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Require Export prosa.analysis.facts.busy_interval.carry_in.[Loading ML file ssrmatching_plugin.cmxs (using legacy method) ... done ] [Loading ML file ssreflect_plugin.cmxs (using legacy method) ... done ] [Loading ML file ring_plugin.cmxs (using legacy method) ... done ] [Loading ML file coq-elpi.elpi ... done ] [Loading ML file zify_plugin.cmxs (using legacy method) ... done ] [Loading ML file micromega_plugin.cmxs (using legacy method) ... done ] [Loading ML file btauto_plugin.cmxs (using legacy method) ... done ] 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.abstract .IBF.task.
Require Export prosa.analysis.facts.interference.
(** Throughout this file, we assume ideal uni-processor schedules. *)
Require Import prosa.model.processor.ideal.
Require Export prosa.analysis.facts.model.ideal.priority_inversion.
(** * JLFP instantiation of Interference and Interfering Workload for ideal uni-processor. *)
(** In this module we instantiate functions Interference and
Interfering Workload for ideal uni-processor schedulers with an
arbitrary JLFP-policy that satisfies the sequential-tasks
hypothesis. We also prove equivalence of Interference and
Interfering Workload to the more conventional notions of service
or workload. *)
Section JLFPInstantiation .
(** Consider any type of tasks ... *)
Context {Task : TaskType}.
Context `{TaskCost Task}.
(** ... and any type of jobs associated with these tasks. *)
Context {Job : JobType}.
Context `{JobTask Job Task}.
Context `{JobArrival Job}.
Context `{JobCost Job}.
(** Consider any valid arrival sequence with consistent arrivals ... *)
Variable arr_seq : arrival_sequence Job.
Hypothesis H_valid_arrival_sequence : valid_arrival_sequence arr_seq.
(** ... and any ideal uni-processor schedule of this arrival
sequence ... *)
Variable sched : schedule (ideal.processor_state Job).
Hypothesis H_jobs_come_from_arrival_sequence :
jobs_come_from_arrival_sequence sched arr_seq.
(** ... where jobs do not execute before their arrival or after
completion. *)
Hypothesis H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched.
Hypothesis H_completed_jobs_dont_execute : completed_jobs_dont_execute sched.
(** Consider a JLFP-policy that indicates a higher-or-equal priority
relation, and assume that this relation is reflexive and
transitive. *)
Context {JLFP : JLFP_policy Job}.
Hypothesis H_priority_is_reflexive : reflexive_job_priorities JLFP.
Hypothesis H_priority_is_transitive : transitive_job_priorities JLFP.
(** We also assume that the policy respects sequential tasks,
meaning that later-arrived jobs of a task don't have higher
priority than earlier-arrived jobs of the same task. *)
Hypothesis H_JLFP_respects_sequential_tasks :
policy_respects_sequential_tasks JLFP.
(** Let [tsk] be any task. *)
Variable tsk : Task.
(** * Interference and Interfering Workload *)
(** In the following, we introduce definitions of interference,
interfering workload and a function that bounds cumulative
interference. *)
(** ** Instantiation of Interference *)
(** Before we define the notion of interference, we need to recall
the definition of priority inversion. We say that job [j] is
incurring a priority inversion at time [t] if there exists a job
with lower priority that executes at time [t]. In order to
simplify things, we ignore the fact that according to this
definition a job can incur priority inversion even before its
release (or after completion). All such (potentially bad) cases
do not cause problems, as each job is analyzed only within the
corresponding busy interval where the priority inversion behaves
in the expected way. *)
(** We say that job [j] incurs interference at time [t] iff it
cannot execute due to a higher-or-equal-priority job being
scheduled, or if it incurs a priority inversion. *)
#[local,program] Instance ideal_jlfp_interference : Interference Job :=
{
interference (j : Job) (t : instant) :=
priority_inversion arr_seq sched j t
|| another_hep_job_interference arr_seq sched j t
}.
(** ** Instantiation of Interfering Workload *)
(** The interfering workload, in turn, is defined as the sum of the
priority inversion predicate and interfering workload of jobs
with higher or equal priority. *)
#[local,program] Instance ideal_jlfp_interfering_workload : InterferingWorkload Job :=
{
interfering_workload (j : Job) (t : instant) :=
priority_inversion arr_seq sched j t
+ other_hep_jobs_interfering_workload arr_seq j t
}.
(** ** Equivalences *)
(** In this section we prove useful equivalences between the
definitions obtained by instantiation of definitions from the
Abstract RTA module (interference and interfering workload) and
definitions corresponding to the conventional concepts.
Instantiated functions of interference and interfering workload
usually do not have any useful lemmas about them. However, it is
possible to prove their equivalence to the more conventional
notions like service or workload. Next, we prove the equivalence
between the instantiations and conventional notions as well as a
few useful rewrite rules. *)
Section Equivalences .
(** In the following subsection, we prove properties of the
introduced functions under the assumption that the schedule is
idle. *)
Section IdleSchedule .
(** Consider a time instant [t] ... *)
Variable t : instant.
(** ... and assume that the schedule is idle at [t]. *)
Hypothesis H_idle : ideal_is_idle sched t.
(** We prove that in this case: ... *)
(** ... there is no interference, ... *)
Lemma no_interference_when_idle :
forall j , ~~ interference j t.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task t : instant H_idle : ideal_is_idle sched t
forall j : Job, ~~ interference j t
Proof .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task t : instant H_idle : ideal_is_idle sched t
forall j : Job, ~~ interference j t
move => j; apply /negP.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task t : instant H_idle : ideal_is_idle sched t j : Job
~ interference j t
rewrite /interference /ideal_jlfp_interference => /orP [PI | HEPI].Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task t : instant H_idle : ideal_is_idle sched t j : Job PI : priority_inversion arr_seq sched j t
False
- Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task t : instant H_idle : ideal_is_idle sched t j : Job PI : priority_inversion arr_seq sched j t
False
have [j' schj'] : exists j' : Job, scheduled_at sched j' t
by exact /priority_inversion_scheduled_at.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task t : instant H_idle : ideal_is_idle sched t j : Job PI : priority_inversion arr_seq sched j t j' : Job schj' : scheduled_at sched j' t
False
exact : ideal_sched_implies_not_idle schj' H_idle.
- Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task t : instant H_idle : ideal_is_idle sched t j : Job HEPI : another_hep_job_interference arr_seq sched j t
False
by move : HEPI => /negPn; rewrite no_hep_job_interference_when_idle // is_idle_def //.
Qed .
(** ... as well as no interference for [tsk]. *)
Lemma no_task_interference_when_idle :
forall j , ~~ task_interference arr_seq sched j t.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task t : instant H_idle : ideal_is_idle sched t
forall j : Job, ~~ task_interference arr_seq sched j t
Proof .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task t : instant H_idle : ideal_is_idle sched t
forall j : Job, ~~ task_interference arr_seq sched j t
move => j; rewrite /task_interference /cond_interference.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task t : instant H_idle : ideal_is_idle sched t j : Job
~~ (nonself arr_seq sched j t && interference j t)
by rewrite negb_and; apply /orP; right ; apply no_interference_when_idle.
Qed .
End IdleSchedule .
(** Next, we prove properties of the introduced functions under
the assumption that the scheduler is not idle. *)
Section ScheduledJob .
(** Consider a job [j] of task [tsk]. In this subsection, job
[j] is deemed to be the main job with respect to which the
functions are computed. *)
Variable j : Job.
Hypothesis H_j_tsk : job_of_task tsk j.
(** Consider a time instant [t]. *)
Variable t : instant.
(** In the next subsection, we consider a case when a job [j']
from the same task (as job [j]) is scheduled. *)
Section FromSameTask .
(** Consider a job [j'] that comes from task [tsk] and is
scheduled at time instant [t]. *)
Variable j' : Job.
Hypothesis H_j'_tsk : job_of_task tsk j'.
Hypothesis H_j'_sched : scheduled_at sched j' t.
(** Similarly, there is no task interference, since in order
to incur the task interference, a job from a distinct task
must be scheduled. *)
Lemma task_interference_eq_false :
~~ task_interference arr_seq sched j t.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job H_j_tsk : job_of_task tsk j t : instant j' : Job H_j'_tsk : job_of_task tsk j' H_j'_sched : scheduled_at sched j' t
~~ task_interference arr_seq sched j t
Proof .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job H_j_tsk : job_of_task tsk j t : instant j' : Job H_j'_tsk : job_of_task tsk j' H_j'_sched : scheduled_at sched j' t
~~ task_interference arr_seq sched j t
apply /negP; move => /andP [+ _]; rewrite /nonself /task_scheduled_at.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job H_j_tsk : job_of_task tsk j t : instant j' : Job H_j'_tsk : job_of_task tsk j' H_j'_sched : scheduled_at sched j' t
~~ task_served_at arr_seq sched (job_task j) t ->
False
rewrite task_served_eq_task_scheduled //=; erewrite job_of_scheduled_task => //.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job H_j_tsk : job_of_task tsk j t : instant j' : Job H_j'_tsk : job_of_task tsk j' H_j'_sched : scheduled_at sched j' t
~~ job_of_task (job_task j) j' -> False
move : H_j_tsk H_j'_tsk; rewrite /job_of_task => /eqP -> /eqP ->.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job H_j_tsk : job_of_task tsk j t : instant j' : Job H_j'_tsk : job_of_task tsk j' H_j'_sched : scheduled_at sched j' t
tsk != tsk -> False
by rewrite eq_refl.
Qed .
End FromSameTask .
(** In the next subsection, we consider a case when a job [j']
from a task other than [j]'s task is scheduled. *)
Section FromDifferentTask .
(** Consider a job [j'] that _does_ _not_ comes from task
[tsk] and is scheduled at time instant [t]. *)
Variable j' : Job.
Hypothesis H_j'_not_tsk : ~~ job_of_task tsk j'.
Hypothesis H_j'_sched : scheduled_at sched j' t.
(** Hence, if we assume that [j'] has higher-or-equal priority, ... *)
Hypothesis H_j'_hep : hep_job j' j.
(** Moreover, in this case, task [tsk] also incurs interference. *)
Lemma sched_athep_implies_task_interference :
task_interference arr_seq sched j t.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job H_j_tsk : job_of_task tsk j t : instant j' : Job H_j'_not_tsk : ~~ job_of_task tsk j' H_j'_sched : scheduled_at sched j' t H_j'_hep : hep_job j' j
task_interference arr_seq sched j t
Proof .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job H_j_tsk : job_of_task tsk j t : instant j' : Job H_j'_not_tsk : ~~ job_of_task tsk j' H_j'_sched : scheduled_at sched j' t H_j'_hep : hep_job j' j
task_interference arr_seq sched j t
apply /andP; split .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job H_j_tsk : job_of_task tsk j t : instant j' : Job H_j'_not_tsk : ~~ job_of_task tsk j' H_j'_sched : scheduled_at sched j' t H_j'_hep : hep_job j' j
nonself arr_seq sched j t
- Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job H_j_tsk : job_of_task tsk j t : instant j' : Job H_j'_not_tsk : ~~ job_of_task tsk j' H_j'_sched : scheduled_at sched j' t H_j'_hep : hep_job j' j
nonself arr_seq sched j t
rewrite /nonself task_served_eq_task_scheduled => //.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job H_j_tsk : job_of_task tsk j t : instant j' : Job H_j'_not_tsk : ~~ job_of_task tsk j' H_j'_sched : scheduled_at sched j' t H_j'_hep : hep_job j' j
~~ task_scheduled_at arr_seq sched (job_task j) t
apply : job_of_other_task_scheduled => //.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job H_j_tsk : job_of_task tsk j t : instant j' : Job H_j'_not_tsk : ~~ job_of_task tsk j' H_j'_sched : scheduled_at sched j' t H_j'_hep : hep_job j' j
~~ job_of_task (job_task j) j'
by move : H_j_tsk => /eqP -> .
- Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job H_j_tsk : job_of_task tsk j t : instant j' : Job H_j'_not_tsk : ~~ job_of_task tsk j' H_j'_sched : scheduled_at sched j' t H_j'_hep : hep_job j' j
interference j t
apply /orP; right .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job H_j_tsk : job_of_task tsk j t : instant j' : Job H_j'_not_tsk : ~~ job_of_task tsk j' H_j'_sched : scheduled_at sched j' t H_j'_hep : hep_job j' j
another_hep_job_interference arr_seq sched j t
apply /hasP; exists j' .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job H_j_tsk : job_of_task tsk j t : instant j' : Job H_j'_not_tsk : ~~ job_of_task tsk j' H_j'_sched : scheduled_at sched j' t H_j'_hep : hep_job j' j
j' \in served_jobs_at arr_seq sched t
+ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job H_j_tsk : job_of_task tsk j t : instant j' : Job H_j'_not_tsk : ~~ job_of_task tsk j' H_j'_sched : scheduled_at sched j' t H_j'_hep : hep_job j' j
j' \in served_jobs_at arr_seq sched t
by apply scheduled_at_implies_in_served_at => //.
+ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job H_j_tsk : job_of_task tsk j t : instant j' : Job H_j'_not_tsk : ~~ job_of_task tsk j' H_j'_sched : scheduled_at sched j' t H_j'_hep : hep_job j' j
another_hep_job j' j
rewrite another_hep_job_diff_task // same_task_sym.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job H_j_tsk : job_of_task tsk j t : instant j' : Job H_j'_not_tsk : ~~ job_of_task tsk j' H_j'_sched : scheduled_at sched j' t H_j'_hep : hep_job j' j
~~ same_task j j'
exact : (diff_task tsk).
Qed .
End FromDifferentTask .
End ScheduledJob .
(** We prove that we can split cumulative interference into two
parts: (1) cumulative priority inversion and (2) cumulative
interference from jobs with higher or equal priority. *)
Lemma cumulative_interference_split :
forall j t1 t2 ,
cumulative_interference j t1 t2
= cumulative_priority_inversion arr_seq sched j t1 t2
+ cumulative_another_hep_job_interference arr_seq sched j t1 t2.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task
forall (j : Job) (t1 t2 : nat),
cumulative_interference j t1 t2 =
cumulative_priority_inversion arr_seq sched j t1 t2 +
cumulative_another_hep_job_interference arr_seq sched
j t1 t2
Proof .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task
forall (j : Job) (t1 t2 : nat),
cumulative_interference j t1 t2 =
cumulative_priority_inversion arr_seq sched j t1 t2 +
cumulative_another_hep_job_interference arr_seq sched
j t1 t2
rewrite /cumulative_interference /cumul_cond_interference /cond_interference /interference.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task
forall (j : Job) (t1 t2 : nat),
\sum_(t1 <= t < t2)
true && ideal_jlfp_interference j t =
cumulative_priority_inversion arr_seq sched j t1 t2 +
cumulative_another_hep_job_interference arr_seq sched
j t1 t2
move => j t1 t2; rewrite -big_split //=.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job t1, t2 : nat
\sum_(t1 <= t < t2) ideal_jlfp_interference j t =
\sum_(t1 <= i < t2)
(priority_inversion arr_seq sched j i +
another_hep_job_interference arr_seq sched j i)
apply /eqP; rewrite eqn_leq; apply /andP; split ; rewrite leq_sum//.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job t1, t2 : nat
forall i : nat,
true ->
ideal_jlfp_interference j i <=
priority_inversion arr_seq sched j i +
another_hep_job_interference arr_seq sched j i
{ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job t1, t2 : nat
forall i : nat,
true ->
ideal_jlfp_interference j i <=
priority_inversion arr_seq sched j i +
another_hep_job_interference arr_seq sched j i
move => t _; unfold ideal_jlfp_interference.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job t1, t2, t : nat
priority_inversion arr_seq sched j t
|| another_hep_job_interference arr_seq sched j t <=
priority_inversion arr_seq sched j t +
another_hep_job_interference arr_seq sched j t
by destruct (priority_inversion _ _ _ _), (another_hep_job_interference arr_seq sched j t).
} Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job t1, t2 : nat
forall i : nat,
true ->
priority_inversion arr_seq sched j i +
another_hep_job_interference arr_seq sched j i <=
ideal_jlfp_interference j i
{ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job t1, t2 : nat
forall i : nat,
true ->
priority_inversion arr_seq sched j i +
another_hep_job_interference arr_seq sched j i <=
ideal_jlfp_interference j i
move => t _; rewrite /ideal_jlfp_interference.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job t1, t2, t : nat
priority_inversion arr_seq sched j t +
another_hep_job_interference arr_seq sched j t <=
priority_inversion arr_seq sched j t
|| another_hep_job_interference arr_seq sched j t
destruct (ideal_proc_model_sched_case_analysis sched t) as [IDLE | [s SCHED]].Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job t1, t2, t : nat IDLE : ideal_is_idle sched t
priority_inversion arr_seq sched j t +
another_hep_job_interference arr_seq sched j t <=
priority_inversion arr_seq sched j t
|| another_hep_job_interference arr_seq sched j t
- Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job t1, t2, t : nat IDLE : ideal_is_idle sched t
priority_inversion arr_seq sched j t +
another_hep_job_interference arr_seq sched j t <=
priority_inversion arr_seq sched j t
|| another_hep_job_interference arr_seq sched j t
have -> : priority_inversion arr_seq sched j t = false
by exact /negbTE/idle_implies_no_priority_inversion.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job t1, t2, t : nat IDLE : ideal_is_idle sched t
false + another_hep_job_interference arr_seq sched j t <=
false
|| another_hep_job_interference arr_seq sched j t
by rewrite add0n.
- Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job t1, t2, t : nat s : Job SCHED : scheduled_at sched s t
priority_inversion arr_seq sched j t +
another_hep_job_interference arr_seq sched j t <=
priority_inversion arr_seq sched j t
|| another_hep_job_interference arr_seq sched j t
destruct (hep_job s j) eqn :PRIO.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job t1, t2, t : nat s : Job SCHED : scheduled_at sched s t PRIO : hep_job s j = true
priority_inversion arr_seq sched j t +
another_hep_job_interference arr_seq sched j t <=
priority_inversion arr_seq sched j t
|| another_hep_job_interference arr_seq sched j t
+ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job t1, t2, t : nat s : Job SCHED : scheduled_at sched s t PRIO : hep_job s j = true
priority_inversion arr_seq sched j t +
another_hep_job_interference arr_seq sched j t <=
priority_inversion arr_seq sched j t
|| another_hep_job_interference arr_seq sched j t
by erewrite sched_hep_implies_no_priority_inversion => //; rewrite add0n.
+ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job t1, t2, t : nat s : Job SCHED : scheduled_at sched s t PRIO : hep_job s j = false
priority_inversion arr_seq sched j t +
another_hep_job_interference arr_seq sched j t <=
priority_inversion arr_seq sched j t
|| another_hep_job_interference arr_seq sched j t
erewrite !sched_lp_implies_priority_inversion => //; last by rewrite PRIO.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job t1, t2, t : nat s : Job SCHED : scheduled_at sched s t PRIO : hep_job s j = false
true + another_hep_job_interference arr_seq sched j t <=
true || another_hep_job_interference arr_seq sched j t
rewrite orTb.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job t1, t2, t : nat s : Job SCHED : scheduled_at sched s t PRIO : hep_job s j = false
true + another_hep_job_interference arr_seq sched j t <=
true
by rewrite add1n ltnS //= leqn0 eqb0; erewrite no_ahep_interference_when_scheduled_lp; last by erewrite PRIO.
}
Qed .
(** Similarly, we prove that we can split cumulative interfering
workload into two parts: (1) cumulative priority inversion and
(2) cumulative interfering workload from jobs with higher or
equal priority. *)
Lemma cumulative_interfering_workload_split :
forall j t1 t2 ,
cumulative_interfering_workload j t1 t2 =
cumulative_priority_inversion arr_seq sched j t1 t2
+ cumulative_other_hep_jobs_interfering_workload arr_seq j t1 t2.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task
forall (j : Job) (t1 t2 : nat),
cumulative_interfering_workload j t1 t2 =
cumulative_priority_inversion arr_seq sched j t1 t2 +
cumulative_other_hep_jobs_interfering_workload arr_seq
j t1 t2
Proof .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task
forall (j : Job) (t1 t2 : nat),
cumulative_interfering_workload j t1 t2 =
cumulative_priority_inversion arr_seq sched j t1 t2 +
cumulative_other_hep_jobs_interfering_workload arr_seq
j t1 t2
rewrite /cumulative_interfering_workload
/cumulative_priority_inversion
/cumulative_other_hep_jobs_interfering_workload.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task
forall (j : Job) (t1 t2 : nat),
\sum_(t1 <= t < t2) interfering_workload j t =
\sum_(t1 <= t < t2)
priority_inversion arr_seq sched j t +
\sum_(t1 <= t < t2)
other_hep_jobs_interfering_workload arr_seq j t
by move => j t1 t2; rewrite -big_split //=.
Qed .
(** Let [j] be any job of task [tsk]. Then the cumulative task
interference received by job [j] is bounded to the sum of the
cumulative priority inversion of job [j] and the cumulative
interference incurred by job [j] due to higher-or-equal
priority jobs from other tasks. *)
Lemma cumulative_task_interference_split :
forall j t1 t2 ,
arrives_in arr_seq j ->
job_of_task tsk j ->
~~ completed_by sched j t2 ->
cumul_task_interference arr_seq sched j t1 t2
<= cumulative_priority_inversion arr_seq sched j t1 t2
+ cumulative_another_task_hep_job_interference arr_seq sched j t1 t2.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task
forall (j : Job) (t1 : nat) (t2 : instant),
arrives_in arr_seq j ->
job_of_task tsk j ->
~~ completed_by sched j t2 ->
cumul_task_interference arr_seq sched j t1 t2 <=
cumulative_priority_inversion arr_seq sched j t1 t2 +
cumulative_another_task_hep_job_interference arr_seq
sched j t1 t2
Proof .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task
forall (j : Job) (t1 : nat) (t2 : instant),
arrives_in arr_seq j ->
job_of_task tsk j ->
~~ completed_by sched j t2 ->
cumul_task_interference arr_seq sched j t1 t2 <=
cumulative_priority_inversion arr_seq sched j t1 t2 +
cumulative_another_task_hep_job_interference arr_seq
sched j t1 t2
move => j t1 R ARR TSK NCOMPL.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job t1 : nat R : instant ARR : arrives_in arr_seq j TSK : job_of_task tsk j NCOMPL : ~~ completed_by sched j R
cumul_task_interference arr_seq sched j t1 R <=
cumulative_priority_inversion arr_seq sched j t1 R +
cumulative_another_task_hep_job_interference arr_seq
sched j t1 R
rewrite /cumul_task_interference /cumul_cond_interference.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job t1 : nat R : instant ARR : arrives_in arr_seq j TSK : job_of_task tsk j NCOMPL : ~~ completed_by sched j R
\sum_(t1 <= t < R)
cond_interference (nonself arr_seq sched) j t <=
cumulative_priority_inversion arr_seq sched j t1 R +
cumulative_another_task_hep_job_interference arr_seq
sched j t1 R
rewrite -big_split //= big_seq_cond [leqRHS]big_seq_cond.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job t1 : nat R : instant ARR : arrives_in arr_seq j TSK : job_of_task tsk j NCOMPL : ~~ completed_by sched j R
\sum_(i <- index_iota t1 R | (i \in index_iota t1 R) &&
true)
cond_interference (nonself arr_seq sched) j i <=
\sum_(i <- index_iota t1 R | (i \in index_iota t1 R) &&
true)
(priority_inversion arr_seq sched j i +
another_task_hep_job_interference arr_seq sched j
i)
apply leq_sum; move => t /andP [IN _].Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job t1 : nat R : instant ARR : arrives_in arr_seq j TSK : job_of_task tsk j NCOMPL : ~~ completed_by sched j R t : Datatypes_nat__canonical__eqtype_Equality IN : t \in index_iota t1 R
cond_interference (nonself arr_seq sched) j t <=
priority_inversion arr_seq sched j t +
another_task_hep_job_interference arr_seq sched j t
rewrite /cond_interference /nonself /interference /ideal_jlfp_interference.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job t1 : nat R : instant ARR : arrives_in arr_seq j TSK : job_of_task tsk j NCOMPL : ~~ completed_by sched j R t : Datatypes_nat__canonical__eqtype_Equality IN : t \in index_iota t1 R
~~ task_served_at arr_seq sched (job_task j) t &&
(priority_inversion arr_seq sched j t
|| another_hep_job_interference arr_seq sched j t) <=
priority_inversion arr_seq sched j t +
another_task_hep_job_interference arr_seq sched j t
have [IDLE|[s SCHEDs]] := ideal_proc_model_sched_case_analysis sched t.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job t1 : nat R : instant ARR : arrives_in arr_seq j TSK : job_of_task tsk j NCOMPL : ~~ completed_by sched j R t : Datatypes_nat__canonical__eqtype_Equality IN : t \in index_iota t1 R IDLE : ideal_is_idle sched t
~~ task_served_at arr_seq sched (job_task j) t &&
(priority_inversion arr_seq sched j t
|| another_hep_job_interference arr_seq sched j t) <=
priority_inversion arr_seq sched j t +
another_task_hep_job_interference arr_seq sched j t
{ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job t1 : nat R : instant ARR : arrives_in arr_seq j TSK : job_of_task tsk j NCOMPL : ~~ completed_by sched j R t : Datatypes_nat__canonical__eqtype_Equality IN : t \in index_iota t1 R IDLE : ideal_is_idle sched t
~~ task_served_at arr_seq sched (job_task j) t &&
(priority_inversion arr_seq sched j t
|| another_hep_job_interference arr_seq sched j t) <=
priority_inversion arr_seq sched j t +
another_task_hep_job_interference arr_seq sched j t
move : (IDLE) => IIDLE; erewrite <-is_idle_def in IDLE => //.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job t1 : nat R : instant ARR : arrives_in arr_seq j TSK : job_of_task tsk j NCOMPL : ~~ completed_by sched j R t : Datatypes_nat__canonical__eqtype_Equality IN : t \in index_iota t1 R IIDLE : ideal_is_idle sched t IDLE : is_idle arr_seq sched t
~~ task_served_at arr_seq sched (job_task j) t &&
(priority_inversion arr_seq sched j t
|| another_hep_job_interference arr_seq sched j t) <=
priority_inversion arr_seq sched j t +
another_task_hep_job_interference arr_seq sched j t
have ->: priority_inversion arr_seq sched j t = false
by apply /eqP; rewrite eqbF_neg; apply : no_priority_inversion_when_idle => //.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job t1 : nat R : instant ARR : arrives_in arr_seq j TSK : job_of_task tsk j NCOMPL : ~~ completed_by sched j R t : Datatypes_nat__canonical__eqtype_Equality IN : t \in index_iota t1 R IIDLE : ideal_is_idle sched t IDLE : is_idle arr_seq sched t
~~ task_served_at arr_seq sched (job_task j) t &&
(false
|| another_hep_job_interference arr_seq sched j t) <=
false +
another_task_hep_job_interference arr_seq sched j t
have ->: another_hep_job_interference arr_seq sched j t = false
by apply /eqP; rewrite eqbF_neg; apply no_hep_job_interference_when_idle.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job t1 : nat R : instant ARR : arrives_in arr_seq j TSK : job_of_task tsk j NCOMPL : ~~ completed_by sched j R t : Datatypes_nat__canonical__eqtype_Equality IN : t \in index_iota t1 R IIDLE : ideal_is_idle sched t IDLE : is_idle arr_seq sched t
~~ task_served_at arr_seq sched (job_task j) t &&
(false || false) <=
false +
another_task_hep_job_interference arr_seq sched j t
have ->: another_task_hep_job_interference arr_seq sched j t = false
by apply /eqP; rewrite eqbF_neg; apply /negP; rewrite_neg @no_hep_task_interference_when_idle.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job t1 : nat R : instant ARR : arrives_in arr_seq j TSK : job_of_task tsk j NCOMPL : ~~ completed_by sched j R t : Datatypes_nat__canonical__eqtype_Equality IN : t \in index_iota t1 R IIDLE : ideal_is_idle sched t IDLE : is_idle arr_seq sched t
~~ task_served_at arr_seq sched (job_task j) t &&
(false || false) <= false + false
by rewrite andbF.
} Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job t1 : nat R : instant ARR : arrives_in arr_seq j TSK : job_of_task tsk j NCOMPL : ~~ completed_by sched j R t : Datatypes_nat__canonical__eqtype_Equality IN : t \in index_iota t1 R s : Job SCHEDs : scheduled_at sched s t
~~ task_served_at arr_seq sched (job_task j) t &&
(priority_inversion arr_seq sched j t
|| another_hep_job_interference arr_seq sched j t) <=
priority_inversion arr_seq sched j t +
another_task_hep_job_interference arr_seq sched j t
{ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job t1 : nat R : instant ARR : arrives_in arr_seq j TSK : job_of_task tsk j NCOMPL : ~~ completed_by sched j R t : Datatypes_nat__canonical__eqtype_Equality IN : t \in index_iota t1 R s : Job SCHEDs : scheduled_at sched s t
~~ task_served_at arr_seq sched (job_task j) t &&
(priority_inversion arr_seq sched j t
|| another_hep_job_interference arr_seq sched j t) <=
priority_inversion arr_seq sched j t +
another_task_hep_job_interference arr_seq sched j t
rewrite (interference_ahep_def _ _ _ _ _ _ _ _ _ _ _ SCHEDs) //.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job t1 : nat R : instant ARR : arrives_in arr_seq j TSK : job_of_task tsk j NCOMPL : ~~ completed_by sched j R t : Datatypes_nat__canonical__eqtype_Equality IN : t \in index_iota t1 R s : Job SCHEDs : scheduled_at sched s t
~~ task_served_at arr_seq sched (job_task j) t &&
(priority_inversion arr_seq sched j t
|| another_hep_job s j) <=
priority_inversion arr_seq sched j t +
another_task_hep_job_interference arr_seq sched j t
rewrite (interference_athep_def _ _ _ _ _ _ _ _ _ _ _ SCHEDs) => //.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job t1 : nat R : instant ARR : arrives_in arr_seq j TSK : job_of_task tsk j NCOMPL : ~~ completed_by sched j R t : Datatypes_nat__canonical__eqtype_Equality IN : t \in index_iota t1 R s : Job SCHEDs : scheduled_at sched s t
~~ task_served_at arr_seq sched (job_task j) t &&
(priority_inversion arr_seq sched j t
|| another_hep_job s j) <=
priority_inversion arr_seq sched j t +
another_task_hep_job s j
rewrite (priority_inversion_equiv_sched_lower_priority _ _ _ _ _ _ _ _ _ SCHEDs) => //.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job t1 : nat R : instant ARR : arrives_in arr_seq j TSK : job_of_task tsk j NCOMPL : ~~ completed_by sched j R t : Datatypes_nat__canonical__eqtype_Equality IN : t \in index_iota t1 R s : Job SCHEDs : scheduled_at sched s t
~~ task_served_at arr_seq sched (job_task j) t &&
(~~ hep_job s j || another_hep_job s j) <=
~~ hep_job s j + another_task_hep_job s j
rewrite task_served_eq_task_scheduled // (job_of_scheduled_task _ _ _ _ _ _ _ _ _ SCHEDs) => //.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job t1 : nat R : instant ARR : arrives_in arr_seq j TSK : job_of_task tsk j NCOMPL : ~~ completed_by sched j R t : Datatypes_nat__canonical__eqtype_Equality IN : t \in index_iota t1 R s : Job SCHEDs : scheduled_at sched s t
~~ job_of_task (job_task j) s &&
(~~ hep_job s j || another_hep_job s j) <=
~~ hep_job s j + another_task_hep_job s j
rewrite /another_hep_job /another_task_hep_job.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job t1 : nat R : instant ARR : arrives_in arr_seq j TSK : job_of_task tsk j NCOMPL : ~~ completed_by sched j R t : Datatypes_nat__canonical__eqtype_Equality IN : t \in index_iota t1 R s : Job SCHEDs : scheduled_at sched s t
~~ job_of_task (job_task j) s &&
(~~ hep_job s j || hep_job s j && (s != j)) <=
~~ hep_job s j +
hep_job s j && (job_task s != job_task j)
have [EQj|NEQj] := eqVneq s j.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job t1 : nat R : instant ARR : arrives_in arr_seq j TSK : job_of_task tsk j NCOMPL : ~~ completed_by sched j R t : Datatypes_nat__canonical__eqtype_Equality IN : t \in index_iota t1 R s : Job SCHEDs : scheduled_at sched s t EQj : s = j
~~ job_of_task (job_task j) s &&
(~~ hep_job s j || hep_job s j && ~~ true) <=
~~ hep_job s j +
hep_job s j && (job_task s != job_task j)
{ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job t1 : nat R : instant ARR : arrives_in arr_seq j TSK : job_of_task tsk j NCOMPL : ~~ completed_by sched j R t : Datatypes_nat__canonical__eqtype_Equality IN : t \in index_iota t1 R s : Job SCHEDs : scheduled_at sched s t EQj : s = j
~~ job_of_task (job_task j) s &&
(~~ hep_job s j || hep_job s j && ~~ true) <=
~~ hep_job s j +
hep_job s j && (job_task s != job_task j)
by subst ; rewrite /job_of_task eq_refl H_priority_is_reflexive. } Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job t1 : nat R : instant ARR : arrives_in arr_seq j TSK : job_of_task tsk j NCOMPL : ~~ completed_by sched j R t : Datatypes_nat__canonical__eqtype_Equality IN : t \in index_iota t1 R s : Job SCHEDs : scheduled_at sched s t NEQj : s != j
~~ job_of_task (job_task j) s &&
(~~ hep_job s j || hep_job s j && ~~ false) <=
~~ hep_job s j +
hep_job s j && (job_task s != job_task j)
have [/eqP EQt|NEQt] := eqVneq (job_task s) (job_task j).Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job t1 : nat R : instant ARR : arrives_in arr_seq j TSK : job_of_task tsk j NCOMPL : ~~ completed_by sched j R t : Datatypes_nat__canonical__eqtype_Equality IN : t \in index_iota t1 R s : Job SCHEDs : scheduled_at sched s t NEQj : s != j EQt : job_task s == job_task j
~~ job_of_task (job_task j) s &&
(~~ hep_job s j || hep_job s j && ~~ false) <=
~~ hep_job s j + hep_job s j && ~~ true
{ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job t1 : nat R : instant ARR : arrives_in arr_seq j TSK : job_of_task tsk j NCOMPL : ~~ completed_by sched j R t : Datatypes_nat__canonical__eqtype_Equality IN : t \in index_iota t1 R s : Job SCHEDs : scheduled_at sched s t NEQj : s != j EQt : job_task s == job_task j
~~ job_of_task (job_task j) s &&
(~~ hep_job s j || hep_job s j && ~~ false) <=
~~ hep_job s j + hep_job s j && ~~ true
apply /eqP; move : (EQt) => /eqP <-.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job t1 : nat R : instant ARR : arrives_in arr_seq j TSK : job_of_task tsk j NCOMPL : ~~ completed_by sched j R t : Datatypes_nat__canonical__eqtype_Equality IN : t \in index_iota t1 R s : Job SCHEDs : scheduled_at sched s t NEQj : s != j EQt : job_task s == job_task j
~~ job_of_task (job_task s) s &&
(~~ hep_job s j || hep_job s j && ~~ false) -
(~~ hep_job s j + hep_job s j && ~~ true) = 0
by rewrite /job_of_task eq_refl //= andbF addn0 eq_sym eqb0; apply /negP => LPs. } Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job t1 : nat R : instant ARR : arrives_in arr_seq j TSK : job_of_task tsk j NCOMPL : ~~ completed_by sched j R t : Datatypes_nat__canonical__eqtype_Equality IN : t \in index_iota t1 R s : Job SCHEDs : scheduled_at sched s t NEQj : s != j NEQt : job_task s != job_task j
~~ job_of_task (job_task j) s &&
(~~ hep_job s j || hep_job s j && ~~ false) <=
~~ hep_job s j + hep_job s j && ~~ false
{ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task j : Job t1 : nat R : instant ARR : arrives_in arr_seq j TSK : job_of_task tsk j NCOMPL : ~~ completed_by sched j R t : Datatypes_nat__canonical__eqtype_Equality IN : t \in index_iota t1 R s : Job SCHEDs : scheduled_at sched s t NEQj : s != j NEQt : job_task s != job_task j
~~ job_of_task (job_task j) s &&
(~~ hep_job s j || hep_job s j && ~~ false) <=
~~ hep_job s j + hep_job s j && ~~ false
by rewrite /job_of_task NEQt //= andbT; case : hep_job. }
}
Qed .
(** In this section, we prove that the (abstract) cumulative
interfering workload is equivalent to the conventional workload,
i.e., the one defined with concrete schedule parameters. *)
Section InstantiatedWorkloadEquivalence .
(** Let <<[t1,t2)>> be any time interval. *)
Variables t1 t2 : instant.
(** Consider any job [j] of [tsk]. *)
Variable j : Job.
Hypothesis H_j_arrives : arrives_in arr_seq j.
Hypothesis H_job_of_tsk : job_of_task tsk j.
(** Then for any job [j], the cumulative interfering workload is
equal to the conventional workload. *)
Lemma cumulative_iw_hep_eq_workload_of_ohep :
cumulative_other_hep_jobs_interfering_workload arr_seq j t1 t2
= workload_of_other_hep_jobs arr_seq j t1 t2.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task t1, t2 : instant j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j
cumulative_other_hep_jobs_interfering_workload arr_seq
j t1 t2 = workload_of_other_hep_jobs arr_seq j t1 t2
Proof .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task t1, t2 : instant j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j
cumulative_other_hep_jobs_interfering_workload arr_seq
j t1 t2 = workload_of_other_hep_jobs arr_seq j t1 t2
rewrite /cumulative_other_hep_jobs_interfering_workload /workload_of_other_hep_jobs.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task t1, t2 : instant j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j
\sum_(t1 <= t < t2)
other_hep_jobs_interfering_workload arr_seq j t =
workload_of_jobs (another_hep_job^~ j)
(arrivals_between arr_seq t1 t2)
case NEQ: (t1 < t2); last first .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task t1, t2 : instant j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j NEQ : (t1 < t2) = false
\sum_(t1 <= t < t2)
other_hep_jobs_interfering_workload arr_seq j t =
workload_of_jobs (another_hep_job^~ j)
(arrivals_between arr_seq t1 t2)
{ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task t1, t2 : instant j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j NEQ : (t1 < t2) = false
\sum_(t1 <= t < t2)
other_hep_jobs_interfering_workload arr_seq j t =
workload_of_jobs (another_hep_job^~ j)
(arrivals_between arr_seq t1 t2)
move : NEQ => /negP /negP; rewrite -leqNgt; move => NEQ.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task t1, t2 : instant j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j NEQ : t2 <= t1
\sum_(t1 <= t < t2)
other_hep_jobs_interfering_workload arr_seq j t =
workload_of_jobs (another_hep_job^~ j)
(arrivals_between arr_seq t1 t2)
rewrite big_geq//.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task t1, t2 : instant j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j NEQ : t2 <= t1
0 =
workload_of_jobs (another_hep_job^~ j)
(arrivals_between arr_seq t1 t2)
rewrite /arrivals_between /arrival_sequence.arrivals_between big_geq//.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task t1, t2 : instant j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j NEQ : t2 <= t1
0 = workload_of_jobs (another_hep_job^~ j) [::]
by rewrite /workload_of_jobs big_nil.
} Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task t1, t2 : instant j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j NEQ : (t1 < t2) = true
\sum_(t1 <= t < t2)
other_hep_jobs_interfering_workload arr_seq j t =
workload_of_jobs (another_hep_job^~ j)
(arrivals_between arr_seq t1 t2)
{ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task t1, t2 : instant j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j NEQ : (t1 < t2) = true
\sum_(t1 <= t < t2)
other_hep_jobs_interfering_workload arr_seq j t =
workload_of_jobs (another_hep_job^~ j)
(arrivals_between arr_seq t1 t2)
unfold other_hep_jobs_interfering_workload, workload_of_jobs.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task t1, t2 : instant j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j NEQ : (t1 < t2) = true
\sum_(t1 <= t < t2)
\sum_(jhp <- arrivals_at arr_seq t | another_hep_job
jhp j)
job_cost jhp =
\sum_(j0 <- arrivals_between arr_seq t1 t2 | another_hep_job
j0 j)
job_cost j0
interval_to_duration t1 t2 k. Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task t1 : instant j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j k : nat
\sum_(t1 <= t < t1 + k)
\sum_(jhp <- arrivals_at arr_seq t | another_hep_job
jhp j)
job_cost jhp =
\sum_(j0 <- arrivals_between arr_seq t1 (t1 + k) |
another_hep_job j0 j) job_cost j0
elim : k => [|k IHk].Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task t1 : instant j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j
\sum_(t1 <= t < t1 + 0 )
\sum_(jhp <- arrivals_at arr_seq t | another_hep_job
jhp j)
job_cost jhp =
\sum_(j0 <- arrivals_between arr_seq t1 (t1 + 0 ) |
another_hep_job j0 j) job_cost j0
- Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task t1 : instant j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j
\sum_(t1 <= t < t1 + 0 )
\sum_(jhp <- arrivals_at arr_seq t | another_hep_job
jhp j)
job_cost jhp =
\sum_(j0 <- arrivals_between arr_seq t1 (t1 + 0 ) |
another_hep_job j0 j) job_cost j0
rewrite !addn0.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task t1 : instant j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j
\sum_(t1 <= t < t1)
\sum_(jhp <- arrivals_at arr_seq t | another_hep_job
jhp j)
job_cost jhp =
\sum_(j0 <- arrivals_between arr_seq t1 t1 | another_hep_job
j0 j)
job_cost j0
rewrite big_geq//.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task t1 : instant j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j
0 =
\sum_(j0 <- arrivals_between arr_seq t1 t1 | another_hep_job
j0 j)
job_cost j0
rewrite /arrivals_between /arrival_sequence.arrivals_between big_geq//.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task t1 : instant j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j
0 =
\sum_(j0 <- [::] | another_hep_job j0 j) job_cost j0
by rewrite /workload_of_jobs big_nil.
- Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task t1 : instant j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j k : nat IHk : \sum_(t1 <= t <
t1 + k)
\sum_(jhp <-
arrivals_at arr_seq t |
another_hep_job jhp j)
job_cost jhp =
\sum_(j0 <-
arrivals_between arr_seq t1 (t1 + k) |
another_hep_job j0 j)
job_cost j0
\sum_(t1 <= t < t1 + k.+1 )
\sum_(jhp <- arrivals_at arr_seq t | another_hep_job
jhp j)
job_cost jhp =
\sum_(j0 <- arrivals_between arr_seq t1 (t1 + k.+1 ) |
another_hep_job j0 j) job_cost j0
rewrite addnS big_nat_recr //=; last by rewrite leq_addr.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task t1 : instant j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j k : nat IHk : \sum_(t1 <= t <
t1 + k)
\sum_(jhp <-
arrivals_at arr_seq t |
another_hep_job jhp j)
job_cost jhp =
\sum_(j0 <-
arrivals_between arr_seq t1 (t1 + k) |
another_hep_job j0 j)
job_cost j0
\sum_(t1 <= i < t1 + k)
\sum_(jhp <- arrivals_at arr_seq i | another_hep_job
jhp j)
job_cost jhp +
\sum_(jhp <- arrivals_at arr_seq (t1 + k) | another_hep_job
jhp j)
job_cost jhp =
\sum_(j0 <- arrivals_between arr_seq t1 (t1 + k).+1 |
another_hep_job j0 j) job_cost j0
rewrite IHk.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task t1 : instant j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j k : nat IHk : \sum_(t1 <= t <
t1 + k)
\sum_(jhp <-
arrivals_at arr_seq t |
another_hep_job jhp j)
job_cost jhp =
\sum_(j0 <-
arrivals_between arr_seq t1 (t1 + k) |
another_hep_job j0 j)
job_cost j0
\sum_(j0 <- arrivals_between arr_seq t1 (t1 + k) |
another_hep_job j0 j) job_cost j0 +
\sum_(jhp <- arrivals_at arr_seq (t1 + k) | another_hep_job
jhp j)
job_cost jhp =
\sum_(j0 <- arrivals_between arr_seq t1 (t1 + k).+1 |
another_hep_job j0 j) job_cost j0
rewrite /arrivals_between /arrival_sequence.arrivals_between big_nat_recr //=.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task t1 : instant j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j k : nat IHk : \sum_(t1 <= t <
t1 + k)
\sum_(jhp <-
arrivals_at arr_seq t |
another_hep_job jhp j)
job_cost jhp =
\sum_(j0 <-
arrivals_between arr_seq t1 (t1 + k) |
another_hep_job j0 j)
job_cost j0
\sum_(j0 <- \cat_(t1<=t<t1 + k)arrivals_at arr_seq t |
another_hep_job j0 j) job_cost j0 +
\sum_(jhp <- arrivals_at arr_seq (t1 + k) | another_hep_job
jhp j)
job_cost jhp =
\sum_(j0 <- (\cat_(t1<=i<t1 + k)arrivals_at arr_seq i ++
arrivals_at arr_seq (t1 + k)) | another_hep_job
j0 j)
job_cost j0
+ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task t1 : instant j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j k : nat IHk : \sum_(t1 <= t <
t1 + k)
\sum_(jhp <-
arrivals_at arr_seq t |
another_hep_job jhp j)
job_cost jhp =
\sum_(j0 <-
arrivals_between arr_seq t1 (t1 + k) |
another_hep_job j0 j)
job_cost j0
\sum_(j0 <- \cat_(t1<=t<t1 + k)arrivals_at arr_seq t |
another_hep_job j0 j) job_cost j0 +
\sum_(jhp <- arrivals_at arr_seq (t1 + k) | another_hep_job
jhp j)
job_cost jhp =
\sum_(j0 <- (\cat_(t1<=i<t1 + k)arrivals_at arr_seq i ++
arrivals_at arr_seq (t1 + k)) | another_hep_job
j0 j)
job_cost j0
by rewrite big_cat.
+ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task t1 : instant j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j k : nat IHk : \sum_(t1 <= t <
t1 + k)
\sum_(jhp <-
arrivals_at arr_seq t |
another_hep_job jhp j)
job_cost jhp =
\sum_(j0 <-
arrivals_between arr_seq t1 (t1 + k) |
another_hep_job j0 j)
job_cost j0
t1 <= t1 + k
by rewrite leq_addr.
}
Qed .
End InstantiatedWorkloadEquivalence .
(** In this section we prove that the abstract definition of busy
interval is equivalent to the conventional, concrete
definition of busy interval for JLFP scheduling. *)
Section BusyIntervalEquivalence .
(** In order to avoid confusion, we denote the notion of a quiet
time in the _classical_ sense as [quiet_time_cl], and the
notion of quiet time in the _abstract_ sense as
[quiet_time_ab]. *)
Let quiet_time_cl := classical.quiet_time arr_seq sched.
Let quiet_time_ab := abstract .definitions.quiet_time sched.
(** Same for the two notions of a busy interval prefix ... *)
Let busy_interval_prefix_cl := classical.busy_interval_prefix arr_seq sched.
Let busy_interval_prefix_ab := abstract .definitions.busy_interval_prefix sched.
(** ... and the two notions of a busy interval. *)
Let busy_interval_cl := classical.busy_interval arr_seq sched.
Let busy_interval_ab := abstract .definitions.busy_interval sched.
(** Consider any job j of [tsk]. *)
Variable j : Job.
Hypothesis H_j_arrives : arrives_in arr_seq j.
Hypothesis H_job_cost_positive : job_cost_positive j.
(** To show the equivalence of the notions of busy intervals, we
first show that the notions of quiet time are also
equivalent. *)
(** First, we show that the classical notion of quiet time
implies the abstract notion of quiet time. *)
Lemma quiet_time_cl_implies_quiet_time_ab :
forall t , quiet_time_cl j t -> quiet_time_ab j t.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j
forall t : instant,
quiet_time_cl j t -> quiet_time_ab j t
Proof .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j
forall t : instant,
quiet_time_cl j t -> quiet_time_ab j t
clear H_JLFP_respects_sequential_tasks.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j
forall t : instant,
quiet_time_cl j t -> quiet_time_ab j t
have zero_is_quiet_time: forall j , quiet_time_cl j 0 .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j
forall j : Job, quiet_time_cl j 0
{ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j
forall j : Job, quiet_time_cl j 0
by move => jhp ARR HP AB; move : AB; rewrite /arrived_before ltn0. } Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j zero_is_quiet_time : forall j : Job, quiet_time_cl j 0
forall t : instant,
quiet_time_cl j t -> quiet_time_ab j t
move => t QT; apply /andP; split ; last first .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j zero_is_quiet_time : forall j : Job, quiet_time_cl j 0 t : instant QT : quiet_time_cl j t
~~ pending_earlier_and_at sched j t
{ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j zero_is_quiet_time : forall j : Job, quiet_time_cl j 0 t : instant QT : quiet_time_cl j t
~~ pending_earlier_and_at sched j t
rewrite negb_and negbK; apply /orP.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j zero_is_quiet_time : forall j : Job, quiet_time_cl j 0 t : instant QT : quiet_time_cl j t
~~ arrived_before j t \/ completed_by sched j t
by case ARR: (arrived_before j t); [right | left ]; [apply QT | ]. } Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j zero_is_quiet_time : forall j : Job, quiet_time_cl j 0 t : instant QT : quiet_time_cl j t
cumulative_interference j 0 t ==
cumulative_interfering_workload j 0 t
{ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j zero_is_quiet_time : forall j : Job, quiet_time_cl j 0 t : instant QT : quiet_time_cl j t
cumulative_interference j 0 t ==
cumulative_interfering_workload j 0 t
erewrite cumulative_interference_split, cumulative_interfering_workload_split; rewrite eqn_add2l.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j zero_is_quiet_time : forall j : Job, quiet_time_cl j 0 t : instant QT : quiet_time_cl j t
cumulative_another_hep_job_interference arr_seq sched
j 0 t ==
cumulative_other_hep_jobs_interfering_workload arr_seq
j 0 t
rewrite cumulative_i_ohep_eq_service_of_ohep//.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j zero_is_quiet_time : forall j : Job, quiet_time_cl j 0 t : instant QT : quiet_time_cl j t
service_of_other_hep_jobs arr_seq sched j 0 t ==
cumulative_other_hep_jobs_interfering_workload arr_seq
j 0 t
rewrite //= cumulative_iw_hep_eq_workload_of_ohep eq_sym; apply /eqP.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j zero_is_quiet_time : forall j : Job, quiet_time_cl j 0 t : instant QT : quiet_time_cl j t
workload_of_other_hep_jobs arr_seq j 0 t =
service_of_other_hep_jobs arr_seq sched j 0 t
apply all_jobs_have_completed_equiv_workload_eq_service => //.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j zero_is_quiet_time : forall j : Job, quiet_time_cl j 0 t : instant QT : quiet_time_cl j t
forall j0 : Job,
j0 \in arrivals_between arr_seq 0 t ->
another_hep_job j0 j -> completed_by sched j0 t
move => j0 IN HEP; apply QT.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j zero_is_quiet_time : forall j : Job, quiet_time_cl j 0 t : instant QT : quiet_time_cl j t j0 : Job IN : j0 \in arrivals_between arr_seq 0 t HEP : another_hep_job j0 j
arrives_in arr_seq j0
- Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j zero_is_quiet_time : forall j : Job, quiet_time_cl j 0 t : instant QT : quiet_time_cl j t j0 : Job IN : j0 \in arrivals_between arr_seq 0 t HEP : another_hep_job j0 j
arrives_in arr_seq j0
by apply in_arrivals_implies_arrived in IN.
- Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j zero_is_quiet_time : forall j : Job, quiet_time_cl j 0 t : instant QT : quiet_time_cl j t j0 : Job IN : j0 \in arrivals_between arr_seq 0 t HEP : another_hep_job j0 j
hep_job j0 j
by move : HEP => /andP [H6 H7].
- Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j zero_is_quiet_time : forall j : Job, quiet_time_cl j 0 t : instant QT : quiet_time_cl j t j0 : Job IN : j0 \in arrivals_between arr_seq 0 t HEP : another_hep_job j0 j
arrived_before j0 t
by apply in_arrivals_implies_arrived_between in IN.
}
Qed .
(** And vice versa, the abstract notion of quiet time implies
the classical notion of quiet time. *)
Lemma quiet_time_ab_implies_quiet_time_cl :
forall t , quiet_time_ab j t -> quiet_time_cl j t.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j
forall t : instant,
quiet_time_ab j t -> quiet_time_cl j t
Proof .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j
forall t : instant,
quiet_time_ab j t -> quiet_time_cl j t
clear H_JLFP_respects_sequential_tasks.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j
forall t : instant,
quiet_time_ab j t -> quiet_time_cl j t
have zero_is_quiet_time: forall j , quiet_time_cl j 0 .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j
forall j : Job, quiet_time_cl j 0
{ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j
forall j : Job, quiet_time_cl j 0
by move => jhp ARR HP AB; move : AB; rewrite /arrived_before ltn0. } Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j zero_is_quiet_time : forall j : Job, quiet_time_cl j 0
forall t : instant,
quiet_time_ab j t -> quiet_time_cl j t
move => t /andP [T0 T1] jhp ARR HP ARB.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j zero_is_quiet_time : forall j : Job, quiet_time_cl j 0 t : instant T0 : cumulative_interference j 0 t ==
cumulative_interfering_workload j 0 t T1 : ~~ pending_earlier_and_at sched j t jhp : Job ARR : arrives_in arr_seq jhp HP : hep_job jhp j ARB : arrived_before jhp t
completed_by sched jhp t
eapply all_jobs_have_completed_equiv_workload_eq_service with
(P := fun jhp => hep_job jhp j) (t1 := 0 ) (t2 := t) => //.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j zero_is_quiet_time : forall j : Job, quiet_time_cl j 0 t : instant T0 : cumulative_interference j 0 t ==
cumulative_interfering_workload j 0 t T1 : ~~ pending_earlier_and_at sched j t jhp : Job ARR : arrives_in arr_seq jhp HP : hep_job jhp j ARB : arrived_before jhp t
workload_of_jobs (hep_job^~ j)
(arrivals_between arr_seq 0 t) =
service_of_jobs sched (hep_job^~ j)
(arrivals_between arr_seq 0 t) 0 t
erewrite service_of_jobs_case_on_pred with (P2 := fun j' => j' != j).Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j zero_is_quiet_time : forall j : Job, quiet_time_cl j 0 t : instant T0 : cumulative_interference j 0 t ==
cumulative_interfering_workload j 0 t T1 : ~~ pending_earlier_and_at sched j t jhp : Job ARR : arrives_in arr_seq jhp HP : hep_job jhp j ARB : arrived_before jhp t
workload_of_jobs (hep_job^~ j)
(arrivals_between arr_seq 0 t) =
service_of_jobs sched
(fun j0 : Job => hep_job j0 j && (j0 != j))
(arrivals_between arr_seq 0 t) 0 t +
service_of_jobs sched
(fun j0 : Job => hep_job j0 j && ~~ (j0 != j))
(arrivals_between arr_seq 0 t) 0 t
erewrite workload_of_jobs_case_on_pred with (P' := fun j' => j' != j) => //.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j zero_is_quiet_time : forall j : Job, quiet_time_cl j 0 t : instant T0 : cumulative_interference j 0 t ==
cumulative_interfering_workload j 0 t T1 : ~~ pending_earlier_and_at sched j t jhp : Job ARR : arrives_in arr_seq jhp HP : hep_job jhp j ARB : arrived_before jhp t
workload_of_jobs
(fun j0 : Job => hep_job j0 j && (j0 != j))
(arrivals_between arr_seq 0 t) +
workload_of_jobs
(fun j0 : Job => hep_job j0 j && ~~ (j0 != j))
(arrivals_between arr_seq 0 t) =
service_of_jobs sched
(fun j0 : Job => hep_job j0 j && (j0 != j))
(arrivals_between arr_seq 0 t) 0 t +
service_of_jobs sched
(fun j0 : Job => hep_job j0 j && ~~ (j0 != j))
(arrivals_between arr_seq 0 t) 0 t
replace ((fun j0 : Job => hep_job j0 j && (j0 != j))) with (another_hep_job^~j); last by rewrite /another_hep_job.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j zero_is_quiet_time : forall j : Job, quiet_time_cl j 0 t : instant T0 : cumulative_interference j 0 t ==
cumulative_interfering_workload j 0 t T1 : ~~ pending_earlier_and_at sched j t jhp : Job ARR : arrives_in arr_seq jhp HP : hep_job jhp j ARB : arrived_before jhp t
workload_of_jobs (another_hep_job^~ j)
(arrivals_between arr_seq 0 t) +
workload_of_jobs
(fun j0 : Job => hep_job j0 j && ~~ (j0 != j))
(arrivals_between arr_seq 0 t) =
service_of_jobs sched (another_hep_job^~ j)
(arrivals_between arr_seq 0 t) 0 t +
service_of_jobs sched
(fun j0 : Job => hep_job j0 j && ~~ (j0 != j))
(arrivals_between arr_seq 0 t) 0 t
rewrite -/(service_of_other_hep_jobs arr_seq _ j 0 t) -cumulative_i_ohep_eq_service_of_ohep => //.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j zero_is_quiet_time : forall j : Job, quiet_time_cl j 0 t : instant T0 : cumulative_interference j 0 t ==
cumulative_interfering_workload j 0 t T1 : ~~ pending_earlier_and_at sched j t jhp : Job ARR : arrives_in arr_seq jhp HP : hep_job jhp j ARB : arrived_before jhp t
workload_of_jobs (another_hep_job^~ j)
(arrivals_between arr_seq 0 t) +
workload_of_jobs
(fun j0 : Job => hep_job j0 j && ~~ (j0 != j))
(arrivals_between arr_seq 0 t) =
cumulative_another_hep_job_interference arr_seq sched
j 0 t +
service_of_jobs sched
(fun j0 : Job => hep_job j0 j && ~~ (j0 != j))
(arrivals_between arr_seq 0 t) 0 t
rewrite -/(workload_of_other_hep_jobs _ j 0 t) -cumulative_iw_hep_eq_workload_of_ohep; eauto .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j zero_is_quiet_time : forall j : Job, quiet_time_cl j 0 t : instant T0 : cumulative_interference j 0 t ==
cumulative_interfering_workload j 0 t T1 : ~~ pending_earlier_and_at sched j t jhp : Job ARR : arrives_in arr_seq jhp HP : hep_job jhp j ARB : arrived_before jhp t
cumulative_other_hep_jobs_interfering_workload arr_seq
j 0 t +
workload_of_jobs
(fun j0 : Job => hep_job j0 j && ~~ (j0 != j))
(arrivals_between arr_seq 0 t) =
cumulative_another_hep_job_interference arr_seq sched
j 0 t +
service_of_jobs sched
(fun j0 : Job => hep_job j0 j && ~~ (j0 != j))
(arrivals_between arr_seq 0 t) 0 t
move : T1; rewrite negb_and => /orP [NA | /negPn COMP].Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j zero_is_quiet_time : forall j : Job, quiet_time_cl j 0 t : instant T0 : cumulative_interference j 0 t ==
cumulative_interfering_workload j 0 t jhp : Job ARR : arrives_in arr_seq jhp HP : hep_job jhp j ARB : arrived_before jhp t NA : ~~ arrived_before j t
cumulative_other_hep_jobs_interfering_workload arr_seq
j 0 t +
workload_of_jobs
(fun j0 : Job => hep_job j0 j && ~~ (j0 != j))
(arrivals_between arr_seq 0 t) =
cumulative_another_hep_job_interference arr_seq sched
j 0 t +
service_of_jobs sched
(fun j0 : Job => hep_job j0 j && ~~ (j0 != j))
(arrivals_between arr_seq 0 t) 0 t
{ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j zero_is_quiet_time : forall j : Job, quiet_time_cl j 0 t : instant T0 : cumulative_interference j 0 t ==
cumulative_interfering_workload j 0 t jhp : Job ARR : arrives_in arr_seq jhp HP : hep_job jhp j ARB : arrived_before jhp t NA : ~~ arrived_before j t
cumulative_other_hep_jobs_interfering_workload arr_seq
j 0 t +
workload_of_jobs
(fun j0 : Job => hep_job j0 j && ~~ (j0 != j))
(arrivals_between arr_seq 0 t) =
cumulative_another_hep_job_interference arr_seq sched
j 0 t +
service_of_jobs sched
(fun j0 : Job => hep_job j0 j && ~~ (j0 != j))
(arrivals_between arr_seq 0 t) 0 t
have PRED__eq: {in arrivals_between arr_seq 0 t, (fun j__copy : Job => hep_job j__copy j && ~~ (j__copy != j)) =1 pred0}.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j zero_is_quiet_time : forall j : Job, quiet_time_cl j 0 t : instant T0 : cumulative_interference j 0 t ==
