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Require Import prosa.analysis.facts.readiness.basic.[Loading ML file ssrmatching_plugin.cmxs (using legacy method) ... done ] [Loading ML file ssreflect_plugin.cmxs (using legacy method) ... done ] [Loading ML file ring_plugin.cmxs (using legacy method) ... done ] [Loading ML file coq-elpi.elpi ... done ] [Loading ML file zify_plugin.cmxs (using legacy method) ... done ] [Loading ML file micromega_plugin.cmxs (using legacy method) ... done ] [Loading ML file btauto_plugin.cmxs (using legacy method) ... done ] Notation "_ + _" was already used in scope nat_scope.
[notation-overridden,parsing,default]Notation "_ - _" was already used in scope nat_scope.
[notation-overridden,parsing,default]Notation "_ <= _" was already used in scope nat_scope.
[notation-overridden,parsing,default]Notation "_ < _" was already used in scope nat_scope.
[notation-overridden,parsing,default]Notation "_ >= _" was already used in scope nat_scope.
[notation-overridden,parsing,default]Notation "_ > _" was already used in scope nat_scope.
[notation-overridden,parsing,default]Notation "_ <= _ <= _" was already used in scope
nat_scope. [notation-overridden,parsing,default]Notation "_ < _ <= _" was already used in scope
nat_scope. [notation-overridden,parsing,default]Notation "_ <= _ < _" was already used in scope
nat_scope. [notation-overridden,parsing,default]Notation "_ < _ < _" was already used in scope
nat_scope. [notation-overridden,parsing,default]Notation "_ * _" was already used in scope nat_scope.
[notation-overridden,parsing,default]
Require Export prosa.analysis.facts.model.restricted_supply.schedule.
Require Export prosa.analysis.abstract .restricted_supply.bounded_bi.jlfp.
Require Export prosa.analysis.abstract .restricted_supply.search_space.fifo.
Require Export prosa.analysis.abstract .restricted_supply.search_space.fifo_fixpoint.
Require Import prosa.analysis.facts.priority.fifo.
Require Export prosa.analysis.facts.priority.fifo_ahep_bound.
Require Export prosa.model.schedule.work_conserving.
(** * RTA for FIFO Scheduling on Restricted-Supply Uniprocessors *)
(** In the following, we derive a response-time analysis for FIFO
schedulers, assuming a workload of sporadic real-time tasks
characterized by arbitrary arrival curves executing upon a
uniprocessor with arbitrary supply restrictions. To this end, we
instantiate the _abstract Restricted-Supply Response-Time
Analysis_ (aRSA) as provided in the
[prosa.analysis.abstract.restricted_supply] module. *)
Section RTAforFullyPreemptiveFIFOModelwithArrivalCurves .
(** ** Defining the System Model *)
(** Before any formal claims can be stated, an initial setup is
needed to define the system model under consideration. To this
end, we next introduce and define the following notions using
Prosa's standard definitions and behavioral semantics:
- processor model,
- tasks, jobs, and their parameters,
- the sequence of job arrivals,
- worst-case execution time (WCET) and the absence of self-suspensions,
- the task under analysis,
- an arbitrary schedule of the task set, and finally,
- a supply-bound function. *)
(** *** Processor Model *)
(** Consider a restricted-supply uniprocessor model. *)
#[local] Existing Instance rs_processor_state .
(** *** Tasks and Jobs *)
(** Consider any type of tasks, each characterized by a WCET
[task_cost], an arrival curve [max_arrivals], a
run-to-completion threshold [task_rtct], and a bound on the the
task's longest non-preemptive segment
[task_max_nonpreemptive_segment] ... *)
Context {Task : TaskType}.
Context `{TaskCost Task}.
Context `{MaxArrivals Task}.
Context `{TaskRunToCompletionThreshold Task}.
Context `{TaskMaxNonpreemptiveSegment Task}.
(** ... and any type of jobs associated with these tasks, where each
job has a task [job_task], a cost [job_cost], an arrival time
[job_arrival], and a predicate indicating when the job is
preemptable [job_preemptable]. *)
Context {Job : JobType}.
Context `{JobTask Job Task}.
Context `{JobCost Job}.
Context `{JobArrival Job}.
Context `{JobPreemptable Job}.
