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Require Export prosa.analysis.facts.blocking_bound.fp.[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.abstract .restricted_supply.abstract_rta.
Require Export prosa.analysis.abstract .restricted_supply.iw_instantiation.
Require Export prosa.analysis.abstract .restricted_supply.bounded_bi.aux.
Require Export prosa.analysis.definitions.sbf.busy.
(** * Sufficient Condition for Bounded Busy Intervals for RS FP *)
(** In this section, we show that the existence of [L] such that [B +
total_hep_rbf L <= SBF L], where [B] is the blocking bound and
[SBF] is a supply-bound function, is a sufficient condition for
the existence of bounded busy intervals under FP scheduling with a
restricted-supply processor model. *)
Section BoundedBusyIntervals .
(** Consider any type of tasks ... *)
Context {Task : TaskType}.
Context `{TaskCost Task}.
(** ... and any type of jobs associated with these tasks. *)
Context {Job : JobType}.
Context `{JobTask Job Task}.
Context `{JobArrival Job}.
Context `{JobCost Job}.
(** Consider any kind of fully supply-consuming unit-supply
uniprocessor model. *)
Context `{PState : ProcessorState Job}.
Hypothesis H_uniprocessor_proc_model : uniprocessor_model PState.
Hypothesis H_unit_supply_proc_model : unit_supply_proc_model PState.
Hypothesis H_consumed_supply_proc_model : fully_consuming_proc_model PState.
(** Consider an FP policy that indicates a higher-or-equal priority
relation, and assume that the relation is reflexive and
transitive. *)
Context {FP : FP_policy Task}.
Hypothesis H_priority_is_reflexive : reflexive_task_priorities FP.
Hypothesis H_priority_is_transitive : transitive_task_priorities FP.
(** Consider any valid arrival sequence. *)
Variable arr_seq : arrival_sequence Job.
Hypothesis H_valid_arrival_sequence : valid_arrival_sequence arr_seq.
(** Next, consider a schedule of this arrival sequence, ... *)
Variable sched : schedule PState.
(** ... allow for any work-bearing notion of job readiness, ... *)
Context `{!JobReady Job PState}.
Hypothesis H_job_ready : work_bearing_readiness arr_seq sched.
(** ... and assume that the schedule is valid. *)
Hypothesis H_sched_valid : valid_schedule sched arr_seq.
(** Assume that jobs have bounded non-preemptive segments. *)
Context `{JobPreemptable Job}.
Context `{TaskMaxNonpreemptiveSegment Task}.
Hypothesis H_valid_preemption_model : valid_preemption_model arr_seq sched.
Hypothesis H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched.
(** Furthermore, assume that the schedule respects the scheduling policy. *)
Hypothesis H_respects_policy : respects_FP_policy_at_preemption_point arr_seq sched FP.
(** Recall that [busy_intervals_are_bounded_by] is an abstract
notion. Hence, we need to introduce interference and interfering
workload. We will use the restricted-supply instantiations. *)
(** We say that job [j] incurs interference at time [t] iff it
cannot execute due to (1) the lack of supply at time [t], (2)
service inversion (i.e., a lower-priority job receiving service
at [t]), or a higher-or-equal-priority job receiving service. *)
#[local] Instance rs_jlfp_interference : Interference Job :=
rs_jlfp_interference arr_seq sched.
(** The interfering workload, in turn, is defined as the sum of the
blackout predicate, service inversion predicate, and the
interfering workload of jobs with higher or equal priority. *)
#[local] Instance rs_jlfp_interfering_workload : InterferingWorkload Job :=
rs_jlfp_interfering_workload arr_seq sched.
(** Assume that the schedule is work-conserving in the abstract sense. *)
Hypothesis H_work_conserving : definitions.work_conserving arr_seq sched.
(** Consider an arbitrary task set [ts], ... *)
Variable ts : list Task.
(** ... assume that all jobs come from the task set, ... *)
Hypothesis H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts.
(** ... and that the cost of a job does not exceed its task's WCET. *)
Hypothesis H_valid_job_cost : arrivals_have_valid_job_costs arr_seq.
(** Let [max_arrivals] be a family of valid arrival curves, i.e.,
for any task [tsk] in [ts], [max_arrival tsk] is (1) an arrival
bound of [tsk], and (2) it is a monotonic function that equals
[0] for the empty interval [delta = 0]. *)
Context `{MaxArrivals Task}.
Hypothesis H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals.
Hypothesis H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts.
(** Let [tsk] be any task in [ts] that is to be analyzed. *)
Variable tsk : Task.
Hypothesis H_tsk_in_ts : tsk \in ts.
(** Consider a unit SBF valid in busy intervals (w.r.t. task
[tsk]). That is, (1) [SBF 0 = 0], (2) for any duration [Δ], the
supply produced during a busy-interval prefix of length [Δ] is
at least [SBF Δ], and (3) [SBF] makes steps of at most one. *)
Context {SBF : SupplyBoundFunction}.
Hypothesis H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF.
Hypothesis H_unit_SBF : unit_supply_bound_function SBF.
(** Let [L] be any positive fixed point of the busy-interval recurrence. *)
Variable L : duration.
Hypothesis H_L_positive : 0 < L.
Hypothesis H_fixed_point :
blocking_bound ts tsk + total_hep_request_bound_function_FP ts tsk L <= SBF L.
(** Next, we provide a step-by-step proof of busy-interval boundedness. *)
Section StepByStepProof .
(** Consider any job [j] of task [tsk] that has a positive job
cost and is in the arrival sequence. *)
Variable j : Job.
Hypothesis H_j_arrives : arrives_in arr_seq j.
Hypothesis H_job_of_tsk : job_of_task tsk j.
Hypothesis H_job_cost_positive : job_cost_positive j.
(** Consider two cases: (1) when the busy-interval prefix
continues until time instant [t1 + L] and (2) when the busy
interval prefix terminates earlier. In either case, we can
show that the busy-interval prefix is bounded. *)
(** We start with the first case, where the busy-interval prefix
continues until time instant [t1 + L]. *)
Section Case1 .
(** Consider a time instant [t1] such that <<[t1, job_arrival
j]>> and <<[t1, t1 + L)>> are both busy-interval prefixes of
job [j].
