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Require Export prosa.model.schedule.priority_driven.[Loading ML file ssrmatching_plugin.cmxs (using legacy method) ... done ] [Loading ML file ssreflect_plugin.cmxs (using legacy method) ... done ] [Loading ML file ring_plugin.cmxs (using legacy method) ... done ] [Loading ML file coq-elpi.elpi ... done ] [Loading ML file zify_plugin.cmxs (using legacy method) ... done ] [Loading ML file micromega_plugin.cmxs (using legacy method) ... done ] [Loading ML file btauto_plugin.cmxs (using legacy method) ... done ] Notation "_ + _" was already used in scope nat_scope.
[notation-overridden,parsing,default]Notation "_ - _" was already used in scope nat_scope.
[notation-overridden,parsing,default]Notation "_ <= _" was already used in scope nat_scope.
[notation-overridden,parsing,default]Notation "_ < _" was already used in scope nat_scope.
[notation-overridden,parsing,default]Notation "_ >= _" was already used in scope nat_scope.
[notation-overridden,parsing,default]Notation "_ > _" was already used in scope nat_scope.
[notation-overridden,parsing,default]Notation "_ <= _ <= _" was already used in scope
nat_scope. [notation-overridden,parsing,default]Notation "_ < _ <= _" was already used in scope
nat_scope. [notation-overridden,parsing,default]Notation "_ <= _ < _" was already used in scope
nat_scope. [notation-overridden,parsing,default]Notation "_ < _ < _" was already used in scope
nat_scope. [notation-overridden,parsing,default]Notation "_ * _" was already used in scope nat_scope.
[notation-overridden,parsing,default]
Require Export prosa.analysis.abstract .ideal.iw_instantiation.
Require Export prosa.analysis.facts.busy_interval.existence.
Require Export prosa.analysis.abstract .ideal.abstract_seq_rta.
Require Export prosa.analysis.facts.model.task_cost.
(** * Abstract RTA for FP-schedulers with Bounded Priority Inversion *)
(** In this module we instantiate the Abstract Response-Time analysis
(aRTA) to FP-schedulers for ideal uni-processor model of
real-time tasks with arbitrary arrival models. *)
(** Given FP priority policy and an ideal uni-processor scheduler
model, we can explicitly specify [interference],
[interfering_workload], and [interference_bound_function]. In this
settings, we can define natural notions of service, workload, busy
interval, etc. The important feature of this instantiation is that
we can induce the meaningful notion of priority
inversion. However, we do not specify the exact cause of priority
inversion (as there may be different reasons for this, like
execution of a non-preemptive segment or blocking due to resource
locking). We only assume that that a priority inversion is
bounded. *)
Section AbstractRTAforFPwithArrivalCurves .
(** Consider any type of tasks ... *)
Context {Task : TaskType}.
Context `{TaskCost Task}.
Context `{TaskRunToCompletionThreshold 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}.
Context `{JobPreemptable Job}.
(** Consider an FP policy that indicates a higher-or-equal priority relation,
and assume that the relation is reflexive. Note that we do not relate
the FP policy with the scheduler. However, we define functions for
Interference and Interfering Workload that actively use the concept of
priorities. We require the FP policy to be reflexive, so a job cannot
cause lower-priority interference (i.e. priority inversion) to itself. *)
Context {FP : FP_policy Task}.
Hypothesis H_priority_is_reflexive : reflexive_task_priorities FP.
(** 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 ideal uni-processor schedule of this arrival sequence, ... *)
Variable sched : schedule (ideal.processor_state Job).
(** ... allow for any work-bearing notion of job readiness, ... *)
Context `{!JobReady Job (ideal.processor_state Job)}.
Hypothesis H_job_ready : work_bearing_readiness arr_seq sched.
(** ... and assume that the schedule is valid. *)
Hypothesis H_sched_valid : valid_schedule sched arr_seq.
(** ** Instantiation of Interference *)
(** We say that job [j] incurs interference at time [t] iff it
cannot execute due to a higher-or-equal-priority job being
scheduled, or if it incurs a priority inversion. *)
#[local] Instance ideal_jlfp_interference : Interference Job :=
ideal_jlfp_interference arr_seq sched.
(** ** Instantiation of Interfering Workload *)
(** The interfering workload, in turn, is defined as the sum of the
priority inversion function and interfering workload of jobs
with higher or equal priority. *)
#[local] Instance ideal_jlfp_interfering_workload : InterferingWorkload Job :=
ideal_jlfp_interfering_workload arr_seq sched.
(** Note that we differentiate between abstract and classical
notions of work conserving schedule. *)
Let work_conserving_ab := definitions.work_conserving arr_seq sched.
Let work_conserving_cl := work_conserving.work_conserving arr_seq sched.
(** We assume that the schedule is a work-conserving schedule in the
_classical_ sense, and later prove that the hypothesis about
abstract work-conservation also holds. *)
Hypothesis H_work_conserving : work_conserving_cl.
(** Assume we have sequential tasks, i.e, jobs from the
same task execute in the order of their arrival. *)
Hypothesis H_sequential_tasks : sequential_tasks arr_seq sched.
(** Assume that a job cost cannot be larger than a task cost. *)
Hypothesis H_valid_job_cost :
arrivals_have_valid_job_costs arr_seq.