(** *** The Job Arrival Sequence *)
(** Consider any arrival sequence [arr_seq] with consistent, non-duplicate arrivals. *)
Variable arr_seq : arrival_sequence Job.
Hypothesis H_valid_arrival_sequence : valid_arrival_sequence arr_seq.
(** *** Absence of Self-Suspensions and WCET Compliance *)
(** We assume the classic (i.e., Liu & Layland) model of readiness
without jitter or self-suspensions, wherein pending jobs are
always ready. *)
#[local] Existing Instance basic_ready_instance .
(** We further require that a job's cost cannot exceed its task's stated WCET. *)
Hypothesis H_valid_job_cost : arrivals_have_valid_job_costs arr_seq.
(** *** The Task Set *)
(** We consider an arbitrary task set [ts] ... *)
Variable ts : seq Task.
(** ... and assume that all jobs stem from tasks in this task set. *)
Hypothesis H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts.
(** We assume that [max_arrivals] is a family of valid arrival
curves that constrains the arrival sequence [arr_seq], i.e., for
any task [tsk] in [ts], [max_arrival tsk] is (1) an arrival
bound of [tsk], and ... *)
Hypothesis H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts.
(** ... (2) a monotonic function that equals 0 for the empty interval [delta = 0]. *)
Hypothesis H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals.
(** *** The Schedule *)
(** Consider any arbitrary, work-conserving, valid restricted-supply
uni-processor schedule of the given arrival sequence [arr_seq]
(and hence the given task set [ts]) ... *)
Variable sched : schedule (rs_processor_state Job).
Hypothesis H_valid_schedule : valid_schedule sched arr_seq.
Hypothesis H_work_conserving : work_conserving arr_seq sched.
(** ... and assume that the schedule respects the FIFO policy. *)
Hypothesis H_respects_policy :
respects_JLFP_policy_at_preemption_point arr_seq sched (FIFO Job).
(** We assume a valid preemption model with bounded non-preemptive
segments. *)
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.
(** *** The Task Under Analysis *)
(** Let [tsk] be any task in [ts] that is to be analyzed. *)
Variable tsk : Task.
Hypothesis H_tsk_in_ts : tsk \in ts.
(** We assume that [tsk] is described by a valid task _run-to-completion
threshold_. That is, there exists a task parameter [task_rtct] such
that [task_rtct tsk] is
- (1) no larger than [tsk]'s WCET, and
- (2) for any job of task [tsk], the job's run-to-completion threshold
[job_rtct] is bounded by [task_rtct tsk]. *)
Hypothesis H_valid_run_to_completion_threshold :
valid_task_run_to_completion_threshold arr_seq tsk.
(** *** Supply-Bound Function *)
(** Assume the minimum amount of supply that any job of task [tsk]
receives is defined by a monotone unit-supply-bound function [SBF]. *)
Context {SBF : SupplyBoundFunction}.
Hypothesis H_SBF_monotone : sbf_is_monotone SBF.
Hypothesis H_unit_SBF : unit_supply_bound_function SBF.
(** Next, we assume that [SBF] properly characterizes all busy
intervals (w.r.t. task [tsk]) in [sched]. That is, (1) [SBF 0 =
0] and (2) for any duration [Δ], at least [SBF Δ] supply is
available in any busy-interval prefix of length [Δ]. *)
Hypothesis H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF.
(** ** Length of Busy Interval *)
(** The next step is to establish a bound on the maximum busy-window
length, which aRSA requires to be given. *)
(** To this end, let [L] be any positive fixed point of the
busy-interval recurrence. As the
[busy_intervals_are_bounded_rs_jlfp] lemma shows, under any
preemptive [JLFP] scheduling policy, this is sufficient to
guarantee that all busy intervals are bounded by [L]. *)
Variable L : duration.
Hypothesis H_L_positive : 0 < L.
Hypothesis H_fixed_point : total_request_bound_function ts L <= SBF L.
(** ** Response-Time Bound *)
(** Having established all necessary preliminaries, it is finally
time to state the claimed response-time bound [R]. *)
(** A value [R] is an RTA-recurrence solution if, for any given
offset [A] in the search space, the response-time bound
recurrence has a solution [F] not exceeding [R]. *)
Definition rta_recurrence_solution R :=
forall (A : duration),
is_in_search_space ts L A ->
exists (F : duration),
A <= F <= A + R
/\ total_request_bound_function ts (A + ε) <= SBF F.
