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Require Export prosa.util.tactics.[Loading ML file ssrmatching_plugin.cmxs (using legacy method) ... done ] [Loading ML file ssreflect_plugin.cmxs (using legacy method) ... done ] [Loading ML file ring_plugin.cmxs (using legacy method) ... done ] [Loading ML file coq-elpi.elpi ... done ]
Require Import prosa.model.readiness.basic.[Loading ML file zify_plugin.cmxs (using legacy method) ... done ] [Loading ML file micromega_plugin.cmxs (using legacy method) ... done ] [Loading ML file btauto_plugin.cmxs (using legacy method) ... done ] Notation "_ + _" was already used in scope nat_scope.
[notation-overridden,parsing,default]Notation "_ - _" was already used in scope nat_scope.
[notation-overridden,parsing,default]Notation "_ <= _" was already used in scope nat_scope.
[notation-overridden,parsing,default]Notation "_ < _" was already used in scope nat_scope.
[notation-overridden,parsing,default]Notation "_ >= _" was already used in scope nat_scope.
[notation-overridden,parsing,default]Notation "_ > _" was already used in scope nat_scope.
[notation-overridden,parsing,default]Notation "_ <= _ <= _" was already used in scope
nat_scope. [notation-overridden,parsing,default]Notation "_ < _ <= _" was already used in scope
nat_scope. [notation-overridden,parsing,default]Notation "_ <= _ < _" was already used in scope
nat_scope. [notation-overridden,parsing,default]Notation "_ < _ < _" was already used in scope
nat_scope. [notation-overridden,parsing,default]Notation "_ * _" was already used in scope nat_scope.
[notation-overridden,parsing,default]
Require Export prosa.analysis.facts.busy_interval.pi_bound.
Require Export prosa.analysis.facts.busy_interval.arrival.
Require Export prosa.results.edf.rta.bounded_pi.
Require Export prosa.model.schedule.work_conserving.
Require Export prosa.analysis.definitions.busy_interval.classical.
Require Export prosa.analysis.facts.blocking_bound.edf.
Require Export prosa.analysis.facts.workload.edf_athep_bound.
(** * 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}.
(** We assume the classic (i.e., Liu & Layland) model of readiness
without jitter or self-suspensions, wherein pending jobs are
always ready. *)
#[local] Existing Instance basic_ready_instance .
(** 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_valid_arrival_sequence : valid_arrival_sequence 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 scheduling policy at every preemption point. *)
Hypothesis H_respects_policy : respects_JLFP_policy_at_preemption_point arr_seq sched EDF.
(** 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.
(** Let's define some local names for clarity. *)
Let response_time_bounded_by := task_response_time_bound arr_seq sched.
(** ** Search Space *)
(** If priority inversion is caused exclusively by non-preemptive sections,
then we do not need to consider the priority-inversion bound in the search
space. Hence we define the following search space, which refines the more
general [bounded_pi.is_in_search_space] for our specific setting. *)
Definition is_in_search_space (L A : duration) :=
(A < L) && (task_rbf_changes_at tsk A
|| bound_on_total_hep_workload_changes_at ts tsk A).
(** For the following proof, we exploit the fact that the blocking bound is
monotonically decreasing in [A], which we note here. *)
Fact blocking_bound_decreasing :
forall A1 A2 ,
A1 <= A2 ->
blocking_bound ts tsk A1 >= blocking_bound ts tsk A2.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_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.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_JLFP_policy_at_preemption_point
arr_seq sched EDF 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 response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop
forall A1 A2 : nat,
A1 <= A2 ->
blocking_bound ts tsk A2 <= blocking_bound ts tsk A1
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_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.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_JLFP_policy_at_preemption_point
arr_seq sched EDF 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 response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop
forall A1 A2 : nat,
A1 <= A2 ->
blocking_bound ts tsk A2 <= blocking_bound ts tsk A1
move => A1 A2 LEQ.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_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.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_JLFP_policy_at_preemption_point
arr_seq sched EDF 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 response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop A1, A2 : nat LEQ : A1 <= A2
blocking_bound ts tsk A2 <= blocking_bound ts tsk A1
rewrite /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_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.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_JLFP_policy_at_preemption_point
arr_seq sched EDF 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 response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop A1, A2 : nat LEQ : A1 <= A2
\max_(tsk_o <- ts | blocking_relevant tsk_o &&
(task_deadline tsk + A2 <
task_deadline tsk_o))
(task_max_nonpreemptive_segment tsk_o - 1 ) <=
\max_(tsk_o <- ts | blocking_relevant tsk_o &&
(task_deadline tsk + A1 <
task_deadline tsk_o))
(task_max_nonpreemptive_segment tsk_o - 1 )
apply : bigmax_subset => tsk_o IN /andP[/andP[OTHER LT] ARR].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_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.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_JLFP_policy_at_preemption_point
arr_seq sched EDF 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 response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop A1, A2 : nat LEQ : A1 <= A2 tsk_o : Task IN : tsk_o \in ts OTHER : 0 < max_arrivals tsk_o 1 LT : 0 < task_cost tsk_oARR : task_deadline tsk + A2 < task_deadline tsk_o
blocking_relevant tsk_o &&
(task_deadline tsk + A1 < task_deadline tsk_o)
by repeat (apply /andP; split ) => //; lia .
