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Require Export prosa.model.schedule.priority_driven.Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]Notation "_ + _" was already used in scope nat_scope.
[notation-overridden,parsing]Notation "_ - _" was already used in scope nat_scope.
[notation-overridden,parsing]Notation "_ <= _" was already used in scope nat_scope.
[notation-overridden,parsing]Notation "_ < _" was already used in scope nat_scope.
[notation-overridden,parsing]Notation "_ >= _" was already used in scope nat_scope.
[notation-overridden,parsing]Notation "_ > _" was already used in scope nat_scope.
[notation-overridden,parsing]Notation "_ <= _ <= _" was already used in scope
nat_scope. [notation-overridden,parsing]Notation "_ < _ <= _" was already used in scope
nat_scope. [notation-overridden,parsing]Notation "_ <= _ < _" was already used in scope
nat_scope. [notation-overridden,parsing]Notation "_ < _ < _" was already used in scope
nat_scope. [notation-overridden,parsing]Notation "_ * _" was already used in scope nat_scope.
[notation-overridden,parsing]
Require Import prosa.analysis.abstract .ideal_jlfp_rta.Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
Require Export prosa.analysis.facts.busy_interval.busy_interval.Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
(** Throughout this file, we assume ideal uni-processor schedules. *)
Require Import prosa.model.processor.ideal.Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ | _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ : _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ | _ ]" was already used
in scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ & _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ | _ ]" was already used in
scope fun_scope. [notation-overridden,parsing]Notation "[ rel _ _ in _ ]" was already used in scope
fun_scope. [notation-overridden,parsing]
(** * 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_policy Task}.
Hypothesis H_priority_is_reflexive : reflexive_priorities.
(** Consider any arrival sequence with consistent, non-duplicate arrivals. *)
Variable arr_seq : arrival_sequence Job.
Hypothesis H_arrival_times_are_consistent : consistent_arrival_times arr_seq.
Hypothesis H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq.
(** Next, consider any 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) _ Cost Arrival}.
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.
(** 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 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).
(** 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. *)
Let interference (j : Job) (t : instant) :=
ideal_jlfp_rta.interference sched j t.
(** 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. *)
Let interfering_workload (j : Job) (t : instant) :=
ideal_jlfp_rta.interfering_workload arr_seq sched j t.
(** Finally, we define the interference bound function ([IBF_other]). [IBF_other] 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 [IBF_other] as the sum of the priority
interference bound and the higher-or-equal-priority workload. *)
Let IBF_other (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 .
(** First, we prove that in the instantiation of interference and interfering workload,
we really take into account everything that can interfere with [tsk]'s jobs, and thus,
the scheduler satisfies the abstract notion of work conserving schedule. *)
Lemma instantiated_i_and_w_are_consistent_with_schedule :
work_conserving_ab tsk interference interfering_workload.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
work_conserving_ab tsk interference
interfering_workload
Proof .Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
work_conserving_ab tsk interference
interfering_workload
intros j t1 t2 t ARR TSK POS BUSY NEQ; split ; intros HYP.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job t1, t2 : instant t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : definitions.busy_interval sched interference
interfering_workload j t1 t2 NEQ : t1 <= t < t2 HYP : ~ interference j t
scheduled_at sched j t
- Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job t1, t2 : instant t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : definitions.busy_interval sched interference
interfering_workload j t1 t2 NEQ : t1 <= t < t2 HYP : ~ interference j t
scheduled_at sched j t
move : HYP => /negP.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job t1, t2 : instant t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : definitions.busy_interval sched interference
interfering_workload j t1 t2 NEQ : t1 <= t < t2
~~ interference j t -> scheduled_at sched j t
rewrite negb_or /is_priority_inversion /is_priority_inversion
/is_interference_from_another_hep_job.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job t1, t2 : instant t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : definitions.busy_interval sched interference
interfering_workload j t1 t2 NEQ : t1 <= t < t2
~~
match sched t with
| Some jlp => ~~ hep_job jlp j
| None => false
end &&
~~
match sched t with
| Some jhp => hep_job jhp j && (jhp != j)
| None => false
end -> scheduled_at sched j t
move => /andP [HYP1 HYP2].Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job t1, t2 : instant t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : definitions.busy_interval sched interference
interfering_workload j t1 t2 NEQ : t1 <= t < t2 HYP1 : ~~
match sched t with
| Some jlp => ~~ hep_job jlp j
| None => false
end HYP2 : ~~
match sched t with
| Some jhp => hep_job jhp j && (jhp != j)
| None => false
end
scheduled_at sched j t
move : H_sched_valid => [CARR MBR].Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job t1, t2 : instant t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : definitions.busy_interval sched interference
interfering_workload j t1 t2 NEQ : t1 <= t < t2 HYP1 : ~~
match sched t with
| Some jlp => ~~ hep_job jlp j
| None => false
end HYP2 : ~~
match sched t with
| Some jhp => hep_job jhp j && (jhp != j)
| None => false
end CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched
scheduled_at sched j t
case SCHED: (sched t) => [s | ].Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job t1, t2 : instant t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : definitions.busy_interval sched interference
interfering_workload j t1 t2 NEQ : t1 <= t < t2 HYP1 : ~~
match sched t with
| Some jlp => ~~ hep_job jlp j
| None => false
end HYP2 : ~~
match sched t with
| Some jhp => hep_job jhp j && (jhp != j)
| None => false
end CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched s : Job SCHED : sched t = Some s
scheduled_at sched j t
+ Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job t1, t2 : instant t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : definitions.busy_interval sched interference
interfering_workload j t1 t2 NEQ : t1 <= t < t2 HYP1 : ~~
match sched t with
| Some jlp => ~~ hep_job jlp j
| None => false
end HYP2 : ~~
match sched t with
| Some jhp => hep_job jhp j && (jhp != j)
| None => false
end CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched s : Job SCHED : sched t = Some s
scheduled_at sched j t
rewrite SCHED in HYP1, HYP2.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job t1, t2 : instant t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : definitions.busy_interval sched interference
