadline sched' j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
PState: ProcessorState Job
sched: schedule PState
H_completed_jobs: completed_jobs_dont_execute sched
H_scheduled_implies_serviced: ideal_progress_proc_model PState
t1, t2: instant
H_well_ordered: t1 <= t2
H_not_EDF: forall j1 j2 : Job, scheduled_at sched j1 t1 -> scheduled_at sched j2 t2 -> job_deadline j2 <= job_deadline j1
H_no_idle_time_at_t2: forall j1 : Job, scheduled_at sched j1 t1 -> exists j2 : Job, scheduled_at sched j2 t2 /\ t2 < job_deadline j2
sched':= swapped sched t1 t2: schedule PState
j: Job
H_deadline_met: job_meets_deadline sched j
AT_t1: scheduled_at sched j t1
j2: Job
AT_t2: scheduled_at sched j2 t2
DL2: t2 < job_deadline j2
DL2_le_DL1: job_deadline j2 <= job_deadline j

job_meets_deadline sched' j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
PState: ProcessorState Job
sched: schedule PState
H_completed_jobs: completed_jobs_dont_execute sched
H_scheduled_implies_serviced: ideal_progress_proc_model PState
t1, t2: instant
H_well_ordered: t1 <= t2
H_not_EDF: forall j1 j2 : Job, scheduled_at sched j1 t1 -> scheduled_at sched j2 t2 -> job_deadline j2 <= job_deadline j1
H_no_idle_time_at_t2: forall j1 : Job, scheduled_at sched j1 t1 -> exists j2 : Job, scheduled_at sched j2 t2 /\ t2 < job_deadline j2
sched':= swapped sched t1 t2: schedule PState
j: Job
H_deadline_met: job_meets_deadline sched j
AT_t1: scheduled_at sched j t1
j2: Job
AT_t2: scheduled_at sched j2 t2
DL2: t2 < job_deadline j2
DL2_le_DL1: job_deadline j2 <= job_deadline j

service sched j (job_deadline j) = service sched' j (job_deadline j)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
PState: ProcessorState Job
sched: schedule PState
H_completed_jobs: completed_jobs_dont_execute sched
H_scheduled_implies_serviced: ideal_progress_proc_model PState
t1, t2: instant
H_well_ordered: t1 <= t2
H_not_EDF: forall j1 j2 : Job, scheduled_at sched j1 t1 -> scheduled_at sched j2 t2 -> job_deadline j2 <= job_deadline j1
H_no_idle_time_at_t2: forall j1 : Job, scheduled_at sched j1 t1 -> exists j2 : Job, scheduled_at sched j2 t2 /\ t2 < job_deadline j2
sched':= swapped sched t1 t2: schedule PState
j: Job
H_deadline_met: job_meets_deadline sched j
AT_t1: scheduled_at sched j t1
j2: Job
AT_t2: scheduled_at sched j2 t2
DL2: t2 < job_deadline j2
DL2_le_DL1: job_deadline j2 <= job_deadline j

t2 < job_deadline j
apply leq_trans with (n := job_deadline j2) => //. Qed. End NoNewDeadlineMissesCases. (** From the above case analysis, we conclude that no deadline misses are introduced in the schedule obtained from swapping the allocations at times t1 and t2. *)
Job: JobType
H: JobCost Job
H0: JobDeadline Job
PState: ProcessorState Job
sched: schedule PState
H_completed_jobs: completed_jobs_dont_execute sched
H_scheduled_implies_serviced: ideal_progress_proc_model PState
t1, t2: instant
H_well_ordered: t1 <= t2
H_not_EDF: forall j1 j2 : Job, scheduled_at sched j1 t1 -> scheduled_at sched j2 t2 -> job_deadline j2 <= job_deadline j1
H_no_idle_time_at_t2: forall j1 : Job, scheduled_at sched j1 t1 -> exists j2 : Job, scheduled_at sched j2 t2 /\ t2 < job_deadline j2
sched':= swapped sched t1 t2: schedule PState