(** Consider an arbitrary task set [ts]. *)
Variable ts : list Task.
(** Next, we assume that all jobs come from the task set. *)
Hypothesis H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts.
(** 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.
(** For clarity, let's define some local names. *)
Let job_pending_at := pending sched.
Let job_scheduled_at := scheduled_at sched.
Let job_completed_by := completed_by sched.
Let job_backlogged_at := backlogged sched.
Let response_time_bounded_by := task_response_time_bound arr_seq sched.
(** We introduce [task_rbf] as an abbreviation of the task request
bound function, which is defined as [task_cost(tsk) ×
max_arrivals(tsk,Δ)]. *)
Let task_rbf := task_request_bound_function tsk.
(** Using the sum of individual request bound functions, we define
the request bound function of all tasks with higher-or-equal
priority (with respect to [tsk]). *)
Let total_hep_rbf := total_hep_request_bound_function_FP ts tsk.
(** Similarly, we define the request bound function of all tasks
other than [tsk] with higher-or-equal priority (with respect to
[tsk]). *)
Let total_ohep_rbf :=
total_ohep_request_bound_function_FP ts tsk.
(** Assume that there exists a constant [priority_inversion_bound]
that bounds the length of any priority inversion experienced by
any job of [tsk]. Since we analyze only task [tsk], we ignore
the lengths of priority inversions incurred by any other
tasks. *)
Variable priority_inversion_bound : duration.
Hypothesis H_priority_inversion_is_bounded :
priority_inversion_is_bounded_by
arr_seq sched tsk (constant priority_inversion_bound).
(** 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 = priority_inversion_bound + total_hep_rbf L.
(** To reduce the time complexity of the analysis, recall the notion
of search space. Intuitively, this corresponds to all
"interesting" arrival offsets that the job under analysis might
have with regard to the beginning of its busy-window. *)
Definition is_in_search_space A := (A < L) && (task_rbf A != task_rbf (A + ε)).
(** Let [R] be a value that upper-bounds the solution of each
response-time recurrence, i.e., for any relative arrival time
[A] in the search space, there exists a corresponding solution
[F] such that [R >= F + (task cost - task lock-in service)]. *)
Variable R : duration.
Hypothesis H_R_is_maximum :
forall (A : duration),
is_in_search_space A ->
exists (F : duration),
A + F >= priority_inversion_bound
+ (task_rbf (A + ε) - (task_cost tsk - task_rtct tsk))
+ total_ohep_rbf (A + F) /\
R >= F + (task_cost tsk - task_rtct tsk).
(** Finally, we define the interference bound function
([task_IBF]). [task_IBF] bounds the interference if tasks are
sequential. Since tasks are sequential, we exclude interference
from other jobs of the same task. For FP, we define [task_IBF]
as the sum of the priority interference bound and the
higher-or-equal-priority workload. *)
Let task_IBF (R : duration) := priority_inversion_bound + total_ohep_rbf R.
(** ** Filling Out Hypotheses Of Abstract RTA Theorem *)
(** In this section we prove that all preconditions necessary to use
the abstract theorem are satisfied. *)
Section FillingOutHypothesesOfAbstractRTATheorem .
(** Recall that [L] is assumed to be a fixed point of the busy
interval recurrence. Thanks to this fact, we can prove that
every busy interval (according to the concrete definition) is
bounded. In addition, we know that the conventional concept of
busy interval and the one obtained from the abstract
definition (with the interference and interfering workload)
coincide. Thus, it follows that any busy interval (in the
abstract sense) is bounded. *)
Lemma instantiated_busy_intervals_are_bounded :
busy_intervals_are_bounded_by arr_seq sched tsk L.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
busy_intervals_are_bounded_by arr_seq sched tsk L
Proof .Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
busy_intervals_are_bounded_by arr_seq sched tsk L
move => j ARR TSK POS.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j POS : 0 < job_cost j
exists t1 t2 : nat,
t1 <= job_arrival j < t2 /\
t2 <= t1 + L /\ busy_interval sched j t1 t2
edestruct (exists_busy_interval) with (delta := L) (priority_inversion_bound := (fun (d : duration) => priority_inversion_bound))
as [t1 [t2 [T1 [T2 BI]]]] => //; last first .Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j POS : 0 < job_cost jt1, t2 : nat T1 : t1 <= job_arrival j < t2 T2 : t2 <= t1 + L BI : classical.busy_interval arr_seq sched j t1 t2
exists t1 t2 : nat,
t1 <= job_arrival j < t2 /\
t2 <= t1 + L /\ busy_interval sched j t1 t2
{ Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j POS : 0 < job_cost jt1, t2 : nat T1 : t1 <= job_arrival j < t2 T2 : t2 <= t1 + L BI : classical.busy_interval arr_seq sched j t1 t2
exists t1 t2 : nat,
t1 <= job_arrival j < t2 /\
t2 <= t1 + L /\ busy_interval sched j t1 t2
exists t1 , t2; split => [//|]; split => [//|].Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j POS : 0 < job_cost jt1, t2 : nat T1 : t1 <= job_arrival j < t2 T2 : t2 <= t1 + L BI : classical.busy_interval arr_seq sched j t1 t2
busy_interval sched j t1 t2
by eapply instantiated_busy_interval_equivalent_busy_interval. } Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j POS : 0 < job_cost j
forall t : nat,
priority_inversion_bound +
workload_of_higher_or_equal_priority_jobs j
(arrivals_between arr_seq t (t + L)) <= L
{ Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j POS : 0 < job_cost j
forall t : nat,
priority_inversion_bound +
workload_of_higher_or_equal_priority_jobs j
(arrivals_between arr_seq t (t + L)) <= L
intros ; rewrite {2 }H_fixed_point leq_add //.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j POS : 0 < job_cost jt : nat
workload_of_higher_or_equal_priority_jobs j
(arrivals_between arr_seq t (t + L)) <=
total_hep_rbf L
rewrite /workload_of_higher_or_equal_priority_jobs /total_hep_rbf
/total_hep_request_bound_function_FP
/workload_of_jobs /hep_job /FP_to_JLFP.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j POS : 0 < job_cost jt : nat
\sum_(j0 <- arrivals_between arr_seq t (t + L) | hep_task
(job_task
j0)
(job_task
j))
job_cost j0 <=
\sum_(tsk_other <- ts | hep_task tsk_other tsk)
task_request_bound_function tsk_other L
move : (TSK) => /eqP ->.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j POS : 0 < job_cost jt : nat
\sum_(j <- arrivals_between arr_seq t (t + L) | hep_task
(job_task
j)
tsk)
job_cost j <=
\sum_(tsk_other <- ts | hep_task tsk_other tsk)
task_request_bound_function tsk_other L
exact : sum_of_jobs_le_sum_rbf. }
Qed .