interfering_workload j t1 t2 NEQ : t1 <= t < t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched s : Job SCHED : sched t = Some s HYP1 : ~~ ~~ hep_job s j HYP2 : ~~ (hep_job s j && (s != j))
scheduled_at sched j t
move : HYP1 HYP2.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job t1, t2 : instant t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : definitions.busy_interval sched interference
interfering_workload j t1 t2 NEQ : t1 <= t < t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched s : Job SCHED : sched t = Some s
~~ ~~ hep_job s j ->
~~ (hep_job s j && (s != j)) -> scheduled_at sched j t
rewrite !Bool.negb_involutive negb_and Bool.negb_involutive.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job t1, t2 : instant t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : definitions.busy_interval sched interference
interfering_workload j t1 t2 NEQ : t1 <= t < t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched s : Job SCHED : sched t = Some s
hep_job s j ->
~~ hep_job s j || (s == j) -> scheduled_at sched j t
move => HYP1 /orP [/negP HYP2| /eqP HYP2].Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job t1, t2 : instant t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : definitions.busy_interval sched interference
interfering_workload j t1 t2 NEQ : t1 <= t < t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched s : Job SCHED : sched t = Some s HYP1 : hep_job s j HYP2 : ~ hep_job s j
scheduled_at sched j t
* Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job t1, t2 : instant t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : definitions.busy_interval sched interference
interfering_workload j t1 t2 NEQ : t1 <= t < t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched s : Job SCHED : sched t = Some s HYP1 : hep_job s j HYP2 : ~ hep_job s j
scheduled_at sched j t
by exfalso .Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job t1, t2 : instant t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : definitions.busy_interval sched interference
interfering_workload j t1 t2 NEQ : t1 <= t < t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched s : Job SCHED : sched t = Some s HYP1 : hep_job s j HYP2 : s = j
scheduled_at sched j t
* Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job t1, t2 : instant t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : definitions.busy_interval sched interference
interfering_workload j t1 t2 NEQ : t1 <= t < t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched s : Job SCHED : sched t = Some s HYP1 : hep_job s j HYP2 : s = j
scheduled_at sched j t
by subst s; rewrite scheduled_at_def //; apply /eqP.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job t1, t2 : instant t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : definitions.busy_interval sched interference
interfering_workload j t1 t2 NEQ : t1 <= t < t2 HYP1 : ~~
match sched t with
| Some jlp => ~~ hep_job jlp j
| None => false
end HYP2 : ~~
match sched t with
| Some jhp => hep_job jhp j && (jhp != j)
| None => false
end CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched SCHED : sched t = None
scheduled_at sched j t
+ Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job t1, t2 : instant t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : definitions.busy_interval sched interference
interfering_workload j t1 t2 NEQ : t1 <= t < t2 HYP1 : ~~
match sched t with
| Some jlp => ~~ hep_job jlp j
| None => false
end HYP2 : ~~
match sched t with
| Some jhp => hep_job jhp j && (jhp != j)
| None => false
end CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched SCHED : sched t = None
scheduled_at sched j t
exfalso ; clear HYP1 HYP2.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job t1, t2 : instant t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : definitions.busy_interval sched interference
interfering_workload j t1 t2 NEQ : t1 <= t < t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched SCHED : sched t = None
False
eapply instantiated_busy_interval_equivalent_busy_interval in BUSY; eauto with basic_facts.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job t1, t2 : instant t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : busy_interval arr_seq sched j t1 t2 NEQ : t1 <= t < t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched SCHED : sched t = None
False
by move : BUSY => [PREF _]; eapply not_quiet_implies_not_idle; eauto 2 with basic_facts; apply /eqP.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job t1, t2 : instant t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : definitions.busy_interval sched interference
interfering_workload j t1 t2 NEQ : t1 <= t < t2 HYP : scheduled_at sched j t
~ interference j t
- Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job t1, t2 : instant t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : definitions.busy_interval sched interference
interfering_workload j t1 t2 NEQ : t1 <= t < t2 HYP : scheduled_at sched j t
~ interference j t
move : (HYP); rewrite scheduled_at_def; move => /eqP HYP2; apply /negP.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job t1, t2 : instant t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : definitions.busy_interval sched interference
interfering_workload j t1 t2 NEQ : t1 <= t < t2 HYP : scheduled_at sched j t HYP2 : sched t = Some j
~~ interference j t
rewrite /interference /ideal_jlfp_rta.interference /is_priority_inversion
/is_interference_from_another_hep_job HYP2 negb_or.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job t1, t2 : instant t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : definitions.busy_interval sched interference
interfering_workload j t1 t2 NEQ : t1 <= t < t2 HYP : scheduled_at sched j t HYP2 : sched t = Some j
~~ ~~ hep_job j j && ~~ (hep_job j j && (j != j))
apply /andP; split .Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job t1, t2 : instant t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : definitions.busy_interval sched interference
interfering_workload j t1 t2 NEQ : t1 <= t < t2 HYP : scheduled_at sched j t HYP2 : sched t = Some j
~~ ~~ hep_job j j
+ Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job t1, t2 : instant t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : definitions.busy_interval sched interference
interfering_workload j t1 t2 NEQ : t1 <= t < t2 HYP : scheduled_at sched j t HYP2 : sched t = Some j
~~ ~~ hep_job j j
rewrite Bool.negb_involutive; eauto 2 .Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job t1, t2 : instant t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : definitions.busy_interval sched interference
interfering_workload j t1 t2 NEQ : t1 <= t < t2 HYP : scheduled_at sched j t HYP2 : sched t = Some j
hep_job j j
by eapply H_priority_is_reflexive with (t := 0 ).Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job t1, t2 : instant t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : definitions.busy_interval sched interference
interfering_workload j t1 t2 NEQ : t1 <= t < t2 HYP : scheduled_at sched j t HYP2 : sched t = Some j
~~ (hep_job j j && (j != j))
+ Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job t1, t2 : instant t : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : definitions.busy_interval sched interference
interfering_workload j t1 t2 NEQ : t1 <= t < t2 HYP : scheduled_at sched j t HYP2 : sched t = Some j
~~ (hep_job j j && (j != j))
by rewrite negb_and Bool.negb_involutive; apply /orP; right .
Qed .