cumulative_interfering_workload j 0 t jhp : Job ARR : arrives_in arr_seq jhp HP : hep_job jhp j ARB : arrived_before jhp t NA : ~~ arrived_before j t
{in arrivals_between arr_seq 0 t,
(fun j__copy : Job =>
hep_job j__copy j && ~~ (j__copy != j)) =1 pred0}
{ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j zero_is_quiet_time : forall j : Job, quiet_time_cl j 0 t : instant T0 : cumulative_interference j 0 t ==
cumulative_interfering_workload j 0 t jhp : Job ARR : arrives_in arr_seq jhp HP : hep_job jhp j ARB : arrived_before jhp t NA : ~~ arrived_before j t
{in arrivals_between arr_seq 0 t,
(fun j__copy : Job =>
hep_job j__copy j && ~~ (j__copy != j)) =1 pred0}
move => j__copy IN; apply negbTE.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j zero_is_quiet_time : forall j : Job, quiet_time_cl j 0 t : instant T0 : cumulative_interference j 0 t ==
cumulative_interfering_workload j 0 t jhp : Job ARR : arrives_in arr_seq jhp HP : hep_job jhp j ARB : arrived_before jhp t NA : ~~ arrived_before j t j__copy : Job IN : j__copy \in arrivals_between arr_seq 0 t
~~ (hep_job j__copy j && ~~ (j__copy != j))
rewrite negb_and; apply /orP; right ; apply /negPn/eqP => EQ; subst j__copy.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j zero_is_quiet_time : forall j : Job, quiet_time_cl j 0 t : instant T0 : cumulative_interference j 0 t ==
cumulative_interfering_workload j 0 t jhp : Job ARR : arrives_in arr_seq jhp HP : hep_job jhp j ARB : arrived_before jhp t NA : ~~ arrived_before j t IN : j \in arrivals_between arr_seq 0 t
False
move : NA => /negP CONTR; apply : CONTR.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j zero_is_quiet_time : forall j : Job, quiet_time_cl j 0 t : instant T0 : cumulative_interference j 0 t ==
cumulative_interfering_workload j 0 t jhp : Job ARR : arrives_in arr_seq jhp HP : hep_job jhp j ARB : arrived_before jhp t IN : j \in arrivals_between arr_seq 0 t
arrived_before j t
by apply in_arrivals_implies_arrived_between in IN. } Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j zero_is_quiet_time : forall j : Job, quiet_time_cl j 0 t : instant T0 : cumulative_interference j 0 t ==
cumulative_interfering_workload j 0 t jhp : Job ARR : arrives_in arr_seq jhp HP : hep_job jhp j ARB : arrived_before jhp t NA : ~~ arrived_before j t PRED__eq : {in arrivals_between arr_seq 0 t,
(fun j__copy : Job =>
hep_job j__copy j && ~~ (j__copy != j)) =1
pred0}
cumulative_other_hep_jobs_interfering_workload arr_seq
j 0 t +
workload_of_jobs
(fun j0 : Job => hep_job j0 j && ~~ (j0 != j))
(arrivals_between arr_seq 0 t) =
cumulative_another_hep_job_interference arr_seq sched
j 0 t +
service_of_jobs sched
(fun j0 : Job => hep_job j0 j && ~~ (j0 != j))
(arrivals_between arr_seq 0 t) 0 t
erewrite service_of_jobs_equiv_pred with (P2 := pred0) => [|//].Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j zero_is_quiet_time : forall j : Job, quiet_time_cl j 0 t : instant T0 : cumulative_interference j 0 t ==
cumulative_interfering_workload j 0 t jhp : Job ARR : arrives_in arr_seq jhp HP : hep_job jhp j ARB : arrived_before jhp t NA : ~~ arrived_before j t PRED__eq : {in arrivals_between arr_seq 0 t,
(fun j__copy : Job =>
hep_job j__copy j && ~~ (j__copy != j)) =1
pred0}
cumulative_other_hep_jobs_interfering_workload arr_seq
j 0 t +
workload_of_jobs
(fun j0 : Job => hep_job j0 j && ~~ (j0 != j))
(arrivals_between arr_seq 0 t) =
cumulative_another_hep_job_interference arr_seq sched
j 0 t +
service_of_jobs sched pred0
(arrivals_between arr_seq 0 t) 0 t
erewrite workload_of_jobs_equiv_pred with (P' := pred0) => [|//].Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j zero_is_quiet_time : forall j : Job, quiet_time_cl j 0 t : instant T0 : cumulative_interference j 0 t ==
cumulative_interfering_workload j 0 t jhp : Job ARR : arrives_in arr_seq jhp HP : hep_job jhp j ARB : arrived_before jhp t NA : ~~ arrived_before j t PRED__eq : {in arrivals_between arr_seq 0 t,
(fun j__copy : Job =>
hep_job j__copy j && ~~ (j__copy != j)) =1
pred0}
cumulative_other_hep_jobs_interfering_workload arr_seq
j 0 t +
workload_of_jobs pred0 (arrivals_between arr_seq 0 t) =
cumulative_another_hep_job_interference arr_seq sched
j 0 t +
service_of_jobs sched pred0
(arrivals_between arr_seq 0 t) 0 t
move : T0; erewrite cumulative_interference_split, cumulative_interfering_workload_split; rewrite eqn_add2l => /eqP EQ.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j zero_is_quiet_time : forall j : Job, quiet_time_cl j 0 t : instant jhp : Job ARR : arrives_in arr_seq jhp HP : hep_job jhp j ARB : arrived_before jhp t NA : ~~ arrived_before j t PRED__eq : {in arrivals_between arr_seq 0 t,
(fun j__copy : Job =>
hep_job j__copy j && ~~ (j__copy != j)) =1
pred0} EQ : cumulative_another_hep_job_interference arr_seq
sched j 0 t =
cumulative_other_hep_jobs_interfering_workload
arr_seq j 0 t
cumulative_other_hep_jobs_interfering_workload arr_seq
j 0 t +
workload_of_jobs pred0 (arrivals_between arr_seq 0 t) =
cumulative_another_hep_job_interference arr_seq sched
j 0 t +
service_of_jobs sched pred0
(arrivals_between arr_seq 0 t) 0 t
rewrite EQ; clear EQ; apply /eqP; rewrite eqn_add2l.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j zero_is_quiet_time : forall j : Job, quiet_time_cl j 0 t : instant jhp : Job ARR : arrives_in arr_seq jhp HP : hep_job jhp j ARB : arrived_before jhp t NA : ~~ arrived_before j t PRED__eq : {in arrivals_between arr_seq 0 t,
(fun j__copy : Job =>
hep_job j__copy j && ~~ (j__copy != j)) =1
pred0}
workload_of_jobs pred0 (arrivals_between arr_seq 0 t) ==
service_of_jobs sched pred0
(arrivals_between arr_seq 0 t) 0 t
by erewrite workload_of_jobs_pred0, service_of_jobs_pred0.
} Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j zero_is_quiet_time : forall j : Job, quiet_time_cl j 0 t : instant T0 : cumulative_interference j 0 t ==
cumulative_interfering_workload j 0 t jhp : Job ARR : arrives_in arr_seq jhp HP : hep_job jhp j ARB : arrived_before jhp t COMP : completed_by sched j t
cumulative_other_hep_jobs_interfering_workload arr_seq
j 0 t +
workload_of_jobs
(fun j0 : Job => hep_job j0 j && ~~ (j0 != j))
(arrivals_between arr_seq 0 t) =
cumulative_another_hep_job_interference arr_seq sched
j 0 t +
service_of_jobs sched
(fun j0 : Job => hep_job j0 j && ~~ (j0 != j))
(arrivals_between arr_seq 0 t) 0 t
{ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j zero_is_quiet_time : forall j : Job, quiet_time_cl j 0 t : instant T0 : cumulative_interference j 0 t ==
cumulative_interfering_workload j 0 t jhp : Job ARR : arrives_in arr_seq jhp HP : hep_job jhp j ARB : arrived_before jhp t COMP : completed_by sched j t
cumulative_other_hep_jobs_interfering_workload arr_seq
j 0 t +
workload_of_jobs
(fun j0 : Job => hep_job j0 j && ~~ (j0 != j))
(arrivals_between arr_seq 0 t) =
cumulative_another_hep_job_interference arr_seq sched
j 0 t +
service_of_jobs sched
(fun j0 : Job => hep_job j0 j && ~~ (j0 != j))
(arrivals_between arr_seq 0 t) 0 t
have PRED__eq: {in arrivals_between arr_seq 0 t, (fun j0 : Job => hep_job j0 j && ~~ (j0 != j)) =1 eq_op j}.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j zero_is_quiet_time : forall j : Job, quiet_time_cl j 0 t : instant T0 : cumulative_interference j 0 t ==
cumulative_interfering_workload j 0 t jhp : Job ARR : arrives_in arr_seq jhp HP : hep_job jhp j ARB : arrived_before jhp t COMP : completed_by sched j t
{in arrivals_between arr_seq 0 t,
(fun j0 : Job => hep_job j0 j && ~~ (j0 != j)) =1
eq_op j}
{ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j zero_is_quiet_time : forall j : Job, quiet_time_cl j 0 t : instant T0 : cumulative_interference j 0 t ==
cumulative_interfering_workload j 0 t jhp : Job ARR : arrives_in arr_seq jhp HP : hep_job jhp j ARB : arrived_before jhp t COMP : completed_by sched j t
{in arrivals_between arr_seq 0 t,
(fun j0 : Job => hep_job j0 j && ~~ (j0 != j)) =1
eq_op j}
move => j__copy IN.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j zero_is_quiet_time : forall j : Job, quiet_time_cl j 0 t : instant T0 : cumulative_interference j 0 t ==
cumulative_interfering_workload j 0 t jhp : Job ARR : arrives_in arr_seq jhp HP : hep_job jhp j ARB : arrived_before jhp t COMP : completed_by sched j t j__copy : Job IN : j__copy \in arrivals_between arr_seq 0 t
hep_job j__copy j && ~~ (j__copy != j) =
(j == j__copy)
replace (~~ (j__copy != j)) with (j__copy == j); last by case : (j__copy == j).Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j zero_is_quiet_time : forall j : Job, quiet_time_cl j 0 t : instant T0 : cumulative_interference j 0 t ==
cumulative_interfering_workload j 0 t jhp : Job ARR : arrives_in arr_seq jhp HP : hep_job jhp j ARB : arrived_before jhp t COMP : completed_by sched j t j__copy : Job IN : j__copy \in arrivals_between arr_seq 0 t
hep_job j__copy j && (j__copy == j) = (j == j__copy)
rewrite eq_sym; destruct (j == j__copy) eqn :EQ; last by rewrite Bool.andb_false_r.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j zero_is_quiet_time : forall j : Job, quiet_time_cl j 0 t : instant T0 : cumulative_interference j 0 t ==
cumulative_interfering_workload j 0 t jhp : Job ARR : arrives_in arr_seq jhp HP : hep_job jhp j ARB : arrived_before jhp t COMP : completed_by sched j t j__copy : Job IN : j__copy \in arrivals_between arr_seq 0 t EQ : (j == j__copy) = true
hep_job j__copy j && true = true
by move : EQ => /eqP EQ; rewrite Bool.andb_true_r; apply /eqP; rewrite eqb_id; subst . } Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j zero_is_quiet_time : forall j : Job, quiet_time_cl j 0 t : instant T0 : cumulative_interference j 0 t ==
cumulative_interfering_workload j 0 t jhp : Job ARR : arrives_in arr_seq jhp HP : hep_job jhp j ARB : arrived_before jhp t COMP : completed_by sched j t PRED__eq : {in arrivals_between arr_seq 0 t,
(fun j0 : Job =>
hep_job j0 j && ~~ (j0 != j)) =1
eq_op j}
cumulative_other_hep_jobs_interfering_workload arr_seq
j 0 t +
workload_of_jobs
(fun j0 : Job => hep_job j0 j && ~~ (j0 != j))
(arrivals_between arr_seq 0 t) =
cumulative_another_hep_job_interference arr_seq sched
j 0 t +
service_of_jobs sched
(fun j0 : Job => hep_job j0 j && ~~ (j0 != j))
(arrivals_between arr_seq 0 t) 0 t
erewrite service_of_jobs_equiv_pred with (P2 := eq_op j) => [|//].Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j zero_is_quiet_time : forall j : Job, quiet_time_cl j 0 t : instant T0 : cumulative_interference j 0 t ==
cumulative_interfering_workload j 0 t jhp : Job ARR : arrives_in arr_seq jhp HP : hep_job jhp j ARB : arrived_before jhp t COMP : completed_by sched j t PRED__eq : {in arrivals_between arr_seq 0 t,
(fun j0 : Job =>
hep_job j0 j && ~~ (j0 != j)) =1
eq_op j}
cumulative_other_hep_jobs_interfering_workload arr_seq
j 0 t +
workload_of_jobs
(fun j0 : Job => hep_job j0 j && ~~ (j0 != j))
(arrivals_between arr_seq 0 t) =
cumulative_another_hep_job_interference arr_seq sched
j 0 t +
service_of_jobs sched (eq_op j)
(arrivals_between arr_seq 0 t) 0 t
erewrite workload_of_jobs_equiv_pred with (P' := eq_op j) => [|//].Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j zero_is_quiet_time : forall j : Job, quiet_time_cl j 0 t : instant T0 : cumulative_interference j 0 t ==
cumulative_interfering_workload j 0 t jhp : Job ARR : arrives_in arr_seq jhp HP : hep_job jhp j ARB : arrived_before jhp t COMP : completed_by sched j t PRED__eq : {in arrivals_between arr_seq 0 t,
(fun j0 : Job =>
hep_job j0 j && ~~ (j0 != j)) =1
eq_op j}
cumulative_other_hep_jobs_interfering_workload arr_seq
j 0 t +
workload_of_jobs (eq_op j)
(arrivals_between arr_seq 0 t) =
cumulative_another_hep_job_interference arr_seq sched
j 0 t +
service_of_jobs sched (eq_op j)
(arrivals_between arr_seq 0 t) 0 t
move : T0; erewrite cumulative_interference_split, cumulative_interfering_workload_split; rewrite eqn_add2l => /eqP EQ.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j zero_is_quiet_time : forall j : Job, quiet_time_cl j 0 t : instant jhp : Job ARR : arrives_in arr_seq jhp HP : hep_job jhp j ARB : arrived_before jhp t COMP : completed_by sched j t PRED__eq : {in arrivals_between arr_seq 0 t,
(fun j0 : Job =>
hep_job j0 j && ~~ (j0 != j)) =1
eq_op j} EQ : cumulative_another_hep_job_interference arr_seq
sched j 0 t =
cumulative_other_hep_jobs_interfering_workload
arr_seq j 0 t
cumulative_other_hep_jobs_interfering_workload arr_seq
j 0 t +
workload_of_jobs (eq_op j)
(arrivals_between arr_seq 0 t) =
cumulative_another_hep_job_interference arr_seq sched
j 0 t +
service_of_jobs sched (eq_op j)
(arrivals_between arr_seq 0 t) 0 t
rewrite EQ; clear EQ; apply /eqP; rewrite eqn_add2l.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j zero_is_quiet_time : forall j : Job, quiet_time_cl j 0 t : instant jhp : Job ARR : arrives_in arr_seq jhp HP : hep_job jhp j ARB : arrived_before jhp t COMP : completed_by sched j t PRED__eq : {in arrivals_between arr_seq 0 t,
(fun j0 : Job =>
hep_job j0 j && ~~ (j0 != j)) =1
eq_op j}
workload_of_jobs (eq_op j)
(arrivals_between arr_seq 0 t) ==
service_of_jobs sched (eq_op j)
(arrivals_between arr_seq 0 t) 0 t
apply /eqP; eapply all_jobs_have_completed_equiv_workload_eq_service with
(P := eq_op j) (t1 := 0 ) (t2 := t) => //.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j zero_is_quiet_time : forall j : Job, quiet_time_cl j 0 t : instant jhp : Job ARR : arrives_in arr_seq jhp HP : hep_job jhp j ARB : arrived_before jhp t COMP : completed_by sched j t PRED__eq : {in arrivals_between arr_seq 0 t,
(fun j0 : Job =>
hep_job j0 j && ~~ (j0 != j)) =1
eq_op j}
forall j0 : Job,
j0 \in arrivals_between arr_seq 0 t ->
j == j0 -> completed_by sched j0 t
by move => j__copy _ /eqP EQ; subst j__copy.