(** Finally, using the abstract restricted-supply analysis, we
establish that any [R] that satisfies the stated equation is a
sound response-time bound for the FIFO scheduling with arbitrary
supply restrictions. *)
Theorem uniprocessor_response_time_bound_fifo :
forall (R : duration),
rta_recurrence_solution R ->
task_response_time_bound arr_seq sched tsk R.Task : TaskType H : TaskCost Task H0 : MaxArrivals Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptable Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (FIFO Job) 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 tsk : Task H_tsk_in_ts : tsk \in ts H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L
forall R : duration,
rta_recurrence_solution R ->
task_response_time_bound arr_seq sched tsk R
Proof .Task : TaskType H : TaskCost Task H0 : MaxArrivals Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptable Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (FIFO Job) 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 tsk : Task H_tsk_in_ts : tsk \in ts H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L
forall R : duration,
rta_recurrence_solution R ->
task_response_time_bound arr_seq sched tsk R
move => R SOL js ARRs TSKs.Task : TaskType H : TaskCost Task H0 : MaxArrivals Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptable Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (FIFO Job) 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 tsk : Task H_tsk_in_ts : tsk \in ts H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js
job_response_time_bound sched js R
have [ZERO|POS] := posnP (job_cost js).Task : TaskType H : TaskCost Task H0 : MaxArrivals Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptable Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (FIFO Job) 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 tsk : Task H_tsk_in_ts : tsk \in ts H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js ZERO : job_cost js = 0
job_response_time_bound sched js R
{ Task : TaskType H : TaskCost Task H0 : MaxArrivals Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptable Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (FIFO Job) 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 tsk : Task H_tsk_in_ts : tsk \in ts H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js ZERO : job_cost js = 0
job_response_time_bound sched js R
by rewrite /job_response_time_bound /completed_by ZERO. } Task : TaskType H : TaskCost Task H0 : MaxArrivals Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptable Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (FIFO Job) 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 tsk : Task H_tsk_in_ts : tsk \in ts H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost js
job_response_time_bound sched js R
have READ : work_bearing_readiness arr_seq sched by done .Task : TaskType H : TaskCost Task H0 : MaxArrivals Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptable Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (FIFO Job) 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 tsk : Task H_tsk_in_ts : tsk \in ts H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched
job_response_time_bound sched js R
eapply uniprocessor_response_time_bound_restricted_supply
with (L := L)
(intra_IBF := fun A Δ => (\sum_(tsko <- ts) task_request_bound_function tsko (A + ε)) - task_cost tsk) => //.Task : TaskType H : TaskCost Task H0 : MaxArrivals Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptable Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (FIFO Job) 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 tsk : Task H_tsk_in_ts : tsk \in ts H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched
definitions.work_conserving arr_seq sched
- Task : TaskType H : TaskCost Task H0 : MaxArrivals Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptable Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (FIFO Job) 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 tsk : Task H_tsk_in_ts : tsk \in ts H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched
definitions.work_conserving arr_seq sched
exact : instantiated_i_and_w_are_coherent_with_schedule.
- Task : TaskType H : TaskCost Task H0 : MaxArrivals Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptable Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (FIFO Job) 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 tsk : Task H_tsk_in_ts : tsk \in ts H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched
busy_intervals_are_bounded_by arr_seq sched tsk L
eapply busy_intervals_are_bounded_rs_jlfp with (blocking_bound := fun _ => 0 )=> //.Task : TaskType H : TaskCost Task H0 : MaxArrivals Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptable Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (FIFO Job) 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 tsk : Task H_tsk_in_ts : tsk \in ts H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched
service_inversion_is_bounded_by arr_seq sched tsk
(fun => 0 )
by apply : FIFO_implies_no_service_inversion.