(** Next, we prove that the interference and interfering workload
functions are consistent with sequential tasks. *)
Lemma instantiated_interference_and_workload_consistent_with_sequential_tasks :
interference_and_workload_consistent_with_sequential_tasks
arr_seq sched tsk interference interfering_workload.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
interference_and_workload_consistent_with_sequential_tasks
arr_seq sched tsk interference interfering_workload
Proof .Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
interference_and_workload_consistent_with_sequential_tasks
arr_seq sched tsk interference interfering_workload
intros j t1 t2 ARR TSK POS BUSY.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job t1, t2 : instant ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : definitions.busy_interval sched interference
interfering_workload j t1 t2
task_workload_between arr_seq tsk 0 t1 =
task_service_of_jobs_in sched tsk
(arrivals_between arr_seq 0 t1) 0 t1
move : H_sched_valid => [CARR MBR].Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job t1, t2 : instant ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : definitions.busy_interval sched interference
interfering_workload j t1 t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched
task_workload_between arr_seq tsk 0 t1 =
task_service_of_jobs_in sched tsk
(arrivals_between arr_seq 0 t1) 0 t1
eapply instantiated_busy_interval_equivalent_busy_interval in BUSY; eauto with basic_facts.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job t1, t2 : instant ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : busy_interval arr_seq sched j t1 t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched
task_workload_between arr_seq tsk 0 t1 =
task_service_of_jobs_in sched tsk
(arrivals_between arr_seq 0 t1) 0 t1
eapply all_jobs_have_completed_equiv_workload_eq_service; eauto with basic_facts.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job t1, t2 : instant ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : busy_interval arr_seq sched j t1 t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched
forall j : Job,
j \in arrivals_between arr_seq 0 t1 ->
job_of_task tsk j -> completed_by sched j t1
intros s 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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job t1, t2 : instant ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : busy_interval arr_seq sched j t1 t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched s : Job ARRs : s \in arrivals_between arr_seq 0 t1 TSKs : job_of_task tsk s
completed_by sched s t1
move : (BUSY) => [[_ [QT _]] _].Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job t1, t2 : instant ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : busy_interval arr_seq sched j t1 t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched s : Job ARRs : s \in arrivals_between arr_seq 0 t1 TSKs : job_of_task tsk s QT : quiet_time arr_seq sched j t1
completed_by sched s t1
apply QT.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job t1, t2 : instant ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : busy_interval arr_seq sched j t1 t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched s : Job ARRs : s \in arrivals_between arr_seq 0 t1 TSKs : job_of_task tsk s QT : quiet_time arr_seq sched j t1
arrives_in arr_seq s
- Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job t1, t2 : instant ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : busy_interval arr_seq sched j t1 t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched s : Job ARRs : s \in arrivals_between arr_seq 0 t1 TSKs : job_of_task tsk s QT : quiet_time arr_seq sched j t1
arrives_in arr_seq s
by apply in_arrivals_implies_arrived in ARRs.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job t1, t2 : instant ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : busy_interval arr_seq sched j t1 t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched s : Job ARRs : s \in arrivals_between arr_seq 0 t1 TSKs : job_of_task tsk s QT : quiet_time arr_seq sched j t1
hep_job s j
- Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job t1, t2 : instant ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : busy_interval arr_seq sched j t1 t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched s : Job ARRs : s \in arrivals_between arr_seq 0 t1 TSKs : job_of_task tsk s QT : quiet_time arr_seq sched j t1
hep_job s j
move : TSKs => /eqP TSKs.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job t1, t2 : instant ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : busy_interval arr_seq sched j t1 t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched s : Job ARRs : s \in arrivals_between arr_seq 0 t1 QT : quiet_time arr_seq sched j t1 TSKs : job_task s = tsk
hep_job s j
rewrite /hep_job /FP_to_JLFP TSK -TSKs; eauto 2 .Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job t1, t2 : instant ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : busy_interval arr_seq sched j t1 t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched s : Job ARRs : s \in arrivals_between arr_seq 0 t1 QT : quiet_time arr_seq sched j t1 TSKs : job_task s = tsk
hep_task (job_task s) (job_task s)
by eapply (H_priority_is_reflexive 0 ); eauto .Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job t1, t2 : instant ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : busy_interval arr_seq sched j t1 t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched s : Job ARRs : s \in arrivals_between arr_seq 0 t1 TSKs : job_of_task tsk s QT : quiet_time arr_seq sched j t1
arrived_before s t1
- Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job t1, t2 : instant ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jBUSY : busy_interval arr_seq sched j t1 t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched s : Job ARRs : s \in arrivals_between arr_seq 0 t1 TSKs : job_of_task tsk s QT : quiet_time arr_seq sched j t1
arrived_before s t1
by eapply in_arrivals_implies_arrived_before; eauto 2 .
Qed .
(** 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 interference interfering_workload L.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
busy_intervals_are_bounded_by arr_seq sched tsk
interference interfering_workload 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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
busy_intervals_are_bounded_by arr_seq sched tsk
interference interfering_workload L
intros 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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost j
exists t1 t2 : nat,
t1 <= job_arrival j < t2 /\
t2 <= t1 + L /\
definitions.busy_interval sched interference
interfering_workload j t1 t2
move : H_sched_valid => [CARR MBR].Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jCARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched
exists t1 t2 : nat,
t1 <= job_arrival j < t2 /\
t2 <= t1 + L /\
definitions.busy_interval sched interference
interfering_workload j t1 t2
edestruct (exists_busy_interval) with (delta := L) as [t1 [t2 [T1 [T2 GGG]]]];
eauto 2 with basic_facts.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jCARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched
forall t : instant,
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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jCARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched
forall t : instant,
priority_inversion_bound +
workload_of_higher_or_equal_priority_jobs j
(arrivals_between arr_seq t (t + L)) <= L
by intros ; rewrite {2 }H_fixed_point leq_add //; apply total_workload_le_total_hep_rbf. } Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jCARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched t1, t2 : nat T1 : t1 <= job_arrival j < t2 T2 : t2 <= t1 + L GGG : busy_interval arr_seq sched j t1 t2
exists t1 t2 : nat,
t1 <= job_arrival j < t2 /\
t2 <= t1 + L /\
definitions.busy_interval sched interference
interfering_workload j t1 t2
exists t1 , t2; 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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jCARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched t1, t2 : nat T1 : t1 <= job_arrival j < t2 T2 : t2 <= t1 + L GGG : busy_interval arr_seq sched j t1 t2
t2 <= t1 + L /\
definitions.busy_interval sched interference
interfering_workload j t1 t2
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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job ARR : arrives_in arr_seq j TSK : job_task j = tsk POS : 0 < job_cost jCARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched t1, t2 : nat T1 : t1 <= job_arrival j < t2 T2 : t2 <= t1 + L GGG : busy_interval arr_seq sched j t1 t2
definitions.busy_interval sched interference
interfering_workload j t1 t2
by eapply instantiated_busy_interval_equivalent_busy_interval; eauto 2 with basic_facts.
Qed .
(** Next, we prove that [IBF_other] 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 t, the relative arrival time of a job j of task t,
and the length of the interval to the maximum amount of interference (for more details see
files limited.abstract_RTA.definitions and limited.abstract_RTA.abstract_seq_rta).
