Built with
Alectryon , running Coq+SerAPI v8.14.0+0.14.0. Bubbles (
) indicate interactive fragments: hover for details, tap to reveal contents. Use
Ctrl+↑ Ctrl+↓ to navigate,
Ctrl+🖱️ to focus. On Mac, use
⌘ instead of
Ctrl .
Require Import prosa.model.priority.edf.Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]Notation "_ + _" was already used in scope nat_scope.
[notation-overridden,parsing]Notation "_ - _" was already used in scope nat_scope.
[notation-overridden,parsing]Notation "_ <= _" was already used in scope nat_scope.
[notation-overridden,parsing]Notation "_ < _" was already used in scope nat_scope.
[notation-overridden,parsing]Notation "_ >= _" was already used in scope nat_scope.
[notation-overridden,parsing]Notation "_ > _" was already used in scope nat_scope.
[notation-overridden,parsing]Notation "_ <= _ <= _" was already used in scope
nat_scope. [notation-overridden,parsing]Notation "_ < _ <= _" was already used in scope
nat_scope. [notation-overridden,parsing]Notation "_ <= _ < _" was already used in scope
nat_scope. [notation-overridden,parsing]Notation "_ < _ < _" was already used in scope
nat_scope. [notation-overridden,parsing]Notation "_ * _" was already used in scope nat_scope.
[notation-overridden,parsing]
Require Export prosa.analysis.facts.model.rbf.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.model.sequential.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.results.edf.rta.bounded_pi.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.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]
(** 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]
(** ... and the classic (i.e., Liu & Layland) model of readiness
without jitter or self-suspensions, wherein pending jobs are
always ready. *)
Require Import prosa.model.readiness.basic.Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
(** * RTA for EDF with Bounded Non-Preemptive Segments *)
(** In this section we instantiate the Abstract RTA for EDF-schedulers
with Bounded Priority Inversion to EDF-schedulers for ideal
uni-processor model of real-time tasks with arbitrary
arrival models _and_ bounded non-preemptive segments. *)
(** Recall that Abstract RTA for EDF-schedulers with Bounded Priority
Inversion does not specify the cause of priority inversion. In
this section, we prove that the priority inversion caused by
execution of non-preemptive segments is bounded. Thus the Abstract
RTA for EDF-schedulers is applicable to this instantiation. *)
Section RTAforEDFwithBoundedNonpreemptiveSegmentsWithArrivalCurves .
(** Consider any type of tasks ... *)
Context {Task : TaskType}.
Context `{TaskCost Task}.
Context `{TaskDeadline Task}.
Context `{TaskRunToCompletionThreshold Task}.
Context `{TaskMaxNonpreemptiveSegment Task}.
(** ... and any type of jobs associated with these tasks. *)
Context {Job : JobType}.
Context `{JobTask Job Task}.
Context `{Arrival : JobArrival Job}.
Context `{Cost : JobCost Job}.
(** For clarity, let's denote the relative deadline of a task as [D]. *)
Let D tsk := task_deadline tsk.
(** Consider the EDF policy that indicates a higher-or-equal priority relation.
Note that we do not relate the EDF policy with the scheduler. However, we
define functions for Interference and Interfering Workload that actively use
the concept of priorities. *)
Let EDF := EDF Job.
(** Consider any arrival sequence with consistent, non-duplicate arrivals. *)
Variable arr_seq : arrival_sequence Job.
Hypothesis H_arrival_times_are_consistent : consistent_arrival_times arr_seq.
Hypothesis H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq.
(** Next, consider any valid ideal uni-processor schedule of this arrival sequence ... *)
Variable sched : schedule (ideal.processor_state Job).
Hypothesis H_sched_valid : valid_schedule sched arr_seq.
(** In addition, we assume the existence of a function mapping jobs
to their preemption points ... *)
Context `{JobPreemptable Job}.
(** ... and assume that it defines a valid preemption model with
bounded non-preemptive segments. *)
Hypothesis H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched.
(** Next, we assume that the schedule is a work-conserving schedule... *)
Hypothesis H_work_conserving : work_conserving arr_seq sched.
(** ... and the schedule respects the policy defined by the [job_preemptable]
function (i.e., jobs have bounded non-preemptive segments). *)
Hypothesis H_respects_policy : respects_policy_at_preemption_point arr_seq sched.
(** Consider an arbitrary task set ts, ... *)
Variable ts : list Task.
(** ... assume that all jobs come from the task set, ... *)
Hypothesis H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts.
(** ... and the cost of a job cannot be larger than the task cost. *)
Hypothesis H_valid_job_cost :
arrivals_have_valid_job_costs arr_seq.
(** Let max_arrivals be a family of valid arrival curves, i.e., for
any task [tsk] in ts [max_arrival tsk] is (1) an arrival bound of
[tsk], and (2) it is a monotonic function that equals 0 for the
empty interval delta = 0. *)
Context `{MaxArrivals Task}.
Hypothesis H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals.
Hypothesis H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts.
(** Let [tsk] be any task in ts that is to be analyzed. *)
Variable tsk : Task.
Hypothesis H_tsk_in_ts : tsk \in ts.
(** Consider a valid preemption model... *)
Hypothesis H_valid_preemption_model :
valid_preemption_model arr_seq sched.
(** ...and a valid task run-to-completion threshold function. That
is, [task_rtct tsk] is (1) no bigger than [tsk]'s cost, (2) for
any job of task [tsk] [job_rtct] is bounded by [task_rtct]. *)
Hypothesis H_valid_run_to_completion_threshold :
valid_task_run_to_completion_threshold arr_seq tsk.
(** We introduce as an abbreviation [rbf] for the task request bound function,
which is defined as [task_cost(T) × max_arrivals(T,Δ)] for a task T. *)
Let rbf := task_request_bound_function.
(** Next, we introduce [task_rbf] as an abbreviation for the task
request bound function of task [tsk]. *)
Let task_rbf := rbf tsk.
