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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 ] [Loading ML file coq-elpi.elpi ... done ] [Loading ML file zify_plugin.cmxs (using legacy method) ... done ] [Loading ML file micromega_plugin.cmxs (using legacy method) ... done ] [Loading ML file btauto_plugin.cmxs (using legacy method) ... done ] Notation "_ + _" was already used in scope nat_scope.
[notation-overridden,parsing,default]Notation "_ - _" was already used in scope nat_scope.
[notation-overridden,parsing,default]Notation "_ <= _" was already used in scope nat_scope.
[notation-overridden,parsing,default]Notation "_ < _" was already used in scope nat_scope.
[notation-overridden,parsing,default]Notation "_ >= _" was already used in scope nat_scope.
[notation-overridden,parsing,default]Notation "_ > _" was already used in scope nat_scope.
[notation-overridden,parsing,default]Notation "_ <= _ <= _" was already used in scope
nat_scope. [notation-overridden,parsing,default]Notation "_ < _ <= _" was already used in scope
nat_scope. [notation-overridden,parsing,default]Notation "_ <= _ < _" was already used in scope
nat_scope. [notation-overridden,parsing,default]Notation "_ < _ < _" was already used in scope
nat_scope. [notation-overridden,parsing,default]Notation "_ * _" was already used in scope nat_scope.
[notation-overridden,parsing,default]
Require Export prosa.analysis.facts.model.restricted_supply.schedule.
Require Export prosa.analysis.facts.preemption.task.preemptive.
Require Export prosa.analysis.facts.preemption.rtc_threshold.preemptive.
Require Export prosa.analysis.abstract .restricted_supply.task_intra_interference_bound.
Require Export prosa.analysis.abstract .restricted_supply.bounded_bi.jlfp.
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.
(** * RTA for Fully Preemptive EDF Scheduling 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 _abstract Sequential Restricted-Supply
Response-Time Analysis_ (aRSA) as provided in the
[prosa.analysis.abstract.restricted_supply] module. *)
Section RTAforFullyPreemptiveEDFModelwithArrivalCurves .
(** ** 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, and, finally,
- an arbitrary schedule of the task set. *)
(** *** Processor Model *)
(** Consider a restricted-supply uniprocessor model, ... *)
#[local] Existing Instance rs_processor_state .
(** ... where the minimum amount of supply is lower-bounded via 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.
(** *** Tasks and Jobs *)
(** Consider any type of tasks, each characterized by a WCET
[task_cost], relative deadline [task_deadline], and an arrival
curve [max_arrivals], ... *)
Context {Task : TaskType}.
Context `{TaskCost Task}.
Context `{TaskDeadline Task}.
Context `{MaxArrivals Task}.
(** ... and any type of jobs associated with these tasks, where each
job has a task [job_task], a cost [job_cost], and an arrival
time [job_arrival]. *)
Context {Job : JobType}.
Context `{JobTask Job Task}.
Context `{JobCost Job}.
Context `{JobArrival Job}.
(** Furthermore, assume that jobs and tasks are fully preemptive. *)
#[local] Existing Instance fully_preemptive_job_model .
#[local] Existing Instance fully_preemptive_task_model .
#[local] Existing Instance fully_preemptive_rtc_threshold .
(** *** 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 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 *)
(** Finally, consider any arbitrary, work-conserving, valid
restricted-supply uni-processor schedule 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.
(** ... and assume that the schedule respects the EDF policy. *)
Hypothesis H_respects_policy :
respects_JLFP_policy_at_preemption_point arr_seq sched (EDF Job).
(** Last but not least, 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 *)
(** For brevity, let's denote the relative deadline of a task as [D]. *)
Let D tsk := task_deadline 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 fixed point of the
busy-interval recurrence. As the
[busy_intervals_are_bounded_rs_jlfp] lemma shows, under any
preemptive [JLFP] scheduling policy, this is sufficient to
guarantee that all busy intervals are bounded by [L]. *)
Variable L : duration.
Hypothesis H_L_positive : 0 < L.
Hypothesis H_fixed_point : total_request_bound_function ts L <= 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]. *)
Variable R : duration.
Hypothesis H_R_is_maximum :
forall (A : duration),
is_in_search_space ts tsk L A ->
exists (F : duration),
A <= F <= A + R
/\ task_request_bound_function tsk (A + ε) + bound_on_athep_workload ts tsk A F <= SBF F.