However, in this module we analyze only one task -- [tsk], therefore it is “hard-coded”
inside the interference bound function [IBF_other]. 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 [IBF_other] signature to
match the required signature in module abstract_seq_RTA, we wrap the [IBF_other] 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 interference interfering_workload (fun t A R => IBF_other R).Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
task_interference_is_bounded_by arr_seq sched tsk
interference interfering_workload
(fun => (fun => [eta IBF_other]))
Proof .Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat
task_interference_is_bounded_by arr_seq sched tsk
interference interfering_workload
(fun => (fun => [eta IBF_other]))
intros ? ? ? ? ARR TSK ? NCOMPL BUSY; simpl .Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job R0, t1, t2 : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk H6 : t1 + R0 < t2 NCOMPL : ~~ completed_by sched j (t1 + R0) BUSY : definitions.busy_interval sched interference
interfering_workload j t1 t2
cumul_task_interference arr_seq sched interference tsk
t2 t1 (t1 + R0) <= IBF_other R0
move : H_sched_valid => [CARR MBR].Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job R0, t1, t2 : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk H6 : t1 + R0 < t2 NCOMPL : ~~ completed_by sched j (t1 + R0) BUSY : definitions.busy_interval sched interference
interfering_workload j t1 t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched
cumul_task_interference arr_seq sched interference tsk
t2 t1 (t1 + R0) <= IBF_other R0
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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job R0, t1, t2 : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk H6 : t1 + R0 < t2 NCOMPL : ~~ completed_by sched j (t1 + R0) BUSY : definitions.busy_interval sched interference
interfering_workload j t1 t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched ZERO : job_cost j = 0
cumul_task_interference arr_seq sched interference tsk
t2 t1 (t1 + R0) <= IBF_other R0
{ Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job R0, t1, t2 : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk H6 : t1 + R0 < t2 NCOMPL : ~~ completed_by sched j (t1 + R0) BUSY : definitions.busy_interval sched interference
interfering_workload j t1 t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched ZERO : job_cost j = 0
cumul_task_interference arr_seq sched interference tsk
t2 t1 (t1 + R0) <= IBF_other R0
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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job R0, t1, t2 : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk H6 : t1 + R0 < t2 NCOMPL : ~~ completed_by sched j (t1 + R0) BUSY : definitions.busy_interval sched interference
interfering_workload j t1 t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched POS : 0 < job_cost j
cumul_task_interference arr_seq sched interference tsk
t2 t1 (t1 + R0) <= IBF_other R0
eapply instantiated_busy_interval_equivalent_busy_interval in BUSY; eauto 2 with basic_facts.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job R0, t1, t2 : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk H6 : t1 + R0 < t2 NCOMPL : ~~ completed_by sched j (t1 + R0) BUSY : busy_interval arr_seq sched j t1 t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched POS : 0 < job_cost j
cumul_task_interference arr_seq sched interference tsk
t2 t1 (t1 + R0) <= IBF_other R0
rewrite /interference; erewrite cumulative_task_interference_split; eauto 2 with basic_facts; 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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job R0, t1, t2 : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk H6 : t1 + R0 < t2 NCOMPL : ~~ completed_by sched j (t1 + R0) BUSY : busy_interval arr_seq sched j t1 t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched POS : 0 < job_cost j
j \in arrivals_before arr_seq t2
{ Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job R0, t1, t2 : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk H6 : t1 + R0 < t2 NCOMPL : ~~ completed_by sched j (t1 + R0) BUSY : busy_interval arr_seq sched j t1 t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched POS : 0 < job_cost j
j \in arrivals_before arr_seq t2
move : BUSY => [[_ [_ [_ /andP [GE LT]]]] _].Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job R0, t1, t2 : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk H6 : t1 + R0 < t2 NCOMPL : ~~ completed_by sched j (t1 + R0) CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched POS : 0 < job_cost jGE : t1 <= job_arrival j LT : job_arrival j < t2
j \in arrivals_before arr_seq t2
by eapply arrived_between_implies_in_arrivals; eauto 2 . } Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job R0, t1, t2 : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk H6 : t1 + R0 < t2 NCOMPL : ~~ completed_by sched j (t1 + R0) BUSY : busy_interval arr_seq sched j t1 t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched POS : 0 < job_cost j
ideal_jlfp_rta.cumulative_priority_inversion sched j
t1 (t1 + R0) +
cumulative_interference_from_hep_jobs_from_other_tasks
sched j t1 (t1 + R0) <= IBF_other R0
unfold IBF_other, interference.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job R0, t1, t2 : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk H6 : t1 + R0 < t2 NCOMPL : ~~ completed_by sched j (t1 + R0) BUSY : busy_interval arr_seq sched j t1 t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched POS : 0 < job_cost j
ideal_jlfp_rta.cumulative_priority_inversion sched j
t1 (t1 + R0) +
cumulative_interference_from_hep_jobs_from_other_tasks
sched j t1 (t1 + R0) <=
priority_inversion_bound + total_ohep_rbf R0
rewrite leq_add; 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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job R0, t1, t2 : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk H6 : t1 + R0 < t2 NCOMPL : ~~ completed_by sched j (t1 + R0) BUSY : busy_interval arr_seq sched j t1 t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched POS : 0 < job_cost j
ideal_jlfp_rta.cumulative_priority_inversion sched j
t1 (t1 + R0) <= 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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job R0, t1, t2 : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk H6 : t1 + R0 < t2 NCOMPL : ~~ completed_by sched j (t1 + R0) BUSY : busy_interval arr_seq sched j t1 t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched POS : 0 < job_cost j
ideal_jlfp_rta.cumulative_priority_inversion sched j
t1 (t1 + R0) <= priority_inversion_bound
move : (H_priority_inversion_is_bounded j ARR TSK) => BOUND.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job R0, t1, t2 : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk H6 : t1 + R0 < t2 NCOMPL : ~~ completed_by sched j (t1 + R0) BUSY : busy_interval arr_seq sched j t1 t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched POS : 0 < job_cost jBOUND : 0 < job_cost j ->
priority_inversion_of_job_is_bounded_by
arr_seq sched j priority_inversion_bound
ideal_jlfp_rta.cumulative_priority_inversion sched j
t1 (t1 + R0) <= priority_inversion_bound
apply leq_trans with (cumulative_priority_inversion sched j t1 (t1 + R0)); 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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job R0, t1, t2 : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk H6 : t1 + R0 < t2 NCOMPL : ~~ completed_by sched j (t1 + R0) BUSY : busy_interval arr_seq sched j t1 t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched POS : 0 < job_cost jBOUND : 0 < job_cost j ->
priority_inversion_of_job_is_bounded_by
arr_seq sched j priority_inversion_bound
cumulative_priority_inversion sched j t1 (t1 + R0) <=
priority_inversion_bound
apply leq_trans with (cumulative_priority_inversion sched j t1 t2); 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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job R0, t1, t2 : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk H6 : t1 + R0 < t2 NCOMPL : ~~ completed_by sched j (t1 + R0) BUSY : busy_interval arr_seq sched j t1 t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched POS : 0 < job_cost jBOUND : 0 < job_cost j ->
priority_inversion_of_job_is_bounded_by
arr_seq sched j priority_inversion_bound
cumulative_priority_inversion sched j t1 t2 <=
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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job R0, t1, t2 : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk H6 : t1 + R0 < t2 NCOMPL : ~~ completed_by sched j (t1 + R0) BUSY : busy_interval arr_seq sched j t1 t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched POS : 0 < job_cost jBOUND : 0 < job_cost j ->