- Task : TaskType H : TaskCost Task H0 : MaxArrivals Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptable Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (FIFO Job) 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 tsk : Task H_tsk_in_ts : tsk \in ts H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched
busy_sbf.valid_busy_sbf arr_seq sched tsk SBF
apply : valid_pred_sbf_switch_predicate; last by exact : H_valid_SBF.Task : TaskType H : TaskCost Task H0 : MaxArrivals Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptable Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (FIFO Job) 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 tsk : Task H_tsk_in_ts : tsk \in ts H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched
forall (j : Job) (t1 t2 : instant),
arrives_in arr_seq j ->
job_of_task tsk j /\
definitions.busy_interval_prefix sched j t1 t2 ->
(fun (j0 : Job) (t3 t4 : instant) =>
job_of_task tsk j0 /\
busy_interval_prefix arr_seq sched j0 t3 t4) j t1 t2
move => ? ? ? ? [? ?]; split => //.Task : TaskType H : TaskCost Task H0 : MaxArrivals Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptable Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (FIFO Job) 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 tsk : Task H_tsk_in_ts : tsk \in ts H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched _j_ : Job _t1_, _t2_ : instant _Hyp_ : arrives_in arr_seq _j_ _a_ : job_of_task tsk _j_ _b_ : definitions.busy_interval_prefix sched _j_ _t1_
_t2_
busy_interval_prefix arr_seq sched _j_ _t1_ _t2_
by apply instantiated_busy_interval_prefix_equivalent_busy_interval_prefix.
- Task : TaskType H : TaskCost Task H0 : MaxArrivals Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptable Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (FIFO Job) 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 tsk : Task H_tsk_in_ts : tsk \in ts H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched
intra_interference_is_bounded_by arr_seq sched tsk
(fun A : nat =>
fun => \sum_(tsko <- ts)
task_request_bound_function tsko (A + 1 ) -
task_cost tsk)
move => t1 t2 Δ j ARR TSK BUSY LT NCOMPL A OFF.Task : TaskType H : TaskCost Task H0 : MaxArrivals Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptable Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (FIFO Job) 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 tsk : Task H_tsk_in_ts : tsk \in ts H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched t1, t2 : instant Δ : nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j BUSY : definitions.busy_interval sched j t1 t2 LT : t1 + Δ < t2 NCOMPL : ~~ completed_by sched j (t1 + Δ) A : nat OFF : relative_arrival_time_of_job_is_A sched j A
cumul_cond_interference (fun => [eta has_supply sched])
j t1 (t1 + Δ) <=
\sum_(tsko <- ts)
task_request_bound_function tsko (A + 1 ) -
task_cost tsk
move : (OFF _ _ BUSY) => EQA; subst A.Task : TaskType H : TaskCost Task H0 : MaxArrivals Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptable Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (FIFO Job) 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 tsk : Task H_tsk_in_ts : tsk \in ts H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched t1, t2 : instant Δ : nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j BUSY : definitions.busy_interval sched j t1 t2 LT : t1 + Δ < t2 NCOMPL : ~~ completed_by sched j (t1 + Δ) OFF : relative_arrival_time_of_job_is_A sched j
(job_arrival j - t1)
cumul_cond_interference (fun => [eta has_supply sched])
j t1 (t1 + Δ) <=
\sum_(tsko <- ts)
task_request_bound_function tsko
(job_arrival j - t1 + 1 ) - task_cost tsk
have [ZERO|POSj] := posnP (job_cost j).Task : TaskType H : TaskCost Task H0 : MaxArrivals Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptable Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (FIFO Job) 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 tsk : Task H_tsk_in_ts : tsk \in ts H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched t1, t2 : instant Δ : nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j BUSY : definitions.busy_interval sched j t1 t2 LT : t1 + Δ < t2 NCOMPL : ~~ completed_by sched j (t1 + Δ) OFF : relative_arrival_time_of_job_is_A sched j