(** Next, we prove that [task_IBF] is indeed an interference
bound.
Recall that in module abstract_seq_RTA hypothesis
[task_interference_is_bounded_by] expects to receive a
function that maps some task [tsk], the relative arrival time of
a job [j] of task [tsk], and the length of the interval to the
maximum amount of interference.
However, in this module we analyze only one task -- [tsk],
therefore it is “hard-coded” inside the interference bound
function [task_IBF]. Moreover, in case of a model with fixed
priorities, interference that some job [j] incurs from
higher-or-equal priority jobs does not depend on the relative
arrival time of job [j]. Therefore, in order for the
[task_IBF] signature to match the required signature in
module [abstract_seq_RTA], we wrap the [task_IBF] function in
a function that accepts, but simply ignores, the task and the
relative arrival time. *)
Lemma instantiated_task_interference_is_bounded :
task_interference_is_bounded_by
arr_seq sched tsk (fun A R => task_IBF R).Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
task_interference_is_bounded_by arr_seq sched tsk
(fun => [eta task_IBF])
Proof .Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
task_interference_is_bounded_by arr_seq sched tsk
(fun => [eta task_IBF])
move => t1 t2 Δ j ARR TSK BUSY LT NCOMPL A OFF.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat t1, t2 : instant Δ : nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j BUSY : busy_interval sched j t1 t2 LT : t1 + Δ < t2 NCOMPL : ~~ completed_by sched j (t1 + Δ) A : nat OFF : relative_arrival_time_of_job_is_A sched j A
cumul_cond_interference (nonself arr_seq sched) j t1
(t1 + Δ) <= task_IBF Δ
move : (posnP (@job_cost _ Cost j)) => [ZERO|POS].Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat t1, t2 : instant Δ : nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j BUSY : busy_interval sched j t1 t2 LT : t1 + Δ < t2 NCOMPL : ~~ completed_by sched j (t1 + Δ) A : nat OFF : relative_arrival_time_of_job_is_A sched j A ZERO : job_cost j = 0
cumul_cond_interference (nonself arr_seq sched) j t1
(t1 + Δ) <= task_IBF Δ
{ Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat t1, t2 : instant Δ : nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j BUSY : busy_interval sched j t1 t2 LT : t1 + Δ < t2 NCOMPL : ~~ completed_by sched j (t1 + Δ) A : nat OFF : relative_arrival_time_of_job_is_A sched j A ZERO : job_cost j = 0
cumul_cond_interference (nonself arr_seq sched) j t1
(t1 + Δ) <= task_IBF Δ
by exfalso ; rewrite /completed_by ZERO in NCOMPL. } Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat t1, t2 : instant Δ : nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j BUSY : busy_interval sched j t1 t2 LT : t1 + Δ < t2 NCOMPL : ~~ completed_by sched j (t1 + Δ) A : nat OFF : relative_arrival_time_of_job_is_A sched j A POS : 0 < job_cost j
cumul_cond_interference (nonself arr_seq sched) j t1
(t1 + Δ) <= task_IBF Δ
rewrite -/(cumul_task_interference _ _ _ _ _).Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat t1, t2 : instant Δ : nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j BUSY : busy_interval sched j t1 t2 LT : t1 + Δ < t2 NCOMPL : ~~ completed_by sched j (t1 + Δ) A : nat OFF : relative_arrival_time_of_job_is_A sched j A POS : 0 < job_cost j
cumul_task_interference arr_seq sched j t1 (t1 + Δ) <=
task_IBF Δ
rewrite (leqRW (cumulative_task_interference_split _ _ _ _ _ _ _ _ _ _ _ _ _)) //=.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat t1, t2 : instant Δ : nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j BUSY : busy_interval sched j t1 t2 LT : t1 + Δ < t2 NCOMPL : ~~ completed_by sched j (t1 + Δ) A : nat OFF : relative_arrival_time_of_job_is_A sched j A POS : 0 < job_cost j
cumulative_priority_inversion arr_seq sched j t1
(t1 + Δ) +
cumulative_another_task_hep_job_interference arr_seq
sched j t1 (t1 + Δ) <= task_IBF Δ
rewrite /task_IBF leq_add//.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat t1, t2 : instant Δ : nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j BUSY : busy_interval sched j t1 t2 LT : t1 + Δ < t2 NCOMPL : ~~ completed_by sched j (t1 + Δ) A : nat OFF : relative_arrival_time_of_job_is_A sched j A POS : 0 < job_cost j
cumulative_priority_inversion arr_seq sched j t1
(t1 + Δ) <= priority_inversion_bound
{ Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat t1, t2 : instant Δ : nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j BUSY : busy_interval sched j t1 t2 LT : t1 + Δ < t2 NCOMPL : ~~ completed_by sched j (t1 + Δ) A : nat OFF : relative_arrival_time_of_job_is_A sched j A POS : 0 < job_cost j
cumulative_priority_inversion arr_seq sched j t1
(t1 + Δ) <= priority_inversion_bound
apply leq_trans with (cumulative_priority_inversion arr_seq sched j t1 (t1 + Δ)); first by done .Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat t1, t2 : instant Δ : nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j BUSY : busy_interval sched j t1 t2 LT : t1 + Δ < t2 NCOMPL : ~~ completed_by sched j (t1 + Δ) A : nat OFF : relative_arrival_time_of_job_is_A sched j A POS : 0 < job_cost j
cumulative_priority_inversion arr_seq sched j t1
(t1 + Δ) <= priority_inversion_bound
apply leq_trans with (cumulative_priority_inversion arr_seq sched j t1 t2);
last by apply : H_priority_inversion_is_bounded => //; eauto 6 with basic_rt_facts.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat t1, t2 : instant Δ : nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j BUSY : busy_interval sched j t1 t2 LT : t1 + Δ < t2 NCOMPL : ~~ completed_by sched j (t1 + Δ) A : nat OFF : relative_arrival_time_of_job_is_A sched j A POS : 0 < job_cost j
cumulative_priority_inversion arr_seq sched j t1
(t1 + Δ) <=
cumulative_priority_inversion arr_seq sched j t1 t2
by rewrite [X in _ <= X](@big_cat_nat _ _ _ (t1 + Δ)) //= leq_addr. } Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat t1, t2 : instant Δ : nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j BUSY : busy_interval sched j t1 t2 LT : t1 + Δ < t2 NCOMPL : ~~ completed_by sched j (t1 + Δ) A : nat OFF : relative_arrival_time_of_job_is_A sched j A POS : 0 < job_cost j
cumulative_another_task_hep_job_interference arr_seq
sched j t1 (t1 + Δ) <= total_ohep_rbf Δ
{ Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat t1, t2 : instant Δ : nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j BUSY : busy_interval sched j t1 t2 LT : t1 + Δ < t2 NCOMPL : ~~ completed_by sched j (t1 + Δ) A : nat OFF : relative_arrival_time_of_job_is_A sched j A POS : 0 < job_cost j
cumulative_another_task_hep_job_interference arr_seq
sched j t1 (t1 + Δ) <= total_ohep_rbf Δ
erewrite cumulative_i_thep_eq_service_of_othep => //;
last by eauto 6 with basic_rt_facts.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat t1, t2 : instant Δ : nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j BUSY : busy_interval sched j t1 t2 LT : t1 + Δ < t2 NCOMPL : ~~ completed_by sched j (t1 + Δ) A : nat OFF : relative_arrival_time_of_job_is_A sched j A POS : 0 < job_cost j
service_of_other_task_hep_jobs arr_seq sched j t1
(t1 + Δ) <= total_ohep_rbf Δ
apply : leq_trans.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat t1, t2 : instant Δ : nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j BUSY : busy_interval sched j t1 t2 LT : t1 + Δ < t2 NCOMPL : ~~ completed_by sched j (t1 + Δ) A : nat OFF : relative_arrival_time_of_job_is_A sched j A POS : 0 < job_cost j
service_of_other_task_hep_jobs arr_seq sched j t1
(t1 + Δ) <= ?Goal
{ Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat t1, t2 : instant Δ : nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j BUSY : busy_interval sched j t1 t2 LT : t1 + Δ < t2 NCOMPL : ~~ completed_by sched j (t1 + Δ) A : nat OFF : relative_arrival_time_of_job_is_A sched j A POS : 0 < job_cost j
service_of_other_task_hep_jobs arr_seq sched j t1
(t1 + Δ) <= ?Goal
apply service_of_jobs_le_workload; first apply ideal_proc_model_provides_unit_service.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat t1, t2 : instant Δ : nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j BUSY : busy_interval sched j t1 t2 LT : t1 + Δ < t2 NCOMPL : ~~ completed_by sched j (t1 + Δ) A : nat OFF : relative_arrival_time_of_job_is_A sched j A POS : 0 < job_cost j
completed_jobs_dont_execute sched
by apply (valid_schedule_implies_completed_jobs_dont_execute sched arr_seq). } Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat t1, t2 : instant Δ : nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j BUSY : busy_interval sched j t1 t2 LT : t1 + Δ < t2 NCOMPL : ~~ completed_by sched j (t1 + Δ) A : nat OFF : relative_arrival_time_of_job_is_A sched j A POS : 0 < job_cost j