(** Using the sum of individual request bound functions, we define the request bound
function of all tasks (total request bound function). *)
Let total_rbf := total_request_bound_function ts.
(** Next, we define an upper bound on interfering workload received from jobs
of other tasks with higher-than-or-equal priority. *)
Let bound_on_total_hep_workload A Δ :=
\sum_(tsk_o <- ts | tsk_o != tsk)
rbf tsk_o (minn ((A + ε) + D tsk - D tsk_o) Δ).
(** Let's define some local names for clarity. *)
Let max_length_of_priority_inversion :=
max_length_of_priority_inversion arr_seq.
Let task_rbf_changes_at A := task_rbf_changes_at tsk A.
Let bound_on_total_hep_workload_changes_at :=
bound_on_total_hep_workload_changes_at ts tsk.
Let response_time_bounded_by := task_response_time_bound arr_seq sched.
Let is_in_search_space := is_in_search_space ts tsk.
(** We also define a bound for the priority inversion caused by jobs with lower priority. *)
Definition blocking_bound :=
\max_(tsk_o <- ts | (tsk_o != tsk) && (D tsk < D tsk_o))
(task_max_nonpreemptive_segment tsk_o - ε).
(** ** Priority inversion is bounded *)
(** In this section, we prove that a priority inversion for task [tsk] is bounded by
the maximum length of non-preemptive segments among the tasks with lower priority. *)
Section PriorityInversionIsBounded .
(** First, we prove that the maximum length of a priority
inversion of job j is bounded by the maximum length of a
non-preemptive section of a task with lower-priority task
(i.e., the blocking term). *)
Lemma priority_inversion_is_bounded_by_blocking :
forall j t ,
arrives_in arr_seq j ->
job_task j = tsk ->
t <= job_arrival j ->
max_length_of_priority_inversion j t <= blocking_bound.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool
forall (j : Job) (t : nat),
arrives_in arr_seq j ->
job_task j = tsk ->
t <= job_arrival j ->
max_length_of_priority_inversion j t <= blocking_bound
Proof .Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool
forall (j : Job) (t : nat),
arrives_in arr_seq j ->
job_task j = tsk ->
t <= job_arrival j ->
max_length_of_priority_inversion j t <= blocking_bound
intros j t ARR TSK LE; unfold max_length_of_priority_inversion, blocking_bound.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk LE : t <= job_arrival j
priority_inversion.max_length_of_priority_inversion
arr_seq j t <=
\max_(tsk_o <- ts | (tsk_o != tsk) &&
(D tsk < D tsk_o))
(task_max_nonpreemptive_segment tsk_o - ε)
apply leq_trans with
(\max_(j_lp <- arrivals_between arr_seq 0 t | ~~ EDF j_lp j)
(task_max_nonpreemptive_segment (job_task j_lp) - ε)).Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk LE : t <= job_arrival j
priority_inversion.max_length_of_priority_inversion
arr_seq j t <=
\max_(j_lp <- arrivals_between arr_seq 0 t | ~~
EDF j_lp
j)
(task_max_nonpreemptive_segment (job_task j_lp) - ε)
- Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk LE : t <= job_arrival j
priority_inversion.max_length_of_priority_inversion
arr_seq j t <=
\max_(j_lp <- arrivals_between arr_seq 0 t | ~~
EDF j_lp
j)
(task_max_nonpreemptive_segment (job_task j_lp) - ε)
apply leq_big_max.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk LE : t <= job_arrival j
forall x : Job,
x \in arrivals_between arr_seq 0 t ->
~~ EDF x j ->
job_max_nonpreemptive_segment x - ε <=
task_max_nonpreemptive_segment (job_task x) - ε
intros j' JINB NOTHEP.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk LE : t <= job_arrival j j' : Job JINB : j' \in arrivals_between arr_seq 0 t NOTHEP : ~~ EDF j' j
job_max_nonpreemptive_segment j' - ε <=
task_max_nonpreemptive_segment (job_task j') - ε
rewrite leq_sub2r //.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk LE : t <= job_arrival j j' : Job JINB : j' \in arrivals_between arr_seq 0 t NOTHEP : ~~ EDF j' j
job_max_nonpreemptive_segment j' <=
task_max_nonpreemptive_segment (job_task j')
apply in_arrivals_implies_arrived in JINB.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk LE : t <= job_arrival j j' : Job JINB : arrives_in arr_seq j' NOTHEP : ~~ EDF j' j
job_max_nonpreemptive_segment j' <=
task_max_nonpreemptive_segment (job_task j')
by apply H_valid_model_with_bounded_nonpreemptive_segments.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk LE : t <= job_arrival j
\max_(j_lp <- arrivals_between arr_seq 0 t | ~~
EDF j_lp
j)
(task_max_nonpreemptive_segment (job_task j_lp) - ε) <=
\max_(tsk_o <- ts | (tsk_o != tsk) &&
(D tsk < D tsk_o))
(task_max_nonpreemptive_segment tsk_o - ε)
- Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk LE : t <= job_arrival j
\max_(j_lp <- arrivals_between arr_seq 0 t | ~~
EDF j_lp
j)
(task_max_nonpreemptive_segment (job_task j_lp) - ε) <=
\max_(tsk_o <- ts | (tsk_o != tsk) &&
(D tsk < D tsk_o))
(task_max_nonpreemptive_segment tsk_o - ε)
apply /bigmax_leq_seqP.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk LE : t <= job_arrival j
forall x : Job,
x \in arrivals_between arr_seq 0 t ->
~~ EDF x j ->
task_max_nonpreemptive_segment (job_task x) - ε <=
\max_(tsk_o <- ts | (tsk_o != tsk) &&
(D tsk < D tsk_o))
(task_max_nonpreemptive_segment tsk_o - ε)
intros j' JINB NOTHEP.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk LE : t <= job_arrival j j' : Job JINB : j' \in arrivals_between arr_seq 0 t NOTHEP : ~~ EDF j' j
task_max_nonpreemptive_segment (job_task j') - ε <=
\max_(tsk_o <- ts | (tsk_o != tsk) &&
(D tsk < D tsk_o))
(task_max_nonpreemptive_segment tsk_o - ε)
apply leq_bigmax_cond_seq with (x := (job_task j')) (F := fun tsk => task_max_nonpreemptive_segment tsk - 1 ).Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk LE : t <= job_arrival j j' : Job JINB : j' \in arrivals_between arr_seq 0 t NOTHEP : ~~ EDF j' j
