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Require Import prosa.analysis.facts.readiness.basic.[Loading ML file ssrmatching_plugin.cmxs (using legacy method) ... done ] [Loading ML file ssreflect_plugin.cmxs (using legacy method) ... done ] [Loading ML file ring_plugin.cmxs (using legacy method) ... done ] Serlib plugin: coq-elpi.elpi is not available: serlib support is missing.
Incremental checking for commands in this plugin will be impacted. [Loading ML file coq-elpi.elpi ... done ] [Loading ML file zify_plugin.cmxs (using legacy method) ... done ] [Loading ML file micromega_core_plugin.cmxs (using legacy method) ... done ] [Loading ML file micromega_plugin.cmxs (using legacy method) ... done ] [Loading ML file btauto_plugin.cmxs (using legacy method) ... done ] Notation "_ + _" was already used in scope nat_scope.
[notation-overridden,parsing,default]Notation "_ - _" was already used in scope nat_scope.
[notation-overridden,parsing,default]Notation "_ <= _" was already used in scope nat_scope.
[notation-overridden,parsing,default]Notation "_ < _" was already used in scope nat_scope.
[notation-overridden,parsing,default]Notation "_ >= _" was already used in scope nat_scope.
[notation-overridden,parsing,default]Notation "_ > _" was already used in scope nat_scope.
[notation-overridden,parsing,default]Notation "_ <= _ <= _" was already used in scope
nat_scope. [notation-overridden,parsing,default]Notation "_ < _ <= _" was already used in scope
nat_scope. [notation-overridden,parsing,default]Notation "_ <= _ < _" was already used in scope
nat_scope. [notation-overridden,parsing,default]Notation "_ < _ < _" was already used in scope
nat_scope. [notation-overridden,parsing,default]Notation "_ * _" was already used in scope nat_scope.
[notation-overridden,parsing,default]
Require Export prosa.analysis.facts.model.restricted_supply.schedule.
Require Export prosa.analysis.facts.preemption.rtc_threshold.limited.
Require Export prosa.analysis.abstract .restricted_supply.task_intra_interference_bound.
Require Export prosa.analysis.abstract .restricted_supply.bounded_bi.edf.
Require Export prosa.analysis.abstract .restricted_supply.search_space.edf.
Require Export prosa.analysis.facts.model.task_cost.
Require Export prosa.analysis.facts.priority.edf.
Require Export prosa.analysis.facts.blocking_bound.edf.
Require Export prosa.analysis.facts.workload.edf_athep_bound.
Require Export prosa.analysis.definitions.sbf.busy.
(** * RTA for EDF Scheduling with Fixed Preemption Points on Restricted-Supply Uniprocessors *)
(** In the following, we derive a response-time analysis for EDF
schedulers, assuming a workload of sporadic real-time tasks
characterized by arbitrary arrival curves executing upon a
uniprocessor with arbitrary supply restrictions. To this end, we
instantiate the sequential variant of _abstract Restricted-Supply
Response-Time Analysis_ (aRSA) as provided in the
[prosa.analysis.abstract.restricted_supply] module. *)
Section RTAforLimitedPreemptiveEDFModelwithArrivalCurves .
(** ** Defining the System Model *)
(** Before any formal claims can be stated, an initial setup is
needed to define the system model under consideration. To this
end, we next introduce and define the following notions using
Prosa's standard definitions and behavioral semantics:
- processor model,
- tasks, jobs, and their parameters,
- the sequence of job arrivals,
- worst-case execution time (WCET) and the absence of self-suspensions,
- the set of tasks under analysis,
- the task under analysis,
- an arbitrary schedule of the task set, and finally,
- a supply-bound function. *)
(** *** Processor Model *)
(** Consider a restricted-supply uniprocessor model. *)
#[local] Existing Instance rs_processor_state .
(** *** Tasks and Jobs *)
(** Consider any type of tasks, each characterized by a WCET
[task_cost], relative deadline [task_deadline], an arrival curve
[max_arrivals], and a predicate indicating task's preemption
points [task_preemption_points], ... *)
Context {Task : TaskType}.
Context `{TaskCost Task}.
Context `{TaskDeadline Task}.
Context `{MaxArrivals Task}.
Context `{TaskPreemptionPoints Task}.
(** ... and any type of jobs associated with these tasks, where each
job has a task [job_task], a cost [job_cost], an arrival time
[job_arrival], and a predicate indicating job's preemption
points [job_preemptive_points]. *)
Context {Job : JobType}.