(job_arrival j - t1) ZERO : job_cost j = 0
cumul_cond_interference (fun => [eta has_supply sched])
j t1 (t1 + Δ) <=
\sum_(tsko <- ts)
task_request_bound_function tsko
(job_arrival j - t1 + 1 ) - task_cost tsk
{ Task : TaskType H : TaskCost Task H0 : MaxArrivals Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptable Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (FIFO Job) 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 tsk : Task H_tsk_in_ts : tsk \in ts H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched t1, t2 : instant Δ : nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j BUSY : definitions.busy_interval sched j t1 t2 LT : t1 + Δ < t2 NCOMPL : ~~ completed_by sched j (t1 + Δ) OFF : relative_arrival_time_of_job_is_A sched j
(job_arrival j - t1) ZERO : job_cost j = 0
cumul_cond_interference (fun => [eta has_supply sched])
j t1 (t1 + Δ) <=
\sum_(tsko <- ts)
task_request_bound_function tsko
(job_arrival j - t1 + 1 ) - task_cost tsk
by exfalso ; rewrite /completed_by ZERO in NCOMPL. } Task : TaskType H : TaskCost Task H0 : MaxArrivals Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptable Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (FIFO Job) 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 tsk : Task H_tsk_in_ts : tsk \in ts H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched t1, t2 : instant Δ : nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j BUSY : definitions.busy_interval sched j t1 t2 LT : t1 + Δ < t2 NCOMPL : ~~ completed_by sched j (t1 + Δ) OFF : relative_arrival_time_of_job_is_A sched j
(job_arrival j - t1) POSj : 0 < job_cost j
cumul_cond_interference (fun => [eta has_supply sched])
j t1 (t1 + Δ) <=
\sum_(tsko <- ts)
task_request_bound_function tsko
(job_arrival j - t1 + 1 ) - task_cost tsk
eapply leq_trans; first by eapply cumulative_intra_interference_split => //.Task : TaskType H : TaskCost Task H0 : MaxArrivals Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptable Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (FIFO Job) 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 tsk : Task H_tsk_in_ts : tsk \in ts H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched t1, t2 : instant Δ : nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j BUSY : definitions.busy_interval sched j t1 t2 LT : t1 + Δ < t2 NCOMPL : ~~ completed_by sched j (t1 + Δ) OFF : relative_arrival_time_of_job_is_A sched j
(job_arrival j - t1) POSj : 0 < job_cost j
cumulative_service_inversion arr_seq sched j t1
(t1 + Δ) +
cumulative_another_hep_job_interference arr_seq sched
j t1 (t1 + Δ) <=
\sum_(tsko <- ts)
task_request_bound_function tsko
(job_arrival j - t1 + 1 ) - task_cost tsk
rewrite -[leqRHS]add0n.Task : TaskType H : TaskCost Task H0 : MaxArrivals Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptable Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (FIFO Job) 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 tsk : Task H_tsk_in_ts : tsk \in ts H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched t1, t2 : instant Δ : nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j BUSY : definitions.busy_interval sched j t1 t2 LT : t1 + Δ < t2 NCOMPL : ~~ completed_by sched j (t1 + Δ) OFF : relative_arrival_time_of_job_is_A sched j
(job_arrival j - t1) POSj : 0 < job_cost j
cumulative_service_inversion arr_seq sched j t1
(t1 + Δ) +
cumulative_another_hep_job_interference arr_seq sched
j t1 (t1 + Δ) <=
0 +
(\sum_(tsko <- ts)
task_request_bound_function tsko
(job_arrival j - t1 + 1 ) - task_cost tsk)
rewrite leq_add//.Task : TaskType H : TaskCost Task H0 : MaxArrivals Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptable Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (FIFO Job) 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 tsk : Task H_tsk_in_ts : tsk \in ts H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched t1, t2 : instant Δ : nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j BUSY : definitions.busy_interval sched j t1 t2 LT : t1 + Δ < t2 NCOMPL : ~~ completed_by sched j (t1 + Δ) OFF : relative_arrival_time_of_job_is_A sched j