Note that at this point we do not necessarily know that
[job_arrival j <= L]; hence, we assume the existence of both
prefixes. *)
Variable t1 : instant.
Hypothesis H_busy_prefix_arr : busy_interval_prefix arr_seq sched j t1 (job_arrival j).+1 .
Hypothesis H_busy_prefix_L : busy_interval_prefix arr_seq sched j t1 (t1 + L).
(** The crucial point to note is that the sum of the job's cost
(represented as [workload_of_job]) and the interfering
workload in the interval <<[t1, t1 + L)>> is bounded by [L]
due to the fixed point [H_fixed_point]. *)
Local Lemma workload_is_bounded :
workload_of_job arr_seq j t1 (t1 + L) + cumulative_interfering_workload j t1 (t1 + L) <= L.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_busy_prefix_L : busy_interval_prefix arr_seq sched j
t1 (t1 + L)
workload_of_job arr_seq j t1 (t1 + L) +
cumulative_interfering_workload j t1 (t1 + L) <= L
Proof .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_busy_prefix_L : busy_interval_prefix arr_seq sched j
t1 (t1 + L)
workload_of_job arr_seq j t1 (t1 + L) +
cumulative_interfering_workload j t1 (t1 + L) <= L
rewrite (cumulative_interfering_workload_split _ _ _).Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_busy_prefix_L : busy_interval_prefix arr_seq sched j
t1 (t1 + L)
workload_of_job arr_seq j t1 (t1 + L) +
(blackout_during sched t1 (t1 + L) +
cumulative_service_inversion arr_seq sched j t1
(t1 + L) +
cumulative_other_hep_jobs_interfering_workload
arr_seq j t1 (t1 + L)) <= L
rewrite (leqRW (blackout_during_bound _ _ _ _ _ _ _ _ (t1 + L) _ _ _)); (try apply H_valid_SBF) => //.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_busy_prefix_L : busy_interval_prefix arr_seq sched j
t1 (t1 + L)
workload_of_job arr_seq j t1 (t1 + L) +
(L - SBF L +
cumulative_service_inversion arr_seq sched j t1
(t1 + L) +
cumulative_other_hep_jobs_interfering_workload
arr_seq j t1 (t1 + L)) <= L
rewrite // addnC -!addnA.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_busy_prefix_L : busy_interval_prefix arr_seq sched j
t1 (t1 + L)
L - SBF L +
(cumulative_service_inversion arr_seq sched j t1
(t1 + L) +
(cumulative_other_hep_jobs_interfering_workload
arr_seq j t1 (t1 + L) +
workload_of_job arr_seq j t1 (t1 + L))) <= L
have E: forall a b c , a <= c -> b <= c - a -> a + b <= c by move => ? ? ? ? ?; lia .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_busy_prefix_L : busy_interval_prefix arr_seq sched j
t1 (t1 + L) E : forall a b c : nat,
a <= c -> b <= c - a -> a + b <= c
L - SBF L +
(cumulative_service_inversion arr_seq sched j t1
(t1 + L) +
(cumulative_other_hep_jobs_interfering_workload
arr_seq j t1 (t1 + L) +
workload_of_job arr_seq j t1 (t1 + L))) <= L
apply : E; first by lia .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_busy_prefix_L : busy_interval_prefix arr_seq sched j
t1 (t1 + L)
cumulative_service_inversion arr_seq sched j t1
(t1 + L) +
(cumulative_other_hep_jobs_interfering_workload
arr_seq j t1 (t1 + L) +
workload_of_job arr_seq j t1 (t1 + L)) <=
L - (L - SBF L)
rewrite subKn; last by apply : sbf_bounded_by_duration => //.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_busy_prefix_L : busy_interval_prefix arr_seq sched j
t1 (t1 + L)
cumulative_service_inversion arr_seq sched j t1
(t1 + L) +
(cumulative_other_hep_jobs_interfering_workload
arr_seq j t1 (t1 + L) +
workload_of_job arr_seq j t1 (t1 + L)) <= SBF L
rewrite -(leqRW H_fixed_point); apply leq_add.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_busy_prefix_L : busy_interval_prefix arr_seq sched j
t1 (t1 + L)
cumulative_service_inversion arr_seq sched j t1
(t1 + L) <= blocking_bound ts tsk
- Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_busy_prefix_L : busy_interval_prefix arr_seq sched j
t1 (t1 + L)
cumulative_service_inversion arr_seq sched j t1
(t1 + L) <= blocking_bound ts tsk
apply : leq_trans; first apply : service_inversion_is_bounded => //.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_busy_prefix_L : busy_interval_prefix arr_seq sched j
t1 (t1 + L)
forall (j0 : Job) (t2 t3 : instant),
arrives_in arr_seq j0 ->
job_of_task tsk j0 ->
busy_interval_prefix arr_seq sched j0 t2 t3 ->
max_lp_nonpreemptive_segment arr_seq j0 t2 <=
?Goal1 (job_arrival j0 - t2)
+ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_busy_prefix_L : busy_interval_prefix arr_seq sched j
t1 (t1 + L)
forall (j0 : Job) (t2 t3 : instant),
arrives_in arr_seq j0 ->
job_of_task tsk j0 ->
busy_interval_prefix arr_seq sched j0 t2 t3 ->
max_lp_nonpreemptive_segment arr_seq j0 t2 <=
?Goal1 (job_arrival j0 - t2)
instantiate (1 := fun _ => blocking_bound ts tsk) => //=.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_busy_prefix_L : busy_interval_prefix arr_seq sched j
t1 (t1 + L)
forall (j : Job) (t1 t2 : instant),
arrives_in arr_seq j ->
job_of_task tsk j ->
busy_interval_prefix arr_seq sched j t1 t2 ->
max_lp_nonpreemptive_segment arr_seq j t1 <=
blocking_bound ts tsk
by move => jo t t' ARRo TSKo PREFo; apply : nonpreemptive_segments_bounded_by_blocking => //.
+ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_busy_prefix_L : busy_interval_prefix arr_seq sched j
t1 (t1 + L)
(fun => blocking_bound ts tsk) (job_arrival j - t1) <=
blocking_bound ts tsk
by done .
- Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_busy_prefix_L : busy_interval_prefix arr_seq sched j
t1 (t1 + L)
cumulative_other_hep_jobs_interfering_workload arr_seq
j t1 (t1 + L) +
workload_of_job arr_seq j t1 (t1 + L) <=
total_hep_request_bound_function_FP ts tsk L
rewrite addnC cumulative_iw_hep_eq_workload_of_ohep workload_job_and_ahep_eq_workload_hep //.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_busy_prefix_L : busy_interval_prefix arr_seq sched j
t1 (t1 + L)
workload_of_hep_jobs arr_seq j t1 (t1 + L) <=
total_hep_request_bound_function_FP ts tsk L
by apply hep_workload_le_total_hep_rbf.
Qed .
(** It follows that [t1 + L] is a quiet time, which means that
the busy prefix ends (i.e., it is bounded). *)
Local Lemma busy_prefix_is_bounded_case1 :
exists t2 ,
job_arrival j < t2
/\ t2 <= t1 + L
/\ busy_interval arr_seq sched j t1 t2.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_busy_prefix_L : busy_interval_prefix arr_seq sched j
t1 (t1 + L)
exists t2 : nat,
job_arrival j < t2 /\
t2 <= t1 + L /\ busy_interval arr_seq sched j t1 t2
Proof .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_busy_prefix_L : busy_interval_prefix arr_seq sched j
t1 (t1 + L)
exists t2 : nat,
job_arrival j < t2 /\
t2 <= t1 + L /\ busy_interval arr_seq sched j t1 t2
have PEND : pending sched j (job_arrival j) by apply job_pending_at_arrival => //.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_busy_prefix_L : busy_interval_prefix arr_seq sched j
t1 (t1 + L) PEND : pending sched j (job_arrival j)
exists t2 : nat,
job_arrival j < t2 /\
t2 <= t1 + L /\ busy_interval arr_seq sched j t1 t2
enough (exists t2 , job_arrival j < t2 /\ t2 <= t1 + L /\ definitions.busy_interval sched j t1 t2) as BUSY.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_busy_prefix_L : busy_interval_prefix arr_seq sched j
t1 (t1 + L) PEND : pending sched j (job_arrival j) BUSY : exists t2 : nat,
job_arrival j < t2 /\
t2 <= t1 + L /\
definitions.busy_interval sched j t1 t2
exists t2 : nat,
job_arrival j < t2 /\
t2 <= t1 + L /\ busy_interval arr_seq sched j t1 t2
{ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_busy_prefix_L : busy_interval_prefix arr_seq sched j
t1 (t1 + L) PEND : pending sched j (job_arrival j) BUSY : exists t2 : nat,
job_arrival j < t2 /\
t2 <= t1 + L /\
definitions.busy_interval sched j t1 t2
exists t2 : nat,
job_arrival j < t2 /\
t2 <= t1 + L /\ busy_interval arr_seq sched j t1 t2
have [t2 [LE1 [LE2 BUSY2]]] := BUSY.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_busy_prefix_L : busy_interval_prefix arr_seq sched j
t1 (t1 + L) PEND : pending sched j (job_arrival j) BUSY : exists t2 : nat,
job_arrival j < t2 /\
t2 <= t1 + L /\
definitions.busy_interval sched j t1 t2t2 : nat LE1 : job_arrival j < t2 LE2 : t2 <= t1 + L BUSY2 : definitions.busy_interval sched j t1 t2
exists t2 : nat,
job_arrival j < t2 /\
t2 <= t1 + L /\ busy_interval arr_seq sched j t1 t2
eexists ; split ; first by exact : LE1.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_busy_prefix_L : busy_interval_prefix arr_seq sched j
t1 (t1 + L) PEND : pending sched j (job_arrival j) BUSY : exists t2 : nat,
job_arrival j < t2 /\
t2 <= t1 + L /\
definitions.busy_interval sched j t1 t2t2 : nat LE1 : job_arrival j < t2 LE2 : t2 <= t1 + L BUSY2 : definitions.busy_interval sched j t1 t2
t2 <= t1 + L /\ busy_interval arr_seq sched j t1 t2
split ; first by done .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_busy_prefix_L : busy_interval_prefix arr_seq sched j
t1 (t1 + L) PEND : pending sched j (job_arrival j) BUSY : exists t2 : nat,
job_arrival j < t2 /\
t2 <= t1 + L /\
definitions.busy_interval sched j t1 t2t2 : nat LE1 : job_arrival j < t2 LE2 : t2 <= t1 + L BUSY2 : definitions.busy_interval sched j t1 t2
busy_interval arr_seq sched j t1 t2
by apply instantiated_busy_interval_equivalent_busy_interval.
} Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_busy_prefix_L : busy_interval_prefix arr_seq sched j
t1 (t1 + L) PEND : pending sched j (job_arrival j)
exists t2 : nat,
job_arrival j < t2 /\
t2 <= t1 + L /\
definitions.busy_interval sched j t1 t2
eapply busy_interval.busy_interval_is_bounded; eauto 2 => //.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_busy_prefix_L : busy_interval_prefix arr_seq sched j
t1 (t1 + L) PEND : pending sched j (job_arrival j)
no_speculative_execution
- Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_busy_prefix_L : busy_interval_prefix arr_seq sched j
t1 (t1 + L) PEND : pending sched j (job_arrival j)
no_speculative_execution
by eapply instantiated_i_and_w_no_speculative_execution; eauto 2 => //.
- Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_busy_prefix_L : busy_interval_prefix arr_seq sched j
t1 (t1 + L) PEND : pending sched j (job_arrival j)
definitions.busy_interval_prefix sched j t1
(job_arrival j).+1
by apply instantiated_busy_interval_prefix_equivalent_busy_interval_prefix => //.
- Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_busy_prefix_L : busy_interval_prefix arr_seq sched j
t1 (t1 + L) PEND : pending sched j (job_arrival j)
workload_of_job arr_seq j t1 (t1 + L) +
cumulative_interfering_workload j t1 (t1 + L) <= L
by apply workload_is_bounded => //.
Qed .
End Case1 .
(** Next, we consider the case when the interval <<[t1, t1 + L)>>
is not a busy-interval prefix. *)
Section Case2 .