job_task j' \in ts
{ Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk LE : t <= job_arrival j j' : Job JINB : j' \in arrivals_between arr_seq 0 t NOTHEP : ~~ EDF j' j
job_task j' \in ts
apply H_all_jobs_from_taskset.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk LE : t <= job_arrival j j' : Job JINB : j' \in arrivals_between arr_seq 0 t NOTHEP : ~~ EDF j' j
arrives_in arr_seq j'
apply mem_bigcat_nat_exists in JINB.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk LE : t <= job_arrival j j' : Job JINB : exists i : nat,
j' \in arrivals_at arr_seq i /\ 0 <= i < tNOTHEP : ~~ EDF j' j
arrives_in arr_seq j'
by inversion JINB as [ta' [JIN' _]]; exists ta' . } Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk LE : t <= job_arrival j j' : Job JINB : j' \in arrivals_between arr_seq 0 t NOTHEP : ~~ EDF j' j
(job_task j' != tsk) && (D tsk < D (job_task j'))
{ Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk LE : t <= job_arrival j j' : Job JINB : j' \in arrivals_between arr_seq 0 t NOTHEP : ~~ EDF j' j
(job_task j' != tsk) && (D tsk < D (job_task j'))
have NINTSK: job_task j' != tsk.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk LE : t <= job_arrival j j' : Job JINB : j' \in arrivals_between arr_seq 0 t NOTHEP : ~~ EDF j' j
job_task j' != tsk
{ Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk LE : t <= job_arrival j j' : Job JINB : j' \in arrivals_between arr_seq 0 t NOTHEP : ~~ EDF j' j
job_task j' != tsk
apply /eqP; intros TSKj'.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk LE : t <= job_arrival j j' : Job JINB : j' \in arrivals_between arr_seq 0 t NOTHEP : ~~ EDF j' j TSKj' : job_task j' = tsk
False
rewrite /EDF -ltnNge in NOTHEP.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk LE : t <= job_arrival j j' : Job JINB : j' \in arrivals_between arr_seq 0 t TSKj' : job_task j' = tsk NOTHEP : job_deadline j < job_deadline j'
False
rewrite /job_deadline /absolute_deadline.job_deadline_from_task_deadline in NOTHEP.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk LE : t <= job_arrival j j' : Job JINB : j' \in arrivals_between arr_seq 0 t TSKj' : job_task j' = tsk NOTHEP : job_arrival j + task_deadline (job_task j) <
job_arrival j' + task_deadline (job_task j')
False
rewrite TSKj' TSK ltn_add2r in NOTHEP.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk LE : t <= job_arrival j j' : Job JINB : j' \in arrivals_between arr_seq 0 t TSKj' : job_task j' = tsk NOTHEP : job_arrival j < job_arrival j'
False
move : NOTHEP; rewrite ltnNge; move => /negP T; apply : T.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk LE : t <= job_arrival j j' : Job JINB : j' \in arrivals_between arr_seq 0 t TSKj' : job_task j' = tsk
job_arrival j' <= job_arrival j
apply leq_trans with t; last by done .Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk LE : t <= job_arrival j j' : Job JINB : j' \in arrivals_between arr_seq 0 t TSKj' : job_task j' = tsk
job_arrival j' <= t
eapply in_arrivals_implies_arrived_between in JINB; last by eauto 2 .Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk LE : t <= job_arrival j j' : Job JINB : arrived_between j' 0 t TSKj' : job_task j' = tsk
job_arrival j' <= t
move : JINB; move => /andP [_ T].Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk LE : t <= job_arrival j j' : Job TSKj' : job_task j' = tsk T : job_arrival j' < t
job_arrival j' <= t
by apply ltnW.
} Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk LE : t <= job_arrival j j' : Job JINB : j' \in arrivals_between arr_seq 0 t NOTHEP : ~~ EDF j' j NINTSK : job_task j' != tsk
(job_task j' != tsk) && (D tsk < D (job_task j'))
apply /andP; split ; first by done .Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk LE : t <= job_arrival j j' : Job JINB : j' \in arrivals_between arr_seq 0 t NOTHEP : ~~ EDF j' j NINTSK : job_task j' != tsk
D tsk < D (job_task j')
rewrite /EDF -ltnNge in NOTHEP.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk LE : t <= job_arrival j j' : Job JINB : j' \in arrivals_between arr_seq 0 t NINTSK : job_task j' != tsk NOTHEP : job_deadline j < job_deadline j'
D tsk < D (job_task j')
rewrite -TSK.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk LE : t <= job_arrival j j' : Job JINB : j' \in arrivals_between arr_seq 0 t NINTSK : job_task j' != tsk NOTHEP : job_deadline j < job_deadline j'
D (job_task j) < D (job_task j')
have ARRLE: job_arrival j' < job_arrival j.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk LE : t <= job_arrival j j' : Job JINB : j' \in arrivals_between arr_seq 0 t NINTSK : job_task j' != tsk NOTHEP : job_deadline j < job_deadline j'
job_arrival j' < job_arrival j
{ Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk LE : t <= job_arrival j j' : Job JINB : j' \in arrivals_between arr_seq 0 t NINTSK : job_task j' != tsk NOTHEP : job_deadline j < job_deadline j'
job_arrival j' < job_arrival j
apply leq_trans with t; last by done .Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk LE : t <= job_arrival j j' : Job JINB : j' \in arrivals_between arr_seq 0 t NINTSK : job_task j' != tsk NOTHEP : job_deadline j < job_deadline j'
job_arrival j' < t
eapply in_arrivals_implies_arrived_between in JINB; last by eauto 2 .Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk LE : t <= job_arrival j j' : Job JINB : arrived_between j' 0 t NINTSK : job_task j' != tsk NOTHEP : job_deadline j < job_deadline j'
job_arrival j' < t
by move : JINB; move => /andP [_ T].
} Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk LE : t <= job_arrival j j' : Job JINB : j' \in arrivals_between arr_seq 0 t NINTSK : job_task j' != tsk NOTHEP : job_deadline j < job_deadline j' ARRLE : job_arrival j' < job_arrival j
D (job_task j) < D (job_task j')
rewrite /job_deadline /absolute_deadline.job_deadline_from_task_deadline in NOTHEP.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk LE : t <= job_arrival j j' : Job JINB : j' \in arrivals_between arr_seq 0 t NINTSK : job_task j' != tsk ARRLE : job_arrival j' < job_arrival j NOTHEP : job_arrival j + task_deadline (job_task j) <
job_arrival j' + task_deadline (job_task j')
D (job_task j) < D (job_task j')
rewrite /D; ssrlia.
}
Qed .
(** Using the lemma above, we prove that the priority inversion of the task is bounded by
the maximum length of a nonpreemptive section of lower-priority tasks. *)
Lemma priority_inversion_is_bounded :
priority_inversion_is_bounded_by arr_seq sched tsk blocking_bound.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool
priority_inversion_is_bounded_by arr_seq sched tsk
blocking_bound
Proof .Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool
priority_inversion_is_bounded_by arr_seq sched tsk
blocking_bound
move => j ARR TSK POS t1 t2 PREF; move : (PREF) => [_ [_ [_ /andP [T _]]]].Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jt1, t2 : instant PREF : busy_interval_prefix arr_seq sched j t1 t2 T : t1 <= job_arrival j
cumulative_priority_inversion sched j t1 t2 <=
blocking_bound
move : H_sched_valid => [COARR MBR].Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jt1, t2 : instant PREF : busy_interval_prefix arr_seq sched j t1 t2 T : t1 <= job_arrival j COARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched
cumulative_priority_inversion sched j t1 t2 <=
blocking_bound
destruct (leqP (t2 - t1) blocking_bound) as [NEQ|NEQ].Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jt1, t2 : instant PREF : busy_interval_prefix arr_seq sched j t1 t2 T : t1 <= job_arrival j COARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched NEQ : t2 - t1 <= blocking_bound
cumulative_priority_inversion sched j t1 t2 <=
blocking_bound
{ Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jt1, t2 : instant PREF : busy_interval_prefix arr_seq sched j t1 t2 T : t1 <= job_arrival j COARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched NEQ : t2 - t1 <= blocking_bound
cumulative_priority_inversion sched j t1 t2 <=
blocking_bound
apply leq_trans with (t2 - t1); last by done .Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jt1, t2 : instant PREF : busy_interval_prefix arr_seq sched j t1 t2 T : t1 <= job_arrival j COARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched NEQ : t2 - t1 <= blocking_bound
cumulative_priority_inversion sched j t1 t2 <= t2 - t1
rewrite /cumulative_priority_inversion /is_priority_inversion.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jt1, t2 : instant PREF : busy_interval_prefix arr_seq sched j t1 t2 T : t1 <= job_arrival j COARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched NEQ : t2 - t1 <= blocking_bound
\sum_(t1 <= t < t2)
match sched t with
| Some jlp => ~~ hep_job jlp j
| None => false
end <= t2 - t1
rewrite -[X in _ <= X]addn0 -[t2 - t1]mul1n -iter_addn -big_const_nat.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jt1, t2 : instant PREF : busy_interval_prefix arr_seq sched j t1 t2 T : t1 <= job_arrival j COARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched NEQ : t2 - t1 <= blocking_bound
\sum_(t1 <= t < t2)
match sched t with
| Some jlp => ~~ hep_job jlp j
| None => false
end <= \sum_(t1 <= i < t2) 1
rewrite leq_sum //.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jt1, t2 : instant PREF : busy_interval_prefix arr_seq sched j t1 t2 T : t1 <= job_arrival j COARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched NEQ : t2 - t1 <= blocking_bound
forall i : nat,
true ->
match sched i with
| Some jlp => ~~ hep_job jlp j
| None => false
end <= 1
intros t _; case : (sched t); last by done .Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jt1, t2 : instant PREF : busy_interval_prefix arr_seq sched j t1 t2 T : t1 <= job_arrival j COARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched NEQ : t2 - t1 <= blocking_bound t : nat
forall a : Job, ~~ hep_job a j <= 1
by intros s; destruct (hep_job s j).
} Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jt1, t2 : instant PREF : busy_interval_prefix arr_seq sched j t1 t2 T : t1 <= job_arrival j COARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched NEQ : blocking_bound < t2 - t1
cumulative_priority_inversion sched j t1 t2 <=
blocking_bound
edestruct @preemption_time_exists as [ppt [PPT NEQ2]]; eauto 2 with basic_facts.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jt1, t2 : instant PREF : busy_interval_prefix arr_seq sched j t1 t2 T : t1 <= job_arrival j COARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched NEQ : blocking_bound < t2 - t1 ppt : instant PPT : preemption_time sched ppt NEQ2 : t1 <= ppt <=
t1 +
priority_inversion.max_length_of_priority_inversion
arr_seq j t1
cumulative_priority_inversion sched j t1 t2 <=
blocking_bound
move : NEQ2 => /andP [GE LE].Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jt1, t2 : instant PREF : busy_interval_prefix arr_seq sched j t1 t2 T : t1 <= job_arrival j COARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched NEQ : blocking_bound < t2 - t1 ppt : instant PPT : preemption_time sched ppt GE : t1 <= ppt LE : ppt <=
t1 +
priority_inversion.max_length_of_priority_inversion
arr_seq j t1
cumulative_priority_inversion sched j t1 t2 <=
blocking_bound
apply leq_trans with (cumulative_priority_inversion sched j t1 ppt);
last apply leq_trans with (ppt - t1).Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jt1, t2 : instant PREF : busy_interval_prefix arr_seq sched j t1 t2 T : t1 <= job_arrival j COARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched NEQ : blocking_bound < t2 - t1 ppt : instant PPT : preemption_time sched ppt GE : t1 <= ppt LE : ppt <=
t1 +
priority_inversion.max_length_of_priority_inversion
arr_seq j t1
cumulative_priority_inversion sched j t1 t2 <=
cumulative_priority_inversion sched j t1 ppt
- Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jt1, t2 : instant PREF : busy_interval_prefix arr_seq sched j t1 t2 T : t1 <= job_arrival j COARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched NEQ : blocking_bound < t2 - t1 ppt : instant PPT : preemption_time sched ppt GE : t1 <= ppt LE : ppt <=
t1 +
priority_inversion.max_length_of_priority_inversion
arr_seq j t1
cumulative_priority_inversion sched j t1 t2 <=
cumulative_priority_inversion sched j t1 ppt
rewrite /cumulative_priority_inversion /is_priority_inversion.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jt1, t2 : instant PREF : busy_interval_prefix arr_seq sched j t1 t2 T : t1 <= job_arrival j COARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched NEQ : blocking_bound < t2 - t1 ppt : instant PPT : preemption_time sched ppt GE : t1 <= ppt LE : ppt <=
t1 +
priority_inversion.max_length_of_priority_inversion
arr_seq j t1
\sum_(t1 <= t < t2)
match sched t with
| Some jlp => ~~ hep_job jlp j
| None => false
end <=
\sum_(t1 <= t < ppt)
match sched t with
| Some jlp => ~~ hep_job jlp j
| None => false
end
rewrite (@big_cat_nat _ _ _ ppt) //=; last first .Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jt1, t2 : instant PREF : busy_interval_prefix arr_seq sched j t1 t2 T : t1 <= job_arrival j COARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched NEQ : blocking_bound < t2 - t1 ppt : instant PPT : preemption_time sched ppt GE : t1 <= ppt LE : ppt <=
t1 +
priority_inversion.max_length_of_priority_inversion
arr_seq j t1
ppt <= t2
{ Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jt1, t2 : instant PREF : busy_interval_prefix arr_seq sched j t1 t2 T : t1 <= job_arrival j COARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched NEQ : blocking_bound < t2 - t1 ppt : instant PPT : preemption_time sched ppt GE : t1 <= ppt LE : ppt <=
t1 +
priority_inversion.max_length_of_priority_inversion
arr_seq j t1
ppt <= t2
rewrite ltn_subRL in NEQ.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jt1, t2 : instant PREF : busy_interval_prefix arr_seq sched j t1 t2 T : t1 <= job_arrival j COARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched ppt : instant PPT : preemption_time sched ppt GE : t1 <= ppt LE : ppt <=
t1 +
priority_inversion.max_length_of_priority_inversion
arr_seq j t1 NEQ : t1 + blocking_bound < t2
ppt <= t2
apply leq_trans with (t1 + blocking_bound); last by apply ltnW.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jt1, t2 : instant PREF : busy_interval_prefix arr_seq sched j t1 t2 T : t1 <= job_arrival j COARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched ppt : instant PPT : preemption_time sched ppt GE : t1 <= ppt LE : ppt <=
t1 +
priority_inversion.max_length_of_priority_inversion
arr_seq j t1 NEQ : t1 + blocking_bound < t2
ppt <= t1 + blocking_bound
apply leq_trans with (t1 + max_length_of_priority_inversion j t1); first by done .Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jt1, t2 : instant PREF : busy_interval_prefix arr_seq sched j t1 t2 T : t1 <= job_arrival j COARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched ppt : instant PPT : preemption_time sched ppt GE : t1 <= ppt LE : ppt <=
t1 +
priority_inversion.max_length_of_priority_inversion
arr_seq j t1 NEQ : t1 + blocking_bound < t2
t1 + max_length_of_priority_inversion j t1 <=
t1 + blocking_bound
by rewrite leq_add2l; eapply priority_inversion_is_bounded_by_blocking; eauto 2 . } Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jt1, t2 : instant PREF : busy_interval_prefix arr_seq sched j t1 t2 T : t1 <= job_arrival j COARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched NEQ : blocking_bound < t2 - t1 ppt : instant PPT : preemption_time sched ppt GE : t1 <= ppt LE : ppt <=
t1 +
priority_inversion.max_length_of_priority_inversion
arr_seq j t1
\sum_(t1 <= i < ppt)
match sched i with
| Some jlp => ~~ hep_job jlp j
| None => false
end +
\sum_(ppt <= i < t2)
match sched i with
| Some jlp => ~~ hep_job jlp j
| None => false
end <=
\sum_(t1 <= t < ppt)
match sched t with
| Some jlp => ~~ hep_job jlp j
| None => false
end