(job_arrival j - t1) POSj : 0 < job_cost j
cumulative_service_inversion arr_seq sched j t1
(t1 + Δ) <= 0
{ Task : TaskType H : TaskCost Task H0 : MaxArrivals Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptable Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (FIFO Job) 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 tsk : Task H_tsk_in_ts : tsk \in ts H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched t1, t2 : instant Δ : nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j BUSY : definitions.busy_interval sched j t1 t2 LT : t1 + Δ < t2 NCOMPL : ~~ completed_by sched j (t1 + Δ) OFF : relative_arrival_time_of_job_is_A sched j
(job_arrival j - t1) POSj : 0 < job_cost j
cumulative_service_inversion arr_seq sched j t1
(t1 + Δ) <= 0
rewrite (leqRW (service_inversion_widen _ _ _ _ _ _ _ _ _)).Task : TaskType H : TaskCost Task H0 : MaxArrivals Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptable Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (FIFO Job) 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 tsk : Task H_tsk_in_ts : tsk \in ts H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched t1, t2 : instant Δ : nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j BUSY : definitions.busy_interval sched j t1 t2 LT : t1 + Δ < t2 NCOMPL : ~~ completed_by sched j (t1 + Δ) OFF : relative_arrival_time_of_job_is_A sched j
(job_arrival j - t1) POSj : 0 < job_cost j
cumulative_service_inversion arr_seq sched j ?Goal0
?Goal1 <= 0
- Task : TaskType H : TaskCost Task H0 : MaxArrivals Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptable Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (FIFO Job) 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 tsk : Task H_tsk_in_ts : tsk \in ts H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched t1, t2 : instant Δ : nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j BUSY : definitions.busy_interval sched j t1 t2 LT : t1 + Δ < t2 NCOMPL : ~~ completed_by sched j (t1 + Δ) OFF : relative_arrival_time_of_job_is_A sched j
(job_arrival j - t1) POSj : 0 < job_cost j
cumulative_service_inversion arr_seq sched j ?Goal0
?Goal1 <= 0
apply : FIFO_implies_no_service_inversion => //.Task : TaskType H : TaskCost Task H0 : MaxArrivals Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptable Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (FIFO Job) 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 tsk : Task H_tsk_in_ts : tsk \in ts H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched t1, t2 : instant Δ : nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j BUSY : definitions.busy_interval sched j t1 t2 LT : t1 + Δ < t2 NCOMPL : ~~ completed_by sched j (t1 + Δ) OFF : relative_arrival_time_of_job_is_A sched j
(job_arrival j - t1) POSj : 0 < job_cost j
busy_interval_prefix arr_seq sched j ?Goal0 ?Goal1
apply instantiated_busy_interval_equivalent_busy_interval => //.
- Task : TaskType H : TaskCost Task H0 : MaxArrivals Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptable Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (FIFO Job) 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 tsk : Task H_tsk_in_ts : tsk \in ts H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched t1, t2 : instant Δ : nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j BUSY : definitions.busy_interval sched j t1 t2 LT : t1 + Δ < t2 NCOMPL : ~~ completed_by sched j (t1 + Δ) OFF : relative_arrival_time_of_job_is_A sched j
(job_arrival j - t1) POSj : 0 < job_cost j
t1 <= t1
by done .
- Task : TaskType H : TaskCost Task H0 : MaxArrivals Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptable Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (FIFO Job) 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 tsk : Task H_tsk_in_ts : tsk \in ts H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched t1, t2 : instant Δ : nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j BUSY : definitions.busy_interval sched j t1 t2 LT : t1 + Δ < t2 NCOMPL : ~~ completed_by sched j (t1 + Δ) OFF : relative_arrival_time_of_job_is_A sched j
(job_arrival j - t1) POSj : 0 < job_cost j
t1 + Δ <= t2
by done .