priority_inversion_of_job_is_bounded_by
arr_seq sched j priority_inversion_bound
cumulative_priority_inversion sched j t1 t2 <=
priority_inversion_bound
by apply BOUND; move : BUSY => [PREF QT2]. } Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job R0, t1, t2 : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk H6 : t1 + R0 < t2 NCOMPL : ~~ completed_by sched j (t1 + R0) BUSY : busy_interval arr_seq sched j t1 t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched POS : 0 < job_cost jBOUND : 0 < job_cost j ->
priority_inversion_of_job_is_bounded_by
arr_seq sched j priority_inversion_bound
cumulative_priority_inversion sched j t1 (t1 + R0) <=
cumulative_priority_inversion sched j t1 t2
rewrite [X in _ <= X](@big_cat_nat _ _ _ (t1 + R0)) //=.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job R0, t1, t2 : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk H6 : t1 + R0 < t2 NCOMPL : ~~ completed_by sched j (t1 + R0) BUSY : busy_interval arr_seq sched j t1 t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched POS : 0 < job_cost jBOUND : 0 < job_cost j ->
priority_inversion_of_job_is_bounded_by
arr_seq sched j priority_inversion_bound
cumulative_priority_inversion sched j t1 (t1 + R0) <=
\sum_(t1 <= i < t1 + R0)
is_priority_inversion sched j i +
\sum_(t1 + R0 <= i < t2)
is_priority_inversion sched j i
- Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job R0, t1, t2 : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk H6 : t1 + R0 < t2 NCOMPL : ~~ completed_by sched j (t1 + R0) BUSY : busy_interval arr_seq sched j t1 t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched POS : 0 < job_cost jBOUND : 0 < job_cost j ->
priority_inversion_of_job_is_bounded_by
arr_seq sched j priority_inversion_bound
cumulative_priority_inversion sched j t1 (t1 + R0) <=
\sum_(t1 <= i < t1 + R0)
is_priority_inversion sched j i +
\sum_(t1 + R0 <= i < t2)
is_priority_inversion sched j i
by rewrite 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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job R0, t1, t2 : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk H6 : t1 + R0 < t2 NCOMPL : ~~ completed_by sched j (t1 + R0) BUSY : busy_interval arr_seq sched j t1 t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched POS : 0 < job_cost jBOUND : 0 < job_cost j ->
priority_inversion_of_job_is_bounded_by
arr_seq sched j priority_inversion_bound
t1 <= t1 + R0
- Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job R0, t1, t2 : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk H6 : t1 + R0 < t2 NCOMPL : ~~ completed_by sched j (t1 + R0) BUSY : busy_interval arr_seq sched j t1 t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched POS : 0 < job_cost jBOUND : 0 < job_cost j ->
priority_inversion_of_job_is_bounded_by
arr_seq sched j priority_inversion_bound
t1 <= t1 + R0
by rewrite 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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job R0, t1, t2 : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk H6 : t1 + R0 < t2 NCOMPL : ~~ completed_by sched j (t1 + R0) BUSY : busy_interval arr_seq sched j t1 t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched POS : 0 < job_cost jBOUND : 0 < job_cost j ->
priority_inversion_of_job_is_bounded_by
arr_seq sched j priority_inversion_bound
t1 + R0 <= t2
- Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job R0, t1, t2 : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk H6 : t1 + R0 < t2 NCOMPL : ~~ completed_by sched j (t1 + R0) BUSY : busy_interval arr_seq sched j t1 t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched POS : 0 < job_cost jBOUND : 0 < job_cost j ->
priority_inversion_of_job_is_bounded_by
arr_seq sched j priority_inversion_bound
t1 + R0 <= t2
by rewrite ltnW.
} Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job R0, t1, t2 : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk H6 : t1 + R0 < t2 NCOMPL : ~~ completed_by sched j (t1 + R0) BUSY : busy_interval arr_seq sched j t1 t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched POS : 0 < job_cost j
cumulative_interference_from_hep_jobs_from_other_tasks
sched j t1 (t1 + R0) <= total_ohep_rbf R0
{ Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job R0, t1, t2 : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk H6 : t1 + R0 < t2 NCOMPL : ~~ completed_by sched j (t1 + R0) BUSY : busy_interval arr_seq sched j t1 t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched POS : 0 < job_cost j
cumulative_interference_from_hep_jobs_from_other_tasks
sched j t1 (t1 + R0) <= total_ohep_rbf R0
erewrite instantiated_cumulative_interference_of_hep_tasks_equal_total_interference_of_hep_tasks;
eauto 2 ; last by unfold quiet_time; move : BUSY => [[_ [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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job R0, t1, t2 : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk H6 : t1 + R0 < t2 NCOMPL : ~~ completed_by sched j (t1 + R0) BUSY : busy_interval arr_seq sched j t1 t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched POS : 0 < job_cost j
service_of_jobs sched
(fun jhp : Job =>
hep_job jhp j && (job_task jhp != job_task j))
(arrivals_between arr_seq t1 (t1 + R0)) t1 (t1 + R0) <=
total_ohep_rbf R0
apply leq_trans with
(workload_of_jobs
(fun jhp : Job => (FP_to_JLFP _ _) jhp j && (job_task jhp != job_task j))
(arrivals_between arr_seq t1 (t1 + R0))).Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job R0, t1, t2 : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk H6 : t1 + R0 < t2 NCOMPL : ~~ completed_by sched j (t1 + R0) BUSY : busy_interval arr_seq sched j t1 t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched POS : 0 < job_cost j
service_of_jobs sched
(fun jhp : Job =>
hep_job jhp j && (job_task jhp != job_task j))
(arrivals_between arr_seq t1 (t1 + R0)) t1 (t1 + R0) <=
workload_of_jobs
(fun jhp : Job =>
FP_to_JLFP Job Task jhp j &&
(job_task jhp != job_task j))
(arrivals_between arr_seq t1 (t1 + R0))
{ Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job R0, t1, t2 : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk H6 : t1 + R0 < t2 NCOMPL : ~~ completed_by sched j (t1 + R0) BUSY : busy_interval arr_seq sched j t1 t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched POS : 0 < job_cost j
service_of_jobs sched
(fun jhp : Job =>
hep_job jhp j && (job_task jhp != job_task j))
(arrivals_between arr_seq t1 (t1 + R0)) t1 (t1 + R0) <=
workload_of_jobs
(fun jhp : Job =>
FP_to_JLFP Job Task jhp j &&
(job_task jhp != job_task j))
(arrivals_between arr_seq t1 (t1 + R0))
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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job R0, t1, t2 : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk H6 : t1 + R0 < t2 NCOMPL : ~~ completed_by sched j (t1 + R0) BUSY : busy_interval arr_seq sched j t1 t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched 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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job R0, t1, t2 : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk H6 : t1 + R0 < t2 NCOMPL : ~~ completed_by sched j (t1 + R0) BUSY : busy_interval arr_seq sched j t1 t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched POS : 0 < job_cost j
workload_of_jobs
(fun jhp : Job =>
FP_to_JLFP Job Task jhp j &&
(job_task jhp != job_task j))
(arrivals_between arr_seq t1 (t1 + R0)) <=
total_ohep_rbf R0
{ Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job R0, t1, t2 : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk H6 : t1 + R0 < t2 NCOMPL : ~~ completed_by sched j (t1 + R0) BUSY : busy_interval arr_seq sched j t1 t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched POS : 0 < job_cost j
workload_of_jobs
(fun jhp : Job =>
FP_to_JLFP Job Task jhp j &&
(job_task jhp != job_task j))
(arrivals_between arr_seq t1 (t1 + R0)) <=
total_ohep_rbf R0
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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job R0, t1, t2 : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk H6 : t1 + R0 < t2 NCOMPL : ~~ completed_by sched j (t1 + R0) BUSY : busy_interval arr_seq sched j t1 t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched POS : 0 < job_cost j
\sum_(j0 <- arrivals_between arr_seq t1 (t1 + R0) |
FP_to_JLFP Job Task j0 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 R0
by rewrite -TSK; apply total_workload_le_total_ohep_rbf. } Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job R0, t1, t2 : nat ARR : arrives_in arr_seq j TSK : job_task j = tsk H6 : t1 + R0 < t2 NCOMPL : ~~ completed_by sched j (t1 + R0) BUSY : busy_interval arr_seq sched j t1 t2 CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched POS : 0 < job_cost j
jobs_must_arrive_to_execute sched
all : eauto 2 using arr_seq with basic_facts. }
Qed .
