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Require Export prosa.analysis.abstract .ideal.abstract_seq_rta.Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]Notation "_ + _" was already used in scope nat_scope.
[notation-overridden,parsing]Notation "_ - _" was already used in scope nat_scope.
[notation-overridden,parsing]Notation "_ <= _" was already used in scope nat_scope.
[notation-overridden,parsing]Notation "_ < _" was already used in scope nat_scope.
[notation-overridden,parsing]Notation "_ >= _" was already used in scope nat_scope.
[notation-overridden,parsing]Notation "_ > _" was already used in scope nat_scope.
[notation-overridden,parsing]Notation "_ <= _ <= _" was already used in scope
nat_scope. [notation-overridden,parsing]Notation "_ < _ <= _" was already used in scope
nat_scope. [notation-overridden,parsing]Notation "_ <= _ < _" was already used in scope
nat_scope. [notation-overridden,parsing]Notation "_ < _ < _" was already used in scope
nat_scope. [notation-overridden,parsing]Notation "_ * _" was already used in scope nat_scope.
[notation-overridden,parsing]
Require Export prosa.analysis.definitions.priority_inversion.Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Require Export prosa.analysis.definitions.work_bearing_readiness.Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Require Export prosa.analysis.facts.busy_interval.carry_in.Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Require Export prosa.analysis.abstract .iw_auxiliary.Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Require Export prosa.analysis.facts.priority.classes.Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
(** Throughout this file, we assume ideal uni-processor schedules. *)
Require Import prosa.model.processor.ideal.Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Require Export prosa.analysis.facts.busy_interval.ideal.priority_inversion.Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
(** * 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_policy Job}.
Hypothesis H_priority_is_reflexive : reflexive_priorities.
Hypothesis H_priority_is_transitive : transitive_priorities.
(** Let [tsk] be any task. *)
Variable tsk : Task.
(** Assume we have sequential tasks, i.e., jobs of the same task
execute in the order of their arrival. *)
Hypothesis H_sequential_tasks : sequential_tasks arr_seq sched.
(** 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.
(** * Interference and Interfering Workload *)
(** In the following, we introduce definitions of interference,
interfering workload and a function that bounds cumulative
interference. *)
(** ** Instantiation of Interference *)
(** We say that job [j] is incurring interference from another
job with higher-or-equal priority at time [t] if there exists a
job [jhp] (different from [j]) with a higher-or-equal priority
that executes at time [t]. *)
Definition another_hep_job_interference (j : Job) (t : instant) :=
exists jhp ,
(jhp \in arrivals_up_to arr_seq t)
/\ another_hep_job jhp j
/\ receives_service_at sched jhp t.
(** In order to use the above definition in aRTA, we need to define
its computational version. *)
Definition another_hep_job_interference_dec (j : Job) (t : instant) :=
has (fun jhp => another_hep_job jhp j && receives_service_at sched jhp t) (arrivals_up_to arr_seq t).
(** Notice that the computational and propositional definitions are
equivalent; ... *)
Lemma another_hep_job_interference_P :
forall j t ,
reflect (another_hep_job_interference j t) (another_hep_job_interference_dec 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
forall (j : Job) (t : instant),
reflect (another_hep_job_interference j t)
(another_hep_job_interference_dec 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
forall (j : Job) (t : instant),
reflect (another_hep_job_interference j t)
(another_hep_job_interference_dec j t)
move => j t; apply /introP.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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant
another_hep_job_interference_dec j t ->
another_hep_job_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant
another_hep_job_interference_dec j t ->
another_hep_job_interference j t
by move => /hasP [jhp ARR /andP [HEP SCHED]]; (exists jhp ; 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant
~~ another_hep_job_interference_dec j t ->
~ another_hep_job_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant
~~ another_hep_job_interference_dec j t ->
~ another_hep_job_interference j t
move => /negP T1; move => [jhp [ARR [HEP 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant T1 : ~ another_hep_job_interference_dec j t jhp : Job ARR : jhp \in arrivals_up_to arr_seq t HEP : another_hep_job jhp j SCHED : receives_service_at sched jhp t
False
apply : T1; apply /hasP.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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant jhp : Job ARR : jhp \in arrivals_up_to arr_seq t HEP : another_hep_job jhp j SCHED : receives_service_at sched jhp t
exists2 x : Job,
x \in arrivals_up_to arr_seq t &
another_hep_job x j && receives_service_at sched x t
exists jhp ; first by done .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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant jhp : Job ARR : jhp \in arrivals_up_to arr_seq t HEP : another_hep_job jhp j SCHED : receives_service_at sched jhp t
another_hep_job jhp j &&
receives_service_at sched jhp t
by apply /andP; split .
Qed .
(** ... for convenience, we prove that their negated counterparts
are equivalent as well. *)
Lemma another_hep_job_interference_negP :
forall j t ,
reflect (~ another_hep_job_interference j t) (~~ another_hep_job_interference_dec 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
forall (j : Job) (t : instant),
reflect (~ another_hep_job_interference j t)
(~~ another_hep_job_interference_dec 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
forall (j : Job) (t : instant),
reflect (~ another_hep_job_interference j t)
(~~ another_hep_job_interference_dec j t)
move => j t; apply /introP.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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant
~~ another_hep_job_interference_dec j t ->
~ another_hep_job_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant
~~ another_hep_job_interference_dec j t ->
~ another_hep_job_interference j t
by move => /negP NDEC NPROP; apply : NDEC; apply /another_hep_job_interference_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant
~~ ~~ another_hep_job_interference_dec j t ->
~ ~ another_hep_job_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant
~~ ~~ another_hep_job_interference_dec j t ->
~ ~ another_hep_job_interference j t
move => /negPn/another_hep_job_interference_P; auto .
Qed .
(** Similarly, we say that job [j] is incurring interference from a
job with higher-or-equal priority of another task at time [t]
if there exists a job [jhp] (of a different task) with
higher-or-equal priority that executes at time [t]. *)
Definition another_task_hep_job_interference (j : Job) (t : instant) :=
exists jhp ,
(jhp \in arrivals_up_to arr_seq t)
/\ another_task_hep_job jhp j
/\ receives_service_at sched jhp t.
(** In order to use the above definition in aRTA, we need to define
its computational version. *)
Definition another_task_hep_job_interference_dec (j : Job) (t : instant) :=
has (fun jhp => another_task_hep_job jhp j && receives_service_at sched jhp t) (arrivals_up_to arr_seq t).
(** We also show that the computational and propositional
definitions are equivalent. *)
Lemma another_task_hep_job_interference_P :
forall j t ,
reflect (another_task_hep_job_interference j t) (another_task_hep_job_interference_dec 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
forall (j : Job) (t : instant),
reflect (another_task_hep_job_interference j t)
(another_task_hep_job_interference_dec 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
forall (j : Job) (t : instant),
reflect (another_task_hep_job_interference j t)
(another_task_hep_job_interference_dec j t)
move => j t; apply /introP.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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant
another_task_hep_job_interference_dec j t ->
another_task_hep_job_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant
another_task_hep_job_interference_dec j t ->
another_task_hep_job_interference j t
by move => /hasP [jhp ARR /andP [HEP SCHED]]; (exists jhp ; 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant
~~ another_task_hep_job_interference_dec j t ->
~ another_task_hep_job_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant
~~ another_task_hep_job_interference_dec j t ->
~ another_task_hep_job_interference j t
move => /negP T1; move => [jhp [ARR [HEP 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant T1 : ~ another_task_hep_job_interference_dec j t jhp : Job ARR : jhp \in arrivals_up_to arr_seq t HEP : another_task_hep_job jhp j SCHED : receives_service_at sched jhp t
False
apply : T1; apply /hasP.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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant jhp : Job ARR : jhp \in arrivals_up_to arr_seq t HEP : another_task_hep_job jhp j SCHED : receives_service_at sched jhp t
exists2 x : Job,
x \in arrivals_up_to arr_seq t &
another_task_hep_job x j &&
receives_service_at sched x t
exists jhp ; first by done .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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant jhp : Job ARR : jhp \in arrivals_up_to arr_seq t HEP : another_task_hep_job jhp j SCHED : receives_service_at sched jhp t
another_task_hep_job jhp j &&
receives_service_at sched jhp t
by apply /andP; split .
Qed .
(** 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_dec arr_seq sched j t || another_hep_job_interference_dec j t
}.
(** ** Instantiation of Interfering Workload *)
(** Now, we define the notion of cumulative interfering workload,
called [other_hep_jobs_interfering_workload], that says how many
units of workload are generated by jobs with higher-or-equal
priority released at time [t]. *)
Definition other_hep_jobs_interfering_workload (j : Job) (t : instant) :=
\sum_(jhp <- arrivals_at arr_seq t | another_hep_job jhp j) job_cost jhp.
(** 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_dec arr_seq sched j t + other_hep_jobs_interfering_workload j t
}.
(** ** Auxiliary definitions *)
(** For each of the concepts defined above, we introduce a
corresponding cumulative function: *)
(** (a) cumulative interference from other jobs with higher-or-equal priority ... *)
Definition cumulative_another_hep_job_interference (j : Job) (t1 t2 : instant) :=
\sum_(t1 <= t < t2) another_hep_job_interference_dec j t.
(** ... (b) and cumulative interference from jobs with higher or
equal priority from other tasks, ... *)
Definition cumulative_another_task_hep_job_interference (j : Job) (t1 t2 : instant) :=
\sum_(t1 <= t < t2) another_task_hep_job_interference_dec j t.
(** ... and (c) cumulative workload from jobs with higher or equal priority. *)
Definition cumulative_other_hep_jobs_interfering_workload (j : Job) (t1 t2 : instant) :=
\sum_(t1 <= t < t2) other_hep_jobs_interfering_workload j t.
(** Instantiated functions usually do not come with any useful lemmas
about them. In order to reuse existing lemmas, we need to prove
equivalence of the instantiated functions to some conventional
notions. The instantiations given in this file are equivalent to
service and workload. Further, we prove these equivalences
formally. *)
(** Before we present the formal proofs of the equivalences, we
recall the notion of workload of higher or equal priority
jobs. *)
Definition workload_of_another_hep_jobs (j : Job) (t1 t2 : instant) :=
workload_of_jobs (fun jhp => another_hep_job jhp j) (arrivals_between arr_seq t1 t2).
(** ... and service of all other jobs with higher or equal priority. *)
Definition service_of_another_hep_jobs (j : Job) (t1 t2 : instant) :=
service_of_jobs sched (fun jhp => another_hep_job jhp j) (arrivals_between arr_seq t1 t2) t1 t2.
(** Similarly, we recall notions of service of higher or equal
priority jobs from other tasks. *)
Definition service_of_another_task_hep_job (j : Job) (t1 t2 : instant) :=
service_of_jobs sched (fun jhp => another_task_hep_job jhp j) (arrivals_between arr_seq t1 t2) t1 t2.
(** ** 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.
As it was mentioned previously, 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. *)
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 : is_idle sched t.
(** We prove that in this case: ... *)
(** ... there is no interference from higher-or-equal priority
jobs, ... *)
Lemma idle_implies_no_hep_job_interference :
forall j , ~ another_hep_job_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks t : instant H_idle : is_idle sched t
forall j : Job, ~ another_hep_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks t : instant H_idle : is_idle sched t
forall j : Job, ~ another_hep_job_interference j t
move => j [j' [ _ [ _ SERV]]].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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks t : instant H_idle : is_idle sched t j, j' : Job SERV : receives_service_at sched j' t
False
by rewrite /receives_service_at ideal_not_idle_implies_sched in SERV.
Qed .
(** ... there is no interference from higher-or-equal priority
jobs from another task, ... *)
Lemma idle_implies_no_hep_task_interference :
forall j , ~ another_task_hep_job_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks t : instant H_idle : is_idle sched t
forall j : Job,
~ another_task_hep_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks t : instant H_idle : is_idle sched t
forall j : Job,
~ another_task_hep_job_interference j t
move => j [j' [ _ [ _ SERV]]].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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks t : instant H_idle : is_idle sched t j, j' : Job SERV : receives_service_at sched j' t
False
by rewrite /receives_service_at ideal_not_idle_implies_sched in SERV.
Qed .
(** ... there is no interference, ... *)
Lemma idle_implies_no_interference :
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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks t : instant H_idle : 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks t : instant H_idle : is_idle sched t
forall j : Job, ~ interference j t
move => 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks t : instant H_idle : 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks t : instant H_idle : is_idle sched t j : Job PI : priority_inversion_dec 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks t : instant H_idle : is_idle sched t j : Job PI : priority_inversion_dec arr_seq sched j t
False
move : PI; move => /priority_inversion_P PI; (feed_n 3 PI; rt_auto).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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks t : instant H_idle : is_idle sched t j : Job PI : priority_inversion sched j t
False
move : PI => [_ [j' /andP [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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks t : instant H_idle : is_idle sched t j, j' : Job SCHED : scheduled_at sched j' t
False
by apply ideal_sched_implies_not_idle in 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks t : instant H_idle : is_idle sched t j : Job HEPI : another_hep_job_interference_dec 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks t : instant H_idle : is_idle sched t j : Job HEPI : another_hep_job_interference_dec j t
False
move : HEPI => /another_hep_job_interference_P 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks t : instant H_idle : is_idle sched t j : Job HEPI : another_hep_job_interference j t
False
by apply idle_implies_no_hep_job_interference in HEPI.
Qed .
(** ... as well as no interference for [tsk]. Recall that the
additional argument [upper_bound] is an artificial horizon
needed needed to make task interference function
constructive. For more details, refer to the original
description of the function. *)
Lemma idle_implies_no_task_interference :
forall upper_bound , ~ task_interference_received_before arr_seq sched tsk upper_bound 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks t : instant H_idle : is_idle sched t
forall upper_bound : instant,
~
task_interference_received_before arr_seq sched tsk
upper_bound 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks t : instant H_idle : is_idle sched t
forall upper_bound : instant,
~
task_interference_received_before arr_seq sched tsk
upper_bound t
move => upp [NSCHEDT [j' [INT TSKBE]]].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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks t : instant H_idle : is_idle sched t upp : instant NSCHEDT : ~ task_scheduled_at sched tsk t j' : Job INT : interference j' t TSKBE : j' \in task_arrivals_before arr_seq tsk upp
False
by apply idle_implies_no_interference in INT.