forall j : Job, job_meets_deadline sched j -> job_meets_deadline sched' j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
PState: ProcessorState Job
sched: schedule PState
H_completed_jobs: completed_jobs_dont_execute sched
H_scheduled_implies_serviced: ideal_progress_proc_model PState
t1, t2: instant
H_well_ordered: t1 <= t2
H_not_EDF: forall j1 j2 : Job, scheduled_at sched j1 t1 -> scheduled_at sched j2 t2 -> job_deadline j2 <= job_deadline j1
H_no_idle_time_at_t2: forall j1 : Job, scheduled_at sched j1 t1 -> exists j2 : Job, scheduled_at sched j2 t2 /\ t2 < job_deadline j2
sched':= swapped sched t1 t2: schedule PState

forall j : Job, job_meets_deadline sched j -> job_meets_deadline sched' j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
PState: ProcessorState Job
sched: schedule PState
H_completed_jobs: completed_jobs_dont_execute sched
H_scheduled_implies_serviced: ideal_progress_proc_model PState
t1, t2: instant
H_well_ordered: t1 <= t2
H_not_EDF: forall j1 j2 : Job, scheduled_at sched j1 t1 -> scheduled_at sched j2 t2 -> job_deadline j2 <= job_deadline j1
H_no_idle_time_at_t2: forall j1 : Job, scheduled_at sched j1 t1 -> exists j2 : Job, scheduled_at sched j2 t2 /\ t2 < job_deadline j2
sched':= swapped sched t1 t2: schedule PState
j: Job
DL_MET: job_meets_deadline sched j

job_meets_deadline sched' j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
PState: ProcessorState Job
sched: schedule PState
H_completed_jobs: completed_jobs_dont_execute sched
H_scheduled_implies_serviced: ideal_progress_proc_model PState
t1, t2: instant
H_well_ordered: t1 <= t2
H_not_EDF: forall j1 j2 : Job, scheduled_at sched j1 t1 -> scheduled_at sched j2 t2 -> job_deadline j2 <= job_deadline j1
H_no_idle_time_at_t2: forall j1 : Job, scheduled_at sched j1 t1 -> exists j2 : Job, scheduled_at sched j2 t2 /\ t2 < job_deadline j2
sched':= swapped sched t1 t2: schedule PState
j: Job
DL_MET: job_meets_deadline sched j
AT_t1: scheduled_at sched j t1

job_meets_deadline sched' j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
PState: ProcessorState Job
sched: schedule PState
H_completed_jobs: completed_jobs_dont_execute sched
H_scheduled_implies_serviced: ideal_progress_proc_model PState
t1, t2: instant
H_well_ordered: t1 <= t2
H_not_EDF: forall j1 j2 : Job, scheduled_at sched j1 t1 -> scheduled_at sched j2 t2 -> job_deadline j2 <= job_deadline j1
H_no_idle_time_at_t2: forall j1 : Job, scheduled_at sched j1 t1 -> exists j2 : Job, scheduled_at sched j2 t2 /\ t2 < job_deadline j2
sched':= swapped sched t1 t2: schedule PState
j: Job
DL_MET: job_meets_deadline sched j
AT_t1: ~~ scheduled_at sched j t1
job_meets_deadline sched' j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
PState: ProcessorState Job
sched: schedule PState
H_completed_jobs: completed_jobs_dont_execute sched
H_scheduled_implies_serviced: ideal_progress_proc_model PState
t1, t2: instant
H_well_ordered: t1 <= t2
H_not_EDF: forall j1 j2 : Job, scheduled_at sched j1 t1 -> scheduled_at sched j2 t2 -> job_deadline j2 <= job_deadline j1
H_no_idle_time_at_t2: forall j1 : Job, scheduled_at sched j1 t1 -> exists j2 : Job, scheduled_at sched j2 t2 /\ t2 < job_deadline j2
sched':= swapped sched t1 t2: schedule PState
j: Job
DL_MET: job_meets_deadline sched j
AT_t1: scheduled_at sched j t1

job_meets_deadline sched' j
by apply moved_later_implies_deadline_met.
Job: JobType
H: JobCost Job
H0: JobDeadline Job
PState: ProcessorState Job
sched: schedule PState
H_completed_jobs: completed_jobs_dont_execute sched
H_scheduled_implies_serviced: ideal_progress_proc_model PState
t1, t2: instant
H_well_ordered: t1 <= t2
H_not_EDF: forall j1 j2 : Job, scheduled_at sched j1 t1 -> scheduled_at sched j2 t2 -> job_deadline j2 <= job_deadline j1
H_no_idle_time_at_t2: forall j1 : Job, scheduled_at sched j1 t1 -> exists j2 : Job, scheduled_at sched j2 t2 /\ t2 < job_deadline j2
sched':= swapped sched t1 t2: schedule PState
j: Job
DL_MET: job_meets_deadline sched j
AT_t1: ~~ scheduled_at sched j t1