workload_of_jobs (another_task_hep_job^~ j)
(arrivals_between arr_seq t1 (t1 + Δ)) <=
total_ohep_rbf Δ
{ Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat t1, t2 : instant Δ : nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j BUSY : busy_interval sched j t1 t2 LT : t1 + Δ < t2 NCOMPL : ~~ completed_by sched j (t1 + Δ) A : nat OFF : relative_arrival_time_of_job_is_A sched j A POS : 0 < job_cost j
workload_of_jobs (another_task_hep_job^~ j)
(arrivals_between arr_seq t1 (t1 + Δ)) <=
total_ohep_rbf Δ
rewrite /workload_of_jobs /total_ohep_rbf /total_ohep_request_bound_function_FP.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat t1, t2 : instant Δ : nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j BUSY : busy_interval sched j t1 t2 LT : t1 + Δ < t2 NCOMPL : ~~ completed_by sched j (t1 + Δ) A : nat OFF : relative_arrival_time_of_job_is_A sched j A POS : 0 < job_cost j
\sum_(j0 <- arrivals_between arr_seq t1 (t1 + Δ) |
another_task_hep_job j0 j) job_cost j0 <=
\sum_(tsk_other <- ts | hep_task tsk_other tsk &&
(tsk_other != tsk))
task_request_bound_function tsk_other Δ
rewrite /another_task_hep_job /hep_job /FP_to_JLFP.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat t1, t2 : instant Δ : nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j BUSY : busy_interval sched j t1 t2 LT : t1 + Δ < t2 NCOMPL : ~~ completed_by sched j (t1 + Δ) A : nat OFF : relative_arrival_time_of_job_is_A sched j A POS : 0 < job_cost j
\sum_(j0 <- arrivals_between arr_seq t1 (t1 + Δ) |
hep_task (job_task j0) (job_task j) &&
(job_task j0 != job_task j)) job_cost j0 <=
\sum_(tsk_other <- ts | hep_task tsk_other tsk &&
(tsk_other != tsk))
task_request_bound_function tsk_other Δ
set (pred_task tsk_other := hep_task tsk_other tsk && (tsk_other != tsk)).Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat t1, t2 : instant Δ : nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j BUSY : busy_interval sched j t1 t2 LT : t1 + Δ < t2 NCOMPL : ~~ completed_by sched j (t1 + Δ) A : nat OFF : relative_arrival_time_of_job_is_A sched j A POS : 0 < job_cost jpred_task := fun tsk_other : Task =>
hep_task tsk_other tsk &&
(tsk_other != tsk): Task -> bool
\sum_(j0 <- arrivals_between arr_seq t1 (t1 + Δ) |
hep_task (job_task j0) (job_task j) &&
(job_task j0 != job_task j)) job_cost j0 <=
\sum_(tsk_other <- ts | hep_task tsk_other tsk &&
(tsk_other != tsk))
task_request_bound_function tsk_other Δ
rewrite (eq_big (fun j => pred_task (job_task j)) job_cost) //;
last by move => j'; rewrite /pred_task; move : TSK => /eqP ->.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat t1, t2 : instant Δ : nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j BUSY : busy_interval sched j t1 t2 LT : t1 + Δ < t2 NCOMPL : ~~ completed_by sched j (t1 + Δ) A : nat OFF : relative_arrival_time_of_job_is_A sched j A POS : 0 < job_cost jpred_task := fun tsk_other : Task =>
hep_task tsk_other tsk &&
(tsk_other != tsk): Task -> bool
\sum_(i <- arrivals_between arr_seq t1 (t1 + Δ) |
pred_task (job_task i)) job_cost i <=
\sum_(tsk_other <- ts | hep_task tsk_other tsk &&
(tsk_other != tsk))
task_request_bound_function tsk_other Δ
erewrite (eq_big pred_task); [|by done |by move => tsk'; eauto ].Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat t1, t2 : instant Δ : nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j BUSY : busy_interval sched j t1 t2 LT : t1 + Δ < t2 NCOMPL : ~~ completed_by sched j (t1 + Δ) A : nat OFF : relative_arrival_time_of_job_is_A sched j A POS : 0 < job_cost jpred_task := fun tsk_other : Task =>
hep_task tsk_other tsk &&
(tsk_other != tsk): Task -> bool
\sum_(i <- arrivals_between arr_seq t1 (t1 + Δ) |
pred_task (job_task i)) job_cost i <=
\sum_(i <- ts | pred_task i)
(task_request_bound_function^~ Δ) i
by apply : sum_of_jobs_le_sum_rbf; eauto . } }
Qed .
(** Finally, we show that there exists a solution for the
response-time recurrence. *)
Section SolutionOfResponseTimeRecurrenceExists .
(** To rule out pathological cases with the concrete search
space, we assume that the task cost is positive and the
arrival curve is non-pathological. *)
Hypothesis H_task_cost_pos : 0 < task_cost tsk.
Hypothesis H_arrival_curve_pos : 0 < max_arrivals tsk ε.