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.
(** First, consider a case when _some_ jobs is scheduled at time [t]. *)
Section SomeJobIsScheduled .
(** Consider a job [j'] (not necessarily distinct from job
[j]) that is scheduled at time [t]. *)
Variable j' : Job.
Hypothesis H_sched : scheduled_at sched j' t.
(** Under the stated assumptions, we show that the
interference from another higher-or-equal priority job is
equivalent to the relation [another_hep_job]. *)
Lemma interference_ahep_equiv_ahep :
another_hep_job_interference j t <-> another_hep_job 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_tsk : job_of_task tsk j t : instant j' : Job H_sched : scheduled_at sched j' t
another_hep_job_interference j t <->
another_hep_job j' j
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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_tsk : job_of_task tsk j t : instant j' : Job H_sched : scheduled_at sched j' t
another_hep_job_interference j t <->
another_hep_job j' j
split ; [move => [jhp [IN [AHEP PSERV]]] | move => AHEP].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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_tsk : job_of_task tsk j t : instant j' : Job H_sched : scheduled_at sched j' t jhp : Job IN : jhp \in arrivals_up_to arr_seq t AHEP : another_hep_job jhp j PSERV : receives_service_at sched jhp t
another_hep_job 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_tsk : job_of_task tsk j t : instant j' : Job H_sched : scheduled_at sched j' t jhp : Job IN : jhp \in arrivals_up_to arr_seq t AHEP : another_hep_job jhp j PSERV : receives_service_at sched jhp t
another_hep_job j' j
apply service_at_implies_scheduled_at in PSERV.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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_tsk : job_of_task tsk j t : instant j' : Job H_sched : scheduled_at sched j' t jhp : Job IN : jhp \in arrivals_up_to arr_seq t AHEP : another_hep_job jhp j PSERV : scheduled_at sched jhp t
another_hep_job j' j
by have -> := ideal_proc_model_is_a_uniprocessor_model _ _ _ _ H_sched PSERV.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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_tsk : job_of_task tsk j t : instant j' : Job H_sched : scheduled_at sched j' t AHEP : another_hep_job j' j
another_hep_job_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_tsk : job_of_task tsk j t : instant j' : Job H_sched : scheduled_at sched j' t AHEP : another_hep_job j' j
another_hep_job_interference j t
exists j' ; repeat 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_tsk : job_of_task tsk j t : instant j' : Job H_sched : scheduled_at sched j' t AHEP : another_hep_job j' j
j' \in arrivals_up_to arr_seq 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_tsk : job_of_task tsk j t : instant j' : Job H_sched : scheduled_at sched j' t AHEP : another_hep_job j' j
j' \in arrivals_up_to arr_seq t
apply arrived_between_implies_in_arrivals; rt_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_tsk : job_of_task tsk j t : instant j' : Job H_sched : scheduled_at sched j' t AHEP : another_hep_job j' j
arrived_between j' 0 t.+1
apply /andP; split ; first by done .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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_tsk : job_of_task tsk j t : instant j' : Job H_sched : scheduled_at sched j' t AHEP : another_hep_job j' j
job_arrival j' < t.+1
by apply H_jobs_must_arrive_to_execute in H_sched; rewrite ltnS.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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_tsk : job_of_task tsk j t : instant j' : Job H_sched : scheduled_at sched j' t AHEP : another_hep_job j' j
another_hep_job 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_tsk : job_of_task tsk j t : instant j' : Job H_sched : scheduled_at sched j' t AHEP : another_hep_job j' j
another_hep_job j' j
by done .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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_tsk : job_of_task tsk j t : instant j' : Job H_sched : scheduled_at sched j' t AHEP : another_hep_job j' 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_tsk : job_of_task tsk j t : instant j' : Job H_sched : scheduled_at sched j' t AHEP : another_hep_job j' j
receives_service_at sched j' t
by rewrite /receives_service_at service_at_is_scheduled_at H_sched.
Qed .
End SomeJobIsScheduled .
(** Next, consider a case when [j] itself is scheduled at [t]. *)
Section JIsScheduled .
(** Assume that [j] is scheduled at time [t]. *)
Hypothesis H_j_sched : scheduled_at sched j t.
(** Then there is no interference from higher-or-equal
priority jobs at time [t]. *)
Lemma interference_ahep_job_eq_false :
~ another_hep_job_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_tsk : job_of_task tsk j t : instant H_j_sched : scheduled_at sched j t
~ another_hep_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_tsk : job_of_task tsk j t : instant H_j_sched : scheduled_at sched j t
~ another_hep_job_interference j t
move => [jhp [IN [AHEP PSERV]]].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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_tsk : job_of_task tsk j t : instant H_j_sched : scheduled_at sched j t jhp : Job IN : jhp \in arrivals_up_to arr_seq t AHEP : another_hep_job jhp j PSERV : receives_service_at sched jhp t
False
apply service_at_implies_scheduled_at in PSERV.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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_tsk : job_of_task tsk j t : instant H_j_sched : scheduled_at sched j t jhp : Job IN : jhp \in arrivals_up_to arr_seq t AHEP : another_hep_job jhp j PSERV : scheduled_at sched jhp t
False
have EQ := ideal_proc_model_is_a_uniprocessor_model _ _ _ _ H_j_sched PSERV; subst jhp.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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_tsk : job_of_task tsk j t : instant H_j_sched, PSERV : scheduled_at sched j t AHEP : another_hep_job j j IN : j \in arrivals_up_to arr_seq t
False
by apply another_hep_job_antireflexive in AHEP.
Qed .
End JIsScheduled .
(** 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.
(** Then we show that interference from higher-or-equal
priority jobs from another task is false. *)
Lemma interference_athep_eq_false :
~ another_task_hep_job_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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
~ another_task_hep_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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
~ another_task_hep_job_interference j t
move => [jhp [IN [AHEP PSERV]]].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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 jhp : Job IN : jhp \in arrivals_up_to arr_seq t AHEP : another_task_hep_job jhp j PSERV : receives_service_at sched jhp t
False
apply service_at_implies_scheduled_at in PSERV.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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 jhp : Job IN : jhp \in arrivals_up_to arr_seq t AHEP : another_task_hep_job jhp j PSERV : scheduled_at sched jhp t
False
have EQ := ideal_proc_model_is_a_uniprocessor_model _ _ _ _ H_j'_sched PSERV; subst jhp.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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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, PSERV : scheduled_at sched j' t AHEP : another_task_hep_job j' j IN : j' \in arrivals_up_to arr_seq t
False
by eapply another_task_hep_job_taskwise_antireflexive in AHEP; rt_eauto.
Qed .
(** 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 :
forall upper_bound , ~ task_interference_received_before arr_seq sched tsk upper_bound 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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
forall upper_bound : instant,
~
task_interference_received_before arr_seq sched tsk
upper_bound 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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
forall upper_bound : instant,
~
task_interference_received_before arr_seq sched tsk
upper_bound t
move => upp [TNSCHED [j'' [INT ARR]]].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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 upp : instant TNSCHED : ~ task_scheduled_at sched tsk t j'' : Job INT : interference j'' t ARR : j'' \in task_arrivals_before arr_seq tsk upp
False
unfold interference, ideal_jlfp_interference 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 upp : instant TNSCHED : ~ task_scheduled_at sched tsk t j'' : Job INT : priority_inversion_dec arr_seq sched j'' t
|| another_hep_job_interference_dec j'' t ARR : j'' \in task_arrivals_before arr_seq tsk upp
False
apply :TNSCHED; rewrite /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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 upp : instant j'' : Job INT : priority_inversion_dec arr_seq sched j'' t
|| another_hep_job_interference_dec j'' t ARR : j'' \in task_arrivals_before arr_seq tsk upp
match sched t with
| Some j => job_task j == tsk
| None => false
end
by move : (H_j'_sched); rewrite scheduled_at_def => /eqP->.
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.
(** We prove that then [j] incurs higher-or-equal priority
interference from another task iff [j'] has
higher-or-equal priority than [j]. *)
Lemma sched_at_implies_interference_athep_eq_hep :
another_task_hep_job_interference j t <-> hep_job 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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
another_task_hep_job_interference j t <-> hep_job j' j
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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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
another_task_hep_job_interference j t <-> hep_job j' j
split ; [move => [j'' [IN [/andP [AHEP FF] RSERV]]] | move => HEP].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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 j'' : Job IN : j'' \in arrivals_up_to arr_seq t AHEP : hep_job j'' j FF : job_task j'' != job_task j RSERV : receives_service_at sched j'' t
hep_job 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 j'' : Job IN : j'' \in arrivals_up_to arr_seq t AHEP : hep_job j'' j FF : job_task j'' != job_task j RSERV : receives_service_at sched j'' t
hep_job j' j
apply service_at_implies_scheduled_at 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 j'' : Job IN : j'' \in arrivals_up_to arr_seq t AHEP : hep_job j'' j FF : job_task j'' != job_task j RSERV : scheduled_at sched j'' t
hep_job j' j
by have EQ := ideal_proc_model_is_a_uniprocessor_model _ _ _ _ H_j'_sched RSERV; subst 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 HEP : hep_job j' j
another_task_hep_job_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 HEP : hep_job j' j
another_task_hep_job_interference j t
exists j' ; repeat 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 HEP : hep_job j' j
j' \in arrivals_up_to arr_seq 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 HEP : hep_job j' j
j' \in arrivals_up_to arr_seq t
apply arrived_between_implies_in_arrivals; rt_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 HEP : hep_job j' j
arrived_between j' 0 t.+1
apply /andP; split ; first by done .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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 HEP : hep_job j' j
job_arrival j' < t.+1
by apply H_jobs_must_arrive_to_execute in H_j'_sched; rewrite ltnS.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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 HEP : hep_job j' j
another_task_hep_job 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 HEP : hep_job j' j
another_task_hep_job j' j
apply /andP; split ; first by done .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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 HEP : hep_job j' j
job_task j' != job_task j
apply /negP => /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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 HEP : hep_job j' j EQ : job_task j' = job_task j
False
by move : (H_j'_not_tsk) => /negP T; apply : T; rewrite /job_of_task 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 HEP : hep_job j' 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 HEP : hep_job j' j
receives_service_at sched j' t
by rewrite /receives_service_at service_at_is_scheduled_at H_j'_sched.
Qed .
(** Hence, if we assume that [j'] has higher-or-equal priority, ... *)
Hypothesis H_j'_hep : hep_job j' j.
(** ... we are able to show that [j] incurs higher-or-equal
priority interference from another task. *)
Lemma sched_athep_implies_interference_athep :
another_task_hep_job_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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_task_hep_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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_task_hep_job_interference j t
by apply sched_at_implies_interference_athep_eq_hep.
Qed .
(** Moreover, in this case, task [tsk] also incurs interference. *)
Lemma sched_athep_implies_task_interference :
forall upper_bound ,
(j \in arrivals_between arr_seq 0 upper_bound) ->
task_interference_received_before arr_seq sched tsk upper_bound 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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
forall upper_bound : instant,
j \in arrivals_between arr_seq 0 upper_bound ->
task_interference_received_before arr_seq sched tsk
upper_bound 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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
forall upper_bound : instant,
j \in arrivals_between arr_seq 0 upper_bound ->
task_interference_received_before arr_seq sched tsk
upper_bound t
move => upp 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 upp : instant IN : j \in arrivals_between arr_seq 0 upp
task_interference_received_before arr_seq sched tsk
upp t
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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 upp : instant IN : j \in arrivals_between arr_seq 0 upp
~ task_scheduled_at sched tsk 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 upp : instant IN : j \in arrivals_between arr_seq 0 upp
~ task_scheduled_at sched tsk t
by move : (H_j'_sched); rewrite /task_scheduled_at scheduled_at_def => /eqP ->; 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 upp : instant IN : j \in arrivals_between arr_seq 0 upp
exists j : Job,
interference j t /\
j \in task_arrivals_before arr_seq tsk upp
- 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 upp : instant IN : j \in arrivals_between arr_seq 0 upp
exists j : Job,
interference j t /\
j \in task_arrivals_before arr_seq tsk upp
exists j ; 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 upp : instant IN : j \in arrivals_between arr_seq 0 upp
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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 upp : instant IN : j \in arrivals_between arr_seq 0 upp
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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 upp : instant IN : j \in arrivals_between arr_seq 0 upp
another_hep_job_interference_dec j t
apply /another_hep_job_interference_P; exists j' ; repeat 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 upp : instant IN : j \in arrivals_between arr_seq 0 upp
j' \in arrivals_up_to arr_seq 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 upp : instant IN : j \in arrivals_between arr_seq 0 upp
j' \in arrivals_up_to arr_seq t
apply arrived_between_implies_in_arrivals; rt_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 upp : instant IN : j \in arrivals_between arr_seq 0 upp
arrived_between j' 0 t.+1
apply /andP; split ; first by done .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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 upp : instant IN : j \in arrivals_between arr_seq 0 upp
job_arrival j' < t.+1
by apply H_jobs_must_arrive_to_execute in H_j'_sched; rewrite ltnS.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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 upp : instant IN : j \in arrivals_between arr_seq 0 upp
another_hep_job 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 upp : instant IN : j \in arrivals_between arr_seq 0 upp
another_hep_job j' j
apply /andP; split ; first by done .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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 upp : instant IN : j \in arrivals_between arr_seq 0 upp
j' != j
apply /negP; move => /eqP EQ; 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_tsk : job_of_task tsk j t : instant H_j'_hep : hep_job j j H_j'_sched : scheduled_at sched j t H_j'_not_tsk : ~~ job_of_task tsk j upp : instant IN : j \in arrivals_between arr_seq 0 upp
False
by move : (H_j'_not_tsk); rewrite H_j_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 upp : instant IN : j \in arrivals_between arr_seq 0 upp
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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 upp : instant IN : j \in arrivals_between arr_seq 0 upp
receives_service_at sched j' t
by rewrite /receives_service_at service_at_is_scheduled_at lt0b.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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 upp : instant IN : j \in arrivals_between arr_seq 0 upp
j \in task_arrivals_before arr_seq tsk upp
+ 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 upp : instant IN : j \in arrivals_between arr_seq 0 upp
j \in task_arrivals_before arr_seq tsk upp
rewrite mem_filter; apply /andP; split ; last by done .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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 upp : instant IN : j \in arrivals_between arr_seq 0 upp
job_of_task tsk j
by rewrite H_j_tsk.