} Task : TaskType H : TaskCost Task H0 : MaxArrivals Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptable Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (FIFO Job) 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 tsk : Task H_tsk_in_ts : tsk \in ts H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched t1, t2 : instant Δ : nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j BUSY : definitions.busy_interval sched j t1 t2 LT : t1 + Δ < t2 NCOMPL : ~~ completed_by sched j (t1 + Δ) OFF : relative_arrival_time_of_job_is_A sched j
(job_arrival j - t1) POSj : 0 < job_cost j
cumulative_another_hep_job_interference arr_seq sched
j t1 (t1 + Δ) <=
\sum_(tsko <- ts)
task_request_bound_function tsko
(job_arrival j - t1 + 1 ) - task_cost tsk
unshelve apply : bound_on_hep_workload; (try apply H_fixed_point).Task : TaskType H : TaskCost Task H0 : MaxArrivals Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptable Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (FIFO Job) 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 tsk : Task H_tsk_in_ts : tsk \in ts H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched t1, t2 : instant Δ : nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j BUSY : definitions.busy_interval sched j t1 t2 LT : t1 + Δ < t2 NCOMPL : ~~ completed_by sched j (t1 + Δ) OFF : relative_arrival_time_of_job_is_A sched j
(job_arrival j - t1) POSj : 0 < job_cost j
instant
all : try apply H_L_positive.Task : TaskType H : TaskCost Task H0 : MaxArrivals Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptable Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (FIFO Job) 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 tsk : Task H_tsk_in_ts : tsk \in ts H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched t1, t2 : instant Δ : nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j BUSY : definitions.busy_interval sched j t1 t2 LT : t1 + Δ < t2 NCOMPL : ~~ completed_by sched j (t1 + Δ) OFF : relative_arrival_time_of_job_is_A sched j
(job_arrival j - t1) POSj : 0 < job_cost j
instant
all : try done .Task : TaskType H : TaskCost Task H0 : MaxArrivals Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptable Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (FIFO Job) 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 tsk : Task H_tsk_in_ts : tsk \in ts H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched t1, t2 : instant Δ : nat j : Job ARR : arrives_in arr_seq j TSK : job_of_task tsk j BUSY : definitions.busy_interval sched j t1 t2 LT : t1 + Δ < t2 NCOMPL : ~~ completed_by sched j (t1 + Δ) OFF : relative_arrival_time_of_job_is_A sched j
(job_arrival j - t1) POSj : 0 < job_cost j
busy_interval arr_seq sched j t1 t2
apply instantiated_busy_interval_equivalent_busy_interval => //.
- Task : TaskType H : TaskCost Task H0 : MaxArrivals Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptable Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (FIFO Job) 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 tsk : Task H_tsk_in_ts : tsk \in ts H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched
forall A : duration,
search_space.is_in_search_space L
(fun A0 Δ : duration =>
Δ - SBF Δ +
(\sum_(tsko <- ts)
task_request_bound_function tsko (A0 + 1 ) -
task_cost tsk)) A ->
exists F : duration,
F <= A + R /\
task_rtct tsk +
(\sum_(tsko <- ts)
task_request_bound_function tsko (A + 1 ) -
task_cost tsk) <= SBF F /\
SBF F + (task_cost tsk - task_rtct tsk) <=
SBF (A + R)
apply : soln_abstract_response_time_recurrence => //.Task : TaskType H : TaskCost Task H0 : MaxArrivals Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptable Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (FIFO Job) 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 tsk : Task H_tsk_in_ts : tsk \in ts H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js TSKs : job_of_task tsk js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched
0 < task_cost tsk
move : TSKs => /eqP <-.Task : TaskType H : TaskCost Task H0 : MaxArrivals Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptable Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (FIFO Job) 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 tsk : Task H_tsk_in_ts : tsk \in ts H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched
0 < task_cost (job_task js)
apply : leq_trans; first by apply POS.Task : TaskType H : TaskCost Task H0 : MaxArrivals Task H1 : TaskRunToCompletionThreshold Task H2 : TaskMaxNonpreemptiveSegment Task Job : JobType H3 : JobTask Job Task H4 : JobCost Job H5 : JobArrival Job H6 : JobPreemptable Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq H_valid_job_cost : arrivals_have_valid_job_costs
arr_seq ts : seq Task H_all_jobs_from_taskset : all_jobs_from_taskset
arr_seq ts H_is_arrival_curve : taskset_respects_max_arrivals
arr_seq ts H_valid_arrival_curve : valid_taskset_arrival_curve ts
max_arrivals sched : schedule (rs_processor_state Job) H_valid_schedule : valid_schedule sched arr_seq H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_JLFP_policy_at_preemption_point
arr_seq sched (FIFO Job) 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 tsk : Task H_tsk_in_ts : tsk \in ts H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk SBF : SupplyBoundFunction H_SBF_monotone : sbf_is_monotone SBF H_unit_SBF : unit_supply_bound_function SBF H_valid_SBF : valid_busy_sbf arr_seq sched tsk SBF L : duration H_L_positive : 0 < LH_fixed_point : total_request_bound_function ts L <=
SBF L R : duration SOL : rta_recurrence_solution R js : Job ARRs : arrives_in arr_seq js POS : 0 < job_cost jsREAD : work_bearing_readiness arr_seq sched
job_cost js <= task_cost (job_task js)
by apply H_valid_job_cost.
Qed .
End RTAforFullyPreemptiveFIFOModelwithArrivalCurves .