job_meets_deadline sched' j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
PState: ProcessorState Job
sched: schedule PState
H_completed_jobs: completed_jobs_dont_execute sched
H_scheduled_implies_serviced: ideal_progress_proc_model PState
t1, t2: instant
H_well_ordered: t1 <= t2
H_not_EDF: forall j1 j2 : Job, scheduled_at sched j1 t1 -> scheduled_at sched j2 t2 -> job_deadline j2 <= job_deadline j1
H_no_idle_time_at_t2: forall j1 : Job, scheduled_at sched j1 t1 -> exists j2 : Job, scheduled_at sched j2 t2 /\ t2 < job_deadline j2
sched':= swapped sched t1 t2: schedule PState
j: Job
DL_MET: job_meets_deadline sched j
AT_t1: ~~ scheduled_at sched j t1
AT_t2: scheduled_at sched j t2

job_meets_deadline sched' j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
PState: ProcessorState Job
sched: schedule PState
H_completed_jobs: completed_jobs_dont_execute sched
H_scheduled_implies_serviced: ideal_progress_proc_model PState
t1, t2: instant
H_well_ordered: t1 <= t2
H_not_EDF: forall j1 j2 : Job, scheduled_at sched j1 t1 -> scheduled_at sched j2 t2 -> job_deadline j2 <= job_deadline j1
H_no_idle_time_at_t2: forall j1 : Job, scheduled_at sched j1 t1 -> exists j2 : Job, scheduled_at sched j2 t2 /\ t2 < job_deadline j2
sched':= swapped sched t1 t2: schedule PState
j: Job
DL_MET: job_meets_deadline sched j
AT_t1: ~~ scheduled_at sched j t1
AT_t2: ~~ scheduled_at sched j t2
job_meets_deadline sched' j
Job: JobType
H: JobCost Job
H0: JobDeadline Job
PState: ProcessorState Job
sched: schedule PState
H_completed_jobs: completed_jobs_dont_execute sched
H_scheduled_implies_serviced: ideal_progress_proc_model PState
t1, t2: instant
H_well_ordered: t1 <= t2
H_not_EDF: forall j1 j2 : Job, scheduled_at sched j1 t1 -> scheduled_at sched j2 t2 -> job_deadline j2 <= job_deadline j1
H_no_idle_time_at_t2: forall j1 : Job, scheduled_at sched j1 t1 -> exists j2 : Job, scheduled_at sched j2 t2 /\ t2 < job_deadline j2
sched':= swapped sched t1 t2: schedule PState
j: Job
DL_MET: job_meets_deadline sched j
AT_t1: ~~ scheduled_at sched j t1
AT_t2: scheduled_at sched j t2

job_meets_deadline sched' j
by apply moved_earlier_implies_deadline_met.
Job: JobType
H: JobCost Job
H0: JobDeadline Job
PState: ProcessorState Job
sched: schedule PState
H_completed_jobs: completed_jobs_dont_execute sched
H_scheduled_implies_serviced: ideal_progress_proc_model PState
t1, t2: instant
H_well_ordered: t1 <= t2
H_not_EDF: forall j1 j2 : Job, scheduled_at sched j1 t1 -> scheduled_at sched j2 t2 -> job_deadline j2 <= job_deadline j1
H_no_idle_time_at_t2: forall j1 : Job, scheduled_at sched j1 t1 -> exists j2 : Job, scheduled_at sched j2 t2 /\ t2 < job_deadline j2
sched':= swapped sched t1 t2: schedule PState
j: Job
DL_MET: job_meets_deadline sched j
AT_t1: ~~ scheduled_at sched j t1
AT_t2: ~~ scheduled_at sched j t2

job_meets_deadline sched' j
by apply uninvolved_implies_deadline_met. Qed. End EDFSwap.