Qed .
End FromDifferentTask .
(** In the last subsection, we consider a case when the
scheduled job [j'] has lower priority than job [j]. *)
Section LowerPriority .
(** Consider a job [j'] that has lower priority than job [j]
and is scheduled at time instant [t]. *)
Variable j' : Job.
Hypothesis H_j'_sched : scheduled_at sched j' t.
Hypothesis H_j'_lp : ~~ hep_job j' j.
Lemma sched_alp_implies_interference_ahep_false :
~ another_hep_job_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_tsk : job_of_task tsk j t : instant j' : Job H_j'_sched : scheduled_at sched j' t H_j'_lp : ~~ hep_job j' j
~ another_hep_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_tsk : job_of_task tsk j t : instant j' : Job H_j'_sched : scheduled_at sched j' t H_j'_lp : ~~ hep_job j' j
~ another_hep_job_interference j t
move => [jlp [IN [AHEP 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_tsk : job_of_task tsk j t : instant j' : Job H_j'_sched : scheduled_at sched j' t H_j'_lp : ~~ hep_job j' j jlp : Job IN : jlp \in arrivals_up_to arr_seq t AHEP : another_hep_job jlp j RSERV : receives_service_at sched jlp t
False
apply service_at_implies_scheduled_at 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_tsk : job_of_task tsk j t : instant j' : Job H_j'_sched : scheduled_at sched j' t H_j'_lp : ~~ hep_job j' j jlp : Job IN : jlp \in arrivals_up_to arr_seq t AHEP : another_hep_job jlp j RSERV : scheduled_at sched jlp t
False
have EQ := ideal_proc_model_is_a_uniprocessor_model _ _ _ _ H_j'_sched RSERV; subst 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_tsk : job_of_task tsk j t : instant jlp : Job H_j'_lp : ~~ hep_job jlp j H_j'_sched : scheduled_at sched jlp t IN : jlp \in arrivals_up_to arr_seq t AHEP : another_hep_job jlp j RSERV : scheduled_at sched jlp t
False
by move : (H_j'_lp) AHEP => LP /andP [HEP A]; rewrite HEP in LP.
Qed .
End LowerPriority .
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 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
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 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
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 j t1 t2
rewrite /cumulative_interference /interference => 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1, t2 : nat
\sum_(t1 <= t < t2) ideal_jlfp_interference j t =
\sum_(t1 <= i < t2)
(priority_inversion_dec arr_seq sched j i +
another_hep_job_interference_dec j i)
apply /eqP; rewrite eqn_leq; apply /andP; split ; rewrite leq_sum; try done .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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1, t2 : nat
forall i : nat,
true ->
ideal_jlfp_interference j i <=
priority_inversion_dec arr_seq sched j i +
another_hep_job_interference_dec 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1, t2 : nat
forall i : nat,
true ->
ideal_jlfp_interference j i <=
priority_inversion_dec arr_seq sched j i +
another_hep_job_interference_dec 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1, t2, t : nat
priority_inversion_dec arr_seq sched j t
|| another_hep_job_interference_dec j t <=
priority_inversion_dec arr_seq sched j t +
another_hep_job_interference_dec j t
by destruct (priority_inversion_dec _ _ _ _), (another_hep_job_interference_dec 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1, t2 : nat
forall i : nat,
true ->
priority_inversion_dec arr_seq sched j i +
another_hep_job_interference_dec 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1, t2 : nat
forall i : nat,
true ->
priority_inversion_dec arr_seq sched j i +
another_hep_job_interference_dec 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1, t2, t : nat
priority_inversion_dec arr_seq sched j t +
another_hep_job_interference_dec j t <=
priority_inversion_dec arr_seq sched j t
|| another_hep_job_interference_dec 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1, t2, t : nat IDLE : is_idle sched t
priority_inversion_dec arr_seq sched j t +
another_hep_job_interference_dec j t <=
priority_inversion_dec arr_seq sched j t
|| another_hep_job_interference_dec 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1, t2, t : nat IDLE : is_idle sched t
priority_inversion_dec arr_seq sched j t +
another_hep_job_interference_dec j t <=
priority_inversion_dec arr_seq sched j t
|| another_hep_job_interference_dec j t
by rewrite idle_implies_no_priority_inversion // 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1, t2, t : nat s : Job SCHED : scheduled_at sched s t
priority_inversion_dec arr_seq sched j t +
another_hep_job_interference_dec j t <=
priority_inversion_dec arr_seq sched j t
|| another_hep_job_interference_dec 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1, t2, t : nat s : Job SCHED : scheduled_at sched s t
priority_inversion_dec arr_seq sched j t +
another_hep_job_interference_dec j t <=
priority_inversion_dec arr_seq sched j t
|| another_hep_job_interference_dec 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1, t2, t : nat s : Job SCHED : scheduled_at sched s t PRIO : hep_job s j = true
priority_inversion_dec arr_seq sched j t +
another_hep_job_interference_dec j t <=
priority_inversion_dec arr_seq sched j t
|| another_hep_job_interference_dec 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1, t2, t : nat s : Job SCHED : scheduled_at sched s t PRIO : hep_job s j = true
priority_inversion_dec arr_seq sched j t +
another_hep_job_interference_dec j t <=
priority_inversion_dec arr_seq sched j t
|| another_hep_job_interference_dec j t
by erewrite sched_hep_implies_no_priority_inversion; rt_eauto; 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1, t2, t : nat s : Job SCHED : scheduled_at sched s t PRIO : hep_job s j = false
priority_inversion_dec arr_seq sched j t +
another_hep_job_interference_dec j t <=
priority_inversion_dec arr_seq sched j t
|| another_hep_job_interference_dec 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1, t2, t : nat s : Job SCHED : scheduled_at sched s t PRIO : hep_job s j = false
priority_inversion_dec arr_seq sched j t +
another_hep_job_interference_dec j t <=
priority_inversion_dec arr_seq sched j t
|| another_hep_job_interference_dec j t
erewrite !sched_lp_implies_priority_inversion; rt_eauto; 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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_dec j t <=
true || another_hep_job_interference_dec 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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_dec j t <= true
destruct (another_hep_job_interference_dec j t) eqn :IAHEP; [exfalso | by done ].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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1, t2, t : nat s : Job SCHED : scheduled_at sched s t PRIO : hep_job s j = false IAHEP : another_hep_job_interference_dec j t = true
False
move : IAHEP => /another_hep_job_interference_P IAHEP.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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1, t2, t : nat s : Job SCHED : scheduled_at sched s t PRIO : hep_job s j = false IAHEP : another_hep_job_interference j t
False
eapply sched_alp_implies_interference_ahep_false; rt_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1, t2, t : nat s : Job SCHED : scheduled_at sched s t PRIO : hep_job s j = false IAHEP : another_hep_job_interference j t
~~ hep_job s j
by rewrite 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 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
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 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
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 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
forall (j : Job) (t1 t2 : nat),
\sum_(t1 <= t < t2) interfering_workload j t =
\sum_(t1 <= t < t2)
priority_inversion_dec arr_seq sched j t +
\sum_(t1 <= t < t2)
other_hep_jobs_interfering_workload j t
by move => j t1 t2; rewrite -big_split //=.
Qed .
(** Before we prove a lemma about the task's interference split,
we show that any job [j] of task [tsk] experiences _either_
priority inversion or task interference if two properties are
satisfied: (1) task [tsk] is not scheduled at a time instant
[t] and (2) there is a job [jo] that experiences interference
at a time [t]. *)
Remark priority_inversion_xor_atask_hep_job_interference :
forall j t ,
job_of_task tsk j ->
~ task_scheduled_at sched tsk t ->
forall jo ,
interference jo t ->
(~~ priority_inversion_dec arr_seq sched j t && another_task_hep_job_interference_dec j t)
|| (priority_inversion_dec arr_seq sched j t && ~~ another_task_hep_job_interference_dec 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
forall (j : Job) (t : instant),
job_of_task tsk j ->
~ task_scheduled_at sched tsk t ->
forall jo : Job,
interference jo t ->
~~ priority_inversion_dec arr_seq sched j t &&
another_task_hep_job_interference_dec j t
|| priority_inversion_dec arr_seq sched j t &&
~~ another_task_hep_job_interference_dec 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
forall (j : Job) (t : instant),
job_of_task tsk j ->
~ task_scheduled_at sched tsk t ->
forall jo : Job,
interference jo t ->
~~ priority_inversion_dec arr_seq sched j t &&
another_task_hep_job_interference_dec j t
|| priority_inversion_dec arr_seq sched j t &&
~~ another_task_hep_job_interference_dec j t
move => j t TSK TNSCHED jo 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant TSK : job_of_task tsk j TNSCHED : ~ task_scheduled_at sched tsk t jo : Job INT : interference jo t
~~ priority_inversion_dec arr_seq sched j t &&
another_task_hep_job_interference_dec j t
|| priority_inversion_dec arr_seq sched j t &&
~~ another_task_hep_job_interference_dec j t
destruct priority_inversion_dec eqn :PI; simpl .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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant TSK : job_of_task tsk j TNSCHED : ~ task_scheduled_at sched tsk t jo : Job INT : interference jo t PI : priority_inversion_dec arr_seq sched j t = true
~~ another_task_hep_job_interference_dec 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant TSK : job_of_task tsk j TNSCHED : ~ task_scheduled_at sched tsk t jo : Job INT : interference jo t PI : priority_inversion_dec arr_seq sched j t = true
~~ another_task_hep_job_interference_dec j t
move : PI => /priority_inversion_negP PI; feed_n 3 PI; rt_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant TSK : job_of_task tsk j TNSCHED : ~ task_scheduled_at sched tsk t jo : Job INT : interference jo t PI : ~ ~ priority_inversion sched j t
~~ another_task_hep_job_interference_dec j t
apply /negP; move => /another_task_hep_job_interference_P [jhp [IN__jhp [/andP [ATHEP__hp BB] RS__jhp]]].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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant TSK : job_of_task tsk j TNSCHED : ~ task_scheduled_at sched tsk t jo : Job INT : interference jo t PI : ~ ~ priority_inversion sched j t jhp : Job IN__jhp : jhp \in arrivals_up_to arr_seq t ATHEP__hp : hep_job jhp j BB : job_task jhp != job_task j RS__jhp : receives_service_at sched jhp t
False
apply : PI; move => [_ [jlp /andP [SCHED__jlp LEP__jlp]]].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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant TSK : job_of_task tsk j TNSCHED : ~ task_scheduled_at sched tsk t jo : Job INT : interference jo t jhp : Job IN__jhp : jhp \in arrivals_up_to arr_seq t ATHEP__hp : hep_job jhp j BB : job_task jhp != job_task j RS__jhp : receives_service_at sched jhp t jlp : Job SCHED__jlp : scheduled_at sched jlp t LEP__jlp : ~~ hep_job jlp j
False
enough (EQ: jlp = jhp); first by (subst ; rewrite ATHEP__hp in LEP__jlp).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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant TSK : job_of_task tsk j TNSCHED : ~ task_scheduled_at sched tsk t jo : Job INT : interference jo t jhp : Job IN__jhp : jhp \in arrivals_up_to arr_seq t ATHEP__hp : hep_job jhp j BB : job_task jhp != job_task j RS__jhp : receives_service_at sched jhp t jlp : Job SCHED__jlp : scheduled_at sched jlp t LEP__jlp : ~~ hep_job jlp j
jlp = jhp
eapply ideal_proc_model_is_a_uniprocessor_model; rt_eauto; move : RS__jhp.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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant TSK : job_of_task tsk j TNSCHED : ~ task_scheduled_at sched tsk t jo : Job INT : interference jo t jhp : Job IN__jhp : jhp \in arrivals_up_to arr_seq t ATHEP__hp : hep_job jhp j BB : job_task jhp != job_task j jlp : Job SCHED__jlp : scheduled_at sched jlp t LEP__jlp : ~~ hep_job jlp j
receives_service_at sched jhp t ->
scheduled_at sched jhp t
by rewrite /receives_service_at service_at_is_scheduled_at lt0b.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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant TSK : job_of_task tsk j TNSCHED : ~ task_scheduled_at sched tsk t jo : Job INT : interference jo t PI : priority_inversion_dec arr_seq sched j t = false
another_task_hep_job_interference_dec 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant TSK : job_of_task tsk j TNSCHED : ~ task_scheduled_at sched tsk t jo : Job INT : interference jo t PI : priority_inversion_dec arr_seq sched j t = false
another_task_hep_job_interference_dec j t || false
destruct another_task_hep_job_interference_dec eqn :IATHEP; try done ; exfalso .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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant TSK : job_of_task tsk j TNSCHED : ~ task_scheduled_at sched tsk t jo : Job INT : interference jo t PI : priority_inversion_dec arr_seq sched j t = false IATHEP : another_task_hep_job_interference_dec j t =
false
False
have L1: interference jo t -> exists jt , scheduled_at sched jt 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant TSK : job_of_task tsk j TNSCHED : ~ task_scheduled_at sched tsk t jo : Job INT : interference jo t PI : priority_inversion_dec arr_seq sched j t = false IATHEP : another_task_hep_job_interference_dec j t =
false
interference jo t ->
exists jt : Job, scheduled_at sched jt 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant TSK : job_of_task tsk j TNSCHED : ~ task_scheduled_at sched tsk t jo : Job INT : interference jo t PI : priority_inversion_dec arr_seq sched j t = false IATHEP : another_task_hep_job_interference_dec j t =
false
interference jo t ->
exists jt : Job, scheduled_at sched jt t
clear PI IATHEP; move => /orP [/priority_inversion_P PI| ].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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant TSK : job_of_task tsk j TNSCHED : ~ task_scheduled_at sched tsk t jo : Job INT : interference jo t PI : jobs_come_from_arrival_sequence sched arr_seq ->
jobs_must_arrive_to_execute sched ->
consistent_arrival_times arr_seq ->
priority_inversion sched jo t
exists jt : Job, scheduled_at sched jt 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant TSK : job_of_task tsk j TNSCHED : ~ task_scheduled_at sched tsk t jo : Job INT : interference jo t PI : jobs_come_from_arrival_sequence sched arr_seq ->
jobs_must_arrive_to_execute sched ->
consistent_arrival_times arr_seq ->
priority_inversion sched jo t
exists jt : Job, scheduled_at sched jt t
by feed_n 3 PI; rt_auto; move : PI => [NSCHED [jt /andP [SCHED _]]]; exists jt . } 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant TSK : job_of_task tsk j TNSCHED : ~ task_scheduled_at sched tsk t jo : Job INT : interference jo t
another_hep_job_interference_dec jo t ->
exists jt : Job, scheduled_at sched jt 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant TSK : job_of_task tsk j TNSCHED : ~ task_scheduled_at sched tsk t jo : Job INT : interference jo t
another_hep_job_interference_dec jo t ->
exists jt : Job, scheduled_at sched jt t
move => /another_hep_job_interference_P [jt [_ [_ 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant TSK : job_of_task tsk j TNSCHED : ~ task_scheduled_at sched tsk t jo : Job INT : interference jo t jt : Job RSERV : receives_service_at sched jt t
exists jt : Job, scheduled_at sched jt t
by exists jt ; move : RSERV; rewrite /receives_service_at service_at_is_scheduled_at lt0b. }
} 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant TSK : job_of_task tsk j TNSCHED : ~ task_scheduled_at sched tsk t jo : Job INT : interference jo t PI : priority_inversion_dec arr_seq sched j t = false IATHEP : another_task_hep_job_interference_dec j t =
false L1 : interference jo t ->
exists jt : Job, scheduled_at sched jt t
False
apply L1 in INT; destruct INT as [jt SCHED]; clear L1.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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant TSK : job_of_task tsk j TNSCHED : ~ task_scheduled_at sched tsk t jo, jt : Job SCHED : scheduled_at sched jt t PI : priority_inversion_dec arr_seq sched j t = false IATHEP : another_task_hep_job_interference_dec j t =
false
False
move : PI => /eqP; rewrite eqbF_neg => /priority_inversion_negP PI; feed_n 3 PI; rt_eauto; apply : PI.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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant TSK : job_of_task tsk j TNSCHED : ~ task_scheduled_at sched tsk t jo, jt : Job SCHED : scheduled_at sched jt t IATHEP : another_task_hep_job_interference_dec j t =
false
priority_inversion sched j t
move : IATHEP => /eqP; rewrite eqbF_neg => /hasPn OH.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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant TSK : job_of_task tsk j TNSCHED : ~ task_scheduled_at sched tsk t jo, jt : Job SCHED : scheduled_at sched jt t OH : {in arrivals_up_to arr_seq t,
forall x : Job,
~~
(another_task_hep_job x j &&
receives_service_at sched x t)}
priority_inversion sched j t
specialize (OH jt); feed OH.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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant TSK : job_of_task tsk j TNSCHED : ~ task_scheduled_at sched tsk t jo, jt : Job SCHED : scheduled_at sched jt t OH : jt \in arrivals_up_to arr_seq t ->
(fun x : Job =>
is_true
(~~
(another_task_hep_job x j &&
receives_service_at sched x t))) jt
jt \in arrivals_up_to arr_seq 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant TSK : job_of_task tsk j TNSCHED : ~ task_scheduled_at sched tsk t jo, jt : Job SCHED : scheduled_at sched jt t OH : jt \in arrivals_up_to arr_seq t ->
(fun x : Job =>
is_true
(~~
(another_task_hep_job x j &&
receives_service_at sched x t))) jt
jt \in arrivals_up_to arr_seq t
apply arrived_between_implies_in_arrivals => //; rt_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant TSK : job_of_task tsk j TNSCHED : ~ task_scheduled_at sched tsk t jo, jt : Job SCHED : scheduled_at sched jt t OH : jt \in arrivals_up_to arr_seq t ->
(fun x : Job =>
is_true
(~~
(another_task_hep_job x j &&
receives_service_at sched x t))) jt
arrived_between jt 0 t.+1
by apply /andP; split ; last (rewrite ltnS //; apply H_jobs_must_arrive_to_execute).