(** Given any job [j] of task [tsk] that arrives exactly [A]
units after the beginning of the busy interval, the bound of
the total interference incurred by [j] within an interval of
length [Δ] is equal to [task_rbf (A + ε) - task_cost tsk +
task_IBF Δ]. *)
Let total_interference_bound A Δ :=
task_rbf (A + ε) - task_cost tsk + task_IBF Δ.
(** Next, consider any [A] from the search space (in the
abstract sense). *)
Variable A : duration.
Hypothesis H_A_is_in_abstract_search_space :
search_space.is_in_search_space L total_interference_bound A.
(** We prove that [A] is also in the concrete search space. *)
Lemma A_is_in_concrete_search_space :
is_in_search_space A.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat H_task_cost_pos : 0 < task_cost tskH_arrival_curve_pos : 0 < max_arrivals tsk 1 total_interference_bound := fun (A : nat) (Δ : duration)
=>
task_rbf (A + 1 ) -
task_cost tsk +
task_IBF Δ: nat -> duration -> nat A : duration H_A_is_in_abstract_search_space : search_space.is_in_search_space
L
total_interference_bound
A
is_in_search_space A
Proof .Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat H_task_cost_pos : 0 < task_cost tskH_arrival_curve_pos : 0 < max_arrivals tsk 1 total_interference_bound := fun (A : nat) (Δ : duration)
=>
task_rbf (A + 1 ) -
task_cost tsk +
task_IBF Δ: nat -> duration -> nat A : duration H_A_is_in_abstract_search_space : search_space.is_in_search_space
L
total_interference_bound
A
is_in_search_space A
move : H_A_is_in_abstract_search_space => [INSP | [/andP [POSA LTL] [x [LTx INSP2]]]].Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat H_task_cost_pos : 0 < task_cost tskH_arrival_curve_pos : 0 < max_arrivals tsk 1 total_interference_bound := fun (A : nat) (Δ : duration)
=>
task_rbf (A + 1 ) -
task_cost tsk +
task_IBF Δ: nat -> duration -> nat A : duration H_A_is_in_abstract_search_space : search_space.is_in_search_space
L
total_interference_bound
A INSP : A = 0
is_in_search_space A
- Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat H_task_cost_pos : 0 < task_cost tskH_arrival_curve_pos : 0 < max_arrivals tsk 1 total_interference_bound := fun (A : nat) (Δ : duration)
=>
task_rbf (A + 1 ) -
task_cost tsk +
task_IBF Δ: nat -> duration -> nat A : duration H_A_is_in_abstract_search_space : search_space.is_in_search_space
L
total_interference_bound
A INSP : A = 0
is_in_search_space A
rewrite INSP.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat H_task_cost_pos : 0 < task_cost tskH_arrival_curve_pos : 0 < max_arrivals tsk 1 total_interference_bound := fun (A : nat) (Δ : duration)
=>
task_rbf (A + 1 ) -
task_cost tsk +
task_IBF Δ: nat -> duration -> nat A : duration H_A_is_in_abstract_search_space : search_space.is_in_search_space
L
total_interference_bound
A INSP : A = 0
is_in_search_space 0
apply /andP; split ; first by done .Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat H_task_cost_pos : 0 < task_cost tskH_arrival_curve_pos : 0 < max_arrivals tsk 1 total_interference_bound := fun (A : nat) (Δ : duration)
=>
task_rbf (A + 1 ) -
task_cost tsk +
task_IBF Δ: nat -> duration -> nat A : duration H_A_is_in_abstract_search_space : search_space.is_in_search_space
L
total_interference_bound
A INSP : A = 0
task_rbf 0 != task_rbf (0 + 1 )
rewrite neq_ltn; apply /orP; left .Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat H_task_cost_pos : 0 < task_cost tskH_arrival_curve_pos : 0 < max_arrivals tsk 1 total_interference_bound := fun (A : nat) (Δ : duration)
=>
task_rbf (A + 1 ) -
task_cost tsk +
task_IBF Δ: nat -> duration -> nat A : duration H_A_is_in_abstract_search_space : search_space.is_in_search_space
L
total_interference_bound
A INSP : A = 0
task_rbf 0 < task_rbf (0 + 1 )
rewrite {1 }/task_rbf; erewrite task_rbf_0_zero; eauto 2 ; try done .Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat H_task_cost_pos : 0 < task_cost tskH_arrival_curve_pos : 0 < max_arrivals tsk 1 total_interference_bound := fun (A : nat) (Δ : duration)
=>
task_rbf (A + 1 ) -
task_cost tsk +
task_IBF Δ: nat -> duration -> nat A : duration H_A_is_in_abstract_search_space : search_space.is_in_search_space
L
total_interference_bound
A INSP : A = 0
0 < task_rbf (0 + 1 )
rewrite add0n /task_rbf; apply leq_trans with (task_cost tsk) => //.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat H_task_cost_pos : 0 < task_cost tskH_arrival_curve_pos : 0 < max_arrivals tsk 1 total_interference_bound := fun (A : nat) (Δ : duration)
=>
task_rbf (A + 1 ) -
task_cost tsk +
task_IBF Δ: nat -> duration -> nat A : duration H_A_is_in_abstract_search_space : search_space.is_in_search_space
L
total_interference_bound
A INSP : A = 0
task_cost tsk <= task_request_bound_function tsk 1
exact : task_rbf_1_ge_task_cost.
- Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat H_task_cost_pos : 0 < task_cost tskH_arrival_curve_pos : 0 < max_arrivals tsk 1 total_interference_bound := fun (A : nat) (Δ : duration)
=>
task_rbf (A + 1 ) -
task_cost tsk +
task_IBF Δ: nat -> duration -> nat A : duration H_A_is_in_abstract_search_space : search_space.is_in_search_space
L
total_interference_bound
A POSA : 0 < ALTL : A < L x : nat LTx : x < L INSP2 : total_interference_bound (A - 1 ) x <>
total_interference_bound A x
is_in_search_space A
apply /andP; split ; first by done .Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat H_task_cost_pos : 0 < task_cost tskH_arrival_curve_pos : 0 < max_arrivals tsk 1 total_interference_bound := fun (A : nat) (Δ : duration)
=>
task_rbf (A + 1 ) -
task_cost tsk +
task_IBF Δ: nat -> duration -> nat A : duration H_A_is_in_abstract_search_space : search_space.is_in_search_space
L
total_interference_bound
A POSA : 0 < ALTL : A < L x : nat LTx : x < L INSP2 : total_interference_bound (A - 1 ) x <>
total_interference_bound A x
task_rbf A != task_rbf (A + 1 )
apply /negP; intros EQ; move : EQ => /eqP EQ.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat H_task_cost_pos : 0 < task_cost tskH_arrival_curve_pos : 0 < max_arrivals tsk 1 total_interference_bound := fun (A : nat) (Δ : duration)
=>
task_rbf (A + 1 ) -
task_cost tsk +
task_IBF Δ: nat -> duration -> nat A : duration H_A_is_in_abstract_search_space : search_space.is_in_search_space
L
total_interference_bound
A POSA : 0 < ALTL : A < L x : nat LTx : x < L INSP2 : total_interference_bound (A - 1 ) x <>
total_interference_bound A x EQ : task_rbf A = task_rbf (A + 1 )
False
by apply INSP2; rewrite /total_interference_bound subn1 addn1 prednK //.
Qed .
(** Then, there exists a solution for the response-time
recurrence (in the abstract sense). *)
Corollary correct_search_space :
exists (F : duration),
A + F >= task_rbf (A + ε) - (task_cost tsk - task_rtct tsk) + task_IBF (A + F) /\
R >= F + (task_cost tsk - task_rtct tsk).Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat H_task_cost_pos : 0 < task_cost tskH_arrival_curve_pos : 0 < max_arrivals tsk 1 total_interference_bound := fun (A : nat) (Δ : duration)
=>
task_rbf (A + 1 ) -
task_cost tsk +
task_IBF Δ: nat -> duration -> nat A : duration H_A_is_in_abstract_search_space : search_space.is_in_search_space
L
total_interference_bound
A
exists F : duration,
task_rbf (A + 1 ) - (task_cost tsk - task_rtct tsk) +
task_IBF (A + F) <= A + F /\
F + (task_cost tsk - task_rtct tsk) <= R
Proof .Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat H_task_cost_pos : 0 < task_cost tskH_arrival_curve_pos : 0 < max_arrivals tsk 1 total_interference_bound := fun (A : nat) (Δ : duration)
=>
task_rbf (A + 1 ) -
task_cost tsk +
task_IBF Δ: nat -> duration -> nat A : duration H_A_is_in_abstract_search_space : search_space.is_in_search_space
L
total_interference_bound
A
exists F : duration,
task_rbf (A + 1 ) - (task_cost tsk - task_rtct tsk) +
task_IBF (A + F) <= A + F /\
F + (task_cost tsk - task_rtct tsk) <= R
move : (H_R_is_maximum A) => FIX.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat H_task_cost_pos : 0 < task_cost tskH_arrival_curve_pos : 0 < max_arrivals tsk 1 total_interference_bound := fun (A : nat) (Δ : duration)
=>
task_rbf (A + 1 ) -
task_cost tsk +
task_IBF Δ: nat -> duration -> nat A : duration H_A_is_in_abstract_search_space : search_space.is_in_search_space
L
total_interference_bound
A FIX : is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <= R
exists F : duration,
task_rbf (A + 1 ) - (task_cost tsk - task_rtct tsk) +
task_IBF (A + F) <= A + F /\
F + (task_cost tsk - task_rtct tsk) <= R
feed FIX; first by apply A_is_in_concrete_search_space. Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat H_task_cost_pos : 0 < task_cost tskH_arrival_curve_pos : 0 < max_arrivals tsk 1 total_interference_bound := fun (A : nat) (Δ : duration)
=>
task_rbf (A + 1 ) -
task_cost tsk +
task_IBF Δ: nat -> duration -> nat A : duration H_A_is_in_abstract_search_space : search_space.is_in_search_space
L
total_interference_bound
A FIX : exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <= R
exists F : duration,
task_rbf (A + 1 ) - (task_cost tsk - task_rtct tsk) +
task_IBF (A + F) <= A + F /\
F + (task_cost tsk - task_rtct tsk) <= R
move : FIX => [F [FIX NEQ]].Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat H_task_cost_pos : 0 < task_cost tskH_arrival_curve_pos : 0 < max_arrivals tsk 1 total_interference_bound := fun (A : nat) (Δ : duration)
=>
task_rbf (A + 1 ) -
task_cost tsk +
task_IBF Δ: nat -> duration -> nat A : duration H_A_is_in_abstract_search_space : search_space.is_in_search_space
L
total_interference_bound
A F : duration FIX : priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F NEQ : F + (task_cost tsk - task_rtct tsk) <= R
exists F : duration,
task_rbf (A + 1 ) - (task_cost tsk - task_rtct tsk) +
task_IBF (A + F) <= A + F /\
F + (task_cost tsk - task_rtct tsk) <= R