(** Consider a time instant [t1] such that <<[t1, job_arrival
j]>> is a busy-interval prefix of [j] and <<[t1, t1 + L)>>
is _not_. *)
Variable t1 : instant.
Hypothesis H_arrives : t1 <= job_arrival j.
Hypothesis H_busy_prefix_arr : busy_interval_prefix arr_seq sched j t1 (job_arrival j).+1 .
Hypothesis H_no_busy_prefix_L : ~ busy_interval_prefix arr_seq sched j t1 (t1 + L).
(** From the properties of busy intervals, one can conclude that
[j]'s arrival time is less than [t1 + L]. *)
Local Lemma job_arrival_is_bounded :
job_arrival j < t1 + L.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L)
job_arrival j < t1 + L
Proof .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L)
job_arrival j < t1 + L
move_neq_up FF. Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L) FF : t1 + L <= job_arrival j
False
move : (H_busy_prefix_arr) (H_busy_prefix_arr) => PREFIX PREFIXA.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L) FF : t1 + L <= job_arrival j PREFIX, PREFIXA : busy_interval_prefix arr_seq sched j t1
(job_arrival j).+1
False
apply instantiated_busy_interval_prefix_equivalent_busy_interval_prefix in PREFIXA => //.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L) FF : t1 + L <= job_arrival j PREFIX : busy_interval_prefix arr_seq sched j t1
(job_arrival j).+1 PREFIXA : definitions.busy_interval_prefix sched j t1
(job_arrival j).+1
False
move : (PREFIXA) => GTC.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L) FF : t1 + L <= job_arrival j PREFIX : busy_interval_prefix arr_seq sched j t1
(job_arrival j).+1 PREFIXA, GTC : definitions.busy_interval_prefix sched j t1
(job_arrival j).+1
False
eapply workload_exceeds_interval with (Δ := L) in PREFIX => //.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L) FF : t1 + L <= job_arrival j PREFIX : L <
workload_of_job arr_seq j t1 (t1 + L) +
cumulative_interfering_workload j t1 (t1 + L) PREFIXA, GTC : definitions.busy_interval_prefix sched j t1
(job_arrival j).+1
False
{ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L) FF : t1 + L <= job_arrival j PREFIX : L <
workload_of_job arr_seq j t1 (t1 + L) +
cumulative_interfering_workload j t1 (t1 + L) PREFIXA, GTC : definitions.busy_interval_prefix sched j t1
(job_arrival j).+1
False
move_neq_down PREFIX. Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L) FF : t1 + L <= job_arrival j PREFIXA, GTC : definitions.busy_interval_prefix sched j t1
(job_arrival j).+1
workload_of_job arr_seq j t1 (t1 + L) +
cumulative_interfering_workload j t1 (t1 + L) <= L
rewrite (cumulative_interfering_workload_split _ _ _).Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L) FF : t1 + L <= job_arrival j PREFIXA, GTC : definitions.busy_interval_prefix sched j t1
(job_arrival j).+1
workload_of_job arr_seq j t1 (t1 + L) +
(blackout_during sched t1 (t1 + L) +
cumulative_service_inversion arr_seq sched j t1
(t1 + L) +
cumulative_other_hep_jobs_interfering_workload
arr_seq j t1 (t1 + L)) <= L
rewrite (leqRW (blackout_during_bound _ _ _ _ _ _ _ _ (job_arrival j).+1 _ _ _)); (try apply H_valid_SBF) => //.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L) FF : t1 + L <= job_arrival j PREFIXA, GTC : definitions.busy_interval_prefix sched j t1
(job_arrival j).+1
workload_of_job arr_seq j t1 (t1 + L) +
(L - SBF L +
cumulative_service_inversion arr_seq sched j t1
(t1 + L) +
cumulative_other_hep_jobs_interfering_workload
arr_seq j t1 (t1 + L)) <= L
rewrite addnC -!addnA.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L) FF : t1 + L <= job_arrival j PREFIXA, GTC : definitions.busy_interval_prefix sched j t1
(job_arrival j).+1
L - SBF L +
(cumulative_service_inversion arr_seq sched j t1
(t1 + L) +
(cumulative_other_hep_jobs_interfering_workload
arr_seq j t1 (t1 + L) +
workload_of_job arr_seq j t1 (t1 + L))) <= L
have E: forall a b c , a <= c -> b <= c - a -> a + b <= c by move => ? ? ? ? ?; lia .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L) FF : t1 + L <= job_arrival j PREFIXA, GTC : definitions.busy_interval_prefix sched j t1
(job_arrival j).+1 E : forall a b c : nat,
a <= c -> b <= c - a -> a + b <= c
L - SBF L +
(cumulative_service_inversion arr_seq sched j t1
(t1 + L) +
(cumulative_other_hep_jobs_interfering_workload
arr_seq j t1 (t1 + L) +
workload_of_job arr_seq j t1 (t1 + L))) <= L
apply : E; first by lia .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L) FF : t1 + L <= job_arrival j PREFIXA, GTC : definitions.busy_interval_prefix sched j t1
(job_arrival j).+1
cumulative_service_inversion arr_seq sched j t1
(t1 + L) +
(cumulative_other_hep_jobs_interfering_workload
arr_seq j t1 (t1 + L) +
workload_of_job arr_seq j t1 (t1 + L)) <=
L - (L - SBF L)