} 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant TSK : job_of_task tsk j TNSCHED : ~ task_scheduled_at sched tsk t jo, jt : Job SCHED : scheduled_at sched jt t OH : (fun x : Job =>
is_true
(~~
(another_task_hep_job x j &&
receives_service_at sched x t))) jt
priority_inversion sched j t
rewrite //= negb_and negb_and in OH.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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant TSK : job_of_task tsk j TNSCHED : ~ task_scheduled_at sched tsk t jo, jt : Job SCHED : scheduled_at sched jt t OH : ~~ hep_job jt j || ~~ (job_task jt != job_task j)
|| ~~ receives_service_at sched jt t
priority_inversion sched j t
move : OH => /orP [/orP [OH11 | OH22] | OH2].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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant TSK : job_of_task tsk j TNSCHED : ~ task_scheduled_at sched tsk t jo, jt : Job SCHED : scheduled_at sched jt t OH11 : ~~ hep_job jt j
priority_inversion 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant TSK : job_of_task tsk j TNSCHED : ~ task_scheduled_at sched tsk t jo, jt : Job SCHED : scheduled_at sched jt t OH11 : ~~ hep_job jt j
priority_inversion sched j t
split ; last by (exists jt ; 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant TSK : job_of_task tsk j TNSCHED : ~ task_scheduled_at sched tsk t jo, jt : Job SCHED : scheduled_at sched jt t OH11 : ~~ hep_job jt j
~~ scheduled_at sched j t
apply /negP => SCHEDj; move : OH11.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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant TSK : job_of_task tsk j TNSCHED : ~ task_scheduled_at sched tsk t jo, jt : Job SCHED : scheduled_at sched jt t SCHEDj : scheduled_at sched j t
~~ hep_job jt j -> False
rewrite (ideal_proc_model_is_a_uniprocessor_model _ _ _ _ SCHEDj 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant TSK : job_of_task tsk j TNSCHED : ~ task_scheduled_at sched tsk t jo, jt : Job SCHED : scheduled_at sched jt t SCHEDj : scheduled_at sched j t
~~ hep_job jt jt -> False
by move => /negP E; apply : E; eapply H_priority_is_reflexive with (t := 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant TSK : job_of_task tsk j TNSCHED : ~ task_scheduled_at sched tsk t jo, jt : Job SCHED : scheduled_at sched jt t OH22 : ~~ (job_task jt != job_task j)
priority_inversion 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant TSK : job_of_task tsk j TNSCHED : ~ task_scheduled_at sched tsk t jo, jt : Job SCHED : scheduled_at sched jt t OH22 : ~~ (job_task jt != job_task j)
priority_inversion sched j t
exfalso ; apply : TNSCHED.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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant TSK : job_of_task tsk j jo, jt : Job SCHED : scheduled_at sched jt t OH22 : ~~ (job_task jt != job_task j)
task_scheduled_at sched tsk t
move : SCHED; rewrite /task_scheduled_at scheduled_at_def => /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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant TSK : job_of_task tsk j jo, jt : Job OH22 : ~~ (job_task jt != job_task j)
job_task jt == tsk
by move : OH22; rewrite Bool.negb_involutive => /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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant TSK : job_of_task tsk j TNSCHED : ~ task_scheduled_at sched tsk t jo, jt : Job SCHED : scheduled_at sched jt t OH2 : ~~ receives_service_at sched jt t
priority_inversion 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant TSK : job_of_task tsk j TNSCHED : ~ task_scheduled_at sched tsk t jo, jt : Job SCHED : scheduled_at sched jt t OH2 : ~~ receives_service_at sched jt t
priority_inversion sched j t
exfalso ; move : OH2 => /negP OH2; apply : OH2.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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t : instant TSK : job_of_task tsk j TNSCHED : ~ task_scheduled_at sched tsk t jo, jt : Job SCHED : scheduled_at sched jt t
receives_service_at sched jt t
by rewrite /receives_service_at service_at_is_scheduled_at lt0b.
Qed .
(** Let [j] be any job of task [tsk], and let [upper_bound] be any
time instant after job [j]'s arrival. Then for any time
interval lying before [upper_bound], the cumulative
interference received by [tsk] is equal to the sum of the
cumulative priority inversion of job [j] and the cumulative
interference incurred by task [tsk] due to other tasks. *)
Lemma cumulative_task_interference_split :
forall j t1 t2 upper_bound ,
arrives_in arr_seq j ->
job_of_task tsk j ->
j \in arrivals_before arr_seq upper_bound ->
~~ completed_by sched j t2 ->
cumul_task_interference arr_seq sched tsk upper_bound t1 t2 =
cumulative_priority_inversion arr_seq sched j t1 t2
+ cumulative_another_task_hep_job_interference 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
forall (j : Job) (t1 : nat) (t2 upper_bound : instant),
arrives_in arr_seq j ->
job_of_task tsk j ->
j \in arrivals_before arr_seq upper_bound ->
~~ completed_by sched j t2 ->
cumul_task_interference arr_seq sched tsk upper_bound
t1 t2 =
cumulative_priority_inversion arr_seq sched j t1 t2 +
cumulative_another_task_hep_job_interference 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks
forall (j : Job) (t1 : nat) (t2 upper_bound : instant),
arrives_in arr_seq j ->
job_of_task tsk j ->
j \in arrivals_before arr_seq upper_bound ->
~~ completed_by sched j t2 ->
cumul_task_interference arr_seq sched tsk upper_bound
t1 t2 =
cumulative_priority_inversion arr_seq sched j t1 t2 +
cumulative_another_task_hep_job_interference j t1 t2
move => j t1 R upp ARRin TSK ARR NCOMPL; rewrite /cumul_task_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1 : nat R, upp : instant ARRin : arrives_in arr_seq j TSK : job_of_task tsk j ARR : j \in arrivals_before arr_seq upp NCOMPL : ~~ completed_by sched j R
\sum_(t1 <= t < R)
task_interference_received_before_dec arr_seq sched
tsk upp t =
cumulative_priority_inversion arr_seq sched j t1 R +
cumulative_another_task_hep_job_interference j t1 R
rewrite -big_split //= big_seq_cond [in X in _ = X]big_seq_cond; apply eq_bigr; 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1 : nat R, upp : instant ARRin : arrives_in arr_seq j TSK : job_of_task tsk j ARR : j \in arrivals_before arr_seq upp NCOMPL : ~~ completed_by sched j R t : nat_eqType IN : t \in index_iota t1 R
task_interference_received_before_dec arr_seq sched
tsk upp t =
priority_inversion_dec arr_seq sched j t +
another_task_hep_job_interference_dec j t
have BinFact: forall (a b c : bool), (a -> (~~ b && c) || (b && ~~c)) -> (b \/ c -> a) -> nat_of_bool a = nat_of_bool b + nat_of_bool c.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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1 : nat R, upp : instant ARRin : arrives_in arr_seq j TSK : job_of_task tsk j ARR : j \in arrivals_before arr_seq upp NCOMPL : ~~ completed_by sched j R t : nat_eqType IN : t \in index_iota t1 R
forall a b c : bool,
(a -> ~~ b && c || b && ~~ c) ->
(b \/ c -> a) -> a = b + c
{ 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1 : nat R, upp : instant ARRin : arrives_in arr_seq j TSK : job_of_task tsk j ARR : j \in arrivals_before arr_seq upp NCOMPL : ~~ completed_by sched j R t : nat_eqType IN : t \in index_iota t1 R
forall a b c : bool,
(a -> ~~ b && c || b && ~~ c) ->
(b \/ c -> a) -> a = b + c
by clear ; move => [] [] []; try compute ; firstorder ; inversion H; inversion H0. } 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1 : nat R, upp : instant ARRin : arrives_in arr_seq j TSK : job_of_task tsk j ARR : j \in arrivals_before arr_seq upp NCOMPL : ~~ completed_by sched j R t : nat_eqType IN : t \in index_iota t1 R BinFact : forall a b c : bool,
(a -> ~~ b && c || b && ~~ c) ->
(b \/ c -> a) -> a = b + c
task_interference_received_before_dec arr_seq sched
tsk upp t =
priority_inversion_dec arr_seq sched j t +
another_task_hep_job_interference_dec j t
apply : BinFact;
[move => /task_interference_received_before_P [TNSCHED [jo [INT TIN]]]
| move => [/priority_inversion_P PRIO | /another_task_hep_job_interference_P [jo [INjo [ATHEP 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1 : nat R, upp : instant ARRin : arrives_in arr_seq j TSK : job_of_task tsk j ARR : j \in arrivals_before arr_seq upp NCOMPL : ~~ completed_by sched j R t : nat_eqType IN : t \in index_iota t1 R TNSCHED : ~ task_scheduled_at sched tsk t jo : Job INT : interference jo t TIN : jo \in task_arrivals_before arr_seq tsk upp
~~ priority_inversion_dec arr_seq sched j t &&
another_task_hep_job_interference_dec j t
|| priority_inversion_dec arr_seq sched j t &&
~~ another_task_hep_job_interference_dec 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1 : nat R, upp : instant ARRin : arrives_in arr_seq j TSK : job_of_task tsk j ARR : j \in arrivals_before arr_seq upp NCOMPL : ~~ completed_by sched j R t : nat_eqType IN : t \in index_iota t1 R TNSCHED : ~ task_scheduled_at sched tsk t jo : Job INT : interference jo t TIN : jo \in task_arrivals_before arr_seq tsk upp
~~ priority_inversion_dec arr_seq sched j t &&
another_task_hep_job_interference_dec j t
|| priority_inversion_dec arr_seq sched j t &&
~~ another_task_hep_job_interference_dec j t
by eapply priority_inversion_xor_atask_hep_job_interference; rt_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1 : nat R, upp : instant ARRin : arrives_in arr_seq j TSK : job_of_task tsk j ARR : j \in arrivals_before arr_seq upp NCOMPL : ~~ completed_by sched j R t : nat_eqType IN : t \in index_iota t1 R PRIO : jobs_come_from_arrival_sequence sched arr_seq ->
jobs_must_arrive_to_execute sched ->
consistent_arrival_times arr_seq ->
priority_inversion sched j t
task_interference_received_before_dec arr_seq sched
tsk upp 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1 : nat R, upp : instant ARRin : arrives_in arr_seq j TSK : job_of_task tsk j ARR : j \in arrivals_before arr_seq upp NCOMPL : ~~ completed_by sched j R t : nat_eqType IN : t \in index_iota t1 R PRIO : jobs_come_from_arrival_sequence sched arr_seq ->
jobs_must_arrive_to_execute sched ->
consistent_arrival_times arr_seq ->
priority_inversion sched j t
task_interference_received_before_dec arr_seq sched
tsk upp t
feed_n 3 PRIO; rt_auto; move : PRIO => [NSCHED [j' /andP [SCHED NHEP]]]. 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1 : nat R, upp : instant ARRin : arrives_in arr_seq j TSK : job_of_task tsk j ARR : j \in arrivals_before arr_seq upp NCOMPL : ~~ completed_by sched j R t : nat_eqType IN : t \in index_iota t1 R NSCHED : ~~ scheduled_at sched j t j' : Job SCHED : scheduled_at sched j' t NHEP : ~~ hep_job j' j
task_interference_received_before_dec arr_seq sched
tsk upp t
apply /task_interference_received_before_P; 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1 : nat R, upp : instant ARRin : arrives_in arr_seq j TSK : job_of_task tsk j ARR : j \in arrivals_before arr_seq upp NCOMPL : ~~ completed_by sched j R t : nat_eqType IN : t \in index_iota t1 R NSCHED : ~~ scheduled_at sched j t j' : Job SCHED : scheduled_at sched j' t NHEP : ~~ hep_job j' j
~ task_scheduled_at sched tsk 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1 : nat R, upp : instant ARRin : arrives_in arr_seq j TSK : job_of_task tsk j ARR : j \in arrivals_before arr_seq upp NCOMPL : ~~ completed_by sched j R t : nat_eqType IN : t \in index_iota t1 R NSCHED : ~~ scheduled_at sched j t j' : Job SCHED : scheduled_at sched j' t NHEP : ~~ hep_job j' j
~ task_scheduled_at sched tsk t
move => TSCHED.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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1 : nat R, upp : instant ARRin : arrives_in arr_seq j TSK : job_of_task tsk j ARR : j \in arrivals_before arr_seq upp NCOMPL : ~~ completed_by sched j R t : nat_eqType IN : t \in index_iota t1 R NSCHED : ~~ scheduled_at sched j t j' : Job SCHED : scheduled_at sched j' t NHEP : ~~ hep_job j' j TSCHED : task_scheduled_at sched tsk t
False
have TSKj' : job_of_task tsk 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1 : nat R, upp : instant ARRin : arrives_in arr_seq j TSK : job_of_task tsk j ARR : j \in arrivals_before arr_seq upp NCOMPL : ~~ completed_by sched j R t : nat_eqType IN : t \in index_iota t1 R NSCHED : ~~ scheduled_at sched j t j' : Job SCHED : scheduled_at sched j' t NHEP : ~~ hep_job j' j TSCHED : task_scheduled_at sched tsk t
job_of_task tsk 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1 : nat R, upp : instant ARRin : arrives_in arr_seq j TSK : job_of_task tsk j ARR : j \in arrivals_before arr_seq upp NCOMPL : ~~ completed_by sched j R t : nat_eqType IN : t \in index_iota t1 R NSCHED : ~~ scheduled_at sched j t j' : Job SCHED : scheduled_at sched j' t NHEP : ~~ hep_job j' j TSCHED : task_scheduled_at sched tsk t
job_of_task tsk j'
move : TSCHED; rewrite /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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1 : nat R, upp : instant ARRin : arrives_in arr_seq j TSK : job_of_task tsk j ARR : j \in arrivals_before arr_seq upp NCOMPL : ~~ completed_by sched j R t : nat_eqType IN : t \in index_iota t1 R NSCHED : ~~ scheduled_at sched j t j' : Job SCHED : scheduled_at sched j' t NHEP : ~~ hep_job j' j
match sched t with
| Some j => job_task j == tsk
| None => false
end -> job_of_task tsk j'
by move : SCHED; rewrite scheduled_at_def => /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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1 : nat R, upp : instant ARRin : arrives_in arr_seq j TSK : job_of_task tsk j ARR : j \in arrivals_before arr_seq upp NCOMPL : ~~ completed_by sched j R t : nat_eqType IN : t \in index_iota t1 R NSCHED : ~~ scheduled_at sched j t j' : Job SCHED : scheduled_at sched j' t NHEP : ~~ hep_job j' j TSCHED : task_scheduled_at sched tsk t TSKj' : job_of_task tsk j'
False