rewrite -[X in _ <= X]addn0 leq_add2l leqn0.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jt1, t2 : instant PREF : busy_interval_prefix arr_seq sched j t1 t2 T : t1 <= job_arrival j COARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched NEQ : blocking_bound < t2 - t1 ppt : instant PPT : preemption_time sched ppt GE : t1 <= ppt LE : ppt <=
t1 +
priority_inversion.max_length_of_priority_inversion
arr_seq j t1
\sum_(ppt <= i < t2)
match sched i with
| Some jlp => ~~ hep_job jlp j
| None => false
end == 0
rewrite big_nat_cond big1 //; move => t /andP [/andP [GEt LTt] _ ].Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jt1, t2 : instant PREF : busy_interval_prefix arr_seq sched j t1 t2 T : t1 <= job_arrival j COARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched NEQ : blocking_bound < t2 - t1 ppt : instant PPT : preemption_time sched ppt GE : t1 <= ppt LE : ppt <=
t1 +
priority_inversion.max_length_of_priority_inversion
arr_seq j t1 t : nat GEt : ppt <= t LTt : t < t2
match sched t with
| Some jlp => ~~ hep_job jlp j
| None => false
end = 0
case SCHED: (sched t) => [s | ]; last by done .Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jt1, t2 : instant PREF : busy_interval_prefix arr_seq sched j t1 t2 T : t1 <= job_arrival j COARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched NEQ : blocking_bound < t2 - t1 ppt : instant PPT : preemption_time sched ppt GE : t1 <= ppt LE : ppt <=
t1 +
priority_inversion.max_length_of_priority_inversion
arr_seq j t1 t : nat GEt : ppt <= t LTt : t < t2 s : Job SCHED : sched t = Some s
~~ hep_job s j = 0
edestruct @not_quiet_implies_exists_scheduled_hp_job
with (K := ppt - t1) (t := t) as [j_hp [ARRB [HP SCHEDHP]]]; eauto 2 with basic_facts.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jt1, t2 : instant PREF : busy_interval_prefix arr_seq sched j t1 t2 T : t1 <= job_arrival j COARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched NEQ : blocking_bound < t2 - t1 ppt : instant PPT : preemption_time sched ppt GE : t1 <= ppt LE : ppt <=
t1 +
priority_inversion.max_length_of_priority_inversion
arr_seq j t1 t : nat GEt : ppt <= t LTt : t < t2 s : Job SCHED : sched t = Some s
exists pr_t : instant,
preemption_time sched pr_t /\
t1 <= pr_t <= t1 + (ppt - t1)
{ Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jt1, t2 : instant PREF : busy_interval_prefix arr_seq sched j t1 t2 T : t1 <= job_arrival j COARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched NEQ : blocking_bound < t2 - t1 ppt : instant PPT : preemption_time sched ppt GE : t1 <= ppt LE : ppt <=
t1 +
priority_inversion.max_length_of_priority_inversion
arr_seq j t1 t : nat GEt : ppt <= t LTt : t < t2 s : Job SCHED : sched t = Some s
exists pr_t : instant,
preemption_time sched pr_t /\
t1 <= pr_t <= t1 + (ppt - t1)
exists ppt ; split .Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jt1, t2 : instant PREF : busy_interval_prefix arr_seq sched j t1 t2 T : t1 <= job_arrival j COARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched NEQ : blocking_bound < t2 - t1 ppt : instant PPT : preemption_time sched ppt GE : t1 <= ppt LE : ppt <=
t1 +
priority_inversion.max_length_of_priority_inversion
arr_seq j t1 t : nat GEt : ppt <= t LTt : t < t2 s : Job SCHED : sched t = Some s
preemption_time sched ppt
by done .Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jt1, t2 : instant PREF : busy_interval_prefix arr_seq sched j t1 t2 T : t1 <= job_arrival j COARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched NEQ : blocking_bound < t2 - t1 ppt : instant PPT : preemption_time sched ppt GE : t1 <= ppt LE : ppt <=
t1 +
priority_inversion.max_length_of_priority_inversion
arr_seq j t1 t : nat GEt : ppt <= t LTt : t < t2 s : Job SCHED : sched t = Some s
t1 <= ppt <= t1 + (ppt - t1)
by rewrite subnKC //; apply /andP; split . } Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jt1, t2 : instant PREF : busy_interval_prefix arr_seq sched j t1 t2 T : t1 <= job_arrival j COARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched NEQ : blocking_bound < t2 - t1 ppt : instant PPT : preemption_time sched ppt GE : t1 <= ppt LE : ppt <=
t1 +
priority_inversion.max_length_of_priority_inversion
arr_seq j t1 t : nat GEt : ppt <= t LTt : t < t2 s : Job SCHED : sched t = Some s
t1 + (ppt - t1) <= t < t2
{ Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jt1, t2 : instant PREF : busy_interval_prefix arr_seq sched j t1 t2 T : t1 <= job_arrival j COARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched NEQ : blocking_bound < t2 - t1 ppt : instant PPT : preemption_time sched ppt GE : t1 <= ppt LE : ppt <=
t1 +
priority_inversion.max_length_of_priority_inversion
arr_seq j t1 t : nat GEt : ppt <= t LTt : t < t2 s : Job SCHED : sched t = Some s
t1 + (ppt - t1) <= t < t2
by rewrite subnKC //; apply /andP; split . } Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jt1, t2 : instant PREF : busy_interval_prefix arr_seq sched j t1 t2 T : t1 <= job_arrival j COARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched NEQ : blocking_bound < t2 - t1 ppt : instant PPT : preemption_time sched ppt GE : t1 <= ppt LE : ppt <=
t1 +
priority_inversion.max_length_of_priority_inversion
arr_seq j t1 t : nat GEt : ppt <= t LTt : t < t2 s : Job SCHED : sched t = Some s j_hp : Job ARRB : arrived_between j_hp t1 t.+1 HP : hep_job j_hp j SCHEDHP : scheduled_at sched j_hp t
~~ hep_job s j = 0
apply /eqP; rewrite eqb0 Bool.negb_involutive.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jt1, t2 : instant PREF : busy_interval_prefix arr_seq sched j t1 t2 T : t1 <= job_arrival j COARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched NEQ : blocking_bound < t2 - t1 ppt : instant PPT : preemption_time sched ppt GE : t1 <= ppt LE : ppt <=
t1 +
priority_inversion.max_length_of_priority_inversion
arr_seq j t1 t : nat GEt : ppt <= t LTt : t < t2 s : Job SCHED : sched t = Some s j_hp : Job ARRB : arrived_between j_hp t1 t.+1 HP : hep_job j_hp j SCHEDHP : scheduled_at sched j_hp t
hep_job s j