rewrite subKn; last by apply : sbf_bounded_by_duration => //.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L) FF : t1 + L <= job_arrival j PREFIXA, GTC : definitions.busy_interval_prefix sched j t1
(job_arrival j).+1
cumulative_service_inversion arr_seq sched j t1
(t1 + L) +
(cumulative_other_hep_jobs_interfering_workload
arr_seq j t1 (t1 + L) +
workload_of_job arr_seq j t1 (t1 + L)) <= SBF L
rewrite -(leqRW H_fixed_point); apply leq_add.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L) FF : t1 + L <= job_arrival j PREFIXA, GTC : definitions.busy_interval_prefix sched j t1
(job_arrival j).+1
cumulative_service_inversion arr_seq sched j t1
(t1 + L) <= blocking_bound ts tsk
{ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L) FF : t1 + L <= job_arrival j PREFIXA, GTC : definitions.busy_interval_prefix sched j t1
(job_arrival j).+1
cumulative_service_inversion arr_seq sched j t1
(t1 + L) <= blocking_bound ts tsk
rewrite (leqRW (service_inversion_widen _ _ _ t1 _ _ (job_arrival j).+1 _ _ )).Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L) FF : t1 + L <= job_arrival j PREFIXA, GTC : definitions.busy_interval_prefix sched j t1
(job_arrival j).+1
cumulative_service_inversion arr_seq sched j ?Goal0
(job_arrival j).+1 <= blocking_bound ts tsk
- Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L) FF : t1 + L <= job_arrival j PREFIXA, GTC : definitions.busy_interval_prefix sched j t1
(job_arrival j).+1
cumulative_service_inversion arr_seq sched j ?Goal0
(job_arrival j).+1 <= blocking_bound ts tsk
apply : leq_trans.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L) FF : t1 + L <= job_arrival j PREFIXA, GTC : definitions.busy_interval_prefix sched j t1
(job_arrival j).+1
cumulative_service_inversion arr_seq sched j ?Goal0
(job_arrival j).+1 <= ?Goal3
+ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L) FF : t1 + L <= job_arrival j PREFIXA, GTC : definitions.busy_interval_prefix sched j t1
(job_arrival j).+1
cumulative_service_inversion arr_seq sched j ?Goal0
(job_arrival j).+1 <= ?Goal3
apply : service_inversion_is_bounded => //.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L) FF : t1 + L <= job_arrival j PREFIXA, GTC : definitions.busy_interval_prefix sched j t1
(job_arrival j).+1
forall (j0 : Job) (t2 t3 : instant),
arrives_in arr_seq j0 ->
job_of_task tsk j0 ->
busy_interval_prefix arr_seq sched j0 t2 t3 ->
max_lp_nonpreemptive_segment arr_seq j0 t2 <=
?Goal3 (job_arrival j0 - t2)
move => *; instantiate (1 := fun _ => blocking_bound ts tsk) => //.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L) FF : t1 + L <= job_arrival j PREFIXA, GTC : definitions.busy_interval_prefix sched j t1
(job_arrival j).+1 _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_ <=
(fun => blocking_bound ts tsk) (job_arrival _j_ - _t1_)
by apply : nonpreemptive_segments_bounded_by_blocking => //.
+ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L) FF : t1 + L <= job_arrival j PREFIXA, GTC : definitions.busy_interval_prefix sched j t1
(job_arrival j).+1
(fun => blocking_bound ts tsk) (job_arrival j - t1) <=
blocking_bound ts tsk
by done .
- Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L) FF : t1 + L <= job_arrival j PREFIXA, GTC : definitions.busy_interval_prefix sched j t1
(job_arrival j).+1
t1 <= t1
by done .
- Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L) FF : t1 + L <= job_arrival j PREFIXA, GTC : definitions.busy_interval_prefix sched j t1
(job_arrival j).+1
t1 + L <= (job_arrival j).+1
lia .
} Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L) FF : t1 + L <= job_arrival j PREFIXA, GTC : definitions.busy_interval_prefix sched j t1
(job_arrival j).+1
cumulative_other_hep_jobs_interfering_workload arr_seq
j t1 (t1 + L) +
workload_of_job arr_seq j t1 (t1 + L) <=
total_hep_request_bound_function_FP ts tsk L
{ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L) FF : t1 + L <= job_arrival j PREFIXA, GTC : definitions.busy_interval_prefix sched j t1
(job_arrival j).+1
cumulative_other_hep_jobs_interfering_workload arr_seq
j t1 (t1 + L) +
workload_of_job arr_seq j t1 (t1 + L) <=
total_hep_request_bound_function_FP ts tsk L
rewrite addnC cumulative_iw_hep_eq_workload_of_ohep workload_job_and_ahep_eq_workload_hep //.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L) FF : t1 + L <= job_arrival j PREFIXA, GTC : definitions.busy_interval_prefix sched j t1
(job_arrival j).+1
workload_of_hep_jobs arr_seq j t1 (t1 + L) <=
total_hep_request_bound_function_FP ts tsk L
by apply hep_workload_le_total_hep_rbf. }
}
Qed .
(** Lemma [job_arrival_is_bounded] implies that the
busy-interval prefix starts at time [t1], continues until
[job_arrival j + 1], and then terminates before [t1 + L].