clear TSCHED.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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1 : nat R, upp : instant ARRin : arrives_in arr_seq j TSK : job_of_task tsk j ARR : j \in arrivals_before arr_seq upp NCOMPL : ~~ completed_by sched j R t : nat_eqType IN : t \in index_iota t1 R NSCHED : ~~ scheduled_at sched j t j' : Job SCHED : scheduled_at sched j' t NHEP : ~~ hep_job j' j TSKj' : job_of_task tsk j'
False
have ARRj': job_arrival j < 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1 : nat R, upp : instant ARRin : arrives_in arr_seq j TSK : job_of_task tsk j ARR : j \in arrivals_before arr_seq upp NCOMPL : ~~ completed_by sched j R t : nat_eqType IN : t \in index_iota t1 R NSCHED : ~~ scheduled_at sched j t j' : Job SCHED : scheduled_at sched j' t NHEP : ~~ hep_job j' j TSKj' : job_of_task tsk j'
job_arrival j < 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1 : nat R, upp : instant ARRin : arrives_in arr_seq j TSK : job_of_task tsk j ARR : j \in arrivals_before arr_seq upp NCOMPL : ~~ completed_by sched j R t : nat_eqType IN : t \in index_iota t1 R NSCHED : ~~ scheduled_at sched j t j' : Job SCHED : scheduled_at sched j' t NHEP : ~~ hep_job j' j TSKj' : job_of_task tsk j'
job_arrival j < job_arrival j'
by move : NHEP; rewrite ltnNge; apply contra, H_JLFP_respects_sequential_tasks; move : 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1 : nat R, upp : instant ARRin : arrives_in arr_seq j TSK : job_of_task tsk j ARR : j \in arrivals_before arr_seq upp NCOMPL : ~~ completed_by sched j R t : nat_eqType IN : t \in index_iota t1 R NSCHED : ~~ scheduled_at sched j t j' : Job SCHED : scheduled_at sched j' t NHEP : ~~ hep_job j' j TSKj' : job_of_task tsk j' ARRj' : job_arrival j < job_arrival j'
False
eapply H_sequential_tasks in ARRj'; rt_eauto; last by rewrite /same_task; move : TSKj' => /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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1 : nat R, upp : instant ARRin : arrives_in arr_seq j TSK : job_of_task tsk j ARR : j \in arrivals_before arr_seq upp NCOMPL : ~~ completed_by sched j R t : nat_eqType IN : t \in index_iota t1 R NSCHED : ~~ scheduled_at sched j t j' : Job SCHED : scheduled_at sched j' t NHEP : ~~ hep_job j' j TSKj' : job_of_task tsk j' ARRj' : scheduled_at sched j' ?t ->
completed_by sched j ?t
False
apply ARRj' in SCHED; clear ARRj'.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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1 : nat R, upp : instant ARRin : arrives_in arr_seq j TSK : job_of_task tsk j ARR : j \in arrivals_before arr_seq upp NCOMPL : ~~ completed_by sched j R t : nat_eqType IN : t \in index_iota t1 R NSCHED : ~~ scheduled_at sched j t j' : Job SCHED : completed_by sched j t NHEP : ~~ hep_job j' j TSKj' : job_of_task tsk j'
False
move : NCOMPL => /negP NCOMPL; apply : 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1 : nat R, upp : instant ARRin : arrives_in arr_seq j TSK : job_of_task tsk j ARR : j \in arrivals_before arr_seq upp t : nat_eqType IN : t \in index_iota t1 R NSCHED : ~~ scheduled_at sched j t j' : Job SCHED : completed_by sched j t NHEP : ~~ hep_job j' j TSKj' : job_of_task tsk j'
completed_by sched j R
apply completion_monotonic with t; rt_auto.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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1 : nat R, upp : instant ARRin : arrives_in arr_seq j TSK : job_of_task tsk j ARR : j \in arrivals_before arr_seq upp t : nat_eqType IN : t \in index_iota t1 R NSCHED : ~~ scheduled_at sched j t j' : Job SCHED : completed_by sched j t NHEP : ~~ hep_job j' j TSKj' : job_of_task tsk j'
t <= R
by move : IN; rewrite mem_iota; clear ; lia .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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1 : nat R, upp : instant ARRin : arrives_in arr_seq j TSK : job_of_task tsk j ARR : j \in arrivals_before arr_seq upp NCOMPL : ~~ completed_by sched j R t : nat_eqType IN : t \in index_iota t1 R NSCHED : ~~ scheduled_at sched j t j' : Job SCHED : scheduled_at sched j' t NHEP : ~~ hep_job j' j
exists j : Job,
interference j t /\
j \in task_arrivals_before arr_seq tsk upp
- 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1 : nat R, upp : instant ARRin : arrives_in arr_seq j TSK : job_of_task tsk j ARR : j \in arrivals_before arr_seq upp NCOMPL : ~~ completed_by sched j R t : nat_eqType IN : t \in index_iota t1 R NSCHED : ~~ scheduled_at sched j t j' : Job SCHED : scheduled_at sched j' t NHEP : ~~ hep_job j' j
exists j : Job,
interference j t /\
j \in task_arrivals_before arr_seq tsk upp
exists j ; 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1 : nat R, upp : instant ARRin : arrives_in arr_seq j TSK : job_of_task tsk j ARR : j \in arrivals_before arr_seq upp NCOMPL : ~~ completed_by sched j R t : nat_eqType IN : t \in index_iota t1 R NSCHED : ~~ scheduled_at sched j t j' : Job SCHED : scheduled_at sched j' t NHEP : ~~ hep_job j' 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1 : nat R, upp : instant ARRin : arrives_in arr_seq j TSK : job_of_task tsk j ARR : j \in arrivals_before arr_seq upp NCOMPL : ~~ completed_by sched j R t : nat_eqType IN : t \in index_iota t1 R NSCHED : ~~ scheduled_at sched j t j' : Job SCHED : scheduled_at sched j' t NHEP : ~~ hep_job j' j
interference j t
apply /orP; left ; apply /priority_inversion_P; rt_auto.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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1 : nat R, upp : instant ARRin : arrives_in arr_seq j TSK : job_of_task tsk j ARR : j \in arrivals_before arr_seq upp NCOMPL : ~~ completed_by sched j R t : nat_eqType IN : t \in index_iota t1 R NSCHED : ~~ scheduled_at sched j t j' : Job SCHED : scheduled_at sched j' t NHEP : ~~ hep_job j' j
priority_inversion sched j t
by split ; last (exists j' ; apply /andP; split ; rt_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1 : nat R, upp : instant ARRin : arrives_in arr_seq j TSK : job_of_task tsk j ARR : j \in arrivals_before arr_seq upp NCOMPL : ~~ completed_by sched j R t : nat_eqType IN : t \in index_iota t1 R NSCHED : ~~ scheduled_at sched j t j' : Job SCHED : scheduled_at sched j' t NHEP : ~~ hep_job j' j
j \in task_arrivals_before arr_seq tsk upp
+ 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1 : nat R, upp : instant ARRin : arrives_in arr_seq j TSK : job_of_task tsk j ARR : j \in arrivals_before arr_seq upp NCOMPL : ~~ completed_by sched j R t : nat_eqType IN : t \in index_iota t1 R NSCHED : ~~ scheduled_at sched j t j' : Job SCHED : scheduled_at sched j' t NHEP : ~~ hep_job j' j
j \in task_arrivals_before arr_seq tsk upp
by rewrite mem_filter; 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1 : nat R, upp : instant ARRin : arrives_in arr_seq j TSK : job_of_task tsk j ARR : j \in arrivals_before arr_seq upp NCOMPL : ~~ completed_by sched j R t : nat_eqType IN : t \in index_iota t1 R jo : Job INjo : jo \in arrivals_up_to arr_seq t ATHEP : another_task_hep_job jo j RSERV : receives_service_at sched jo t
task_interference_received_before_dec arr_seq sched
tsk upp 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1 : nat R, upp : instant ARRin : arrives_in arr_seq j TSK : job_of_task tsk j ARR : j \in arrivals_before arr_seq upp NCOMPL : ~~ completed_by sched j R t : nat_eqType IN : t \in index_iota t1 R jo : Job INjo : jo \in arrivals_up_to arr_seq t ATHEP : another_task_hep_job jo j RSERV : receives_service_at sched jo t
task_interference_received_before_dec arr_seq sched
tsk upp t
apply /task_interference_received_before_P; 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1 : nat R, upp : instant ARRin : arrives_in arr_seq j TSK : job_of_task tsk j ARR : j \in arrivals_before arr_seq upp NCOMPL : ~~ completed_by sched j R t : nat_eqType IN : t \in index_iota t1 R jo : Job INjo : jo \in arrivals_up_to arr_seq t ATHEP : another_task_hep_job jo j RSERV : receives_service_at sched jo t
~ task_scheduled_at sched tsk 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1 : nat R, upp : instant ARRin : arrives_in arr_seq j TSK : job_of_task tsk j ARR : j \in arrivals_before arr_seq upp NCOMPL : ~~ completed_by sched j R t : nat_eqType IN : t \in index_iota t1 R jo : Job INjo : jo \in arrivals_up_to arr_seq t ATHEP : another_task_hep_job jo j RSERV : receives_service_at sched jo t
~ task_scheduled_at sched tsk t
move => TSCHED; move : ATHEP => /andP [_ /negP EQ]; apply : 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1 : nat R, upp : instant ARRin : arrives_in arr_seq j TSK : job_of_task tsk j ARR : j \in arrivals_before arr_seq upp NCOMPL : ~~ completed_by sched j R t : nat_eqType IN : t \in index_iota t1 R jo : Job INjo : jo \in arrivals_up_to arr_seq t RSERV : receives_service_at sched jo t TSCHED : task_scheduled_at sched tsk t
job_task jo == job_task j
move : TSCHED; rewrite /task_scheduled_at; move : 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1 : nat R, upp : instant ARRin : arrives_in arr_seq j TSK : job_of_task tsk j ARR : j \in arrivals_before arr_seq upp NCOMPL : ~~ completed_by sched j R t : nat_eqType IN : t \in index_iota t1 R jo : Job INjo : jo \in arrivals_up_to arr_seq t
receives_service_at sched jo t ->
match sched t with
| Some j => job_task j == tsk
| None => false
end -> job_task jo == job_task j
rewrite /receives_service_at service_at_is_scheduled_at lt0b scheduled_at_def => /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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1 : nat R, upp : instant ARRin : arrives_in arr_seq j TSK : job_of_task tsk j ARR : j \in arrivals_before arr_seq upp NCOMPL : ~~ completed_by sched j R t : nat_eqType IN : t \in index_iota t1 R jo : Job INjo : jo \in arrivals_up_to arr_seq t
tsk == job_task j
by rewrite eq_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1 : nat R, upp : instant ARRin : arrives_in arr_seq j TSK : job_of_task tsk j ARR : j \in arrivals_before arr_seq upp NCOMPL : ~~ completed_by sched j R t : nat_eqType IN : t \in index_iota t1 R jo : Job INjo : jo \in arrivals_up_to arr_seq t ATHEP : another_task_hep_job jo j RSERV : receives_service_at sched jo t
exists j : Job,
interference j t /\
j \in task_arrivals_before arr_seq tsk upp
exists j ; 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1 : nat R, upp : instant ARRin : arrives_in arr_seq j TSK : job_of_task tsk j ARR : j \in arrivals_before arr_seq upp NCOMPL : ~~ completed_by sched j R t : nat_eqType IN : t \in index_iota t1 R jo : Job INjo : jo \in arrivals_up_to arr_seq t ATHEP : another_task_hep_job jo j RSERV : receives_service_at sched jo 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1 : nat R, upp : instant ARRin : arrives_in arr_seq j TSK : job_of_task tsk j ARR : j \in arrivals_before arr_seq upp NCOMPL : ~~ completed_by sched j R t : nat_eqType IN : t \in index_iota t1 R jo : Job INjo : jo \in arrivals_up_to arr_seq t ATHEP : another_task_hep_job jo j RSERV : receives_service_at sched jo t
interference j t
apply /orP; right ; apply /another_hep_job_interference_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1 : nat R, upp : instant ARRin : arrives_in arr_seq j TSK : job_of_task tsk j ARR : j \in arrivals_before arr_seq upp NCOMPL : ~~ completed_by sched j R t : nat_eqType IN : t \in index_iota t1 R jo : Job INjo : jo \in arrivals_up_to arr_seq t ATHEP : another_task_hep_job jo j RSERV : receives_service_at sched jo t
another_hep_job_interference j t
exists jo ; split ; first (by done ); split ; last by done .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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1 : nat R, upp : instant ARRin : arrives_in arr_seq j TSK : job_of_task tsk j ARR : j \in arrivals_before arr_seq upp NCOMPL : ~~ completed_by sched j R t : nat_eqType IN : t \in index_iota t1 R jo : Job INjo : jo \in arrivals_up_to arr_seq t ATHEP : another_task_hep_job jo j RSERV : receives_service_at sched jo t
another_hep_job jo j
move : ATHEP => /andP [A B]; apply /andP; split ; first by done .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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1 : nat R, upp : instant ARRin : arrives_in arr_seq j TSK : job_of_task tsk j ARR : j \in arrivals_before arr_seq upp NCOMPL : ~~ completed_by sched j R t : nat_eqType IN : t \in index_iota t1 R jo : Job INjo : jo \in arrivals_up_to arr_seq t RSERV : receives_service_at sched jo t A : hep_job jo j B : job_task jo != job_task j
jo != j
by apply /negP; move => /eqP EQ; subst jo; rewrite eq_refl in B.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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1 : nat R, upp : instant ARRin : arrives_in arr_seq j TSK : job_of_task tsk j ARR : j \in arrivals_before arr_seq upp NCOMPL : ~~ completed_by sched j R t : nat_eqType IN : t \in index_iota t1 R jo : Job INjo : jo \in arrivals_up_to arr_seq t ATHEP : another_task_hep_job jo j RSERV : receives_service_at sched jo t
j \in task_arrivals_before arr_seq tsk upp
- 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job t1 : nat R, upp : instant ARRin : arrives_in arr_seq j TSK : job_of_task tsk j ARR : j \in arrivals_before arr_seq upp NCOMPL : ~~ completed_by sched j R t : nat_eqType IN : t \in index_iota t1 R jo : Job INjo : jo \in arrivals_up_to arr_seq t ATHEP : another_task_hep_job jo j RSERV : receives_service_at sched jo t
j \in task_arrivals_before arr_seq tsk upp
by rewrite mem_filter; apply /andP; split . }
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 j t1 t2
= workload_of_another_hep_jobs 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 j t1 t2 =
workload_of_another_hep_jobs 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 j t1 t2 =
workload_of_another_hep_jobs j t1 t2
rewrite /cumulative_other_hep_jobs_interfering_workload /workload_of_another_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 j t =
workload_of_jobs (another_hep_job^~ j)
(arrivals_between arr_seq t1 t2)
rewrite big_geq; last by done .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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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; last by done .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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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