enough (EQ : s = j_hp); first by subst .Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jt1, t2 : instant PREF : busy_interval_prefix arr_seq sched j t1 t2 T : t1 <= job_arrival j COARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched NEQ : blocking_bound < t2 - t1 ppt : instant PPT : preemption_time sched ppt GE : t1 <= ppt LE : ppt <=
t1 +
priority_inversion.max_length_of_priority_inversion
arr_seq j t1 t : nat GEt : ppt <= t LTt : t < t2 s : Job SCHED : sched t = Some s j_hp : Job ARRB : arrived_between j_hp t1 t.+1 HP : hep_job j_hp j SCHEDHP : scheduled_at sched j_hp t
s = j_hp
move : SCHED => /eqP SCHED; rewrite -scheduled_at_def in SCHED.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jt1, t2 : instant PREF : busy_interval_prefix arr_seq sched j t1 t2 T : t1 <= job_arrival j COARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched NEQ : blocking_bound < t2 - t1 ppt : instant PPT : preemption_time sched ppt GE : t1 <= ppt LE : ppt <=
t1 +
priority_inversion.max_length_of_priority_inversion
arr_seq j t1 t : nat GEt : ppt <= t LTt : t < t2 s, j_hp : Job ARRB : arrived_between j_hp t1 t.+1 HP : hep_job j_hp j SCHEDHP : scheduled_at sched j_hp t SCHED : scheduled_at sched s t
s = j_hp
by eapply ideal_proc_model_is_a_uniprocessor_model; [exact SCHED | exact SCHEDHP].Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jt1, t2 : instant PREF : busy_interval_prefix arr_seq sched j t1 t2 T : t1 <= job_arrival j COARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched NEQ : blocking_bound < t2 - t1 ppt : instant PPT : preemption_time sched ppt GE : t1 <= ppt LE : ppt <=
t1 +
priority_inversion.max_length_of_priority_inversion
arr_seq j t1
cumulative_priority_inversion sched j t1 ppt <=
ppt - t1
- Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jt1, t2 : instant PREF : busy_interval_prefix arr_seq sched j t1 t2 T : t1 <= job_arrival j COARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched NEQ : blocking_bound < t2 - t1 ppt : instant PPT : preemption_time sched ppt GE : t1 <= ppt LE : ppt <=
t1 +
priority_inversion.max_length_of_priority_inversion
arr_seq j t1
cumulative_priority_inversion sched j t1 ppt <=
ppt - t1
rewrite /cumulative_priority_inversion /is_priority_inversion.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jt1, t2 : instant PREF : busy_interval_prefix arr_seq sched j t1 t2 T : t1 <= job_arrival j COARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched NEQ : blocking_bound < t2 - t1 ppt : instant PPT : preemption_time sched ppt GE : t1 <= ppt LE : ppt <=
t1 +
priority_inversion.max_length_of_priority_inversion
arr_seq j t1
\sum_(t1 <= t < ppt)
match sched t with
| Some jlp => ~~ hep_job jlp j
| None => false
end <= ppt - t1
rewrite -[X in _ <= X]addn0 -[ppt - t1]mul1n -iter_addn -big_const_nat.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jt1, t2 : instant PREF : busy_interval_prefix arr_seq sched j t1 t2 T : t1 <= job_arrival j COARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched NEQ : blocking_bound < t2 - t1 ppt : instant PPT : preemption_time sched ppt GE : t1 <= ppt LE : ppt <=
t1 +
priority_inversion.max_length_of_priority_inversion
arr_seq j t1
\sum_(t1 <= t < ppt)
match sched t with
| Some jlp => ~~ hep_job jlp j
| None => false
end <= \sum_(t1 <= i < ppt) 1
rewrite leq_sum //.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jt1, t2 : instant PREF : busy_interval_prefix arr_seq sched j t1 t2 T : t1 <= job_arrival j COARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched NEQ : blocking_bound < t2 - t1 ppt : instant PPT : preemption_time sched ppt GE : t1 <= ppt LE : ppt <=
t1 +
priority_inversion.max_length_of_priority_inversion
arr_seq j t1
forall i : nat,
true ->
match sched i with
| Some jlp => ~~ hep_job jlp j
| None => false
end <= 1
intros t _; case : (sched t); last by done .Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jt1, t2 : instant PREF : busy_interval_prefix arr_seq sched j t1 t2 T : t1 <= job_arrival j COARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched NEQ : blocking_bound < t2 - t1 ppt : instant PPT : preemption_time sched ppt GE : t1 <= ppt LE : ppt <=
t1 +
priority_inversion.max_length_of_priority_inversion
arr_seq j t1 t : nat
forall a : Job, ~~ hep_job a j <= 1
by intros s; destruct (hep_job s j).Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jt1, t2 : instant PREF : busy_interval_prefix arr_seq sched j t1 t2 T : t1 <= job_arrival j COARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched NEQ : blocking_bound < t2 - t1 ppt : instant PPT : preemption_time sched ppt GE : t1 <= ppt LE : ppt <=
t1 +
priority_inversion.max_length_of_priority_inversion
arr_seq j t1
ppt - t1 <= blocking_bound
- Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jt1, t2 : instant PREF : busy_interval_prefix arr_seq sched j t1 t2 T : t1 <= job_arrival j COARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched NEQ : blocking_bound < t2 - t1 ppt : instant PPT : preemption_time sched ppt GE : t1 <= ppt LE : ppt <=
t1 +
priority_inversion.max_length_of_priority_inversion
arr_seq j t1
ppt - t1 <= blocking_bound
rewrite leq_subLR.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jt1, t2 : instant PREF : busy_interval_prefix arr_seq sched j t1 t2 T : t1 <= job_arrival j COARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched NEQ : blocking_bound < t2 - t1 ppt : instant PPT : preemption_time sched ppt GE : t1 <= ppt LE : ppt <=
t1 +
priority_inversion.max_length_of_priority_inversion
arr_seq j t1
ppt <= t1 + blocking_bound
apply leq_trans with (t1 + max_length_of_priority_inversion j t1); first by done .Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jt1, t2 : instant PREF : busy_interval_prefix arr_seq sched j t1 t2 T : t1 <= job_arrival j COARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched NEQ : blocking_bound < t2 - t1 ppt : instant PPT : preemption_time sched ppt GE : t1 <= ppt LE : ppt <=
t1 +
priority_inversion.max_length_of_priority_inversion
arr_seq j t1
t1 + max_length_of_priority_inversion j t1 <=
t1 + blocking_bound
by rewrite leq_add2l; eapply priority_inversion_is_bounded_by_blocking; eauto 2 .