Or, in other words, there is point in time [t2] such that
(1) [j]'s arrival is bounded by [t2], (2) [t2] is bounded by
[t1 + L], and (3) <<[t1, t2)>> is busy interval of job
[j]. *)
Local Lemma busy_prefix_is_bounded_case2 :
exists t2 , job_arrival j < t2 /\ t2 <= t1 + L /\ busy_interval arr_seq sched j t1 t2.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L)
exists t2 : nat,
job_arrival j < t2 /\
t2 <= t1 + L /\ busy_interval arr_seq sched j t1 t2
Proof .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L)
exists t2 : nat,
job_arrival j < t2 /\
t2 <= t1 + L /\ busy_interval arr_seq sched j t1 t2
have LE: job_arrival j < t1 + L
by apply job_arrival_is_bounded => //; try apply instantiated_busy_interval_prefix_equivalent_busy_interval_prefix.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L) LE : job_arrival j < t1 + L
exists t2 : nat,
job_arrival j < t2 /\
t2 <= t1 + L /\ busy_interval arr_seq sched j t1 t2
move : (H_busy_prefix_arr) => PREFIX.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L) LE : job_arrival j < t1 + L PREFIX : busy_interval_prefix arr_seq sched j t1
(job_arrival j).+1
exists t2 : nat,
job_arrival j < t2 /\
t2 <= t1 + L /\ busy_interval arr_seq sched j t1 t2
move : (H_no_busy_prefix_L) => NOPREF.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L) LE : job_arrival j < t1 + L PREFIX : busy_interval_prefix arr_seq sched j t1
(job_arrival j).+1 NOPREF : ~
busy_interval_prefix arr_seq sched j t1
(t1 + L)
exists t2 : nat,
job_arrival j < t2 /\
t2 <= t1 + L /\ busy_interval arr_seq sched j t1 t2
apply instantiated_busy_interval_prefix_equivalent_busy_interval_prefix in PREFIX => //.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L) LE : job_arrival j < t1 + L PREFIX : definitions.busy_interval_prefix sched j t1
(job_arrival j).+1 NOPREF : ~
busy_interval_prefix arr_seq sched j t1
(t1 + L)
exists t2 : nat,
job_arrival j < t2 /\
t2 <= t1 + L /\ busy_interval arr_seq sched j t1 t2
have BUSY := terminating_busy_prefix_is_busy_interval _ _ _ _ _ _ _ PREFIX.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L) LE : job_arrival j < t1 + L PREFIX : definitions.busy_interval_prefix sched j t1
(job_arrival j).+1 NOPREF : ~
busy_interval_prefix arr_seq sched j t1
(t1 + L) BUSY : job_cost_positive j ->
forall n : nat,
job_arrival j < n ->
~ definitions.busy_interval_prefix sched j t1 n ->
exists t2'' : instant,
definitions.busy_interval sched j t1 t2''
exists t2 : nat,
job_arrival j < t2 /\
t2 <= t1 + L /\ busy_interval arr_seq sched j t1 t2
edestruct BUSY as [t2 BUS]; clear BUSY; try apply TSK; eauto 2 => //.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L) LE : job_arrival j < t1 + L PREFIX : definitions.busy_interval_prefix sched j t1
(job_arrival j).+1 NOPREF : ~
busy_interval_prefix arr_seq sched j t1
(t1 + L)
~ definitions.busy_interval_prefix sched j t1 (t1 + L)
{ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L) LE : job_arrival j < t1 + L PREFIX : definitions.busy_interval_prefix sched j t1
(job_arrival j).+1 NOPREF : ~
busy_interval_prefix arr_seq sched j t1
(t1 + L)
~ definitions.busy_interval_prefix sched j t1 (t1 + L)
move => T; apply : NOPREF.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L) LE : job_arrival j < t1 + L PREFIX : definitions.busy_interval_prefix sched j t1
(job_arrival j).+1 T : definitions.busy_interval_prefix sched j t1
(t1 + L)
busy_interval_prefix arr_seq sched j t1 (t1 + L)
by apply instantiated_busy_interval_prefix_equivalent_busy_interval_prefix in T => //.
} Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L) LE : job_arrival j < t1 + L PREFIX : definitions.busy_interval_prefix sched j t1
(job_arrival j).+1 NOPREF : ~
busy_interval_prefix arr_seq sched j t1
(t1 + L) t2 : instant BUS : definitions.busy_interval sched j t1 t2
exists t2 : nat,
job_arrival j < t2 /\
t2 <= t1 + L /\ busy_interval arr_seq sched j t1 t2
exists t2 ; split ; last split .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L) LE : job_arrival j < t1 + L PREFIX : definitions.busy_interval_prefix sched j t1
(job_arrival j).+1 NOPREF : ~
busy_interval_prefix arr_seq sched j t1
(t1 + L) t2 : instant BUS : definitions.busy_interval sched j t1 t2
job_arrival j < t2
{ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L) LE : job_arrival j < t1 + L PREFIX : definitions.busy_interval_prefix sched j t1
(job_arrival j).+1 NOPREF : ~
busy_interval_prefix arr_seq sched j t1
(t1 + L) t2 : instant BUS : definitions.busy_interval sched j t1 t2
job_arrival j < t2
move : BUS => [[A _] _]; lia . } Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L) LE : job_arrival j < t1 + L PREFIX : definitions.busy_interval_prefix sched j t1
(job_arrival j).+1 NOPREF : ~
busy_interval_prefix arr_seq sched j t1
(t1 + L) t2 : instant BUS : definitions.busy_interval sched j t1 t2
t2 <= t1 + L
{ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L) LE : job_arrival j < t1 + L PREFIX : definitions.busy_interval_prefix sched j t1
(job_arrival j).+1 NOPREF : ~
busy_interval_prefix arr_seq sched j t1
(t1 + L) t2 : instant BUS : definitions.busy_interval sched j t1 t2
t2 <= t1 + L
move_neq_up FA. Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L) LE : job_arrival j < t1 + L PREFIX : definitions.busy_interval_prefix sched j t1
(job_arrival j).+1 NOPREF : ~
busy_interval_prefix arr_seq sched j t1
(t1 + L) t2 : instant BUS : definitions.busy_interval sched j t1 t2 FA : t1 + L < t2
False
apply : NOPREF; split ; [lia | split ; first by apply H_busy_prefix_arr].Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L) LE : job_arrival j < t1 + L PREFIX : definitions.busy_interval_prefix sched j t1
(job_arrival j).+1 t2 : instant BUS : definitions.busy_interval sched j t1 t2 FA : t1 + L < t2
(forall t : nat,
t1 < t < t1 + L -> ~ quiet_time arr_seq sched j t) /\
t1 <= job_arrival j < t1 + L
split .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L) LE : job_arrival j < t1 + L PREFIX : definitions.busy_interval_prefix sched j t1
(job_arrival j).+1 t2 : instant BUS : definitions.busy_interval sched j t1 t2 FA : t1 + L < t2
forall t : nat,
t1 < t < t1 + L -> ~ quiet_time arr_seq sched j t
- Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L) LE : job_arrival j < t1 + L PREFIX : definitions.busy_interval_prefix sched j t1
(job_arrival j).+1 t2 : instant BUS : definitions.busy_interval sched j t1 t2 FA : t1 + L < t2
forall t : nat,
t1 < t < t1 + L -> ~ quiet_time arr_seq sched j t
move => t NEQ.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L) LE : job_arrival j < t1 + L PREFIX : definitions.busy_interval_prefix sched j t1
(job_arrival j).+1 t2 : instant BUS : definitions.busy_interval sched j t1 t2 FA : t1 + L < t2 t : nat NEQ : t1 < t < t1 + L
~ quiet_time arr_seq sched j t
apply abstract_busy_interval_classic_busy_interval_prefix in BUS => //.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L) LE : job_arrival j < t1 + L PREFIX : definitions.busy_interval_prefix sched j t1
(job_arrival j).+1 t2 : instant BUS : busy_interval_prefix arr_seq sched j t1 t2 FA : t1 + L < t2 t : nat NEQ : t1 < t < t1 + L
~ quiet_time arr_seq sched j t
by apply BUS; lia .
- Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L) LE : job_arrival j < t1 + L PREFIX : definitions.busy_interval_prefix sched j t1
(job_arrival j).+1 t2 : instant BUS : definitions.busy_interval sched j t1 t2 FA : t1 + L < t2
t1 <= job_arrival j < t1 + L
lia .
} Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L) LE : job_arrival j < t1 + L PREFIX : definitions.busy_interval_prefix sched j t1
(job_arrival j).+1 NOPREF : ~
busy_interval_prefix arr_seq sched j t1
(t1 + L) t2 : instant BUS : definitions.busy_interval sched j t1 t2
busy_interval arr_seq sched j t1 t2
{ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job H_j_arrives : arrives_in arr_seq j H_job_of_tsk : job_of_task tsk j H_job_cost_positive : job_cost_positive j t1 : instant H_arrives : t1 <= job_arrival j H_busy_prefix_arr : busy_interval_prefix arr_seq sched
j t1 (job_arrival j).+1 H_no_busy_prefix_L : ~
busy_interval_prefix arr_seq
sched j t1
(t1 + L) LE : job_arrival j < t1 + L PREFIX : definitions.busy_interval_prefix sched j t1
(job_arrival j).+1 NOPREF : ~
busy_interval_prefix arr_seq sched j t1
(t1 + L) t2 : instant BUS : definitions.busy_interval sched j t1 t2
busy_interval arr_seq sched j t1 t2
by apply instantiated_busy_interval_equivalent_busy_interval => //. }
Qed .
End Case2 .
End StepByStepProof .
(** Combining the cases analyzed above, we conclude that busy
intervals of jobs released by task [tsk] are bounded by [L]. *)
Lemma busy_intervals_are_bounded_rs_fp :
busy_intervals_are_bounded_by arr_seq sched tsk L.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L
busy_intervals_are_bounded_by arr_seq sched tsk L
Proof .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L
busy_intervals_are_bounded_by arr_seq sched tsk L
move => j ARR TSK POS.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j POS : 0 < job_cost j
exists t1 t2 : nat,
t1 <= job_arrival j < t2 /\
t2 <= t1 + L /\
definitions.busy_interval sched j t1 t2
have PEND : pending sched j (job_arrival j) by apply job_pending_at_arrival => //.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j POS : 0 < job_cost jPEND : pending sched j (job_arrival j)
exists t1 t2 : nat,
t1 <= job_arrival j < t2 /\
t2 <= t1 + L /\
definitions.busy_interval sched j t1 t2
have [t1 [GE PREFIX]]:
exists t1 , t1 <= job_arrival j
/\ busy_interval_prefix arr_seq sched j t1 (job_arrival j).+1
by apply : busy_interval_prefix_exists.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j POS : 0 < job_cost jPEND : pending sched j (job_arrival j) t1 : nat GE : t1 <= job_arrival j PREFIX : busy_interval_prefix arr_seq sched j t1
(job_arrival j).+1
exists t1 t2 : nat,
t1 <= job_arrival j < t2 /\
t2 <= t1 + L /\
definitions.busy_interval sched j t1 t2
exists t1 .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j POS : 0 < job_cost jPEND : pending sched j (job_arrival j) t1 : nat GE : t1 <= job_arrival j PREFIX : busy_interval_prefix arr_seq sched j t1
(job_arrival j).+1
exists t2 : nat,
t1 <= job_arrival j < t2 /\
t2 <= t1 + L /\
definitions.busy_interval sched j t1 t2
enough (exists t2 , job_arrival j < t2 /\ t2 <= t1 + L /\ busy_interval arr_seq sched j t1 t2) as BUSY.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j POS : 0 < job_cost jPEND : pending sched j (job_arrival j) t1 : nat GE : t1 <= job_arrival j PREFIX : busy_interval_prefix arr_seq sched j t1
(job_arrival j).+1 BUSY : exists t2 : nat,
job_arrival j < t2 /\
t2 <= t1 + L /\
busy_interval arr_seq sched j t1 t2
exists t2 : nat,
t1 <= job_arrival j < t2 /\
t2 <= t1 + L /\
definitions.busy_interval sched j t1 t2
{ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j POS : 0 < job_cost jPEND : pending sched j (job_arrival j) t1 : nat GE : t1 <= job_arrival j PREFIX : busy_interval_prefix arr_seq sched j t1
(job_arrival j).+1 BUSY : exists t2 : nat,
job_arrival j < t2 /\
t2 <= t1 + L /\
busy_interval arr_seq sched j t1 t2
exists t2 : nat,
t1 <= job_arrival j < t2 /\
t2 <= t1 + L /\
definitions.busy_interval sched j t1 t2
move : BUSY => [t2 [LT [LE BUSY]]]; eexists ; split ; last first .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j POS : 0 < job_cost jPEND : pending sched j (job_arrival j) t1 : nat GE : t1 <= job_arrival j PREFIX : busy_interval_prefix arr_seq sched j t1
(job_arrival j).+1 t2 : nat LT : job_arrival j < t2 LE : t2 <= t1 + L BUSY : busy_interval arr_seq sched j t1 t2
?t2 <= t1 + L /\
definitions.busy_interval sched j t1 ?t2
{ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j POS : 0 < job_cost jPEND : pending sched j (job_arrival j) t1 : nat GE : t1 <= job_arrival j PREFIX : busy_interval_prefix arr_seq sched j t1
(job_arrival j).+1 t2 : nat LT : job_arrival j < t2 LE : t2 <= t1 + L BUSY : busy_interval arr_seq sched j t1 t2
?t2 <= t1 + L /\
definitions.busy_interval sched j t1 ?t2
split ; first by exact : LE.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j POS : 0 < job_cost jPEND : pending sched j (job_arrival j) t1 : nat GE : t1 <= job_arrival j PREFIX : busy_interval_prefix arr_seq sched j t1
(job_arrival j).+1 t2 : nat LT : job_arrival j < t2 LE : t2 <= t1 + L BUSY : busy_interval arr_seq sched j t1 t2
definitions.busy_interval sched j t1 t2
by apply instantiated_busy_interval_equivalent_busy_interval. } Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j POS : 0 < job_cost jPEND : pending sched j (job_arrival j) t1 : nat GE : t1 <= job_arrival j PREFIX : busy_interval_prefix arr_seq sched j t1