(** Finally, we show that there exists a solution for the response-time recurrence. *)
Section SolutionOfResponseTimeRecurrenceExists .
(** Consider any job [j] of [tsk]. *)
Variable j : Job.
Hypothesis H_j_arrives : arrives_in arr_seq j.
Hypothesis H_job_of_tsk : job_task j = tsk.
Hypothesis H_job_cost_positive : job_cost_positive j.
(** 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 + IBF_other Δ]. *)
Let total_interference_bound tsk A Δ :=
task_rbf (A + ε) - task_cost tsk + IBF_other 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 tsk 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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_task j = tsk H_job_cost_positive : job_cost_positive j total_interference_bound := fun (tsk : Task)
(A : nat)
(Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk +
IBF_other Δ: Task ->
nat -> duration -> nat A : duration H_A_is_in_abstract_search_space : search_space.is_in_search_space
tsk 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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_task j = tsk H_job_cost_positive : job_cost_positive j total_interference_bound := fun (tsk : Task)
(A : nat)
(Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk +
IBF_other Δ: Task ->
nat -> duration -> nat A : duration H_A_is_in_abstract_search_space : search_space.is_in_search_space
tsk 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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_task j = tsk H_job_cost_positive : job_cost_positive j total_interference_bound := fun (tsk : Task)
(A : nat)
(Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk +
IBF_other Δ: Task ->
nat -> duration -> nat A : duration H_A_is_in_abstract_search_space : search_space.is_in_search_space
tsk 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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_task j = tsk H_job_cost_positive : job_cost_positive j total_interference_bound := fun (tsk : Task)
(A : nat)
(Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk +
IBF_other Δ: Task ->
nat -> duration -> nat A : duration H_A_is_in_abstract_search_space : search_space.is_in_search_space
tsk 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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_task j = tsk H_job_cost_positive : job_cost_positive j total_interference_bound := fun (tsk : Task)
(A : nat)
(Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk +
IBF_other Δ: Task ->
nat -> duration -> nat A : duration H_A_is_in_abstract_search_space : search_space.is_in_search_space
tsk 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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_task j = tsk H_job_cost_positive : job_cost_positive j total_interference_bound := fun (tsk : Task)
(A : nat)
(Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk +
IBF_other Δ: Task ->
nat -> duration -> nat A : duration H_A_is_in_abstract_search_space : search_space.is_in_search_space
tsk L
total_interference_bound
A INSP : A = 0
task_rbf 0 != task_rbf (0 + ε)
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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_task j = tsk H_job_cost_positive : job_cost_positive j total_interference_bound := fun (tsk : Task)
(A : nat)
(Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk +
IBF_other Δ: Task ->
nat -> duration -> nat A : duration H_A_is_in_abstract_search_space : search_space.is_in_search_space
tsk L
total_interference_bound
A INSP : A = 0
task_rbf 0 < task_rbf (0 + ε)
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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_task j = tsk H_job_cost_positive : job_cost_positive j total_interference_bound := fun (tsk : Task)
(A : nat)
(Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk +
IBF_other Δ: Task ->
nat -> duration -> nat A : duration H_A_is_in_abstract_search_space : search_space.is_in_search_space
tsk L
total_interference_bound
A INSP : A = 0
0 < task_rbf (0 + ε)
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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_task j = tsk H_job_cost_positive : job_cost_positive j total_interference_bound := fun (tsk : Task)
(A : nat)
(Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk +
IBF_other Δ: Task ->
nat -> duration -> nat A : duration H_A_is_in_abstract_search_space : search_space.is_in_search_space
tsk L
total_interference_bound
A INSP : A = 0
0 < 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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_task j = tsk H_job_cost_positive : job_cost_positive j total_interference_bound := fun (tsk : Task)
(A : nat)
(Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk +
IBF_other Δ: Task ->
nat -> duration -> nat A : duration H_A_is_in_abstract_search_space : search_space.is_in_search_space
tsk L
total_interference_bound
A INSP : A = 0
0 < task_cost tsk
by apply leq_trans with (job_cost j); rewrite -?H_job_of_tsk ; eauto 2 .Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_task j = tsk H_job_cost_positive : job_cost_positive j total_interference_bound := fun (tsk : Task)
(A : nat)
(Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk +
IBF_other Δ: Task ->
nat -> duration -> nat A : duration H_A_is_in_abstract_search_space : search_space.is_in_search_space
tsk L
total_interference_bound
A INSP : A = 0
task_cost tsk <= task_request_bound_function tsk ε
+ Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_task j = tsk H_job_cost_positive : job_cost_positive j total_interference_bound := fun (tsk : Task)
(A : nat)
(Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk +
IBF_other Δ: Task ->
nat -> duration -> nat A : duration H_A_is_in_abstract_search_space : search_space.is_in_search_space
tsk L
total_interference_bound
A INSP : A = 0
task_cost tsk <= task_request_bound_function tsk ε
eapply task_rbf_1_ge_task_cost; eauto 2 .Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_task j = tsk H_job_cost_positive : job_cost_positive j total_interference_bound := fun (tsk : Task)
(A : nat)
(Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk +
IBF_other Δ: Task ->
nat -> duration -> nat A : duration H_A_is_in_abstract_search_space : search_space.is_in_search_space
tsk L
total_interference_bound
A POSA : 0 < ALTL : A < L x : nat LTx : x < L INSP2 : total_interference_bound tsk (A - ε) x <>
total_interference_bound tsk A x
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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_task j = tsk H_job_cost_positive : job_cost_positive j total_interference_bound := fun (tsk : Task)
(A : nat)
(Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk +
IBF_other Δ: Task ->
nat -> duration -> nat A : duration H_A_is_in_abstract_search_space : search_space.is_in_search_space
tsk L
total_interference_bound
A POSA : 0 < ALTL : A < L x : nat LTx : x < L INSP2 : total_interference_bound tsk (A - ε) x <>
total_interference_bound tsk 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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_task j = tsk H_job_cost_positive : job_cost_positive j total_interference_bound := fun (tsk : Task)