Context `{JobTask Job Task}.
Context `{JobCost Job}.
Context `{JobArrival Job}.
Context `{JobPreemptionPoints Job}.
(** We assume that jobs are limited-preemptive. *)
#[local] Existing Instance limited_preemptive_job_model .
(** *** The Job Arrival Sequence *)
(** Consider any arrival sequence [arr_seq] with consistent, non-duplicate arrivals. *)
Variable arr_seq : arrival_sequence Job.
Hypothesis H_valid_arrival_sequence : valid_arrival_sequence arr_seq.
(** *** Absence of Self-Suspensions and WCET Compliance *)
(** We assume the classic (i.e., Liu & Layland) model of readiness
without jitter or self-suspensions, wherein pending jobs are
always ready. *)
#[local] Existing Instance basic_ready_instance .
(** We further require that a job's cost cannot exceed its task's stated WCET. *)
Hypothesis H_valid_job_cost : arrivals_have_valid_job_costs arr_seq.
(** *** The Task Set *)
(** We consider an arbitrary task set [ts] ... *)
Variable ts : seq Task.
(** ... and assume that all jobs stem from tasks in this task set. *)
Hypothesis H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts.
(** We assume a model with fixed preemption points. I.e., each task
is divided into a number of non-preemptive segments by inserting
statically predefined preemption points. *)
Hypothesis H_valid_model_with_fixed_preemption_points :
valid_fixed_preemption_points_model arr_seq ts.
(** We assume that [max_arrivals] is a family of valid arrival
curves that constrains the arrival sequence [arr_seq], i.e., for
any task [tsk] in [ts], [max_arrival tsk] is (1) an arrival
bound of [tsk], and ... *)
Hypothesis H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts.
(** ... (2) a monotonic function that equals 0 for the empty interval [delta = 0]. *)
Hypothesis H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals.
(** *** The Task Under Analysis *)
(** Let [tsk] be any task in [ts] that is to be analyzed. *)
Variable tsk : Task.
Hypothesis H_tsk_in_ts : tsk \in ts.
(** *** The Schedule *)
(** Consider any arbitrary, work-conserving, valid restricted-supply
uni-processor schedule with limited preemptions of the given
arrival sequence [arr_seq] (and hence the given task set [ts]). *)
Variable sched : schedule (rs_processor_state Job).
Hypothesis H_valid_schedule : valid_schedule sched arr_seq.
Hypothesis H_work_conserving : work_conserving arr_seq sched.
Hypothesis H_schedule_with_limited_preemptions :
schedule_respects_preemption_model arr_seq sched.
(** Assume that the schedule respects the EDF policy. *)
Hypothesis H_respects_policy :
respects_JLFP_policy_at_preemption_point arr_seq sched (EDF Job).
(** *** Supply-Bound Function *)
(** Assume the minimum amount of supply that any job of task [tsk]
receives is defined by a monotone unit-supply-bound function [SBF]. *)
Context {SBF : SupplyBoundFunction}.
Hypothesis H_SBF_monotone : sbf_is_monotone SBF.
Hypothesis H_unit_SBF : unit_supply_bound_function SBF.
(** We assume that [SBF] properly characterizes all busy intervals
(w.r.t. task [tsk]) in [sched]. That is, (1) [SBF 0 = 0] and (2)
for any duration [Δ], at least [SBF Δ] supply is available in
any busy-interval prefix of length [Δ]. *)
Hypothesis H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF.
(** ** Workload Abbreviation *)
(** Let's denote the relative deadline of a task as [D]. *)
Let D tsk := task_deadline tsk.
(** We introduce [task_rbf] as an abbreviation
for the task request bound function of task [tsk]. *)
Let task_rbf := task_request_bound_function tsk.
(** ** Length of Busy Interval *)
(** The next step is to establish a bound on the maximum busy-window
length, which aRSA requires to be given. *)
(** To this end, let [L] be any positive constant such that ... *)
Variable L : duration.
Hypothesis H_L_positive : 0 < L.
(** ... [L] satisfies a fixed-point recurrence for the
busy-interval-length bound (i.e., [total_RBF ts L <= SBF L] ... *)
Hypothesis H_fixed_point : total_request_bound_function ts L <= SBF L.