}
Qed .
(** The equivalence trivially follows from the lemmas above. *)
Corollary instantiated_quiet_time_equivalent_quiet_time :
forall t ,
quiet_time_cl j t <-> quiet_time_ab j t.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j
forall t : instant,
quiet_time_cl j t <-> quiet_time_ab j t
Proof .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j
forall t : instant,
quiet_time_cl j t <-> quiet_time_ab j t
clear H_JLFP_respects_sequential_tasks.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j
forall t : instant,
quiet_time_cl j t <-> quiet_time_ab j t
move => ?; split .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j _t_ : instant
quiet_time_cl j _t_ -> quiet_time_ab j _t_
- Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j _t_ : instant
quiet_time_cl j _t_ -> quiet_time_ab j _t_
by apply quiet_time_cl_implies_quiet_time_ab.
- Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j _t_ : instant
quiet_time_ab j _t_ -> quiet_time_cl j _t_
by apply quiet_time_ab_implies_quiet_time_cl.
Qed .
(** Based on that, we prove that the concept of busy interval
prefix obtained by instantiating the abstract definition of
busy interval prefix coincides with the conventional
definition of busy interval prefix. *)
Lemma instantiated_busy_interval_prefix_equivalent_busy_interval_prefix :
forall t1 t2 , busy_interval_prefix_cl j t1 t2 <-> busy_interval_prefix_ab j t1 t2.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j
forall t1 t2 : instant,
busy_interval_prefix_cl j t1 t2 <->
busy_interval_prefix_ab j t1 t2
Proof .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j
forall t1 t2 : instant,
busy_interval_prefix_cl j t1 t2 <->
busy_interval_prefix_ab j t1 t2
move => t1 t2; split .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j t1, t2 : instant
busy_interval_prefix_cl j t1 t2 ->
busy_interval_prefix_ab j t1 t2
{ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j t1, t2 : instant
busy_interval_prefix_cl j t1 t2 ->
busy_interval_prefix_ab j t1 t2
move => [NEQ [QTt1 [NQT REL]]].Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j t1, t2 : instant NEQ : t1 < t2 QTt1 : quiet_time arr_seq sched j t1 NQT : forall t : nat,
t1 < t < t2 -> ~ quiet_time arr_seq sched j tREL : t1 <= job_arrival j < t2
busy_interval_prefix_ab j t1 t2
split => [//|]; split .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j t1, t2 : instant NEQ : t1 < t2 QTt1 : quiet_time arr_seq sched j t1 NQT : forall t : nat,
t1 < t < t2 -> ~ quiet_time arr_seq sched j tREL : t1 <= job_arrival j < t2
definitions.quiet_time sched j t1
- Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j t1, t2 : instant NEQ : t1 < t2 QTt1 : quiet_time arr_seq sched j t1 NQT : forall t : nat,
t1 < t < t2 -> ~ quiet_time arr_seq sched j tREL : t1 <= job_arrival j < t2
definitions.quiet_time sched j t1
by apply instantiated_quiet_time_equivalent_quiet_time in QTt1.
- Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j t1, t2 : instant NEQ : t1 < t2 QTt1 : quiet_time arr_seq sched j t1 NQT : forall t : nat,
t1 < t < t2 -> ~ quiet_time arr_seq sched j tREL : t1 <= job_arrival j < t2
forall t : nat,
t1 < t < t2 -> ~ definitions.quiet_time sched j t
by move => t NE QT; eapply NQT; eauto 2 ; apply instantiated_quiet_time_equivalent_quiet_time.
} Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j t1, t2 : instant
busy_interval_prefix_ab j t1 t2 ->
busy_interval_prefix_cl j t1 t2
{ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j t1, t2 : instant
busy_interval_prefix_ab j t1 t2 ->
busy_interval_prefix_cl j t1 t2
move => [/andP [NEQ1 NEQ2] [QTt1 NQT]].Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j t1, t2 : instant NEQ1 : t1 <= job_arrival j NEQ2 : job_arrival j < t2 QTt1 : definitions.quiet_time sched j t1 NQT : forall t : nat,
t1 < t < t2 ->
~ definitions.quiet_time sched j t
busy_interval_prefix_cl j t1 t2
split ; [ | split ; [ |split ] ].Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j t1, t2 : instant NEQ1 : t1 <= job_arrival j NEQ2 : job_arrival j < t2 QTt1 : definitions.quiet_time sched j t1 NQT : forall t : nat,
t1 < t < t2 ->
~ definitions.quiet_time sched j t
t1 < t2
- Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j t1, t2 : instant NEQ1 : t1 <= job_arrival j NEQ2 : job_arrival j < t2 QTt1 : definitions.quiet_time sched j t1 NQT : forall t : nat,
t1 < t < t2 ->
~ definitions.quiet_time sched j t
t1 < t2
by apply leq_ltn_trans with (job_arrival j).
- Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j t1, t2 : instant NEQ1 : t1 <= job_arrival j NEQ2 : job_arrival j < t2 QTt1 : definitions.quiet_time sched j t1 NQT : forall t : nat,
t1 < t < t2 ->
~ definitions.quiet_time sched j t
quiet_time arr_seq sched j t1
by eapply instantiated_quiet_time_equivalent_quiet_time.
- Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j t1, t2 : instant NEQ1 : t1 <= job_arrival j NEQ2 : job_arrival j < t2 QTt1 : definitions.quiet_time sched j t1 NQT : forall t : nat,
t1 < t < t2 ->
~ definitions.quiet_time sched j t
forall t : nat,
t1 < t < t2 -> ~ quiet_time arr_seq sched j t
by move => t NEQ QT; eapply NQT; eauto 2 ; eapply instantiated_quiet_time_equivalent_quiet_time in QT.
- Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j t1, t2 : instant NEQ1 : t1 <= job_arrival j NEQ2 : job_arrival j < t2 QTt1 : definitions.quiet_time sched j t1 NQT : forall t : nat,
t1 < t < t2 ->
~ definitions.quiet_time sched j t
t1 <= job_arrival j < t2
by apply /andP.
}
Qed .
(** Similarly, we prove that the concept of busy interval
obtained by instantiating the abstract definition of busy
interval coincides with the conventional definition of busy
interval. *)
Lemma instantiated_busy_interval_equivalent_busy_interval :
forall t1 t2 , busy_interval_cl j t1 t2 <-> busy_interval_ab j t1 t2.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j
forall t1 t2 : instant,
busy_interval_cl j t1 t2 <-> busy_interval_ab j t1 t2
Proof .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j
forall t1 t2 : instant,
busy_interval_cl j t1 t2 <-> busy_interval_ab j t1 t2
move => t1 t2; split .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j t1, t2 : instant
busy_interval_cl j t1 t2 -> busy_interval_ab j t1 t2
{ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j t1, t2 : instant
busy_interval_cl j t1 t2 -> busy_interval_ab j t1 t2
move => [PREF QTt2]; split .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j t1, t2 : instant PREF : busy_interval_prefix arr_seq sched j t1 t2 QTt2 : quiet_time arr_seq sched j t2
definitions.busy_interval_prefix sched j t1 t2
- Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j t1, t2 : instant PREF : busy_interval_prefix arr_seq sched j t1 t2 QTt2 : quiet_time arr_seq sched j t2
definitions.busy_interval_prefix sched j t1 t2
by apply instantiated_busy_interval_prefix_equivalent_busy_interval_prefix.
- Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j t1, t2 : instant PREF : busy_interval_prefix arr_seq sched j t1 t2 QTt2 : quiet_time arr_seq sched j t2
definitions.quiet_time sched j t2
by eapply instantiated_quiet_time_equivalent_quiet_time in QTt2.
} Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j t1, t2 : instant
busy_interval_ab j t1 t2 -> busy_interval_cl j t1 t2
{ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j t1, t2 : instant
busy_interval_ab j t1 t2 -> busy_interval_cl j t1 t2
move => [PREF QTt2]; split .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j t1, t2 : instant PREF : definitions.busy_interval_prefix sched j t1 t2 QTt2 : definitions.quiet_time sched j t2
busy_interval_prefix arr_seq sched j t1 t2
- Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j t1, t2 : instant PREF : definitions.busy_interval_prefix sched j t1 t2 QTt2 : definitions.quiet_time sched j t2
busy_interval_prefix arr_seq sched j t1 t2
by apply instantiated_busy_interval_prefix_equivalent_busy_interval_prefix.
- Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j t1, t2 : instant PREF : definitions.busy_interval_prefix sched j t1 t2 QTt2 : definitions.quiet_time sched j t2
quiet_time arr_seq sched j t2
by eapply instantiated_quiet_time_equivalent_quiet_time.
}
Qed .
(** For the sake of proof automation, we note the frequently needed
special case of an abstract busy window implying the existence of a
classic quiet time. *)
Fact abstract_busy_interval_classic_quiet_time :
forall t1 t2 ,
busy_interval_ab j t1 t2 -> quiet_time_cl j t1.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j
forall t1 t2 : instant,
busy_interval_ab j t1 t2 -> quiet_time_cl j t1
Proof .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j
forall t1 t2 : instant,
busy_interval_ab j t1 t2 -> quiet_time_cl j t1
by move => ? ? /instantiated_busy_interval_equivalent_busy_interval [[_ [? _]] _].
Qed .
(** Also for automation, we note a similar fact about classic busy-window prefixes. *)
Fact abstract_busy_interval_classic_busy_interval_prefix :
forall t1 t2 ,
busy_interval_ab j t1 t2 -> busy_interval_prefix_cl j t1 t2.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j
forall t1 t2 : instant,
busy_interval_ab j t1 t2 ->
busy_interval_prefix_cl j t1 t2
Proof .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> bool busy_interval_prefix_cl := busy_interval_prefix arr_seq
sched : Job ->
instant -> instant -> Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop busy_interval_cl := busy_interval arr_seq sched : Job -> instant -> instant -> Prop busy_interval_ab := definitions.busy_interval sched : Job -> instant -> instant -> Prop j : Job H_j_arrives : arrives_in arr_seq j H_job_cost_positive : job_cost_positive j
forall t1 t2 : instant,
busy_interval_ab j t1 t2 ->
busy_interval_prefix_cl j t1 t2
by move => ? ? /instantiated_busy_interval_equivalent_busy_interval [+ _]. Qed .
End BusyIntervalEquivalence .
End Equivalences .
(** In this section we prove some properties about the interference
and interfering workload as defined in this file. *)
Section I_IW_correctness .
(** Consider work-bearing readiness. *)
Context `{!JobReady Job (ideal.processor_state Job)}.
Hypothesis H_work_bearing_readiness : work_bearing_readiness arr_seq sched.
(** Assume that the schedule is valid and work-conserving. *)
Hypothesis H_sched_valid : valid_schedule sched arr_seq.
(** Note that we differentiate between abstract and classical
notions of work-conserving schedule. *)
Let work_conserving_ab := definitions.work_conserving arr_seq sched.
Let work_conserving_cl := work_conserving.work_conserving arr_seq sched.
Let busy_interval_prefix_ab := definitions.busy_interval_prefix sched.
(** We assume that the schedule is a work-conserving schedule in
the _classical_ sense, and later prove that the hypothesis
about abstract work-conservation also holds. *)
Hypothesis H_work_conserving : work_conserving_cl.