Qed .
(** To use the refined search space with the abstract theorem, we must show
that it still includes all relevant points. To this end, we first observe
that a step in the blocking bound implies the existence of a task that
could release a job with an absolute deadline equal to the absolute
deadline of the job under analysis. *)
Lemma task_with_equal_deadline_exists :
forall {A },
priority_inversion_changes_at (blocking_bound ts tsk) A ->
exists tsk_o , (tsk_o \in ts)
&& (blocking_relevant tsk_o)
&& (tsk_o != tsk)
&& (D tsk_o == D tsk + A).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_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.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_JLFP_policy_at_preemption_point
arr_seq sched EDF 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 response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop
forall A : duration,
priority_inversion_changes_at (blocking_bound ts tsk)
A ->
exists tsk_o : Task,
(tsk_o \in ts) && blocking_relevant tsk_o &&
(tsk_o != tsk) && (D tsk_o == D tsk + A)
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_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.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_JLFP_policy_at_preemption_point
arr_seq sched EDF 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 response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop
forall A : duration,
priority_inversion_changes_at (blocking_bound ts tsk)
A ->
exists tsk_o : Task,
(tsk_o \in ts) && blocking_relevant tsk_o &&
(tsk_o != tsk) && (D tsk_o == D tsk + A)
move => A.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_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.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_JLFP_policy_at_preemption_point
arr_seq sched EDF 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 response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop A : duration
priority_inversion_changes_at (blocking_bound ts tsk)
A ->
exists tsk_o : Task,
(tsk_o \in ts) && blocking_relevant tsk_o &&
(tsk_o != tsk) && (D tsk_o == D tsk + A)
rewrite /priority_inversion_changes_at => 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_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.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_JLFP_policy_at_preemption_point
arr_seq sched EDF 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 response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop A : duration NEQ : blocking_bound ts tsk (A - 1 )
!= blocking_bound ts tsk A
exists tsk_o : Task,
(tsk_o \in ts) && blocking_relevant tsk_o &&
(tsk_o != tsk) && (D tsk_o == D tsk + A)
have LEQ: blocking_bound ts tsk A <= blocking_bound ts tsk (A - ε) by apply : blocking_bound_decreasing; lia .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_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.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_JLFP_policy_at_preemption_point
arr_seq sched EDF 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 response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop A : duration NEQ : blocking_bound ts tsk (A - 1 )
!= blocking_bound ts tsk A LEQ : blocking_bound ts tsk A <=
blocking_bound ts tsk (A - 1 )
exists tsk_o : Task,
(tsk_o \in ts) && blocking_relevant tsk_o &&
(tsk_o != tsk) && (D tsk_o == D tsk + A)
have LT: blocking_bound ts tsk A < blocking_bound ts tsk (A - ε) by lia .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_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.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_JLFP_policy_at_preemption_point
arr_seq sched EDF 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 response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop A : duration NEQ : blocking_bound ts tsk (A - 1 )
!= blocking_bound ts tsk A LEQ : blocking_bound ts tsk A <=
blocking_bound ts tsk (A - 1 ) LT : blocking_bound ts tsk A <
blocking_bound ts tsk (A - 1 )
exists tsk_o : Task,
(tsk_o \in ts) && blocking_relevant tsk_o &&
(tsk_o != tsk) && (D tsk_o == D tsk + A)
move : LT; rewrite /blocking_bound => LT {LEQ} {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_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.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_JLFP_policy_at_preemption_point