(A : nat)
(Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk +
IBF_other Δ: Task ->
nat -> duration -> nat A : duration H_A_is_in_abstract_search_space : search_space.is_in_search_space
tsk L
total_interference_bound
A POSA : 0 < ALTL : A < L x : nat LTx : x < L INSP2 : total_interference_bound tsk (A - ε) x <>
total_interference_bound tsk A x
task_rbf A != task_rbf (A + ε)
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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_task j = tsk H_job_cost_positive : job_cost_positive j total_interference_bound := fun (tsk : Task)
(A : nat)
(Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk +
IBF_other Δ: Task ->
nat -> duration -> nat A : duration H_A_is_in_abstract_search_space : search_space.is_in_search_space
tsk L
total_interference_bound
A POSA : 0 < ALTL : A < L x : nat LTx : x < L INSP2 : total_interference_bound tsk (A - ε) x <>
total_interference_bound tsk A x EQ : task_rbf A = task_rbf (A + ε)
False
apply INSP2.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_task j = tsk H_job_cost_positive : job_cost_positive j total_interference_bound := fun (tsk : Task)
(A : nat)
(Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk +
IBF_other Δ: Task ->
nat -> duration -> nat A : duration H_A_is_in_abstract_search_space : search_space.is_in_search_space
tsk L
total_interference_bound
A POSA : 0 < ALTL : A < L x : nat LTx : x < L INSP2 : total_interference_bound tsk (A - ε) x <>
total_interference_bound tsk A x EQ : task_rbf A = task_rbf (A + ε)
total_interference_bound tsk (A - ε) x =
total_interference_bound tsk A x
unfold total_interference_bound in *.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_task j = tsk H_job_cost_positive : job_cost_positive j total_interference_bound := fun (tsk : Task)
(A : nat)
(Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk +
IBF_other Δ: Task ->
nat -> duration -> nat A : duration H_A_is_in_abstract_search_space : search_space.is_in_search_space
tsk L
(fun (tsk : Task)
(A : nat)
(Δ : duration)
=>
task_rbf (A + ε) -
task_cost tsk +
IBF_other Δ) A POSA : 0 < ALTL : A < L x : nat LTx : x < L INSP2 : task_rbf (A - ε + ε) - task_cost tsk +
IBF_other x <>
task_rbf (A + ε) - task_cost tsk + IBF_other x EQ : task_rbf A = task_rbf (A + ε)
task_rbf (A - ε + ε) - task_cost tsk + IBF_other x =
task_rbf (A + ε) - task_cost tsk + IBF_other x
rewrite subn1 addn1 prednK; 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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_task j = tsk H_job_cost_positive : job_cost_positive j total_interference_bound := fun (tsk : Task)
(A : nat)
(Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk +
IBF_other Δ: Task ->
nat -> duration -> nat A : duration H_A_is_in_abstract_search_space : search_space.is_in_search_space
tsk L
(fun (tsk : Task)
(A : nat)
(Δ : duration)
=>
task_rbf (A + ε) -
task_cost tsk +
IBF_other Δ) A POSA : 0 < ALTL : A < L x : nat LTx : x < L INSP2 : task_rbf (A - ε + ε) - task_cost tsk +
IBF_other x <>
task_rbf (A + ε) - task_cost tsk + IBF_other x EQ : task_rbf A = task_rbf (A + ε)
task_rbf A - task_cost tsk + IBF_other x =
task_rbf (A + ε) - task_cost tsk + IBF_other x
by rewrite -EQ.
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) + IBF_other (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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_task j = tsk H_job_cost_positive : job_cost_positive j total_interference_bound := fun (tsk : Task)
(A : nat)
(Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk +
IBF_other Δ: Task ->
nat -> duration -> nat A : duration H_A_is_in_abstract_search_space : search_space.is_in_search_space
tsk L
total_interference_bound
A
exists F : duration,
task_rbf (A + ε) - (task_cost tsk - task_rtct tsk) +
IBF_other (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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_task j = tsk H_job_cost_positive : job_cost_positive j total_interference_bound := fun (tsk : Task)
(A : nat)
(Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk +
IBF_other Δ: Task ->
nat -> duration -> nat A : duration H_A_is_in_abstract_search_space : search_space.is_in_search_space
tsk L
total_interference_bound
A
exists F : duration,
task_rbf (A + ε) - (task_cost tsk - task_rtct tsk) +
IBF_other (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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_task j = tsk H_job_cost_positive : job_cost_positive j total_interference_bound := fun (tsk : Task)
(A : nat)
(Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk +
IBF_other Δ: Task ->
nat -> duration -> nat A : duration H_A_is_in_abstract_search_space : search_space.is_in_search_space
tsk L
total_interference_bound
A FIX : is_in_search_space A ->
exists F : duration,
priority_inversion_bound +
(task_rbf (A + ε) -
(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 + ε) - (task_cost tsk - task_rtct tsk) +
IBF_other (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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_task j = tsk H_job_cost_positive : job_cost_positive j total_interference_bound := fun (tsk : Task)
(A : nat)
(Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk +
IBF_other Δ: Task ->
nat -> duration -> nat A : duration H_A_is_in_abstract_search_space : search_space.is_in_search_space
tsk L
total_interference_bound
A FIX : exists F : duration,
priority_inversion_bound +
(task_rbf (A + ε) -
(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 + ε) - (task_cost tsk - task_rtct tsk) +
IBF_other (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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_task j = tsk H_job_cost_positive : job_cost_positive j total_interference_bound := fun (tsk : Task)
(A : nat)
(Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk +
IBF_other Δ: Task ->
nat -> duration -> nat A : duration H_A_is_in_abstract_search_space : search_space.is_in_search_space
tsk L
total_interference_bound
A F : duration FIX : priority_inversion_bound +
(task_rbf (A + ε) -
(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 + ε) - (task_cost tsk - task_rtct tsk) +
IBF_other (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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_task j = tsk H_job_cost_positive : job_cost_positive j total_interference_bound := fun (tsk : Task)
(A : nat)
(Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk +
IBF_other Δ: Task ->
nat -> duration -> nat A : duration H_A_is_in_abstract_search_space : search_space.is_in_search_space
tsk L
total_interference_bound
A F : duration FIX : priority_inversion_bound +
(task_rbf (A + ε) -
(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 + ε) - (task_cost tsk - task_rtct tsk) +
IBF_other (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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_task j = tsk H_job_cost_positive : job_cost_positive j total_interference_bound := fun (tsk : Task)
(A : nat)
(Δ : duration) =>
task_rbf (A + ε) -
task_cost tsk +
IBF_other Δ: Task ->