(** Assume the scheduling policy under consideration is reflexive. *)
Hypothesis H_policy_reflexive : reflexive_job_priorities JLFP.
(** In this section, we prove the correctness of interference
inside the busy interval, i.e., we prove that if interference
for a job is [false] then the job is scheduled and vice versa.
This property is referred to as abstract work conservation. *)
Section Abstract_Work_Conservation .
(** Consider a job [j] that is in the arrival sequence
and has a positive job cost. *)
Variable j : Job.
Hypothesis H_arrives : arrives_in arr_seq j.
Hypothesis H_job_cost_positive : 0 < job_cost j.
(** Let the busy interval of the job be <<[t1,t2)>>. *)
Variable t1 t2 : instant.
Hypothesis H_busy_interval_prefix : busy_interval_prefix_ab j t1 t2.
(** Consider a time [t] inside the busy interval of the job. *)
Variable t : instant.
Hypothesis H_t_in_busy_interval : t1 <= t < t2.
(** First, we prove that if interference is [false] at a time
[t] then the job is scheduled. *)
Lemma not_interference_implies_scheduled :
~ interference j t -> receives_service_at sched j t.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP j : Job H_arrives : arrives_in arr_seq j H_job_cost_positive : 0 < job_cost jt1, t2 : instant H_busy_interval_prefix : busy_interval_prefix_ab j t1
t2 t : instant H_t_in_busy_interval : t1 <= t < t2
~ interference j t -> receives_service_at sched j t
Proof .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP j : Job H_arrives : arrives_in arr_seq j H_job_cost_positive : 0 < job_cost jt1, t2 : instant H_busy_interval_prefix : busy_interval_prefix_ab j t1
t2 t : instant H_t_in_busy_interval : t1 <= t < t2
~ interference j t -> receives_service_at sched j t
move => /negP HYP; move : HYP.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP j : Job H_arrives : arrives_in arr_seq j H_job_cost_positive : 0 < job_cost jt1, t2 : instant H_busy_interval_prefix : busy_interval_prefix_ab j t1
t2 t : instant H_t_in_busy_interval : t1 <= t < t2
~~ interference j t -> receives_service_at sched j t
rewrite negb_or /another_hep_job_interference.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP j : Job H_arrives : arrives_in arr_seq j H_job_cost_positive : 0 < job_cost jt1, t2 : instant H_busy_interval_prefix : busy_interval_prefix_ab j t1
t2 t : instant H_t_in_busy_interval : t1 <= t < t2
~~ priority_inversion arr_seq sched j t &&
~~
has (another_hep_job^~ j)
(served_jobs_at arr_seq sched t) ->
receives_service_at sched j t
move => /andP [HYP1 HYP2].Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP j : Job H_arrives : arrives_in arr_seq j H_job_cost_positive : 0 < job_cost jt1, t2 : instant H_busy_interval_prefix : busy_interval_prefix_ab j t1
t2 t : instant H_t_in_busy_interval : t1 <= t < t2 HYP1 : ~~ priority_inversion arr_seq sched j t HYP2 : ~~
has (another_hep_job^~ j)
(served_jobs_at arr_seq sched t)
receives_service_at sched j t
have [Idle|[jo Sched_jo]] := ideal_proc_model_sched_case_analysis sched t.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP j : Job H_arrives : arrives_in arr_seq j H_job_cost_positive : 0 < job_cost jt1, t2 : instant H_busy_interval_prefix : busy_interval_prefix_ab j t1
t2 t : instant H_t_in_busy_interval : t1 <= t < t2 HYP1 : ~~ priority_inversion arr_seq sched j t HYP2 : ~~
has (another_hep_job^~ j)
(served_jobs_at arr_seq sched t) Idle : ideal_is_idle sched t
receives_service_at sched j t
{ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP j : Job H_arrives : arrives_in arr_seq j H_job_cost_positive : 0 < job_cost jt1, t2 : instant H_busy_interval_prefix : busy_interval_prefix_ab j t1
t2 t : instant H_t_in_busy_interval : t1 <= t < t2 HYP1 : ~~ priority_inversion arr_seq sched j t HYP2 : ~~
has (another_hep_job^~ j)
(served_jobs_at arr_seq sched t) Idle : ideal_is_idle sched t
receives_service_at sched j t
exfalso ; clear HYP1 HYP2.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP j : Job H_arrives : arrives_in arr_seq j H_job_cost_positive : 0 < job_cost jt1, t2 : instant H_busy_interval_prefix : busy_interval_prefix_ab j t1
t2 t : instant H_t_in_busy_interval : t1 <= t < t2 Idle : ideal_is_idle sched t
False
eapply instantiated_busy_interval_prefix_equivalent_busy_interval_prefix in H_busy_interval_prefix => //.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP j : Job H_arrives : arrives_in arr_seq j H_job_cost_positive : 0 < job_cost jt1, t2 : instant H_busy_interval_prefix : busy_interval_prefix arr_seq
sched j t1 t2 t : instant H_t_in_busy_interval : t1 <= t < t2 Idle : ideal_is_idle sched t
False
apply : not_quiet_implies_not_idle => //.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP j : Job H_arrives : arrives_in arr_seq j H_job_cost_positive : 0 < job_cost jt1, t2 : instant H_busy_interval_prefix : busy_interval_prefix arr_seq
sched j t1 t2 t : instant H_t_in_busy_interval : t1 <= t < t2 Idle : ideal_is_idle sched t
is_idle arr_seq sched t
by rewrite is_idle_def.
} Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP j : Job H_arrives : arrives_in arr_seq j H_job_cost_positive : 0 < job_cost jt1, t2 : instant H_busy_interval_prefix : busy_interval_prefix_ab j t1
t2 t : instant H_t_in_busy_interval : t1 <= t < t2 HYP1 : ~~ priority_inversion arr_seq sched j t HYP2 : ~~
has (another_hep_job^~ j)
(served_jobs_at arr_seq sched t) jo : Job Sched_jo : scheduled_at sched jo t
receives_service_at sched j t
{ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP j : Job H_arrives : arrives_in arr_seq j H_job_cost_positive : 0 < job_cost jt1, t2 : instant H_busy_interval_prefix : busy_interval_prefix_ab j t1
t2 t : instant H_t_in_busy_interval : t1 <= t < t2 HYP1 : ~~ priority_inversion arr_seq sched j t HYP2 : ~~
has (another_hep_job^~ j)
(served_jobs_at arr_seq sched t) jo : Job Sched_jo : scheduled_at sched jo t
receives_service_at sched j t
have PINV := negP HYP1.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP j : Job H_arrives : arrives_in arr_seq j H_job_cost_positive : 0 < job_cost jt1, t2 : instant H_busy_interval_prefix : busy_interval_prefix_ab j t1
t2 t : instant H_t_in_busy_interval : t1 <= t < t2 HYP1 : ~~ priority_inversion arr_seq sched j t HYP2 : ~~
has (another_hep_job^~ j)
(served_jobs_at arr_seq sched t) jo : Job Sched_jo : scheduled_at sched jo t PINV : ~ priority_inversion arr_seq sched j t
receives_service_at sched j t
have : ~~ another_hep_job jo j by apply : contraNN HYP2; erewrite <-interference_ahep_def => //.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP j : Job H_arrives : arrives_in arr_seq j H_job_cost_positive : 0 < job_cost jt1, t2 : instant H_busy_interval_prefix : busy_interval_prefix_ab j t1
t2 t : instant H_t_in_busy_interval : t1 <= t < t2 HYP1 : ~~ priority_inversion arr_seq sched j t HYP2 : ~~
has (another_hep_job^~ j)
(served_jobs_at arr_seq sched t) jo : Job Sched_jo : scheduled_at sched jo t PINV : ~ priority_inversion arr_seq sched j t
~~ another_hep_job jo j ->
receives_service_at sched j t
rewrite negb_and => /orP [NHEP | EQ].Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP j : Job H_arrives : arrives_in arr_seq j H_job_cost_positive : 0 < job_cost jt1, t2 : instant H_busy_interval_prefix : busy_interval_prefix_ab j t1
t2 t : instant H_t_in_busy_interval : t1 <= t < t2 HYP1 : ~~ priority_inversion arr_seq sched j t HYP2 : ~~
has (another_hep_job^~ j)
(served_jobs_at arr_seq sched t) jo : Job Sched_jo : scheduled_at sched jo t PINV : ~ priority_inversion arr_seq sched j t NHEP : ~~ hep_job jo j
receives_service_at sched j t
- Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP j : Job H_arrives : arrives_in arr_seq j H_job_cost_positive : 0 < job_cost jt1, t2 : instant H_busy_interval_prefix : busy_interval_prefix_ab j t1
t2 t : instant H_t_in_busy_interval : t1 <= t < t2 HYP1 : ~~ priority_inversion arr_seq sched j t HYP2 : ~~
has (another_hep_job^~ j)
(served_jobs_at arr_seq sched t) jo : Job Sched_jo : scheduled_at sched jo t PINV : ~ priority_inversion arr_seq sched j t NHEP : ~~ hep_job jo j
receives_service_at sched j t
rewrite /receives_service_at; move_neq_up ZS; move : ZS; rewrite leqn0 => /eqP ZS.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP j : Job H_arrives : arrives_in arr_seq j H_job_cost_positive : 0 < job_cost jt1, t2 : instant H_busy_interval_prefix : busy_interval_prefix_ab j t1
t2 t : instant H_t_in_busy_interval : t1 <= t < t2 HYP1 : ~~ priority_inversion arr_seq sched j t HYP2 : ~~
has (another_hep_job^~ j)
(served_jobs_at arr_seq sched t) jo : Job Sched_jo : scheduled_at sched jo t PINV : ~ priority_inversion arr_seq sched j t NHEP : ~~ hep_job jo j ZS : service_at sched j t = 0
False
apply no_service_not_scheduled in ZS => //; apply : PINV.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP j : Job H_arrives : arrives_in arr_seq j H_job_cost_positive : 0 < job_cost jt1, t2 : instant H_busy_interval_prefix : busy_interval_prefix_ab j t1
t2 t : instant H_t_in_busy_interval : t1 <= t < t2 HYP1 : ~~ priority_inversion arr_seq sched j t HYP2 : ~~
has (another_hep_job^~ j)
(served_jobs_at arr_seq sched t) jo : Job Sched_jo : scheduled_at sched jo t NHEP : ~~ hep_job jo j ZS : ~~ scheduled_at sched j t
priority_inversion arr_seq sched j t
exact /uni_priority_inversion_P.
- Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP j : Job H_arrives : arrives_in arr_seq j H_job_cost_positive : 0 < job_cost jt1, t2 : instant H_busy_interval_prefix : busy_interval_prefix_ab j t1
t2 t : instant H_t_in_busy_interval : t1 <= t < t2 HYP1 : ~~ priority_inversion arr_seq sched j t HYP2 : ~~
has (another_hep_job^~ j)
(served_jobs_at arr_seq sched t) jo : Job Sched_jo : scheduled_at sched jo t PINV : ~ priority_inversion arr_seq sched j t EQ : ~~ (jo != j)
receives_service_at sched j t
apply negbNE in EQ; move : EQ => /eqP EQ; subst jo.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP j : Job H_arrives : arrives_in arr_seq j H_job_cost_positive : 0 < job_cost jt1, t2 : instant H_busy_interval_prefix : busy_interval_prefix_ab j t1
t2 t : instant H_t_in_busy_interval : t1 <= t < t2 HYP1 : ~~ priority_inversion arr_seq sched j t HYP2 : ~~
has (another_hep_job^~ j)
(served_jobs_at arr_seq sched t) Sched_jo : scheduled_at sched j t PINV : ~ priority_inversion arr_seq sched j t
receives_service_at sched j t
by rewrite /receives_service_at service_at_is_scheduled_at Sched_jo.
}
Qed .
(** Conversely, if the job is scheduled at [t] then interference is [false]. *)
Lemma scheduled_implies_no_interference :
receives_service_at sched j t -> ~ interference j t.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP j : Job H_arrives : arrives_in arr_seq j H_job_cost_positive : 0 < job_cost jt1, t2 : instant H_busy_interval_prefix : busy_interval_prefix_ab j t1
t2 t : instant H_t_in_busy_interval : t1 <= t < t2
receives_service_at sched j t -> ~ interference j t
Proof .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP j : Job H_arrives : arrives_in arr_seq j H_job_cost_positive : 0 < job_cost jt1, t2 : instant H_busy_interval_prefix : busy_interval_prefix_ab j t1
t2 t : instant H_t_in_busy_interval : t1 <= t < t2
receives_service_at sched j t -> ~ interference j t
move => RSERV /orP[PINV | INT].Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP j : Job H_arrives : arrives_in arr_seq j H_job_cost_positive : 0 < job_cost jt1, t2 : instant H_busy_interval_prefix : busy_interval_prefix_ab j t1
t2 t : instant H_t_in_busy_interval : t1 <= t < t2 RSERV : receives_service_at sched j t PINV : priority_inversion arr_seq sched j t
False
- Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP j : Job H_arrives : arrives_in arr_seq j H_job_cost_positive : 0 < job_cost jt1, t2 : instant H_busy_interval_prefix : busy_interval_prefix_ab j t1
t2 t : instant H_t_in_busy_interval : t1 <= t < t2 RSERV : receives_service_at sched j t PINV : priority_inversion arr_seq sched j t
False
rewrite (sched_hep_implies_no_priority_inversion _ _ _ _ _ _ _ _ j) in PINV => //.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP j : Job H_arrives : arrives_in arr_seq j H_job_cost_positive : 0 < job_cost jt1, t2 : instant H_busy_interval_prefix : busy_interval_prefix_ab j t1
t2 t : instant H_t_in_busy_interval : t1 <= t < t2 RSERV : receives_service_at sched j t
scheduled_at sched j t
by rewrite /receives_service_at service_at_is_scheduled_at lt0b in RSERV.
- Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP j : Job H_arrives : arrives_in arr_seq j H_job_cost_positive : 0 < job_cost jt1, t2 : instant H_busy_interval_prefix : busy_interval_prefix_ab j t1
t2 t : instant H_t_in_busy_interval : t1 <= t < t2 RSERV : receives_service_at sched j t INT : another_hep_job_interference arr_seq sched j t
False
rewrite /receives_service_at service_at_is_scheduled_at lt0b in RSERV.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP j : Job H_arrives : arrives_in arr_seq j H_job_cost_positive : 0 < job_cost jt1, t2 : instant H_busy_interval_prefix : busy_interval_prefix_ab j t1
t2 t : instant H_t_in_busy_interval : t1 <= t < t2 INT : another_hep_job_interference arr_seq sched j t RSERV : scheduled_at sched j t
False
by move : INT; rewrite_neg @no_ahep_interference_when_scheduled.
Qed .
End Abstract_Work_Conservation .