arr_seq sched EDF 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 response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop A : duration LT : \max_(tsk_o <- ts |
blocking_relevant tsk_o &&
(task_deadline tsk + A < task_deadline tsk_o))
(task_max_nonpreemptive_segment tsk_o - 1 ) <
\max_(tsk_o <- ts |
blocking_relevant tsk_o &&
(task_deadline tsk + (A - 1 ) <
task_deadline tsk_o))
(task_max_nonpreemptive_segment tsk_o - 1 )
exists tsk_o : Task,
(tsk_o \in ts) && blocking_relevant tsk_o &&
(tsk_o != tsk) && (D tsk_o == D tsk + A)
move : (bigmax_witness_diff LT) => [tsk_o [IN [NOT HOLDS]]].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_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.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_JLFP_policy_at_preemption_point
arr_seq sched EDF 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 response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop A : duration LT : \max_(tsk_o <- ts |
blocking_relevant tsk_o &&
(task_deadline tsk + A < task_deadline tsk_o))
(task_max_nonpreemptive_segment tsk_o - 1 ) <
\max_(tsk_o <- ts |
blocking_relevant tsk_o &&
(task_deadline tsk + (A - 1 ) <
task_deadline tsk_o))
(task_max_nonpreemptive_segment tsk_o - 1 ) tsk_o : Task IN : tsk_o \in ts NOT : ~~
(blocking_relevant tsk_o &&
(task_deadline tsk + A < task_deadline tsk_o)) HOLDS : blocking_relevant tsk_o &&
(task_deadline tsk + (A - 1 ) <
task_deadline tsk_o)
exists tsk_o : Task,
(tsk_o \in ts) && blocking_relevant tsk_o &&
(tsk_o != tsk) && (D tsk_o == D tsk + A)
move : HOLDS => /andP[REL LTeps].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_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.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_JLFP_policy_at_preemption_point
arr_seq sched EDF 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 response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop A : duration LT : \max_(tsk_o <- ts |
blocking_relevant tsk_o &&
(task_deadline tsk + A < task_deadline tsk_o))
(task_max_nonpreemptive_segment tsk_o - 1 ) <
\max_(tsk_o <- ts |
blocking_relevant tsk_o &&
(task_deadline tsk + (A - 1 ) <
task_deadline tsk_o))
(task_max_nonpreemptive_segment tsk_o - 1 ) tsk_o : Task IN : tsk_o \in ts NOT : ~~
(blocking_relevant tsk_o &&
(task_deadline tsk + A < task_deadline tsk_o)) REL : blocking_relevant tsk_o LTeps : task_deadline tsk + (A - 1 ) <
task_deadline tsk_o
exists tsk_o : Task,
(tsk_o \in ts) && blocking_relevant tsk_o &&
(tsk_o != tsk) && (D tsk_o == D tsk + A)
exists tsk_o ; repeat (apply /andP; split ) => //;
first by apply /eqP => EQ; move : LTeps; rewrite EQ; lia .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_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.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_JLFP_policy_at_preemption_point
arr_seq sched EDF 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 response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop A : duration LT : \max_(tsk_o <- ts |
blocking_relevant tsk_o &&
(task_deadline tsk + A < task_deadline tsk_o))
(task_max_nonpreemptive_segment tsk_o - 1 ) <
\max_(tsk_o <- ts |
blocking_relevant tsk_o &&
(task_deadline tsk + (A - 1 ) <
task_deadline tsk_o))
(task_max_nonpreemptive_segment tsk_o - 1 ) tsk_o : Task IN : tsk_o \in ts NOT : ~~
(blocking_relevant tsk_o &&
(task_deadline tsk + A < task_deadline tsk_o)) REL : blocking_relevant tsk_o LTeps : task_deadline tsk + (A - 1 ) <
task_deadline tsk_o
D tsk_o == D tsk + A
move : NOT; rewrite negb_and => /orP[/negP // |]; unfold D.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_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.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_JLFP_policy_at_preemption_point
arr_seq sched EDF 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 response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop A : duration LT : \max_(tsk_o <- ts |
blocking_relevant tsk_o &&
(task_deadline tsk + A < task_deadline tsk_o))
(task_max_nonpreemptive_segment tsk_o - 1 ) <
\max_(tsk_o <- ts |
blocking_relevant tsk_o &&
(task_deadline tsk + (A - 1 ) <
task_deadline tsk_o))
(task_max_nonpreemptive_segment tsk_o - 1 ) tsk_o : Task IN : tsk_o \in ts REL : blocking_relevant tsk_o LTeps : task_deadline tsk + (A - 1 ) <
task_deadline tsk_o
~~ (task_deadline tsk + A < task_deadline tsk_o) ->
task_deadline tsk_o == task_deadline tsk + A
by lia .