induction 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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; last by done .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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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; last by done .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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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
- 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 //=;
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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks 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 //=.
}
Qed .
End InstantiatedWorkloadEquivalence .
(** In this section, we prove that the (abstract) cumulative
interference of jobs with higher or equal priority is equal to
total service of jobs with higher or equal priority. *)
Section InstantiatedServiceEquivalences .
(** 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.
(** We consider an arbitrary time interval <<[t1, t)>> that
starts with a quiet time. *)
Variable t1 t : instant.
Hypothesis H_quiet_time : busy_interval.quiet_time arr_seq sched j t1.
(** Then for job [j], the (abstract) instantiated function of
interference is equal to the total service of jobs with
higher or equal priority. *)
Lemma cumulative_i_ohep_eq_service_of_ohep :
cumulative_another_hep_job_interference j t1 t
= service_of_another_hep_jobs j t1 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1
cumulative_another_hep_job_interference j t1 t =
service_of_another_hep_jobs j t1 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1
cumulative_another_hep_job_interference j t1 t =
service_of_another_hep_jobs j t1 t
clear H_job_of_tsk; rewrite /service_of_another_hep_jobs /service_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1
cumulative_another_hep_job_interference j t1 t =
\sum_(j0 <- arrivals_between arr_seq t1 t | another_hep_job
j0 j)
service_during sched j0 t1 t
rewrite /cumulative_another_hep_job_interference /another_hep_job_interference_dec.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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1
\sum_(t1 <= t < t)
has
(fun jhp : Job =>
another_hep_job jhp j &&
receives_service_at sched jhp t)
(arrivals_up_to arr_seq t) =
\sum_(j0 <- arrivals_between arr_seq t1 t | another_hep_job
j0 j)
service_during sched j0 t1 t
rewrite exchange_big //= big_nat_cond [in X in _ = X]big_nat_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1
\sum_(t1 <= i < t | (t1 <= i < t) && true)
has
(fun jhp : Job =>
another_hep_job jhp j &&
receives_service_at sched jhp i)
(arrivals_up_to arr_seq i) =
\sum_(t1 <= i < t | (t1 <= i < t) && true)
\sum_(i0 <- arrivals_between arr_seq t1 t | another_hep_job
i0 j)
service_at sched i0 i
apply eq_bigr => x /andP [/andP [Ge Le] _].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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t
has
(fun jhp : Job =>
another_hep_job jhp j &&
receives_service_at sched jhp x)
(arrivals_up_to arr_seq x) =
\sum_(i <- arrivals_between arr_seq t1 t | another_hep_job
i j)
service_at sched i x
ideal_proc_model_sched_case_analysis_eq sched x 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t Idle : is_idle sched x EqIdle : sched x = None
has
(fun jhp : Job =>
another_hep_job jhp j &&
receives_service_at sched jhp x)
(arrivals_up_to arr_seq x) =
\sum_(i <- arrivals_between arr_seq t1 t | another_hep_job
i j)
service_at sched i x
{ 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t Idle : is_idle sched x EqIdle : sched x = None
has
(fun jhp : Job =>
another_hep_job jhp j &&
receives_service_at sched jhp x)
(arrivals_up_to arr_seq x) =
\sum_(i <- arrivals_between arr_seq t1 t | another_hep_job
i j)
service_at sched i x
erewrite eq_in_has; [erewrite has_pred0; symmetry | ].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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t Idle : is_idle sched x EqIdle : sched x = None
\sum_(i <- arrivals_between arr_seq t1 t | another_hep_job
i j)
service_at sched i x = 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t Idle : is_idle sched x EqIdle : sched x = None
\sum_(i <- arrivals_between arr_seq t1 t | another_hep_job
i j)
service_at sched i x = false
by apply big1 => j' _; rewrite ideal_not_idle_implies_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t Idle : is_idle sched x EqIdle : sched x = None
{in arrivals_up_to arr_seq x,
(fun jhp : Job =>
another_hep_job jhp j &&
receives_service_at sched jhp x) =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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t Idle : is_idle sched x EqIdle : sched x = None
{in arrivals_up_to arr_seq x,
(fun jhp : Job =>
another_hep_job jhp j &&
receives_service_at sched jhp x) =1 pred0}
by move => j' _; apply Bool.andb_false_intro2; rewrite /receives_service_at ideal_not_idle_implies_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x EqSched_jo : #|[pred x0 |
let
'FiniteQuant .Quantified F :=
FiniteQuant.ex (T:=Core)
(, scheduled_on jo (sched x) x0)
x0 x0 in F]| <> 0
has
(fun jhp : Job =>
another_hep_job jhp j &&
receives_service_at sched jhp x)
(arrivals_up_to arr_seq x) =
\sum_(i <- arrivals_between arr_seq t1 t | another_hep_job
i j)
service_at sched i x
{ 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x EqSched_jo : #|[pred x0 |
let
'FiniteQuant .Quantified F :=
FiniteQuant.ex (T:=Core)
(, scheduled_on jo (sched x) x0)
x0 x0 in F]| <> 0
has
(fun jhp : Job =>
another_hep_job jhp j &&
receives_service_at sched jhp x)
(arrivals_up_to arr_seq x) =
\sum_(i <- arrivals_between arr_seq t1 t | another_hep_job
i j)
service_at sched i x
have ARRIN: arrives_in arr_seq jo by apply H_jobs_come_from_arrival_sequence with x.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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x EqSched_jo : #|[pred x0 |
let
'FiniteQuant .Quantified F :=
FiniteQuant.ex (T:=Core)
(, scheduled_on jo (sched x) x0)
x0 x0 in F]| <> 0 ARRIN : arrives_in arr_seq jo
has
(fun jhp : Job =>
another_hep_job jhp j &&
receives_service_at sched jhp x)
(arrivals_up_to arr_seq x) =
\sum_(i <- arrivals_between arr_seq t1 t | another_hep_job
i j)
service_at sched i x
clear EqSched_jo; destruct (another_hep_job jo 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_hep_job jo j = true
has
(fun jhp : Job =>
another_hep_job jhp j &&
receives_service_at sched jhp x)
(arrivals_up_to arr_seq x) =
\sum_(i <- arrivals_between arr_seq t1 t | another_hep_job
i j)
service_at sched i x
{ 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_hep_job jo j = true
has
(fun jhp : Job =>
another_hep_job jhp j &&
receives_service_at sched jhp x)
(arrivals_up_to arr_seq x) =
\sum_(i <- arrivals_between arr_seq t1 t | another_hep_job
i j)
service_at sched i x
replace (has _ _) with true; symmetry ; 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_hep_job jo j = true
has
(fun jhp : Job =>
another_hep_job jhp j &&
receives_service_at sched jhp x)
(arrivals_up_to arr_seq x) = 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_hep_job jo j = true
has
(fun jhp : Job =>
another_hep_job jhp j &&
receives_service_at sched jhp x)
(arrivals_up_to arr_seq x) = true
apply /hasP; exists 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_hep_job jo j = true
jo \in arrivals_up_to arr_seq x
- 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_hep_job jo j = true
jo \in arrivals_up_to arr_seq x
apply arrived_between_implies_in_arrivals => //; rt_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_hep_job jo j = true
arrived_between jo 0 x.+1
by apply /andP; split ; last (rewrite ltnS //; apply H_jobs_must_arrive_to_execute).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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_hep_job jo j = true
another_hep_job jo j && receives_service_at sched jo x
- 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_hep_job jo j = true
another_hep_job jo j && receives_service_at sched jo x
by rewrite PRIO //= /receives_service_at service_at_is_scheduled_at Sched_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_hep_job jo j = true
\sum_(i <- arrivals_between arr_seq t1 t | another_hep_job
i j)
service_at sched i x = 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_hep_job jo j = true
\sum_(i <- arrivals_between arr_seq t1 t | another_hep_job
i j)
service_at sched i x = true
apply /eqP; rewrite eqn_leq; apply /andP; split ; rt_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_hep_job jo j = true
\sum_(i <- arrivals_between arr_seq t1 t | another_hep_job
i j)
service_at sched i x <= 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_hep_job jo j = true
\sum_(i <- arrivals_between arr_seq t1 t | another_hep_job
i j)
service_at sched i x <= true
eapply service_of_jobs_le_1; rt_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_hep_job jo j = true
true <=
\sum_(i <- arrivals_between arr_seq t1 t | another_hep_job
i j)
service_at sched i x
- 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_hep_job jo j = true
true <=
\sum_(i <- arrivals_between arr_seq t1 t | another_hep_job
i j)
service_at sched i x
rewrite sum_nat_gt0; apply /hasP; exists jo ; last by rewrite service_at_is_scheduled_at Sched_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_hep_job jo j = true
jo
\in [seq x <- arrivals_between arr_seq t1 t
| another_hep_job x j]
rewrite mem_filter PRIO //=; apply arrived_between_implies_in_arrivals => //; rt_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_hep_job jo j = true
arrived_between jo t1 t
apply /negPn; rewrite negb_and -ltnNge -leqNgt; apply /negP => /orP [LT|GE]; first last .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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_hep_job jo j = true GE : t <= job_arrival jo
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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_hep_job jo j = true GE : t <= job_arrival jo
False
by apply H_jobs_must_arrive_to_execute in Sched_jo; unfold has_arrived in *; lia .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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_hep_job jo j = true LT : job_arrival jo < t1
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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_hep_job jo j = true LT : job_arrival jo < t1
False
move : Sched_jo; rewrite -[scheduled_at _ _ _]Bool.negb_involutive => /negP SCH; apply : SCH.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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job ARRIN : arrives_in arr_seq jo PRIO : another_hep_job jo j = true LT : job_arrival jo < t1
~~ scheduled_at sched jo x
eapply completed_implies_not_scheduled => //; apply completion_monotonic with t1 => //; apply H_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job ARRIN : arrives_in arr_seq jo PRIO : another_hep_job jo j = true LT : job_arrival jo < t1
hep_job jo j
by move : PRIO => /andP [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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_hep_job jo j = false
has
(fun jhp : Job =>
another_hep_job jhp j &&
receives_service_at sched jhp x)
(arrivals_up_to arr_seq x) =
\sum_(i <- arrivals_between arr_seq t1 t | another_hep_job
i j)
service_at sched i x
{ 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_hep_job jo j = false
has
(fun jhp : Job =>
another_hep_job jhp j &&
receives_service_at sched jhp x)
(arrivals_up_to arr_seq x) =
\sum_(i <- arrivals_between arr_seq t1 t | another_hep_job
i j)
service_at sched i x
erewrite eq_in_has; [erewrite has_pred0; symmetry | ].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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_hep_job jo j = false
\sum_(i <- arrivals_between arr_seq t1 t | another_hep_job
i j)
service_at sched i x = 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_hep_job jo j = false
\sum_(i <- arrivals_between arr_seq t1 t | another_hep_job
i j)
service_at sched i x = false
apply big1 => j' AHEP; apply /eqP; rewrite service_at_is_scheduled_at //= eqb0; apply /negP => 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_hep_job jo j = false j' : Job AHEP : another_hep_job j' j SCHED : scheduled_at sched j' x
False
enough (EQ : jo = j'); first by subst ; rewrite AHEP in 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_hep_job jo j = false j' : Job AHEP : another_hep_job j' j SCHED : scheduled_at sched j' x
jo = j'
by eapply ideal_proc_model_is_a_uniprocessor_model; rt_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_hep_job jo j = false
{in arrivals_up_to arr_seq x,
(fun jhp : Job =>
another_hep_job jhp j &&
receives_service_at sched jhp x) =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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_hep_job jo j = false
{in arrivals_up_to arr_seq x,
(fun jhp : Job =>
another_hep_job jhp j &&
receives_service_at sched jhp x) =1 pred0}
move => j' IN => //=; 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_hep_job jo j = false j' : Job IN : j' \in arrivals_up_to arr_seq x
another_hep_job j' j && receives_service_at sched j' x ==
false
rewrite eqbF_neg negb_and Bool.orb_comm -implyNb Bool.negb_involutive; apply /implyP => 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_hep_job jo j = false j' : Job IN : j' \in arrivals_up_to arr_seq x SCHED : receives_service_at sched j' x
~~ another_hep_job j' j
rewrite /receives_service_at service_at_is_scheduled_at lt0b in 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_hep_job jo j = false j' : Job IN : j' \in arrivals_up_to arr_seq x SCHED : scheduled_at sched j' x
~~ another_hep_job j' j
enough (EQ : jo = j'); first by subst ; 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_hep_job jo j = false j' : Job IN : j' \in arrivals_up_to arr_seq x SCHED : scheduled_at sched j' x
jo = j'
by eapply ideal_proc_model_is_a_uniprocessor_model; rt_eauto.