(job_arrival j).+1 t2 : nat LT : job_arrival j < t2 LE : t2 <= t1 + L BUSY : busy_interval arr_seq sched j t1 t2
t1 <= job_arrival j < t2
{ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j POS : 0 < job_cost jPEND : pending sched j (job_arrival j) t1 : nat GE : t1 <= job_arrival j PREFIX : busy_interval_prefix arr_seq sched j t1
(job_arrival j).+1 t2 : nat LT : job_arrival j < t2 LE : t2 <= t1 + L BUSY : busy_interval arr_seq sched j t1 t2
t1 <= job_arrival j < t2
by apply /andP; split . }
} Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j POS : 0 < job_cost jPEND : pending sched j (job_arrival j) t1 : nat GE : t1 <= job_arrival j PREFIX : busy_interval_prefix arr_seq sched j t1
(job_arrival j).+1
exists t2 : nat,
job_arrival j < t2 /\
t2 <= t1 + L /\ busy_interval arr_seq sched j t1 t2
{ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j POS : 0 < job_cost jPEND : pending sched j (job_arrival j) t1 : nat GE : t1 <= job_arrival j PREFIX : busy_interval_prefix arr_seq sched j t1
(job_arrival j).+1
exists t2 : nat,
job_arrival j < t2 /\
t2 <= t1 + L /\ busy_interval arr_seq sched j t1 t2
have [LPREF|NOPREF] := busy_interval_prefix_case ltac :(eauto ) j t1 (t1 + L).Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j POS : 0 < job_cost jPEND : pending sched j (job_arrival j) t1 : nat GE : t1 <= job_arrival j PREFIX : busy_interval_prefix arr_seq sched j t1
(job_arrival j).+1 LPREF : definitions.busy_interval_prefix sched j t1
(t1 + L)
exists t2 : nat,
job_arrival j < t2 /\
t2 <= t1 + L /\ busy_interval arr_seq sched j t1 t2
{ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j POS : 0 < job_cost jPEND : pending sched j (job_arrival j) t1 : nat GE : t1 <= job_arrival j PREFIX : busy_interval_prefix arr_seq sched j t1
(job_arrival j).+1 LPREF : definitions.busy_interval_prefix sched j t1
(t1 + L)
exists t2 : nat,
job_arrival j < t2 /\
t2 <= t1 + L /\ busy_interval arr_seq sched j t1 t2
apply busy_prefix_is_bounded_case1 => //.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j POS : 0 < job_cost jPEND : pending sched j (job_arrival j) t1 : nat GE : t1 <= job_arrival j PREFIX : busy_interval_prefix arr_seq sched j t1
(job_arrival j).+1 LPREF : definitions.busy_interval_prefix sched j t1
(t1 + L)
busy_interval_prefix arr_seq sched j t1 (t1 + L)
by apply instantiated_busy_interval_prefix_equivalent_busy_interval_prefix => //. } Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j POS : 0 < job_cost jPEND : pending sched j (job_arrival j) t1 : nat GE : t1 <= job_arrival j PREFIX : busy_interval_prefix arr_seq sched j t1
(job_arrival j).+1 NOPREF : ~
definitions.busy_interval_prefix sched j t1
(t1 + L)
exists t2 : nat,
job_arrival j < t2 /\
t2 <= t1 + L /\ busy_interval arr_seq sched j t1 t2
{ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j POS : 0 < job_cost jPEND : pending sched j (job_arrival j) t1 : nat GE : t1 <= job_arrival j PREFIX : busy_interval_prefix arr_seq sched j t1
(job_arrival j).+1 NOPREF : ~
definitions.busy_interval_prefix sched j t1
(t1 + L)
exists t2 : nat,
job_arrival j < t2 /\
t2 <= t1 + L /\ busy_interval arr_seq sched j t1 t2
apply busy_prefix_is_bounded_case2=> //.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j POS : 0 < job_cost jPEND : pending sched j (job_arrival j) t1 : nat GE : t1 <= job_arrival j PREFIX : busy_interval_prefix arr_seq sched j t1
(job_arrival j).+1 NOPREF : ~
definitions.busy_interval_prefix sched j t1
(t1 + L)
~ busy_interval_prefix arr_seq sched j t1 (t1 + L)
move => NP; apply : NOPREF.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job PState : ProcessorState Job H_uniprocessor_proc_model : uniprocessor_model PState H_unit_supply_proc_model : unit_supply_proc_model
PState H_consumed_supply_proc_model : fully_consuming_proc_model PState FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq sched : schedule PState JobReady0 : JobReady Job PState H_job_ready : work_bearing_readiness arr_seq sched H_sched_valid : valid_schedule sched arr_seq H3 : JobPreemptable Job H4 : TaskMaxNonpreemptiveSegment Task H_valid_preemption_model : valid_preemption_model
arr_seq sched H_valid_model_with_bounded_nonpreemptive_segments : valid_model_with_bounded_nonpreemptive_segments
arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP H_work_conserving : definitions.work_conserving
arr_seq sched ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq H5 : MaxArrivals Task H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts tsk : Task H_tsk_in_ts : tsk \in ts SBF : SupplyBoundFunction H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF H_unit_SBF : unit_supply_bound_function SBF L : duration H_L_positive : 0 < LH_fixed_point : blocking_bound ts tsk +
total_hep_request_bound_function_FP ts
tsk L <=
SBF L j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j POS : 0 < job_cost jPEND : pending sched j (job_arrival j) t1 : nat GE : t1 <= job_arrival j PREFIX : busy_interval_prefix arr_seq sched j t1
(job_arrival j).+1 NP : busy_interval_prefix arr_seq sched j t1 (t1 + L)
definitions.busy_interval_prefix sched j t1 (t1 + L)
by apply instantiated_busy_interval_prefix_equivalent_busy_interval_prefix => //.
}
}
Qed .
End BoundedBusyIntervals .