Qed .
(** With the above setup in place, we can show that the search space defined
above by [is_in_search_space] covers the the more abstract search space
defined by [bounded_pi.is_in_search_space]. *)
Lemma search_space_inclusion :
forall {A L },
bounded_pi.is_in_search_space ts tsk (blocking_bound ts tsk) L A ->
is_in_search_space L A.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_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.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_JLFP_policy_at_preemption_point
arr_seq sched EDF 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 response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop
forall A L : duration,
bounded_pi.is_in_search_space ts tsk
(blocking_bound ts tsk) L A ->
is_in_search_space L A
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_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.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_JLFP_policy_at_preemption_point
arr_seq sched EDF 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 response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop
forall A L : duration,
bounded_pi.is_in_search_space ts tsk
(blocking_bound ts tsk) L A ->
is_in_search_space L A
move => A L /andP[BOUND STEP].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_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.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_JLFP_policy_at_preemption_point
arr_seq sched EDF 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 response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop A, L : duration BOUND : A < L STEP : priority_inversion_changes_at
(blocking_bound ts tsk) A
|| task_rbf_changes_at tsk A
|| bound_on_total_hep_workload_changes_at ts
tsk A
is_in_search_space L A
apply /andP; split => //; apply /orP.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_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.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_JLFP_policy_at_preemption_point
arr_seq sched EDF 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 response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop A, L : duration BOUND : A < L STEP : priority_inversion_changes_at
(blocking_bound ts tsk) A
|| task_rbf_changes_at tsk A
|| bound_on_total_hep_workload_changes_at ts
tsk A
task_rbf_changes_at tsk A \/
bound_on_total_hep_workload_changes_at ts tsk A
move : STEP => /orP[/orP[STEP|RBF] | IBF]; [right | by left | by right ].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_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.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_JLFP_policy_at_preemption_point
arr_seq sched EDF 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 response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop A, L : duration BOUND : A < L STEP : priority_inversion_changes_at
(blocking_bound ts tsk) A
bound_on_total_hep_workload_changes_at ts tsk A
move : (task_with_equal_deadline_exists STEP) => [tsk_o /andP[/andP[/andP[IN REL] OTHER] EQ]].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_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.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_JLFP_policy_at_preemption_point
arr_seq sched EDF 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 response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop A, L : duration BOUND : A < L STEP : priority_inversion_changes_at
(blocking_bound ts tsk) A tsk_o : Task IN : tsk_o \in ts REL : blocking_relevant tsk_o OTHER : tsk_o != tsk EQ : D tsk_o == D tsk + A
bound_on_total_hep_workload_changes_at ts tsk A
rewrite /bound_on_total_hep_workload_changes_at.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_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.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_JLFP_policy_at_preemption_point
arr_seq sched EDF 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 response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop A, L : duration BOUND : A < L STEP : priority_inversion_changes_at
(blocking_bound ts tsk) A tsk_o : Task IN : tsk_o \in ts REL : blocking_relevant tsk_o OTHER : tsk_o != tsk EQ : D tsk_o == D tsk + A
has
(fun tsko : Task =>
(tsk != tsko) &&
(task_request_bound_function tsko
(A + task_deadline tsk - task_deadline tsko)
!= task_request_bound_function tsko
(A + 1 + task_deadline tsk -
task_deadline tsko))) ts
apply /hasP; exists tsk_o => //.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_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.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_JLFP_policy_at_preemption_point
arr_seq sched EDF 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 response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop A, L : duration BOUND : A < L STEP : priority_inversion_changes_at
(blocking_bound ts tsk) A tsk_o : Task IN : tsk_o \in ts REL : blocking_relevant tsk_o OTHER : tsk_o != tsk EQ : D tsk_o == D tsk + A
(tsk != tsk_o) &&
(task_request_bound_function tsk_o
(A + task_deadline tsk - task_deadline tsk_o)
!= task_request_bound_function tsk_o
(A + 1 + task_deadline tsk - task_deadline tsk_o))
apply /andP; split ; first by rewrite eq_sym.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_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.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_JLFP_policy_at_preemption_point
arr_seq sched EDF 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 response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop A, L : duration BOUND : A < L STEP : priority_inversion_changes_at