Qed .
End PriorityInversionIsBounded .
(** ** Response-Time Bound *)
(** In this section, we prove that the maximum among the solutions of the response-time
bound recurrence is a response-time bound for [tsk]. *)
Section ResponseTimeBound .
(** Let L be any positive fixed point of the busy interval recurrence. *)
Variable L : duration.
Hypothesis H_L_positive : L > 0 .
Hypothesis H_fixed_point : L = total_rbf L.
(** Consider any value [R], and assume that for any given arrival
offset [A] in the search space, there is a solution of the
response-time bound recurrence which is bounded by [R]. *)
Variable R : duration.
Hypothesis H_R_is_maximum :
forall (A : duration),
is_in_search_space L A ->
exists (F : duration),
A + F >= blocking_bound
+ (task_rbf (A + ε) - (task_cost tsk - task_rtct tsk))
+ bound_on_total_hep_workload A (A + F) /\
R >= F + (task_cost tsk - task_rtct tsk).
(** Then, using the results for the general RTA for EDF-schedulers, we establish a
response-time bound for the more concrete model of bounded nonpreemptive segments.
Note that in case of the general RTA for EDF-schedulers, we just _assume_ that
the priority inversion is bounded. In this module we provide the preemption model
with bounded nonpreemptive segments and _prove_ that the priority inversion is
bounded. *)
Theorem uniprocessor_response_time_bound_edf_with_bounded_nonpreemptive_segments :
response_time_bounded_by tsk R.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool L : duration H_L_positive : 0 < LH_fixed_point : L = total_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space L A ->
exists F : duration,
blocking_bound +
(task_rbf (A + ε) -
(task_cost tsk - task_rtct tsk)) +
bound_on_total_hep_workload A
(A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
R
response_time_bounded_by tsk R
Proof .Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool L : duration H_L_positive : 0 < LH_fixed_point : L = total_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space L A ->
exists F : duration,
blocking_bound +
(task_rbf (A + ε) -
(task_cost tsk - task_rtct tsk)) +
bound_on_total_hep_workload A
(A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
R
response_time_bounded_by tsk R
eapply uniprocessor_response_time_bound_edf; eauto 2 with basic_facts.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool L : duration H_L_positive : 0 < LH_fixed_point : L = total_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space L A ->
exists F : duration,
blocking_bound +
(task_rbf (A + ε) -
(task_cost tsk - task_rtct tsk)) +
bound_on_total_hep_workload A
(A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
R
sequential_tasks arr_seq sched
- Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool L : duration H_L_positive : 0 < LH_fixed_point : L = total_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space L A ->
exists F : duration,
blocking_bound +
(task_rbf (A + ε) -
(task_cost tsk - task_rtct tsk)) +
bound_on_total_hep_workload A
(A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
R
sequential_tasks arr_seq sched
eapply EDF_implies_sequential_tasks; eauto 2 .Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool L : duration H_L_positive : 0 < LH_fixed_point : L = total_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space L A ->
exists F : duration,
blocking_bound +
(task_rbf (A + ε) -
(task_cost tsk - task_rtct tsk)) +
bound_on_total_hep_workload A
(A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
R
work_bearing_readiness arr_seq sched
+ Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool L : duration H_L_positive : 0 < LH_fixed_point : L = total_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space L A ->
exists F : duration,
blocking_bound +
(task_rbf (A + ε) -
(task_cost tsk - task_rtct tsk)) +
bound_on_total_hep_workload A
(A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
R
work_bearing_readiness arr_seq sched
by apply basic.basic_readiness_is_work_bearing_readiness, EDF_is_reflexive.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool L : duration H_L_positive : 0 < LH_fixed_point : L = total_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space L A ->
exists F : duration,
blocking_bound +
(task_rbf (A + ε) -
(task_cost tsk - task_rtct tsk)) +
bound_on_total_hep_workload A
(A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
R
priority_inversion_is_bounded_by arr_seq sched tsk
blocking_bound
- Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job D := [eta task_deadline] : Task -> duration EDF := edf.EDF Job : JLFP_policy Job arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H_sched_valid : valid_schedule sched arr_seq H4 : JobPreemptable Job H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_policy_at_preemption_point
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_rbf := total_request_bound_function ts : duration -> nat bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + D tsk -
D tsk_o) Δ): nat -> nat -> nat max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq : Job ->
instant -> nat task_rbf_changes_at := [eta bounded_pi.task_rbf_changes_at
tsk] : duration -> bool bound_on_total_hep_workload_changes_at := bounded_pi.bound_on_total_hep_workload_changes_at
ts tsk : nat -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop is_in_search_space := bounded_pi.is_in_search_space ts
tsk : duration -> duration -> bool L : duration H_L_positive : 0 < LH_fixed_point : L = total_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space L A ->
exists F : duration,
blocking_bound +
(task_rbf (A + ε) -
(task_cost tsk - task_rtct tsk)) +
bound_on_total_hep_workload A
(A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
R
priority_inversion_is_bounded_by arr_seq sched tsk
blocking_bound
by apply priority_inversion_is_bounded.
Qed .
End ResponseTimeBound .
End RTAforEDFwithBoundedNonpreemptiveSegmentsWithArrivalCurves .