}
}
Qed .
(** The same applies to the alternative definition of interference. *)
Lemma cumulative_i_thep_eq_service_of_othep :
cumulative_another_task_hep_job_interference j t1 t
= service_of_another_task_hep_job j t1 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1
cumulative_another_task_hep_job_interference j t1 t =
service_of_another_task_hep_job j t1 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1
cumulative_another_task_hep_job_interference j t1 t =
service_of_another_task_hep_job j t1 t
rewrite /service_of_another_task_hep_job /service_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1
cumulative_another_task_hep_job_interference j t1 t =
\sum_(j0 <- arrivals_between arr_seq t1 t | another_task_hep_job
j0 j)
service_during sched j0 t1 t
rewrite /cumulative_another_task_hep_job_interference /another_task_hep_job_interference_dec.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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1
\sum_(t1 <= t < t)
has
(fun jhp : Job =>
another_task_hep_job jhp j &&
receives_service_at sched jhp t)
(arrivals_up_to arr_seq t) =
\sum_(j0 <- arrivals_between arr_seq t1 t | another_task_hep_job
j0 j)
service_during sched j0 t1 t
rewrite exchange_big //= big_nat_cond [in X in _ = X]big_nat_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1
\sum_(t1 <= i < t | (t1 <= i < t) && true)
has
(fun jhp : Job =>
another_task_hep_job jhp j &&
receives_service_at sched jhp i)
(arrivals_up_to arr_seq i) =
\sum_(t1 <= i < t | (t1 <= i < t) && true)
\sum_(i0 <- arrivals_between arr_seq t1 t | another_task_hep_job
i0 j)
service_at sched i0 i
apply eq_bigr => x /andP [/andP [Ge Le] _].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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t
has
(fun jhp : Job =>
another_task_hep_job jhp j &&
receives_service_at sched jhp x)
(arrivals_up_to arr_seq x) =
\sum_(i <- arrivals_between arr_seq t1 t | another_task_hep_job
i j)
service_at sched i x
ideal_proc_model_sched_case_analysis_eq sched x 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t Idle : is_idle sched x EqIdle : sched x = None
has
(fun jhp : Job =>
another_task_hep_job jhp j &&
receives_service_at sched jhp x)
(arrivals_up_to arr_seq x) =
\sum_(i <- arrivals_between arr_seq t1 t | another_task_hep_job
i j)
service_at sched i x
{ 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t Idle : is_idle sched x EqIdle : sched x = None
has
(fun jhp : Job =>
another_task_hep_job jhp j &&
receives_service_at sched jhp x)
(arrivals_up_to arr_seq x) =
\sum_(i <- arrivals_between arr_seq t1 t | another_task_hep_job
i j)
service_at sched i x
erewrite eq_in_has; [erewrite has_pred0; symmetry | ].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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t Idle : is_idle sched x EqIdle : sched x = None
\sum_(i <- arrivals_between arr_seq t1 t | another_task_hep_job
i j)
service_at sched i x = 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t Idle : is_idle sched x EqIdle : sched x = None
\sum_(i <- arrivals_between arr_seq t1 t | another_task_hep_job
i j)
service_at sched i x = false
by apply big1 => j' _; rewrite ideal_not_idle_implies_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t Idle : is_idle sched x EqIdle : sched x = None
{in arrivals_up_to arr_seq x,
(fun jhp : Job =>
another_task_hep_job jhp j &&
receives_service_at sched jhp x) =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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t Idle : is_idle sched x EqIdle : sched x = None
{in arrivals_up_to arr_seq x,
(fun jhp : Job =>
another_task_hep_job jhp j &&
receives_service_at sched jhp x) =1 pred0}
by move => j' _; apply Bool.andb_false_intro2; rewrite /receives_service_at ideal_not_idle_implies_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x EqSched_jo : #|[pred x0 |
let
'FiniteQuant .Quantified F :=
FiniteQuant.ex (T:=Core)
(, scheduled_on jo (sched x) x0)
x0 x0 in F]| <> 0
has
(fun jhp : Job =>
another_task_hep_job jhp j &&
receives_service_at sched jhp x)
(arrivals_up_to arr_seq x) =
\sum_(i <- arrivals_between arr_seq t1 t | another_task_hep_job
i j)
service_at sched i x
{ 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x EqSched_jo : #|[pred x0 |
let
'FiniteQuant .Quantified F :=
FiniteQuant.ex (T:=Core)
(, scheduled_on jo (sched x) x0)
x0 x0 in F]| <> 0
has
(fun jhp : Job =>
another_task_hep_job jhp j &&
receives_service_at sched jhp x)
(arrivals_up_to arr_seq x) =
\sum_(i <- arrivals_between arr_seq t1 t | another_task_hep_job
i j)
service_at sched i x
have ARRIN: arrives_in arr_seq jo by apply H_jobs_come_from_arrival_sequence with x.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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x EqSched_jo : #|[pred x0 |
let
'FiniteQuant .Quantified F :=
FiniteQuant.ex (T:=Core)
(, scheduled_on jo (sched x) x0)
x0 x0 in F]| <> 0 ARRIN : arrives_in arr_seq jo
has
(fun jhp : Job =>
another_task_hep_job jhp j &&
receives_service_at sched jhp x)
(arrivals_up_to arr_seq x) =
\sum_(i <- arrivals_between arr_seq t1 t | another_task_hep_job
i j)
service_at sched i x
clear EqSched_jo; destruct (another_task_hep_job jo 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_task_hep_job jo j = true
has
(fun jhp : Job =>
another_task_hep_job jhp j &&
receives_service_at sched jhp x)
(arrivals_up_to arr_seq x) =
\sum_(i <- arrivals_between arr_seq t1 t | another_task_hep_job
i j)
service_at sched i x
{ 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_task_hep_job jo j = true
has
(fun jhp : Job =>
another_task_hep_job jhp j &&
receives_service_at sched jhp x)
(arrivals_up_to arr_seq x) =
\sum_(i <- arrivals_between arr_seq t1 t | another_task_hep_job
i j)
service_at sched i x
replace (has _ _) with true; symmetry ; 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_task_hep_job jo j = true
has
(fun jhp : Job =>
another_task_hep_job jhp j &&
receives_service_at sched jhp x)
(arrivals_up_to arr_seq x) = 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_task_hep_job jo j = true
has
(fun jhp : Job =>
another_task_hep_job jhp j &&
receives_service_at sched jhp x)
(arrivals_up_to arr_seq x) = true
apply /hasP; exists 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_task_hep_job jo j = true
jo \in arrivals_up_to arr_seq x
- 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_task_hep_job jo j = true
jo \in arrivals_up_to arr_seq x
apply arrived_between_implies_in_arrivals => //; rt_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_task_hep_job jo j = true
arrived_between jo 0 x.+1
by apply /andP; split ; last (rewrite ltnS //; apply H_jobs_must_arrive_to_execute).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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_task_hep_job jo j = true
another_task_hep_job jo j &&
receives_service_at sched jo x
- 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_task_hep_job jo j = true
another_task_hep_job jo j &&
receives_service_at sched jo x
by rewrite PRIO //= /receives_service_at service_at_is_scheduled_at Sched_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_task_hep_job jo j = true
\sum_(i <- arrivals_between arr_seq t1 t | another_task_hep_job
i j)
service_at sched i x = 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_task_hep_job jo j = true
\sum_(i <- arrivals_between arr_seq t1 t | another_task_hep_job
i j)
service_at sched i x = true
apply /eqP; rewrite eqn_leq; apply /andP; split ; rt_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_task_hep_job jo j = true
\sum_(i <- arrivals_between arr_seq t1 t | another_task_hep_job
i j)
service_at sched i x <= 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_task_hep_job jo j = true
\sum_(i <- arrivals_between arr_seq t1 t | another_task_hep_job
i j)
service_at sched i x <= true
eapply service_of_jobs_le_1; rt_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_task_hep_job jo j = true
true <=
\sum_(i <- arrivals_between arr_seq t1 t | another_task_hep_job
i j)
service_at sched i x
- 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_task_hep_job jo j = true
true <=
\sum_(i <- arrivals_between arr_seq t1 t | another_task_hep_job
i j)
service_at sched i x
rewrite sum_nat_gt0; apply /hasP; exists jo ; last by rewrite service_at_is_scheduled_at Sched_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_task_hep_job jo j = true
jo
\in [seq x <- arrivals_between arr_seq t1 t
| another_task_hep_job x j]
rewrite mem_filter PRIO //=; apply arrived_between_implies_in_arrivals => //; rt_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_task_hep_job jo j = true
arrived_between jo t1 t
apply /negPn; rewrite negb_and -ltnNge -leqNgt; apply /negP => /orP [LT|GE]; first last .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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_task_hep_job jo j = true GE : t <= job_arrival jo
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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_task_hep_job jo j = true GE : t <= job_arrival jo
False
by apply H_jobs_must_arrive_to_execute in Sched_jo; unfold has_arrived in *; lia .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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_task_hep_job jo j = true LT : job_arrival jo < t1
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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_task_hep_job jo j = true LT : job_arrival jo < t1
False
move : Sched_jo; rewrite -[scheduled_at _ _ _]Bool.negb_involutive => /negP SCH; apply : SCH.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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job ARRIN : arrives_in arr_seq jo PRIO : another_task_hep_job jo j = true LT : job_arrival jo < t1
~~ scheduled_at sched jo x
eapply completed_implies_not_scheduled => //; apply completion_monotonic with t1 => //; apply H_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job ARRIN : arrives_in arr_seq jo PRIO : another_task_hep_job jo j = true LT : job_arrival jo < t1
hep_job jo j
by move : PRIO => /andP [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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_task_hep_job jo j = false
has
(fun jhp : Job =>
another_task_hep_job jhp j &&
receives_service_at sched jhp x)
(arrivals_up_to arr_seq x) =
\sum_(i <- arrivals_between arr_seq t1 t | another_task_hep_job
i j)
service_at sched i x
{ 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_task_hep_job jo j = false
has
(fun jhp : Job =>
another_task_hep_job jhp j &&
receives_service_at sched jhp x)
(arrivals_up_to arr_seq x) =
\sum_(i <- arrivals_between arr_seq t1 t | another_task_hep_job
i j)
service_at sched i x
erewrite eq_in_has; [erewrite has_pred0; symmetry | ].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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_task_hep_job jo j = false
\sum_(i <- arrivals_between arr_seq t1 t | another_task_hep_job
i j)
service_at sched i x = 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_task_hep_job jo j = false
\sum_(i <- arrivals_between arr_seq t1 t | another_task_hep_job
i j)
service_at sched i x = false
apply big1 => j' AHEP; apply /eqP; rewrite service_at_is_scheduled_at //= eqb0; apply /negP => 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_task_hep_job jo j = false j' : Job AHEP : another_task_hep_job j' j SCHED : scheduled_at sched j' x
False
enough (EQ : jo = j'); first by subst ; rewrite AHEP in 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_task_hep_job jo j = false j' : Job AHEP : another_task_hep_job j' j SCHED : scheduled_at sched j' x
jo = j'
by eapply ideal_proc_model_is_a_uniprocessor_model; rt_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_task_hep_job jo j = false
{in arrivals_up_to arr_seq x,
(fun jhp : Job =>
another_task_hep_job jhp j &&
receives_service_at sched jhp x) =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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_task_hep_job jo j = false
{in arrivals_up_to arr_seq x,
(fun jhp : Job =>
another_task_hep_job jhp j &&
receives_service_at sched jhp x) =1 pred0}
move => j' IN => //=; 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_task_hep_job jo j = false j' : Job IN : j' \in arrivals_up_to arr_seq x
another_task_hep_job j' j &&
receives_service_at sched j' x == false
rewrite eqbF_neg negb_and Bool.orb_comm -implyNb Bool.negb_involutive; apply /implyP => 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_task_hep_job jo j = false j' : Job IN : j' \in arrivals_up_to arr_seq x SCHED : receives_service_at sched j' x
~~ another_task_hep_job j' j
rewrite /receives_service_at service_at_is_scheduled_at lt0b in 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_task_hep_job jo j = false j' : Job IN : j' \in arrivals_up_to arr_seq x SCHED : scheduled_at sched j' x
~~ another_task_hep_job j' j
enough (EQ : jo = j'); first by subst ; 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j t1, t : instant H_quiet_time : quiet_time arr_seq sched j t1 x : nat Ge : t1 <= x Le : x < t jo : Job Sched_jo : scheduled_at sched jo x ARRIN : arrives_in arr_seq jo PRIO : another_task_hep_job jo j = false j' : Job IN : j' \in arrivals_up_to arr_seq x SCHED : scheduled_at sched j' x
jo = j'
by eapply ideal_proc_model_is_a_uniprocessor_model; rt_eauto.