(blocking_bound ts tsk) A tsk_o : Task IN : tsk_o \in ts REL : blocking_relevant tsk_o OTHER : tsk_o != tsk EQ : D tsk_o == D tsk + A
task_request_bound_function tsk_o
(A + task_deadline tsk - task_deadline tsk_o)
!= task_request_bound_function tsk_o
(A + 1 + task_deadline tsk - task_deadline tsk_o)
move : EQ.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_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.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_JLFP_policy_at_preemption_point
arr_seq sched EDF 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 response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop A, L : duration BOUND : A < L STEP : priority_inversion_changes_at
(blocking_bound ts tsk) A tsk_o : Task IN : tsk_o \in ts REL : blocking_relevant tsk_o OTHER : tsk_o != tsk
D tsk_o == D tsk + A ->
task_request_bound_function tsk_o
(A + task_deadline tsk - task_deadline tsk_o)
!= task_request_bound_function tsk_o
(A + 1 + task_deadline tsk - task_deadline tsk_o)
rewrite /D => /eqP EQ.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_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.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_JLFP_policy_at_preemption_point
arr_seq sched EDF 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 response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop A, L : duration BOUND : A < L STEP : priority_inversion_changes_at
(blocking_bound ts tsk) A tsk_o : Task IN : tsk_o \in ts REL : blocking_relevant tsk_o OTHER : tsk_o != tsk EQ : task_deadline tsk_o = task_deadline tsk + A
task_request_bound_function tsk_o
(A + task_deadline tsk - task_deadline tsk_o)
!= task_request_bound_function tsk_o
(A + 1 + task_deadline tsk - task_deadline tsk_o)
rewrite /task_request_bound_function EQ.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_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.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_JLFP_policy_at_preemption_point
arr_seq sched EDF 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 response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop A, L : duration BOUND : A < L STEP : priority_inversion_changes_at
(blocking_bound ts tsk) A tsk_o : Task IN : tsk_o \in ts REL : blocking_relevant tsk_o OTHER : tsk_o != tsk EQ : task_deadline tsk_o = task_deadline tsk + A
task_cost tsk_o *
max_arrivals tsk_o
(A + task_deadline tsk - (task_deadline tsk + A))
!= task_cost tsk_o *
max_arrivals tsk_o
(A + 1 + task_deadline tsk -
(task_deadline tsk + A))
move : REL; rewrite /blocking_relevant => /andP [ARRIVES COST].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_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.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_JLFP_policy_at_preemption_point
arr_seq sched EDF 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 response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop A, L : duration BOUND : A < L STEP : priority_inversion_changes_at
(blocking_bound ts tsk) A tsk_o : Task IN : tsk_o \in ts OTHER : tsk_o != tsk EQ : task_deadline tsk_o = task_deadline tsk + A ARRIVES : 0 < max_arrivals tsk_o 1 COST : 0 < task_cost tsk_o
task_cost tsk_o *
max_arrivals tsk_o
(A + task_deadline tsk - (task_deadline tsk + A))
!= task_cost tsk_o *
max_arrivals tsk_o
(A + 1 + task_deadline tsk -
(task_deadline tsk + A))
rewrite eqn_pmul2l //.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_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.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_JLFP_policy_at_preemption_point
arr_seq sched EDF 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 response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop A, L : duration BOUND : A < L STEP : priority_inversion_changes_at
(blocking_bound ts tsk) A tsk_o : Task IN : tsk_o \in ts OTHER : tsk_o != tsk EQ : task_deadline tsk_o = task_deadline tsk + A ARRIVES : 0 < max_arrivals tsk_o 1 COST : 0 < task_cost tsk_o
max_arrivals tsk_o
(A + task_deadline tsk - (task_deadline tsk + A))
!= max_arrivals tsk_o
(A + 1 + task_deadline tsk -
(task_deadline tsk + A))
have -> : A + task_deadline tsk - (task_deadline tsk + A)
= 0 by lia .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_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.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_JLFP_policy_at_preemption_point
arr_seq sched EDF 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 response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop A, L : duration BOUND : A < L STEP : priority_inversion_changes_at
(blocking_bound ts tsk) A tsk_o : Task IN : tsk_o \in ts OTHER : tsk_o != tsk EQ : task_deadline tsk_o = task_deadline tsk + A ARRIVES : 0 < max_arrivals tsk_o 1 COST : 0 < task_cost tsk_o
max_arrivals tsk_o 0
!= max_arrivals tsk_o
(A + 1 + task_deadline tsk -
(task_deadline tsk + A))
have -> : A + ε + task_deadline tsk - (task_deadline tsk + A)
= ε by lia .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_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.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_JLFP_policy_at_preemption_point
arr_seq sched EDF 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 response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop A, L : duration BOUND : A < L STEP : priority_inversion_changes_at
(blocking_bound ts tsk) A tsk_o : Task IN : tsk_o \in ts OTHER : tsk_o != tsk EQ : task_deadline tsk_o = task_deadline tsk + A ARRIVES : 0 < max_arrivals tsk_o 1 COST : 0 < task_cost tsk_o
max_arrivals tsk_o 0 != max_arrivals tsk_o 1
by move : (H_valid_arrival_curve tsk_o IN) => [-> _]; lia .