(** ... and [SBF L] bounds [longest_busy_interval_with_pi ts tsk]. *)
Hypothesis H_L_bounds_bi_with_pi :
longest_busy_interval_with_pi ts tsk <= SBF L.
(** ** Response-Time Bound *)
(** Having established all necessary preliminaries, it is finally
time to state the claimed response-time bound [R].
A value [R] is a response-time bound if, for any given offset
[A] in the search space, the response-time bound recurrence has
a solution [F] not exceeding [R]. *)
Definition rta_recurrence_solution R :=
forall (A : duration),
is_in_search_space ts tsk L A ->
exists (F : duration),
A <= F <= A + R
/\ blocking_bound ts tsk A
+ (task_rbf (A + ε) - (task_last_nonpr_segment tsk - ε))
+ bound_on_athep_workload ts tsk A F
<= SBF F
/\ SBF F + (task_last_nonpr_segment tsk - ε) <= SBF (A + R).
(** Finally, using the sequential variant of abstract
restricted-supply analysis, we establish that any such [R] is a
sound response-time bound for the concrete model of EDF
scheduling with limited preemptions with arbitrary supply
restrictions. *)
Theorem uniprocessor_response_time_bound_limited_edf :
forall (R : duration),
rta_recurrence_solution R ->
task_response_time_bound arr_seq sched tsk R.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task H2 : TaskPreemptionPoints Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptionPoints Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals tsk : Task H_tsk_in_ts : tsk \in ts sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched
(EDF Job) SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration task_rbf := task_request_bound_function tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L H_L_bounds_bi_with_pi : longest_busy_interval_with_pi
ts tsk <=
SBF L
forall R : duration,
rta_recurrence_solution R ->
task_response_time_bound arr_seq sched tsk R
Proof .Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task H2 : TaskPreemptionPoints Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptionPoints Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals tsk : Task H_tsk_in_ts : tsk \in ts sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched
(EDF Job) SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration task_rbf := task_request_bound_function tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L H_L_bounds_bi_with_pi : longest_busy_interval_with_pi
ts tsk <=
SBF L
forall R : duration,
rta_recurrence_solution R ->
task_response_time_bound arr_seq sched tsk R
move => R SOL js ARRs TSKs.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task H2 : TaskPreemptionPoints Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptionPoints Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals tsk : Task H_tsk_in_ts : tsk \in ts sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched
(EDF Job) SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration task_rbf := task_request_bound_function tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L H_L_bounds_bi_with_pi : longest_busy_interval_with_pi
ts tsk <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js
job_response_time_bound sched js R
have [ZERO|POS] := posnP (job_cost js); first by rewrite /job_response_time_bound /completed_by ZERO.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task H2 : TaskPreemptionPoints Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptionPoints Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals tsk : Task H_tsk_in_ts : tsk \in ts sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched
(EDF Job) SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration task_rbf := task_request_bound_function tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L H_L_bounds_bi_with_pi : longest_busy_interval_with_pi
ts tsk <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost js
job_response_time_bound sched js R
have VAL1 : valid_preemption_model arr_seq sched.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task H2 : TaskPreemptionPoints Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptionPoints Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals tsk : Task H_tsk_in_ts : tsk \in ts sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched
(EDF Job) SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration task_rbf := task_request_bound_function tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L H_L_bounds_bi_with_pi : longest_busy_interval_with_pi
ts tsk <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost js
valid_preemption_model arr_seq sched
{ Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task H2 : TaskPreemptionPoints Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptionPoints Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals tsk : Task H_tsk_in_ts : tsk \in ts sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched
(EDF Job) SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration task_rbf := task_request_bound_function tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L H_L_bounds_bi_with_pi : longest_busy_interval_with_pi
ts tsk <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost js
valid_preemption_model arr_seq sched
apply valid_fixed_preemption_points_model_lemma => //.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task H2 : TaskPreemptionPoints Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptionPoints Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals tsk : Task H_tsk_in_ts : tsk \in ts sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched
(EDF Job) SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration task_rbf := task_request_bound_function tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L H_L_bounds_bi_with_pi : longest_busy_interval_with_pi
ts tsk <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost js
valid_limited_preemptions_job_model arr_seq