(** Using the above two lemmas, we can prove that abstract work
conservation always holds for these instantiations of [I] and
[IW]. *)
Corollary instantiated_i_and_w_are_coherent_with_schedule :
work_conserving_ab.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP
work_conserving_ab
Proof .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP
work_conserving_ab
move => j t1 t2 t ARR POS BUSY NEQ; split .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP j : Job t1, t2 : instant t : nat ARR : arrives_in arr_seq j POS : 0 < job_cost jBUSY : definitions.busy_interval_prefix sched j t1 t2 NEQ : t1 <= t < t2
~ interference j t -> receives_service_at sched j t
- Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP j : Job t1, t2 : instant t : nat ARR : arrives_in arr_seq j POS : 0 < job_cost jBUSY : definitions.busy_interval_prefix sched j t1 t2 NEQ : t1 <= t < t2
~ interference j t -> receives_service_at sched j t
exact : (not_interference_implies_scheduled j ARR POS).
- Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP j : Job t1, t2 : instant t : nat ARR : arrives_in arr_seq j POS : 0 < job_cost jBUSY : definitions.busy_interval_prefix sched j t1 t2 NEQ : t1 <= t < t2
receives_service_at sched j t -> ~ interference j t
exact : scheduled_implies_no_interference.
Qed .
(** Next, in order to prove that these definitions of [I] and [IW]
are consistent with sequential tasks, we need to assume that
the policy under consideration respects sequential tasks. *)
Hypothesis H_policy_respects_sequential_tasks : policy_respects_sequential_tasks JLFP.
(** We prove that these definitions of [I] and [IW] are consistent
with sequential tasks. *)
Lemma instantiated_interference_and_workload_consistent_with_sequential_tasks :
interference_and_workload_consistent_with_sequential_tasks arr_seq sched tsk.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP H_policy_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP
interference_and_workload_consistent_with_sequential_tasks
arr_seq sched tsk
Proof .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP H_policy_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP
interference_and_workload_consistent_with_sequential_tasks
arr_seq sched tsk
move => j t1 t2 ARR /eqP TSK POS BUSY.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP H_policy_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP j : Job t1, t2 : instant ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : definitions.busy_interval sched j t1 t2
task_workload_between arr_seq tsk 0 t1 =
task_service_of_jobs_in sched tsk
(arrivals_between arr_seq 0 t1) 0 t1
eapply instantiated_busy_interval_equivalent_busy_interval in BUSY => //.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP H_policy_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP j : Job t1, t2 : instant ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : busy_interval arr_seq sched j t1 t2
task_workload_between arr_seq tsk 0 t1 =
task_service_of_jobs_in sched tsk
(arrivals_between arr_seq 0 t1) 0 t1
eapply all_jobs_have_completed_equiv_workload_eq_service => //.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP H_policy_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP j : Job t1, t2 : instant ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : busy_interval arr_seq sched j t1 t2
forall j : Job,
j \in arrivals_between arr_seq 0 t1 ->
job_of_task tsk j -> completed_by sched j t1
move => s INs /eqP TSKs.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP H_policy_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP j : Job t1, t2 : instant ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : busy_interval arr_seq sched j t1 t2 s : Job INs : s \in arrivals_between arr_seq 0 t1 TSKs : job_task s = tsk
completed_by sched s t1
move : (INs) => NEQ.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP H_policy_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP j : Job t1, t2 : instant ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : busy_interval arr_seq sched j t1 t2 s : Job INs : s \in arrivals_between arr_seq 0 t1 TSKs : job_task s = tsk NEQ : s \in arrivals_between arr_seq 0 t1
completed_by sched s t1
eapply in_arrivals_implies_arrived_between in NEQ => //.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP H_policy_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP j : Job t1, t2 : instant ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : busy_interval arr_seq sched j t1 t2 s : Job INs : s \in arrivals_between arr_seq 0 t1 TSKs : job_task s = tsk NEQ : arrived_between s 0 t1
completed_by sched s t1
move : NEQ => /andP [_ JAs].Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP H_policy_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP j : Job t1, t2 : instant ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : busy_interval arr_seq sched j t1 t2 s : Job INs : s \in arrivals_between arr_seq 0 t1 TSKs : job_task s = tsk JAs : job_arrival s < t1
completed_by sched s t1
move : (BUSY) => [[ _ [QT [_ /andP [JAj _]]] _]].Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP H_policy_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP j : Job t1, t2 : instant ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : busy_interval arr_seq sched j t1 t2 s : Job INs : s \in arrivals_between arr_seq 0 t1 TSKs : job_task s = tsk JAs : job_arrival s < t1 QT : quiet_time arr_seq sched j t1 JAj : t1 <= job_arrival j
completed_by sched s t1
apply QT => //; first exact : in_arrivals_implies_arrived.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP H_policy_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP j : Job t1, t2 : instant ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : busy_interval arr_seq sched j t1 t2 s : Job INs : s \in arrivals_between arr_seq 0 t1 TSKs : job_task s = tsk JAs : job_arrival s < t1 QT : quiet_time arr_seq sched j t1 JAj : t1 <= job_arrival j
hep_job s j
apply H_policy_respects_sequential_tasks; first by rewrite TSK TSKs.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP H_policy_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP j : Job t1, t2 : instant ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : busy_interval arr_seq sched j t1 t2 s : Job INs : s \in arrivals_between arr_seq 0 t1 TSKs : job_task s = tsk JAs : job_arrival s < t1 QT : quiet_time arr_seq sched j t1 JAj : t1 <= job_arrival j
job_arrival s <= job_arrival j
by apply leq_trans with t1; [lia |].
Qed .
(** Since interfering and interfering workload are sufficient to define the busy window,
next, we reason about the bound on the length of the busy window. *)
Section BusyWindowBound .
(** Consider an arrival curve. *)
Context `{MaxArrivals Task}.
(** Consider a set of tasks that respects the arrival curve. *)
Variable ts : list Task.
Hypothesis H_taskset_respects_max_arrivals : taskset_respects_max_arrivals arr_seq ts.
(** Assume that all jobs come from this task set. *)
Hypothesis H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts.
(** Consider a constant [L] such that... *)
Variable L : duration.
(** ... [L] is greater than [0], and... *)
Hypothesis H_L_positive : L > 0 .
(** [L] is the fixed point of the following equation. *)
Hypothesis H_fixed_point : L = total_request_bound_function ts L.
(** Assume all jobs have a valid job cost. *)
Hypothesis H_arrivals_have_valid_job_costs : arrivals_have_valid_job_costs arr_seq.
(** Then, we prove that [L] is a bound on the length of the busy window. *)
Lemma instantiated_busy_intervals_are_bounded :
busy_intervals_are_bounded_by arr_seq sched tsk L.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP H_policy_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP H3 : MaxArrivals Task ts : seq Task H_taskset_respects_max_arrivals : taskset_respects_max_arrivals
arr_seq ts H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts L : duration H_L_positive : 0 < LH_fixed_point : L = total_request_bound_function ts L H_arrivals_have_valid_job_costs : arrivals_have_valid_job_costs
arr_seq
busy_intervals_are_bounded_by arr_seq sched tsk L
Proof .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP H_policy_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP H3 : MaxArrivals Task ts : seq Task H_taskset_respects_max_arrivals : taskset_respects_max_arrivals
arr_seq ts H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts L : duration H_L_positive : 0 < LH_fixed_point : L = total_request_bound_function ts L H_arrivals_have_valid_job_costs : arrivals_have_valid_job_costs
arr_seq
busy_intervals_are_bounded_by arr_seq sched tsk L
move => j ARR TSK POS.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP H_policy_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP H3 : MaxArrivals Task ts : seq Task H_taskset_respects_max_arrivals : taskset_respects_max_arrivals
arr_seq ts H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts L : duration H_L_positive : 0 < LH_fixed_point : L = total_request_bound_function ts L H_arrivals_have_valid_job_costs : arrivals_have_valid_job_costs
arr_seq j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j POS : 0 < job_cost j
exists t1 t2 : nat,
t1 <= job_arrival j < t2 /\
t2 <= t1 + L /\
definitions.busy_interval sched j t1 t2
edestruct busy_interval_from_total_workload_bound
with (Δ := L) as [t1 [t2 [T1 [T2 GGG]]]] => //.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP H_policy_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP H3 : MaxArrivals Task ts : seq Task H_taskset_respects_max_arrivals : taskset_respects_max_arrivals
arr_seq ts H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts L : duration H_L_positive : 0 < LH_fixed_point : L = total_request_bound_function ts L H_arrivals_have_valid_job_costs : arrivals_have_valid_job_costs
arr_seq j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j POS : 0 < job_cost j
forall t : instant,
no_carry_in arr_seq sched t ->
blackout_during sched t (t + L) +
workload_of_jobs predT
(arrivals_between arr_seq t (t + L)) <= L
{ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP H_policy_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP H3 : MaxArrivals Task ts : seq Task H_taskset_respects_max_arrivals : taskset_respects_max_arrivals
arr_seq ts H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts L : duration H_L_positive : 0 < LH_fixed_point : L = total_request_bound_function ts L H_arrivals_have_valid_job_costs : arrivals_have_valid_job_costs
arr_seq j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j POS : 0 < job_cost j
forall t : instant,
no_carry_in arr_seq sched t ->
blackout_during sched t (t + L) +
workload_of_jobs predT
(arrivals_between arr_seq t (t + L)) <= L
move => t _; rewrite {3 }H_fixed_point.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP H_policy_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP H3 : MaxArrivals Task ts : seq Task H_taskset_respects_max_arrivals : taskset_respects_max_arrivals
arr_seq ts H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts L : duration H_L_positive : 0 < LH_fixed_point : L = total_request_bound_function ts L H_arrivals_have_valid_job_costs : arrivals_have_valid_job_costs
arr_seq j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j POS : 0 < job_cost jt : instant
blackout_during sched t (t + L) +
workload_of_jobs predT
(arrivals_between arr_seq t (t + L)) <=
total_request_bound_function ts L
have ->: blackout_during sched t (t + L) = 0 .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP H_policy_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP H3 : MaxArrivals Task ts : seq Task H_taskset_respects_max_arrivals : taskset_respects_max_arrivals
arr_seq ts H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts L : duration H_L_positive : 0 < LH_fixed_point : L = total_request_bound_function ts L H_arrivals_have_valid_job_costs : arrivals_have_valid_job_costs
arr_seq j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j POS : 0 < job_cost jt : instant
blackout_during sched t (t + L) = 0
{ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP H_policy_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP H3 : MaxArrivals Task ts : seq Task H_taskset_respects_max_arrivals : taskset_respects_max_arrivals
arr_seq ts H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts L : duration H_L_positive : 0 < LH_fixed_point : L = total_request_bound_function ts L H_arrivals_have_valid_job_costs : arrivals_have_valid_job_costs
arr_seq j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j POS : 0 < job_cost jt : instant
blackout_during sched t (t + L) = 0
apply /eqP; rewrite /blackout_during big1 // => l _.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP H_policy_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP H3 : MaxArrivals Task ts : seq Task H_taskset_respects_max_arrivals : taskset_respects_max_arrivals
arr_seq ts H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts L : duration H_L_positive : 0 < LH_fixed_point : L = total_request_bound_function ts L H_arrivals_have_valid_job_costs : arrivals_have_valid_job_costs
arr_seq j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j POS : 0 < job_cost jt : instant l : nat
is_blackout sched l = 0
by rewrite /is_blackout ideal_proc_has_supply. } Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP H_policy_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP H3 : MaxArrivals Task ts : seq Task H_taskset_respects_max_arrivals : taskset_respects_max_arrivals
arr_seq ts H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts L : duration H_L_positive : 0 < LH_fixed_point : L = total_request_bound_function ts L H_arrivals_have_valid_job_costs : arrivals_have_valid_job_costs
arr_seq j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j POS : 0 < job_cost jt : instant
0 +
workload_of_jobs predT
(arrivals_between arr_seq t (t + L)) <=
total_request_bound_function ts L
by rewrite add0n; apply total_workload_le_total_rbf. } Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP H_policy_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP H3 : MaxArrivals Task ts : seq Task H_taskset_respects_max_arrivals : taskset_respects_max_arrivals
arr_seq ts H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts L : duration H_L_positive : 0 < LH_fixed_point : L = total_request_bound_function ts L H_arrivals_have_valid_job_costs : arrivals_have_valid_job_costs
arr_seq j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j POS : 0 < job_cost jt1, t2 : nat T1 : t1 <= job_arrival j < t2 T2 : t2 <= t1 + L GGG : busy_interval arr_seq sched j t1 t2
exists t1 t2 : nat,
t1 <= job_arrival j < t2 /\
t2 <= t1 + L /\
definitions.busy_interval sched j t1 t2
exists t1 , t2; split => [//|]; split => [//|].Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (processor_state Job) H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute
sched H_completed_jobs_dont_execute : completed_jobs_dont_execute
sched JLFP : JLFP_policy Job H_priority_is_reflexive : reflexive_job_priorities
JLFP H_priority_is_transitive : transitive_job_priorities
JLFP H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP tsk : Task JobReady0 : JobReady Job (processor_state Job) H_work_bearing_readiness : work_bearing_readiness
arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop busy_interval_prefix_ab := definitions.busy_interval_prefix
sched : Job ->
instant -> instant -> Prop H_work_conserving : work_conserving_cl H_policy_reflexive : reflexive_job_priorities JLFP H_policy_respects_sequential_tasks : policy_respects_sequential_tasks
JLFP H3 : MaxArrivals Task ts : seq Task H_taskset_respects_max_arrivals : taskset_respects_max_arrivals
arr_seq ts H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts L : duration H_L_positive : 0 < LH_fixed_point : L = total_request_bound_function ts L H_arrivals_have_valid_job_costs : arrivals_have_valid_job_costs
arr_seq j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j POS : 0 < job_cost jt1, t2 : nat T1 : t1 <= job_arrival j < t2 T2 : t2 <= t1 + L GGG : busy_interval arr_seq sched j t1 t2
definitions.busy_interval sched j t1 t2
by apply instantiated_busy_interval_equivalent_busy_interval.
Qed .
End BusyWindowBound .
End I_IW_correctness .
End JLFPInstantiation .
(** To preserve modularity and hide the implementation details of a
technical definition presented in this file, we make the
definition opaque. This way, we ensure that the system will treat
each of these definitions as a single entity. *)
Global Opaque another_hep_job_interference
another_task_hep_job_interference
ideal_jlfp_interference
ideal_jlfp_interfering_workload
cumulative_another_hep_job_interference
cumulative_another_task_hep_job_interference
cumulative_other_hep_jobs_interfering_workload
other_hep_jobs_interfering_workload.
(** We add some facts into the "Hint Database" basic_rt_facts, so Coq will be
able to apply them automatically where needed. *)
Global Hint Resolve
abstract_busy_interval_classic_quiet_time
abstract_busy_interval_classic_busy_interval_prefix
: basic_rt_facts.