Qed .
(** ** 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 ts tsk A
+ (task_rbf (A + ε) - (task_cost tsk - task_rtct tsk))
+ bound_on_athep_workload ts tsk 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_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.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_JLFP_policy_at_preemption_point
arr_seq sched EDF 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 response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop 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 ts tsk A +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
bound_on_athep_workload ts tsk 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_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.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_JLFP_policy_at_preemption_point
arr_seq sched EDF 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 response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop 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 ts tsk A +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
bound_on_athep_workload ts tsk A
(A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
R
response_time_bounded_by tsk R
apply : uniprocessor_response_time_bound_edf; eauto 4 with basic_rt_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_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.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_JLFP_policy_at_preemption_point
arr_seq sched EDF 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 response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop 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 ts tsk A +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
bound_on_athep_workload ts tsk A
(A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
R
priority_inversion_is_bounded_by arr_seq sched tsk
?Goal
{ 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_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.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_JLFP_policy_at_preemption_point
arr_seq sched EDF 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 response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop 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 ts tsk A +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
bound_on_athep_workload ts tsk A
(A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
R
priority_inversion_is_bounded_by arr_seq sched tsk
?Goal
apply : priority_inversion_is_bounded => //.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_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.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_JLFP_policy_at_preemption_point
arr_seq sched EDF 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 response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop 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 ts tsk A +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
bound_on_athep_workload ts tsk A
(A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
R
forall (j : Job) (t1 t2 : instant),
arrives_in arr_seq j ->
job_of_task tsk j ->
busy_interval_prefix arr_seq sched j t1 t2 ->
max_lp_nonpreemptive_segment arr_seq j t1 <=
?Goal (job_arrival j - t1)
by move => *; apply : nonpreemptive_segments_bounded_by_blocking. } 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_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.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_JLFP_policy_at_preemption_point
arr_seq sched EDF 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 response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop 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 ts tsk A +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
bound_on_athep_workload ts tsk A
(A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
R
forall A : duration,
bounded_pi.is_in_search_space ts tsk
(blocking_bound ts tsk) L A ->
exists F : duration,
blocking_bound ts tsk A +
(task_request_bound_function tsk (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
bound_on_athep_workload ts tsk A (A + F) <= A + F /\
F + (task_cost tsk - task_rtct 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_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.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_JLFP_policy_at_preemption_point
arr_seq sched EDF 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 response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop 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 ts tsk A +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
bound_on_athep_workload ts tsk A
(A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
R
forall A : duration,
bounded_pi.is_in_search_space ts tsk
(blocking_bound ts tsk) L A ->
exists F : duration,
blocking_bound ts tsk A +
(task_request_bound_function tsk (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
bound_on_athep_workload ts tsk A (A + F) <= A + F /\
F + (task_cost tsk - task_rtct tsk) <= R
move => A BPI_SP.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_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.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_JLFP_policy_at_preemption_point
arr_seq sched EDF 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 response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop 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 ts tsk A +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
bound_on_athep_workload ts tsk A
(A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
RA : duration BPI_SP : bounded_pi.is_in_search_space ts tsk
(blocking_bound ts tsk) L A
exists F : duration,
blocking_bound ts tsk A +
(task_request_bound_function tsk (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
bound_on_athep_workload ts tsk A (A + F) <= A + F /\
F + (task_cost tsk - task_rtct tsk) <= R
by apply H_R_is_maximum, search_space_inclusion.
Qed .
End ResponseTimeBound .
End RTAforEDFwithBoundedNonpreemptiveSegmentsWithArrivalCurves .