by apply H_valid_model_with_fixed_preemption_points. } Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task H2 : TaskPreemptionPoints Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptionPoints Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals tsk : Task H_tsk_in_ts : tsk \in ts sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched
(EDF Job) SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration task_rbf := task_request_bound_function tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L H_L_bounds_bi_with_pi : longest_busy_interval_with_pi
ts tsk <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsVAL1 : valid_preemption_model arr_seq sched
job_response_time_bound sched js R
have READ : work_bearing_readiness arr_seq sched by done .Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task H2 : TaskPreemptionPoints Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptionPoints Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals tsk : Task H_tsk_in_ts : tsk \in ts sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched
(EDF Job) SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration task_rbf := task_request_bound_function tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L H_L_bounds_bi_with_pi : longest_busy_interval_with_pi
ts tsk <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsVAL1 : valid_preemption_model arr_seq sched READ : work_bearing_readiness arr_seq sched
job_response_time_bound sched js R
eapply uniprocessor_response_time_bound_restricted_supply_seq with (L := L) => //.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task H2 : TaskPreemptionPoints Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptionPoints Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals tsk : Task H_tsk_in_ts : tsk \in ts sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched
(EDF Job) SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration task_rbf := task_request_bound_function tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L H_L_bounds_bi_with_pi : longest_busy_interval_with_pi
ts tsk <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsVAL1 : valid_preemption_model arr_seq sched READ : work_bearing_readiness arr_seq sched
definitions.work_conserving arr_seq sched
- Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task H2 : TaskPreemptionPoints Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptionPoints Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals tsk : Task H_tsk_in_ts : tsk \in ts sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched
(EDF Job) SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration task_rbf := task_request_bound_function tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L H_L_bounds_bi_with_pi : longest_busy_interval_with_pi
ts tsk <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsVAL1 : valid_preemption_model arr_seq sched READ : work_bearing_readiness arr_seq sched
definitions.work_conserving arr_seq sched
exact : instantiated_i_and_w_are_coherent_with_schedule.
- Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task H2 : TaskPreemptionPoints Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptionPoints Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals tsk : Task H_tsk_in_ts : tsk \in ts sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched
(EDF Job) SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration task_rbf := task_request_bound_function tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L H_L_bounds_bi_with_pi : longest_busy_interval_with_pi
ts tsk <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsVAL1 : valid_preemption_model arr_seq sched READ : work_bearing_readiness arr_seq sched
sequential_tasks arr_seq sched
exact : EDF_implies_sequential_tasks.
- Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task H2 : TaskPreemptionPoints Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptionPoints Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals tsk : Task H_tsk_in_ts : tsk \in ts sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched
(EDF Job) SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration task_rbf := task_request_bound_function tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L H_L_bounds_bi_with_pi : longest_busy_interval_with_pi
ts tsk <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsVAL1 : valid_preemption_model arr_seq sched READ : work_bearing_readiness arr_seq sched
interference_and_workload_consistent_with_sequential_tasks
arr_seq sched tsk
exact : instantiated_interference_and_workload_consistent_with_sequential_tasks.
- Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task H2 : TaskPreemptionPoints Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptionPoints Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals tsk : Task H_tsk_in_ts : tsk \in ts sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched
(EDF Job) SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration task_rbf := task_request_bound_function tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L H_L_bounds_bi_with_pi : longest_busy_interval_with_pi
ts tsk <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsVAL1 : valid_preemption_model arr_seq sched READ : work_bearing_readiness arr_seq sched
busy_intervals_are_bounded_by arr_seq sched tsk L
apply : busy_intervals_are_bounded_rs_edf => //.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task H2 : TaskPreemptionPoints Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptionPoints Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals tsk : Task H_tsk_in_ts : tsk \in ts sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched
(EDF Job) SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration task_rbf := task_request_bound_function tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L H_L_bounds_bi_with_pi : longest_busy_interval_with_pi
ts tsk <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsVAL1 : valid_preemption_model arr_seq sched READ : work_bearing_readiness arr_seq sched
definitions.work_conserving arr_seq sched
by apply : instantiated_i_and_w_are_coherent_with_schedule.
- Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task H2 : TaskPreemptionPoints Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptionPoints Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals tsk : Task H_tsk_in_ts : tsk \in ts sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched
(EDF Job) SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration task_rbf := task_request_bound_function tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L H_L_bounds_bi_with_pi : longest_busy_interval_with_pi
ts tsk <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsVAL1 : valid_preemption_model arr_seq sched READ : work_bearing_readiness arr_seq sched
busy_sbf.valid_busy_sbf arr_seq sched tsk SBF
apply : valid_pred_sbf_switch_predicate; last by exact : H_valid_SBF.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task H2 : TaskPreemptionPoints Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptionPoints Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals tsk : Task H_tsk_in_ts : tsk \in ts sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched
(EDF Job) SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration task_rbf := task_request_bound_function tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L H_L_bounds_bi_with_pi : longest_busy_interval_with_pi
ts tsk <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsVAL1 : valid_preemption_model arr_seq sched READ : work_bearing_readiness arr_seq sched
forall (j : Job) (t1 t2 : instant),
arrives_in arr_seq j ->
job_of_task tsk j /\
definitions.busy_interval_prefix sched j t1 t2 ->
(fun (j0 : Job) (t3 t4 : instant) =>
job_of_task tsk j0 /\
busy_interval_prefix arr_seq sched j0 t3 t4) j t1 t2
move => ? ? ? ? [? ?]; split => //.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task H2 : TaskPreemptionPoints Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptionPoints Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals tsk : Task H_tsk_in_ts : tsk \in ts sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched
(EDF Job) SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration task_rbf := task_request_bound_function tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L H_L_bounds_bi_with_pi : longest_busy_interval_with_pi
ts tsk <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsVAL1 : valid_preemption_model arr_seq sched READ : work_bearing_readiness arr_seq sched _j_ : Job _t1_, _t2_ : instant _Hyp_ : arrives_in arr_seq _j_ _a_ : job_of_task tsk _j_ _b_ : definitions.busy_interval_prefix sched _j_ _t1_
_t2_
busy_interval_prefix arr_seq sched _j_ _t1_ _t2_
by apply instantiated_busy_interval_prefix_equivalent_busy_interval_prefix.
- Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task H2 : TaskPreemptionPoints Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptionPoints Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals tsk : Task H_tsk_in_ts : tsk \in ts sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched
(EDF Job) SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration task_rbf := task_request_bound_function tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L H_L_bounds_bi_with_pi : longest_busy_interval_with_pi
ts tsk <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsVAL1 : valid_preemption_model arr_seq sched READ : work_bearing_readiness arr_seq sched
task_intra_interference_is_bounded_by arr_seq sched
tsk ?task_intra_IBF
apply : instantiated_task_intra_interference_is_bounded; eauto 1 => //; first last .Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task H2 : TaskPreemptionPoints Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptionPoints Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals tsk : Task H_tsk_in_ts : tsk \in ts sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched
(EDF Job) SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration task_rbf := task_request_bound_function tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L H_L_bounds_bi_with_pi : longest_busy_interval_with_pi
ts tsk <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsVAL1 : valid_preemption_model arr_seq sched READ : work_bearing_readiness arr_seq sched
athep_workload_is_bounded arr_seq sched tsk ?Goal2
+ Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task H2 : TaskPreemptionPoints Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptionPoints Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals tsk : Task H_tsk_in_ts : tsk \in ts sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched
(EDF Job) SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration task_rbf := task_request_bound_function tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L H_L_bounds_bi_with_pi : longest_busy_interval_with_pi
ts tsk <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsVAL1 : valid_preemption_model arr_seq sched READ : work_bearing_readiness arr_seq sched
athep_workload_is_bounded arr_seq sched tsk ?Goal2
by (apply : bound_on_athep_workload_is_valid; try apply H_fixed_point) => //.
+ Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task H2 : TaskPreemptionPoints Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptionPoints Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals tsk : Task H_tsk_in_ts : tsk \in ts sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched
(EDF Job) SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration task_rbf := task_request_bound_function tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L H_L_bounds_bi_with_pi : longest_busy_interval_with_pi
ts tsk <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsVAL1 : valid_preemption_model arr_seq sched READ : work_bearing_readiness arr_seq sched
service_inversion_is_bounded_by arr_seq sched tsk
?Goal0
apply : service_inversion_is_bounded => // => jo t1 t2 ARRo TSKo BUSYo.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task H2 : TaskPreemptionPoints Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptionPoints Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals tsk : Task H_tsk_in_ts : tsk \in ts sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched
(EDF Job) SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration task_rbf := task_request_bound_function tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L H_L_bounds_bi_with_pi : longest_busy_interval_with_pi
ts tsk <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsVAL1 : valid_preemption_model arr_seq sched READ : work_bearing_readiness arr_seq sched jo : Job t1, t2 : instant ARRo : arrives_in arr_seq jo TSKo : job_of_task tsk jo BUSYo : busy_interval_prefix arr_seq sched jo t1 t2
max_lp_nonpreemptive_segment arr_seq jo t1 <=
?Goal0 (job_arrival jo - t1)
by apply : nonpreemptive_segments_bounded_by_blocking => //.
- Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task H2 : TaskPreemptionPoints Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptionPoints Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals tsk : Task H_tsk_in_ts : tsk \in ts sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched
(EDF Job) SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration task_rbf := task_request_bound_function tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L H_L_bounds_bi_with_pi : longest_busy_interval_with_pi
ts tsk <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsVAL1 : valid_preemption_model arr_seq sched READ : work_bearing_readiness arr_seq sched
forall A : duration,
search_space.is_in_search_space L
(fun A0 Δ : duration =>
task_request_bound_function tsk (A0 + 1 ) -
task_cost tsk +
task_intra_IBF (blocking_bound ts tsk)
(bound_on_athep_workload ts tsk) A0 Δ) A ->
exists F : duration,
F <= A + R /\
task_request_bound_function tsk (A + 1 ) -
(task_cost tsk - task_rtct tsk) +
task_intra_IBF (blocking_bound ts tsk)
(bound_on_athep_workload ts tsk) A F <= SBF F /\
SBF F + (task_cost tsk - task_rtct tsk) <=
SBF (A + R)
move => A SP.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task H2 : TaskPreemptionPoints Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptionPoints Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals tsk : Task H_tsk_in_ts : tsk \in ts sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched
(EDF Job) SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration task_rbf := task_request_bound_function tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L H_L_bounds_bi_with_pi : longest_busy_interval_with_pi
ts tsk <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsVAL1 : valid_preemption_model arr_seq sched READ : work_bearing_readiness arr_seq sched A : duration SP : search_space.is_in_search_space L
(fun A Δ : duration =>
task_request_bound_function tsk (A + 1 ) -
task_cost tsk +
task_intra_IBF
(blocking_bound ts tsk)
(bound_on_athep_workload ts tsk) A Δ) A
exists F : duration,
F <= A + R /\
task_request_bound_function tsk (A + 1 ) -
(task_cost tsk - task_rtct tsk) +
task_intra_IBF (blocking_bound ts tsk)
(bound_on_athep_workload ts tsk) A F <= SBF F /\
SBF F + (task_cost tsk - task_rtct tsk) <=
SBF (A + R)
move : (SOL A) => [].Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task H2 : TaskPreemptionPoints Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptionPoints Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals tsk : Task H_tsk_in_ts : tsk \in ts sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched
(EDF Job) SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration task_rbf := task_request_bound_function tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L H_L_bounds_bi_with_pi : longest_busy_interval_with_pi
ts tsk <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsVAL1 : valid_preemption_model arr_seq sched READ : work_bearing_readiness arr_seq sched A : duration SP : search_space.is_in_search_space L
(fun A Δ : duration =>
task_request_bound_function tsk (A + 1 ) -
task_cost tsk +
task_intra_IBF
(blocking_bound ts tsk)
(bound_on_athep_workload ts tsk) A Δ) A
is_in_search_space ts tsk L A
{ Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task H2 : TaskPreemptionPoints Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptionPoints Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals tsk : Task H_tsk_in_ts : tsk \in ts sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched
(EDF Job) SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration task_rbf := task_request_bound_function tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L H_L_bounds_bi_with_pi : longest_busy_interval_with_pi
ts tsk <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsVAL1 : valid_preemption_model arr_seq sched READ : work_bearing_readiness arr_seq sched A : duration SP : search_space.is_in_search_space L
(fun A Δ : duration =>
task_request_bound_function tsk (A + 1 ) -
task_cost tsk +
task_intra_IBF
(blocking_bound ts tsk)
(bound_on_athep_workload ts tsk) A Δ) A
is_in_search_space ts tsk L A
by apply : search_space_sub => //; apply : search_space_switch_IBF. } Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task H2 : TaskPreemptionPoints Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptionPoints Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals tsk : Task H_tsk_in_ts : tsk \in ts sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched
(EDF Job) SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration task_rbf := task_request_bound_function tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L H_L_bounds_bi_with_pi : longest_busy_interval_with_pi