}
}
Qed .
End InstantiatedServiceEquivalences .
(** 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 := busy_interval.quiet_time arr_seq sched.
Let quiet_time_ab := definitions.quiet_time sched.
(** Same for the two notions of a busy interval prefix ... *)
Let busy_interval_prefix_cl := busy_interval.busy_interval_prefix arr_seq sched.
Let busy_interval_prefix_ab := definitions.busy_interval_prefix sched.
(** ... and the two notions of a busy interval. *)
Let busy_interval_cl := busy_interval.busy_interval arr_seq sched.
Let busy_interval_ab := 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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; 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 Bool.negb_involutive; 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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
case ARR: (arrived_before j t); [right | left ]; [apply QT | ]; 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 ARR : arrived_before j t = true
hep_job j j
by apply H_priority_is_reflexive with (t := 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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; 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 j 0 t ==
cumulative_other_hep_jobs_interfering_workload j 0 t
rewrite cumulative_i_ohep_eq_service_of_ohep; rt_eauto; first last .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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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
quiet_time arr_seq sched 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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
quiet_time arr_seq sched j 0
by move => ? _ _ ; unfold arrived_before; lia . } 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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_another_hep_jobs j 0 t ==
cumulative_other_hep_jobs_interfering_workload 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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_another_hep_jobs j 0 t =
service_of_another_hep_jobs j 0 t
apply all_jobs_have_completed_equiv_workload_eq_service; rt_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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
- 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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
- 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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; rt_eauto.
}
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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 [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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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); rt_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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); rt_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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); rt_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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_another_hep_jobs j 0 t) -cumulative_i_ohep_eq_service_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 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_another_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 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 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 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 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 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 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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; rt_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 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 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); last by done .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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 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 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); last by done .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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 j 0 t +
workload_of_jobs pred0 (arrivals_between arr_seq 0 t) =
cumulative_another_hep_job_interference 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 => /eqP; 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 j 0 t =
cumulative_other_hep_jobs_interfering_workload j
0 t
cumulative_other_hep_jobs_interfering_workload j 0 t +
workload_of_jobs pred0 (arrivals_between arr_seq 0 t) =
cumulative_another_hep_job_interference 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 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 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 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 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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
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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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__copy : Job H_job_cost_positive : job_cost_positive j__copy H_j_arrives : arrives_in arr_seq j__copy zero_is_quiet_time : forall j : Job, quiet_time_cl j 0 t : instant T0 : cumulative_interference j__copy 0 t =
cumulative_interfering_workload j__copy 0 t jhp : Job ARR : arrives_in arr_seq jhp HP : hep_job jhp j__copy ARB : arrived_before jhp t COMP : completed_by sched j__copy t IN : j__copy \in arrivals_between arr_seq 0 t
hep_job j__copy j__copy
by eapply (H_priority_is_reflexive 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 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 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); last by done .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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 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 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); last by done .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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 j 0 t +
workload_of_jobs (eq_op j)
(arrivals_between arr_seq 0 t) =
cumulative_another_hep_job_interference 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 => /eqP; 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 j 0 t =
cumulative_other_hep_jobs_interfering_workload j
0 t
cumulative_other_hep_jobs_interfering_workload j 0 t +
workload_of_jobs (eq_op j)
(arrivals_between arr_seq 0 t) =
cumulative_another_hep_job_interference 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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); rt_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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.
} 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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
quiet_time arr_seq sched j 0
move => ? _ _ ; unfold arrived_before; lia .
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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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_
- 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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
t1 <= job_arrival j < 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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
t1 <= job_arrival j < t2
by done .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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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
- 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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
- 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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; rt_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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
- 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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; rt_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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
- 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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; split .
}
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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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
- 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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
- 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks quiet_time_cl := quiet_time arr_seq sched : Job -> instant -> Prop quiet_time_ab := definitions.quiet_time sched : Job -> instant -> Prop 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; rt_eauto.
}
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 Job (ideal.processor_state Job) 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 policy_reflexive : reflexive_priorities.
(** 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities 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_dec arr_seq sched j t &&
~~ another_hep_job_interference_dec j 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities 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_dec arr_seq sched j t HYP2 : ~~ another_hep_job_interference_dec j t
receives_service_at sched j t
ideal_proc_model_sched_case_analysis_eq sched t 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities 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_dec arr_seq sched j t HYP2 : ~~ another_hep_job_interference_dec j t Idle : is_idle sched t EqIdle : sched t = None
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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities 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_dec arr_seq sched j t HYP2 : ~~ another_hep_job_interference_dec j t Idle : is_idle sched t EqIdle : sched t = None
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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities 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 : is_idle sched t EqIdle : sched t = None
False
eapply instantiated_busy_interval_prefix_equivalent_busy_interval_prefix in H_busy_interval_prefix; rt_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities 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 : is_idle sched t EqIdle : sched t = None
False
by eapply not_quiet_implies_not_idle; rt_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities 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_dec arr_seq sched j t HYP2 : ~~ another_hep_job_interference_dec j t jo : Job Sched_jo : scheduled_at sched jo t EqSched_jo : #|[pred x |
let
'FiniteQuant .Quantified F :=
FiniteQuant.ex (T:=Core)
(, scheduled_on jo (sched t) x) x
x in F]| <> 0
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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities 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_dec arr_seq sched j t HYP2 : ~~ another_hep_job_interference_dec j t jo : Job Sched_jo : scheduled_at sched jo t EqSched_jo : #|[pred x |
let
'FiniteQuant .Quantified F :=
FiniteQuant.ex (T:=Core)
(, scheduled_on jo (sched t) x) x
x in F]| <> 0
receives_service_at sched j t
move : HYP1 => /priority_inversion_negP PINV; feed_n 3 PINV; rt_auto.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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities 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 HYP2 : ~~ another_hep_job_interference_dec j t jo : Job Sched_jo : scheduled_at sched jo t EqSched_jo : #|[pred x |
let
'FiniteQuant .Quantified F :=
FiniteQuant.ex (T:=Core)
(, scheduled_on jo (sched t) x) x
x in F]| <> 0 PINV : ~ priority_inversion sched j t
receives_service_at sched j t
move : HYP2 => /another_hep_job_interference_negP 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities 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 jo : Job Sched_jo : scheduled_at sched jo t EqSched_jo : #|[pred x |
let
'FiniteQuant .Quantified F :=
FiniteQuant.ex (T:=Core)
(, scheduled_on jo (sched t) x) x
x in F]| <> 0 PINV : ~ priority_inversion sched j t INT : ~ another_hep_job_interference j t
receives_service_at sched j t
eapply iffLRn in INT; last apply interference_ahep_equiv_ahep; rt_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities 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 jo : Job Sched_jo : scheduled_at sched jo t EqSched_jo : #|[pred x |
let
'FiniteQuant .Quantified F :=
FiniteQuant.ex (T:=Core)
(, scheduled_on jo (sched t) x) x
x in F]| <> 0 PINV : ~ priority_inversion sched j t INT : ~ another_hep_job jo j
receives_service_at sched j t
move : INT => /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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities 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 jo : Job Sched_jo : scheduled_at sched jo t EqSched_jo : #|[pred x |
let
'FiniteQuant .Quantified F :=
FiniteQuant.ex (T:=Core)
(, scheduled_on jo (sched t) x) x
x in F]| <> 0 PINV : ~ priority_inversion 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities 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 jo : Job Sched_jo : scheduled_at sched jo t EqSched_jo : #|[pred x |
let
'FiniteQuant .Quantified F :=
FiniteQuant.ex (T:=Core)
(, scheduled_on jo (sched t) x) x
x in F]| <> 0 PINV : ~ priority_inversion 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities 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 jo : Job Sched_jo : scheduled_at sched jo t EqSched_jo : #|[pred x |
let
'FiniteQuant .Quantified F :=
FiniteQuant.ex (T:=Core)
(, scheduled_on jo (sched t) x) x
x in F]| <> 0 PINV : ~ priority_inversion 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities 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 jo : Job Sched_jo : scheduled_at sched jo t EqSched_jo : #|[pred x |
let
'FiniteQuant .Quantified F :=
FiniteQuant.ex (T:=Core)
(, scheduled_on jo (sched t) x) x
x in F]| <> 0 PINV : ~ priority_inversion sched j t NHEP : ~~ hep_job jo j ZS : service_at sched j t = 0
False
apply no_service_not_scheduled in ZS; rt_auto; 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities 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 jo : Job Sched_jo : scheduled_at sched jo t EqSched_jo : #|[pred x |
let
'FiniteQuant .Quantified F :=
FiniteQuant.ex (T:=Core)
(, scheduled_on jo (sched t) x) x
x in F]| <> 0 NHEP : ~~ hep_job jo j ZS : ~~ scheduled_at sched j t
priority_inversion sched j t
by split ; rt_auto; exists jo ; apply /andP; split ; rt_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities 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 jo : Job Sched_jo : scheduled_at sched jo t EqSched_jo : #|[pred x |
let
'FiniteQuant .Quantified F :=
FiniteQuant.ex (T:=Core)
(, scheduled_on jo (sched t) x) x
x in F]| <> 0 PINV : ~ priority_inversion sched j t EQ : ~~ (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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities 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 jo : Job Sched_jo : scheduled_at sched jo t EqSched_jo : #|[pred x |
let
'FiniteQuant .Quantified F :=
FiniteQuant.ex (T:=Core)
(, scheduled_on jo (sched t) x) x
x in F]| <> 0 PINV : ~ priority_inversion 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities 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 EqSched_jo : #|[pred x |
let
'FiniteQuant .Quantified F :=
FiniteQuant.ex (T:=Core)
(, scheduled_on j (sched t) x) x x
in F]| <> 0 Sched_jo : scheduled_at sched j t PINV : ~ priority_inversion 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities 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|/another_hep_job_interference_P 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities 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_dec 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities 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_dec arr_seq sched j t
False
rewrite (sched_hep_implies_no_priority_inversion _ _ _ _ j) in PINV; rt_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities 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 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities 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 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities 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 j t RSERV : scheduled_at sched j t
False
by apply interference_ahep_job_eq_false in RSERV.
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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities
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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities
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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities 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
by eapply (not_interference_implies_scheduled j ARR POS); rt_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities 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
- 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities 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
by apply (scheduled_implies_no_interference j t ).
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.
(** 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities H_policy_respects_sequential_tasks : policy_respects_sequential_tasks
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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities H_policy_respects_sequential_tasks : policy_respects_sequential_tasks
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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities H_policy_respects_sequential_tasks : policy_respects_sequential_tasks 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; rt_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities H_policy_respects_sequential_tasks : policy_respects_sequential_tasks 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; rt_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities H_policy_respects_sequential_tasks : policy_respects_sequential_tasks 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities H_policy_respects_sequential_tasks : policy_respects_sequential_tasks 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities H_policy_respects_sequential_tasks : policy_respects_sequential_tasks 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; rt_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities H_policy_respects_sequential_tasks : policy_respects_sequential_tasks 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities H_policy_respects_sequential_tasks : policy_respects_sequential_tasks 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities H_policy_respects_sequential_tasks : policy_respects_sequential_tasks 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; try done ; first by eapply in_arrivals_implies_arrived; eauto 2 .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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities H_policy_respects_sequential_tasks : policy_respects_sequential_tasks 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities H_policy_respects_sequential_tasks : policy_respects_sequential_tasks 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 | done ].
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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities H_policy_respects_sequential_tasks : policy_respects_sequential_tasks H5 : 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities H_policy_respects_sequential_tasks : policy_respects_sequential_tasks H5 : 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities H_policy_respects_sequential_tasks : policy_respects_sequential_tasks H5 : 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 exists_busy_interval_from_total_workload_bound
with (Δ := L) as [t1 [t2 [T1 [T2 GGG]]]]; rt_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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities H_policy_respects_sequential_tasks : policy_respects_sequential_tasks H5 : 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,
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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities H_policy_respects_sequential_tasks : policy_respects_sequential_tasks H5 : 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,
workload_of_jobs predT
(arrivals_between arr_seq t (t + L)) <= L
move => t; rewrite {2 }H_fixed_point; apply total_workload_le_total_rbf; try by done . } 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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities H_policy_respects_sequential_tasks : policy_respects_sequential_tasks H5 : 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 ; first by done .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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities H_policy_respects_sequential_tasks : policy_respects_sequential_tasks H5 : 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
t2 <= t1 + L /\
definitions.busy_interval sched j t1 t2
split ; first by done .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 H3 : JLFP_policy Job H_priority_is_reflexive : reflexive_priorities H_priority_is_transitive : transitive_priorities tsk : Task H_sequential_tasks : sequential_tasks arr_seq sched H_JLFP_respects_sequential_tasks : policy_respects_sequential_tasks H4 : 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 policy_reflexive : reflexive_priorities H_policy_respects_sequential_tasks : policy_respects_sequential_tasks H5 : 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_hep_job_interference_dec
another_task_hep_job_interference
another_task_hep_job_interference_dec
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.