exists F ; split ; last by done .Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat H_task_cost_pos : 0 < task_cost tskH_arrival_curve_pos : 0 < max_arrivals tsk 1 total_interference_bound := fun (A : nat) (Δ : duration)
=>
task_rbf (A + 1 ) -
task_cost tsk +
task_IBF Δ: nat -> duration -> nat A : duration H_A_is_in_abstract_search_space : search_space.is_in_search_space
L
total_interference_bound
A F : duration FIX : priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F NEQ : F + (task_cost tsk - task_rtct tsk) <= R
task_rbf (A + 1 ) - (task_cost tsk - task_rtct tsk) +
task_IBF (A + F) <= A + F
rewrite -{2 }(leqRW FIX).Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat H_task_cost_pos : 0 < task_cost tskH_arrival_curve_pos : 0 < max_arrivals tsk 1 total_interference_bound := fun (A : nat) (Δ : duration)
=>
task_rbf (A + 1 ) -
task_cost tsk +
task_IBF Δ: nat -> duration -> nat A : duration H_A_is_in_abstract_search_space : search_space.is_in_search_space
L
total_interference_bound
A F : duration FIX : priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F NEQ : F + (task_cost tsk - task_rtct tsk) <= R
task_rbf (A + 1 ) - (task_cost tsk - task_rtct tsk) +
task_IBF (A + F) <=
priority_inversion_bound +
(task_rbf (A + 1 ) - (task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F)
by rewrite addnA [_ + priority_inversion_bound]addnC -!addnA.
Qed .
End SolutionOfResponseTimeRecurrenceExists .
End FillingOutHypothesesOfAbstractRTATheorem .
(** ** Final Theorem *)
(** Based on the properties established above, we apply the abstract
analysis framework to infer that [R] is a response-time bound
for [tsk]. *)
Theorem uniprocessor_response_time_bound_fp :
response_time_bounded_by tsk R.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
response_time_bounded_by tsk R
Proof .Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
response_time_bounded_by tsk R
intros js ARRs TSKs.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js
job_response_time_bound sched js R
move : (posnP (@job_cost _ Cost js)) => [ZERO|POS].Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js ZERO : job_cost js = 0
job_response_time_bound sched js R
{ Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js ZERO : job_cost js = 0
job_response_time_bound sched js R
by rewrite /job_response_time_bound /completed_by ZERO. } Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost js
job_response_time_bound sched js R
eapply uniprocessor_response_time_bound_seq => //.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost js
work_conserving arr_seq sched
- Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost js
work_conserving arr_seq sched
exact : instantiated_i_and_w_are_coherent_with_schedule.
- Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost js
interference_and_workload_consistent_with_sequential_tasks
arr_seq sched tsk
exact : instantiated_interference_and_workload_consistent_with_sequential_tasks.
- Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost js
busy_intervals_are_bounded_by arr_seq sched tsk ?L
exact : instantiated_busy_intervals_are_bounded.
- Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost js
task_interference_is_bounded_by arr_seq sched tsk
?task_IBF
exact : instantiated_task_interference_is_bounded.
- Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule (ideal.processor_state Job) JobReady0 : JobReady Job (ideal.processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := work_conserving arr_seq sched : Prop work_conserving_cl := work_conserving.work_conserving
arr_seq sched : Prop H_work_conserving : work_conserving_cl H_sequential_tasks : sequential_tasks arr_seq sched H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H3 : 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 job_pending_at := pending sched : Job -> instant -> bool job_scheduled_at := scheduled_at sched : Job -> instant -> bool job_completed_by := completed_by sched : Job -> instant -> bool job_backlogged_at := backlogged sched : Job -> instant -> bool response_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop task_rbf := task_request_bound_function tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat priority_inversion_bound : duration H_priority_inversion_is_bounded : priority_inversion_is_bounded_by
arr_seq sched tsk
(constant
priority_inversion_bound) L : duration H_L_positive : 0 < LH_fixed_point : L =
priority_inversion_bound +
total_hep_rbf L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rtask_IBF := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost js
forall A : duration,
search_space.is_in_search_space L
(fun A0 Δ : duration =>
task_request_bound_function tsk (A0 + 1 ) -
task_cost tsk + (fun => [eta task_IBF]) A0 Δ) A ->
exists F : duration,
task_request_bound_function tsk (A + 1 ) -
(task_cost tsk - task_rtct tsk) +
(fun => [eta task_IBF]) A (A + F) <= A + F /\
F + (task_cost tsk - task_rtct tsk) <= R
exact : correct_search_space.
Qed .
End AbstractRTAforFPwithArrivalCurves .