nat -> duration -> nat A : duration H_A_is_in_abstract_search_space : search_space.is_in_search_space
tsk L
total_interference_bound
A F : duration FIX : priority_inversion_bound +
(task_rbf (A + ε) -
(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 + ε) - (task_cost tsk - task_rtct tsk) +
IBF_other (A + F) <=
priority_inversion_bound +
(task_rbf (A + ε) - (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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := 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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := 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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat js : Job ARRs : arrives_in arr_seq js TSKs : job_task js = tsk
job_response_time_bound sched js R
move : H_sched_valid => [CARR MBR].Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat js : Job ARRs : arrives_in arr_seq js TSKs : job_task js = tsk CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched
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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat js : Job ARRs : arrives_in arr_seq js TSKs : job_task js = tsk CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched 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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat js : Job ARRs : arrives_in arr_seq js TSKs : job_task js = tsk CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched 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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat js : Job ARRs : arrives_in arr_seq js TSKs : job_task js = tsk CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched POS : 0 < job_cost js
job_response_time_bound sched js R
eapply uniprocessor_response_time_bound_seq; eauto 2 with basic_facts.Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat js : Job ARRs : arrives_in arr_seq js TSKs : job_task js = tsk CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched POS : 0 < job_cost js
definitions.work_conserving arr_seq sched tsk
?interference ?interfering_workload
- Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat js : Job ARRs : arrives_in arr_seq js TSKs : job_task js = tsk CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched POS : 0 < job_cost js
definitions.work_conserving arr_seq sched tsk
?interference ?interfering_workload
by apply instantiated_i_and_w_are_consistent_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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat js : Job ARRs : arrives_in arr_seq js TSKs : job_task js = tsk CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched POS : 0 < job_cost js
interference_and_workload_consistent_with_sequential_tasks
arr_seq sched tsk interference interfering_workload
- Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat js : Job ARRs : arrives_in arr_seq js TSKs : job_task js = tsk CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched POS : 0 < job_cost js
interference_and_workload_consistent_with_sequential_tasks
arr_seq sched tsk interference interfering_workload
by apply 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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat js : Job ARRs : arrives_in arr_seq js TSKs : job_task js = tsk CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched POS : 0 < job_cost js
busy_intervals_are_bounded_by arr_seq sched tsk
interference interfering_workload ?L
- Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat js : Job ARRs : arrives_in arr_seq js TSKs : job_task js = tsk CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched POS : 0 < job_cost js
busy_intervals_are_bounded_by arr_seq sched tsk
interference interfering_workload ?L
by apply 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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat js : Job ARRs : arrives_in arr_seq js TSKs : job_task js = tsk CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched POS : 0 < job_cost js
task_interference_is_bounded_by arr_seq sched tsk
interference interfering_workload
?task_interference_bound_function
- Task : TaskType H : TaskCost Task H0 : TaskRunToCompletionThreshold Task Job : JobType H1 : JobTask Job Task Arrival : JobArrival Job Cost : JobCost Job H2 : JobPreemptable Job H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat js : Job ARRs : arrives_in arr_seq js TSKs : job_task js = tsk CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched POS : 0 < job_cost js
task_interference_is_bounded_by arr_seq sched tsk
interference interfering_workload
?task_interference_bound_function
by apply 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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat js : Job ARRs : arrives_in arr_seq js TSKs : job_task js = tsk CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched POS : 0 < job_cost js
forall A : duration,
search_space.is_in_search_space tsk L
(fun (tsk0 : Task) (A0 Δ : duration) =>
task_request_bound_function tsk (A0 + ε) -
task_cost tsk0 +
(fun => (fun => [eta IBF_other])) tsk0 A0 Δ) A ->
exists F : duration,
task_request_bound_function tsk (A + ε) -
(task_cost tsk - task_rtct tsk) +
(fun => (fun => [eta IBF_other])) tsk A (A + F) <=
A + F /\ F + (task_cost tsk - task_rtct 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 H3 : FP_policy Task H_priority_is_reflexive : reflexive_priorities arr_seq : arrival_sequence Job H_arrival_times_are_consistent : consistent_arrival_times
arr_seq H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq sched : schedule (processor_state Job) H4 : JobReady Job (processor_state Job) H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq work_conserving_ab := definitions.work_conserving
arr_seq sched : Task ->
(Job -> instant -> bool) ->
(Job -> instant -> duration) ->
Prop work_conserving_cl := 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 H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk 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
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 + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_rbf (A + F) <=
A + F /\
F + (task_cost tsk - task_rtct tsk) <=
Rinterference := fun j : Job =>
[eta ideal_jlfp_rta.interference sched j]: Job -> instant -> bool interfering_workload := fun j : Job =>
[eta ideal_jlfp_rta.interfering_workload
arr_seq sched j]: Job -> instant -> nat IBF_other := fun R : duration =>
priority_inversion_bound + total_ohep_rbf R: duration -> nat js : Job ARRs : arrives_in arr_seq js TSKs : job_task js = tsk CARR : jobs_come_from_arrival_sequence sched arr_seq MBR : jobs_must_be_ready_to_execute sched POS : 0 < job_cost js
forall A : duration,
search_space.is_in_search_space tsk L
(fun (tsk0 : Task) (A0 Δ : duration) =>
task_request_bound_function tsk (A0 + ε) -
task_cost tsk0 +
(fun => (fun => [eta IBF_other])) tsk0 A0 Δ) A ->
exists F : duration,
task_request_bound_function tsk (A + ε) -
(task_cost tsk - task_rtct tsk) +
(fun => (fun => [eta IBF_other])) tsk A (A + F) <=
A + F /\ F + (task_cost tsk - task_rtct tsk) <= R
by eapply correct_search_space; eauto 2 .
Qed .
End AbstractRTAforFPwithArrivalCurves .