(** 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
fully-preemptive EDF scheduling with arbitrary supply
restrictions. *)
Theorem uniprocessor_response_time_bound_fully_preemptive_edf :
task_response_time_bound arr_seq sched tsk R.SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task Job : JobType H2 : JobTask Job Task H3 : JobCost Job H4 : JobArrival 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_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_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (EDF Job) H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space ts tsk L A ->
exists F : duration,
A <= F <= A + R /\
task_request_bound_function tsk
(A + 1 ) +
bound_on_athep_workload ts tsk A F <=
SBF F
task_response_time_bound arr_seq sched tsk R
Proof .SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task Job : JobType H2 : JobTask Job Task H3 : JobCost Job H4 : JobArrival 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_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_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (EDF Job) H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space ts tsk L A ->
exists F : duration,
A <= F <= A + R /\
task_request_bound_function tsk
(A + 1 ) +
bound_on_athep_workload ts tsk A F <=
SBF F
task_response_time_bound arr_seq sched tsk R
move => js ARRs TSKs.SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task Job : JobType H2 : JobTask Job Task H3 : JobCost Job H4 : JobArrival 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_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_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (EDF Job) H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space ts tsk L A ->
exists F : duration,
A <= F <= A + R /\
task_request_bound_function tsk
(A + 1 ) +
bound_on_athep_workload ts tsk A F <=
SBF Fjs : 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.SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task Job : JobType H2 : JobTask Job Task H3 : JobCost Job H4 : JobArrival 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_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_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (EDF Job) H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space ts tsk L A ->
exists F : duration,
A <= F <= A + R /\
task_request_bound_function tsk
(A + 1 ) +
bound_on_athep_workload ts tsk A F <=
SBF Fjs : 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 READ : work_bearing_readiness arr_seq sched by done .SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task Job : JobType H2 : JobTask Job Task H3 : JobCost Job H4 : JobArrival 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_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_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (EDF Job) H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space ts tsk L A ->
exists F : duration,
A <= F <= A + R /\
task_request_bound_function tsk
(A + 1 ) +
bound_on_athep_workload ts tsk A F <=
SBF Fjs : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched
job_response_time_bound sched js R
have BLOCK: forall tsk A , blocking_bound ts tsk A = 0 .SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task Job : JobType H2 : JobTask Job Task H3 : JobCost Job H4 : JobArrival 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_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_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (EDF Job) H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space ts tsk L A ->
exists F : duration,
A <= F <= A + R /\
task_request_bound_function tsk
(A + 1 ) +
bound_on_athep_workload ts tsk A F <=
SBF Fjs : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched
forall (tsk : Task) (A : duration),
blocking_bound ts tsk A = 0
{ SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task Job : JobType H2 : JobTask Job Task H3 : JobCost Job H4 : JobArrival 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_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_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (EDF Job) H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space ts tsk L A ->
exists F : duration,
A <= F <= A + R /\
task_request_bound_function tsk
(A + 1 ) +
bound_on_athep_workload ts tsk A F <=
SBF Fjs : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched
forall (tsk : Task) (A : duration),
blocking_bound ts tsk A = 0
by move => A tsk2; rewrite /blocking_bound /parameters.task_max_nonpreemptive_segment
/fully_preemptive_task_model subnn big1_eq. } SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task Job : JobType H2 : JobTask Job Task H3 : JobCost Job H4 : JobArrival 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_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_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (EDF Job) H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space ts tsk L A ->
exists F : duration,
A <= F <= A + R /\
task_request_bound_function tsk
(A + 1 ) +
bound_on_athep_workload ts tsk A F <=
SBF Fjs : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched BLOCK : forall (tsk : Task) (A : duration),
blocking_bound ts tsk A = 0
job_response_time_bound sched js R
eapply uniprocessor_response_time_bound_restricted_supply_seq with (L := L) => //.SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task Job : JobType H2 : JobTask Job Task H3 : JobCost Job H4 : JobArrival 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_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_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (EDF Job) H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space ts tsk L A ->
exists F : duration,
A <= F <= A + R /\
task_request_bound_function tsk
(A + 1 ) +
bound_on_athep_workload ts tsk A F <=
SBF Fjs : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched BLOCK : forall (tsk : Task) (A : duration),
blocking_bound ts tsk A = 0
definitions.work_conserving arr_seq sched
- SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task Job : JobType H2 : JobTask Job Task H3 : JobCost Job H4 : JobArrival 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_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_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (EDF Job) H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space ts tsk L A ->
exists F : duration,
A <= F <= A + R /\
task_request_bound_function tsk
(A + 1 ) +
bound_on_athep_workload ts tsk A F <=
SBF Fjs : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched BLOCK : forall (tsk : Task) (A : duration),
blocking_bound ts tsk A = 0
definitions.work_conserving arr_seq sched
exact : instantiated_i_and_w_are_coherent_with_schedule.
- SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task Job : JobType H2 : JobTask Job Task H3 : JobCost Job H4 : JobArrival 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_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_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (EDF Job) H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space ts tsk L A ->
exists F : duration,
A <= F <= A + R /\
task_request_bound_function tsk
(A + 1 ) +
bound_on_athep_workload ts tsk A F <=
SBF Fjs : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched BLOCK : forall (tsk : Task) (A : duration),
blocking_bound ts tsk A = 0
sequential_tasks arr_seq sched
exact : EDF_implies_sequential_tasks.
- SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task Job : JobType H2 : JobTask Job Task H3 : JobCost Job H4 : JobArrival 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_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_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (EDF Job) H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space ts tsk L A ->
exists F : duration,
A <= F <= A + R /\
task_request_bound_function tsk
(A + 1 ) +
bound_on_athep_workload ts tsk A F <=
SBF Fjs : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched BLOCK : forall (tsk : Task) (A : duration),
blocking_bound ts tsk A = 0
interference_and_workload_consistent_with_sequential_tasks
arr_seq sched tsk
exact : instantiated_interference_and_workload_consistent_with_sequential_tasks.
- SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task Job : JobType H2 : JobTask Job Task H3 : JobCost Job H4 : JobArrival 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_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_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (EDF Job) H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space ts tsk L A ->
exists F : duration,
A <= F <= A + R /\
task_request_bound_function tsk
(A + 1 ) +
bound_on_athep_workload ts tsk A F <=
SBF Fjs : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched BLOCK : forall (tsk : Task) (A : duration),
blocking_bound ts tsk A = 0
busy_intervals_are_bounded_by arr_seq sched tsk L
eapply busy_intervals_are_bounded_rs_jlfp; try done .SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task Job : JobType H2 : JobTask Job Task H3 : JobCost Job H4 : JobArrival 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_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_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (EDF Job) H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space ts tsk L A ->
exists F : duration,
A <= F <= A + R /\
task_request_bound_function tsk
(A + 1 ) +
bound_on_athep_workload ts tsk A F <=
SBF Fjs : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched BLOCK : forall (tsk : Task) (A : duration),
blocking_bound ts tsk A = 0
service_inversion_is_bounded_by arr_seq sched tsk
(blocking_bound ts ?tsk )
+ SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task Job : JobType H2 : JobTask Job Task H3 : JobCost Job H4 : JobArrival 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_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_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (EDF Job) H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space ts tsk L A ->
exists F : duration,
A <= F <= A + R /\
task_request_bound_function tsk
(A + 1 ) +
bound_on_athep_workload ts tsk A F <=
SBF Fjs : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched BLOCK : forall (tsk : Task) (A : duration),
blocking_bound ts tsk A = 0
service_inversion_is_bounded_by arr_seq sched tsk
(blocking_bound ts ?tsk )
apply : service_inversion_is_bounded => // => ? ? ? ? ? ?.SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task Job : JobType H2 : JobTask Job Task H3 : JobCost Job H4 : JobArrival 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_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_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (EDF Job) H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space ts tsk L A ->
exists F : duration,
A <= F <= A + R /\
task_request_bound_function tsk
(A + 1 ) +
bound_on_athep_workload ts tsk A F <=
SBF Fjs : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched BLOCK : forall (tsk : Task) (A : duration),
blocking_bound ts tsk A = 0 _j_ : Job _t1_, _t2_ : instant _Hyp_ : arrives_in arr_seq _j_ _Hyp1_ : job_of_task tsk _j_ _Hyp2_ : busy_interval_prefix arr_seq sched _j_ _t1_
_t2_
max_lp_nonpreemptive_segment arr_seq _j_ _t1_ <=
blocking_bound ts ?tsk (job_arrival _j_ - _t1_)
exact : nonpreemptive_segments_bounded_by_blocking.
+ SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task Job : JobType H2 : JobTask Job Task H3 : JobCost Job H4 : JobArrival 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_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_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (EDF Job) H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space ts tsk L A ->
exists F : duration,
A <= F <= A + R /\
task_request_bound_function tsk
(A + 1 ) +
bound_on_athep_workload ts tsk A F <=
SBF Fjs : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched BLOCK : forall (tsk : Task) (A : duration),
blocking_bound ts tsk A = 0
blocking_bound ts tsk 0 +
total_request_bound_function ts L <= SBF L
by rewrite BLOCK add0n; apply H_fixed_point.
- SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task Job : JobType H2 : JobTask Job Task H3 : JobCost Job H4 : JobArrival 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_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_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (EDF Job) H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space ts tsk L A ->
exists F : duration,
A <= F <= A + R /\
task_request_bound_function tsk
(A + 1 ) +
bound_on_athep_workload ts tsk A F <=
SBF Fjs : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched BLOCK : forall (tsk : Task) (A : duration),
blocking_bound ts tsk A = 0
busy_sbf.valid_busy_sbf arr_seq sched tsk SBF
apply : valid_pred_sbf_switch_predicate; last by exact : H_valid_SBF.SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task Job : JobType H2 : JobTask Job Task H3 : JobCost Job H4 : JobArrival 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_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_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (EDF Job) H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space ts tsk L A ->
exists F : duration,
A <= F <= A + R /\
task_request_bound_function tsk
(A + 1 ) +
bound_on_athep_workload ts tsk A F <=
SBF Fjs : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched BLOCK : forall (tsk : Task) (A : duration),
blocking_bound ts tsk A = 0
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 => //.SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task Job : JobType H2 : JobTask Job Task H3 : JobCost Job H4 : JobArrival 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_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_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (EDF Job) H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space ts tsk L A ->
exists F : duration,
A <= F <= A + R /\
task_request_bound_function tsk
(A + 1 ) +
bound_on_athep_workload ts tsk A F <=
SBF Fjs : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched BLOCK : forall (tsk : Task) (A : duration),
blocking_bound ts tsk A = 0 _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.
- SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task Job : JobType H2 : JobTask Job Task H3 : JobCost Job H4 : JobArrival 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_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_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (EDF Job) H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space ts tsk L A ->
exists F : duration,
A <= F <= A + R /\
task_request_bound_function tsk
(A + 1 ) +
bound_on_athep_workload ts tsk A F <=
SBF Fjs : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched BLOCK : forall (tsk : Task) (A : duration),
blocking_bound ts tsk A = 0
task_intra_interference_is_bounded_by arr_seq sched
tsk ?task_intra_IBF
apply : instantiated_task_intra_interference_is_bounded; eauto 1 => //; first last .SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task Job : JobType H2 : JobTask Job Task H3 : JobCost Job H4 : JobArrival 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_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_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (EDF Job) H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space ts tsk L A ->
exists F : duration,
A <= F <= A + R /\
task_request_bound_function tsk
(A + 1 ) +
bound_on_athep_workload ts tsk A F <=
SBF Fjs : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched BLOCK : forall (tsk : Task) (A : duration),
blocking_bound ts tsk A = 0
athep_workload_is_bounded arr_seq sched tsk ?Goal2
+ SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task Job : JobType H2 : JobTask Job Task H3 : JobCost Job H4 : JobArrival 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_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_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (EDF Job) H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space ts tsk L A ->
exists F : duration,
A <= F <= A + R /\
task_request_bound_function tsk
(A + 1 ) +
bound_on_athep_workload ts tsk A F <=
SBF Fjs : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched BLOCK : forall (tsk : Task) (A : duration),
blocking_bound ts tsk A = 0
athep_workload_is_bounded arr_seq sched tsk ?Goal2
by (apply : bound_on_athep_workload_is_valid; try apply H_fixed_point) => //.
+ SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task Job : JobType H2 : JobTask Job Task H3 : JobCost Job H4 : JobArrival 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_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_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (EDF Job) H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space ts tsk L A ->
exists F : duration,
A <= F <= A + R /\
task_request_bound_function tsk
(A + 1 ) +
bound_on_athep_workload ts tsk A F <=
SBF Fjs : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched BLOCK : forall (tsk : Task) (A : duration),
blocking_bound ts tsk A = 0
service_inversion_is_bounded_by arr_seq sched tsk
?Goal0
apply : service_inversion_is_bounded => // => jo t1 t2 ARRo TSKo BUSYo.SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task Job : JobType H2 : JobTask Job Task H3 : JobCost Job H4 : JobArrival 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_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_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (EDF Job) H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space ts tsk L A ->
exists F : duration,
A <= F <= A + R /\
task_request_bound_function tsk
(A + 1 ) +
bound_on_athep_workload ts tsk A F <=
SBF Fjs : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched BLOCK : forall (tsk : Task) (A : duration),
blocking_bound ts tsk A = 0 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 => //.
- SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task Job : JobType H2 : JobTask Job Task H3 : JobCost Job H4 : JobArrival 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_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_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (EDF Job) H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space ts tsk L A ->
exists F : duration,
A <= F <= A + R /\
task_request_bound_function tsk
(A + 1 ) +
bound_on_athep_workload ts tsk A F <=
SBF Fjs : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched BLOCK : forall (tsk : Task) (A : duration),
blocking_bound ts tsk A = 0
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.SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task Job : JobType H2 : JobTask Job Task H3 : JobCost Job H4 : JobArrival 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_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_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (EDF Job) H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space ts tsk L A ->
exists F : duration,
A <= F <= A + R /\
task_request_bound_function tsk
(A + 1 ) +
bound_on_athep_workload ts tsk A F <=
SBF Fjs : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched BLOCK : forall (tsk : Task) (A : duration),
blocking_bound ts tsk A = 0 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 : (H_R_is_maximum A) => [].SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task Job : JobType H2 : JobTask Job Task H3 : JobCost Job H4 : JobArrival 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_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_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (EDF Job) H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space ts tsk L A ->
exists F : duration,
A <= F <= A + R /\
task_request_bound_function tsk
(A + 1 ) +
bound_on_athep_workload ts tsk A F <=
SBF Fjs : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched BLOCK : forall (tsk : Task) (A : duration),
blocking_bound ts tsk A = 0 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
+ SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task Job : JobType H2 : JobTask Job Task H3 : JobCost Job H4 : JobArrival 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_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_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (EDF Job) H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space ts tsk L A ->
exists F : duration,
A <= F <= A + R /\
task_request_bound_function tsk
(A + 1 ) +
bound_on_athep_workload ts tsk A F <=
SBF Fjs : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched BLOCK : forall (tsk : Task) (A : duration),
blocking_bound ts tsk A = 0 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 => //.
+ SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task Job : JobType H2 : JobTask Job Task H3 : JobCost Job H4 : JobArrival 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_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_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (EDF Job) H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space ts tsk L A ->
exists F : duration,
A <= F <= A + R /\
task_request_bound_function tsk
(A + 1 ) +
bound_on_athep_workload ts tsk A F <=
SBF Fjs : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched BLOCK : forall (tsk : Task) (A : duration),
blocking_bound ts tsk A = 0 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 /\
task_request_bound_function tsk (A + 1 ) +
bound_on_athep_workload ts tsk A x <= SBF x ->
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 => F [/andP [_ LE] FIX]; exists F ; split => //.SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task Job : JobType H2 : JobTask Job Task H3 : JobCost Job H4 : JobArrival 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_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_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (EDF Job) H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space ts tsk L A ->
exists F : duration,
A <= F <= A + R /\
task_request_bound_function tsk
(A + 1 ) +
bound_on_athep_workload ts tsk A F <=
SBF Fjs : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched BLOCK : forall (tsk : Task) (A : duration),
blocking_bound ts tsk A = 0 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 F : duration LE : F <= A + R FIX : task_request_bound_function tsk (A + 1 ) +
bound_on_athep_workload ts tsk A F <=
SBF F
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)
rewrite /task_intra_IBF /task_rtct /fully_preemptive_rtc_threshold.SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF Task : TaskType H : TaskCost Task H0 : TaskDeadline Task H1 : MaxArrivals Task Job : JobType H2 : JobTask Job Task H3 : JobCost Job H4 : JobArrival 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_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_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (EDF Job) H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF D := [eta task_deadline] : Task -> duration L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration H_R_is_maximum : forall A : duration,
is_in_search_space ts tsk L A ->
exists F : duration,
A <= F <= A + R /\
task_request_bound_function tsk
(A + 1 ) +
bound_on_athep_workload ts tsk A F <=
SBF Fjs : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched BLOCK : forall (tsk : Task) (A : duration),
blocking_bound ts tsk A = 0 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 F : duration LE : F <= A + R FIX : task_request_bound_function tsk (A + 1 ) +
bound_on_athep_workload ts tsk A F <=
SBF F
task_request_bound_function tsk (A + 1 ) -
(task_cost tsk - task_cost tsk) +
(blocking_bound ts tsk A +
bound_on_athep_workload ts tsk A F) <= SBF F /\
SBF F + (task_cost tsk - task_cost tsk) <= SBF (A + R)
by rewrite BLOCK subnn //= add0n addn0 subn0.
Qed .
End RTAforFullyPreemptiveEDFModelwithArrivalCurves .