ts tsk <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsVAL1 : valid_preemption_model arr_seq sched READ : work_bearing_readiness arr_seq sched A : duration SP : search_space.is_in_search_space L
(fun A Δ : duration =>
task_request_bound_function tsk (A + 1 ) -
task_cost tsk +
task_intra_IBF
(blocking_bound ts tsk)
(bound_on_athep_workload ts tsk) A Δ) A
forall x : duration,
A <= x <= A + R /\
blocking_bound ts tsk A +
(task_rbf (A + 1 ) - (task_last_nonpr_segment tsk - 1 )) +
bound_on_athep_workload ts tsk A x <= SBF x /\
SBF x + (task_last_nonpr_segment tsk - 1 ) <=
SBF (A + R) ->
exists F : duration,
F <= A + R /\
task_request_bound_function tsk (A + 1 ) -
(task_cost tsk - task_rtct tsk) +
task_intra_IBF (blocking_bound ts tsk)
(bound_on_athep_workload ts tsk) A F <= SBF F /\
SBF F + (task_cost tsk - task_rtct tsk) <=
SBF (A + R)
move => FF [EQ1 [EQ2 EQ3]].Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task H2 : TaskPreemptionPoints Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptionPoints Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals tsk : Task H_tsk_in_ts : tsk \in ts sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched
(EDF Job) SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration task_rbf := task_request_bound_function tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L H_L_bounds_bi_with_pi : longest_busy_interval_with_pi
ts tsk <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsVAL1 : valid_preemption_model arr_seq sched READ : work_bearing_readiness arr_seq sched A : duration SP : search_space.is_in_search_space L
(fun A Δ : duration =>
task_request_bound_function tsk (A + 1 ) -
task_cost tsk +
task_intra_IBF
(blocking_bound ts tsk)
(bound_on_athep_workload ts tsk) A Δ) A FF : duration EQ1 : A <= FF <= A + R EQ2 : blocking_bound ts tsk A +
(task_rbf (A + 1 ) -
(task_last_nonpr_segment tsk - 1 )) +
bound_on_athep_workload ts tsk A FF <=
SBF FF EQ3 : SBF FF + (task_last_nonpr_segment tsk - 1 ) <=
SBF (A + R)
exists F : duration,
F <= A + R /\
task_request_bound_function tsk (A + 1 ) -
(task_cost tsk - task_rtct tsk) +
task_intra_IBF (blocking_bound ts tsk)
(bound_on_athep_workload ts tsk) A F <= SBF F /\
SBF F + (task_cost tsk - task_rtct tsk) <=
SBF (A + R)
exists FF ; split ; last split .Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task H2 : TaskPreemptionPoints Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptionPoints Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals tsk : Task H_tsk_in_ts : tsk \in ts sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched
(EDF Job) SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration task_rbf := task_request_bound_function tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L H_L_bounds_bi_with_pi : longest_busy_interval_with_pi
ts tsk <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsVAL1 : valid_preemption_model arr_seq sched READ : work_bearing_readiness arr_seq sched A : duration SP : search_space.is_in_search_space L
(fun A Δ : duration =>
task_request_bound_function tsk (A + 1 ) -
task_cost tsk +
task_intra_IBF
(blocking_bound ts tsk)
(bound_on_athep_workload ts tsk) A Δ) A FF : duration EQ1 : A <= FF <= A + R EQ2 : blocking_bound ts tsk A +
(task_rbf (A + 1 ) -
(task_last_nonpr_segment tsk - 1 )) +
bound_on_athep_workload ts tsk A FF <=
SBF FF EQ3 : SBF FF + (task_last_nonpr_segment tsk - 1 ) <=
SBF (A + R)
FF <= A + R
+ Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task H2 : TaskPreemptionPoints Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptionPoints Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals tsk : Task H_tsk_in_ts : tsk \in ts sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched
(EDF Job) SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration task_rbf := task_request_bound_function tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L H_L_bounds_bi_with_pi : longest_busy_interval_with_pi
ts tsk <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsVAL1 : valid_preemption_model arr_seq sched READ : work_bearing_readiness arr_seq sched A : duration SP : search_space.is_in_search_space L
(fun A Δ : duration =>
task_request_bound_function tsk (A + 1 ) -
task_cost tsk +
task_intra_IBF
(blocking_bound ts tsk)
(bound_on_athep_workload ts tsk) A Δ) A FF : duration EQ1 : A <= FF <= A + R EQ2 : blocking_bound ts tsk A +
(task_rbf (A + 1 ) -
(task_last_nonpr_segment tsk - 1 )) +
bound_on_athep_workload ts tsk A FF <=
SBF FF EQ3 : SBF FF + (task_last_nonpr_segment tsk - 1 ) <=
SBF (A + R)
FF <= A + R
lia .
+ Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task H2 : TaskPreemptionPoints Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptionPoints Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals tsk : Task H_tsk_in_ts : tsk \in ts sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched
(EDF Job) SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration task_rbf := task_request_bound_function tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L H_L_bounds_bi_with_pi : longest_busy_interval_with_pi
ts tsk <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsVAL1 : valid_preemption_model arr_seq sched READ : work_bearing_readiness arr_seq sched A : duration SP : search_space.is_in_search_space L
(fun A Δ : duration =>
task_request_bound_function tsk (A + 1 ) -
task_cost tsk +
task_intra_IBF
(blocking_bound ts tsk)
(bound_on_athep_workload ts tsk) A Δ) A FF : duration EQ1 : A <= FF <= A + R EQ2 : blocking_bound ts tsk A +
(task_rbf (A + 1 ) -
(task_last_nonpr_segment tsk - 1 )) +
bound_on_athep_workload ts tsk A FF <=
SBF FF EQ3 : SBF FF + (task_last_nonpr_segment tsk - 1 ) <=
SBF (A + R)
task_request_bound_function tsk (A + 1 ) -
(task_cost tsk - task_rtct tsk) +
task_intra_IBF (blocking_bound ts tsk)
(bound_on_athep_workload ts tsk) A FF <= SBF FF
move : EQ2; rewrite /task_intra_IBF -/task_rbf.Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task H2 : TaskPreemptionPoints Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptionPoints Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals tsk : Task H_tsk_in_ts : tsk \in ts sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched
(EDF Job) SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration task_rbf := task_request_bound_function tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L H_L_bounds_bi_with_pi : longest_busy_interval_with_pi
ts tsk <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsVAL1 : valid_preemption_model arr_seq sched READ : work_bearing_readiness arr_seq sched A : duration SP : search_space.is_in_search_space L
(fun A Δ : duration =>
task_request_bound_function tsk (A + 1 ) -
task_cost tsk +
task_intra_IBF
(blocking_bound ts tsk)
(bound_on_athep_workload ts tsk) A Δ) A FF : duration EQ1 : A <= FF <= A + R EQ3 : SBF FF + (task_last_nonpr_segment tsk - 1 ) <=
SBF (A + R)
blocking_bound ts tsk A +
(task_rbf (A + 1 ) - (task_last_nonpr_segment tsk - 1 )) +
bound_on_athep_workload ts tsk A FF <= SBF FF ->
task_rbf (A + 1 ) - (task_cost tsk - task_rtct tsk) +
(blocking_bound ts tsk A +
bound_on_athep_workload ts tsk A FF) <= SBF FF
by erewrite last_segment_eq_cost_minus_rtct => //; lia .
+ Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task H2 : TaskPreemptionPoints Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptionPoints Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals tsk : Task H_tsk_in_ts : tsk \in ts sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched
(EDF Job) SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration task_rbf := task_request_bound_function tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L H_L_bounds_bi_with_pi : longest_busy_interval_with_pi
ts tsk <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsVAL1 : valid_preemption_model arr_seq sched READ : work_bearing_readiness arr_seq sched A : duration SP : search_space.is_in_search_space L
(fun A Δ : duration =>
task_request_bound_function tsk (A + 1 ) -
task_cost tsk +
task_intra_IBF
(blocking_bound ts tsk)
(bound_on_athep_workload ts tsk) A Δ) A FF : duration EQ1 : A <= FF <= A + R EQ2 : blocking_bound ts tsk A +
(task_rbf (A + 1 ) -
(task_last_nonpr_segment tsk - 1 )) +
bound_on_athep_workload ts tsk A FF <=
SBF FF EQ3 : SBF FF + (task_last_nonpr_segment tsk - 1 ) <=
SBF (A + R)
SBF FF + (task_cost tsk - task_rtct tsk) <=
SBF (A + R)
by erewrite last_segment_eq_cost_minus_rtct.
Qed .
End RTAforLimitedPreemptiveEDFModelwithArrivalCurves .