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Require Export prosa.results.fixed_priority.rta.bounded_nps.[Loading ML file ssrmatching_plugin.cmxs (using legacy method) ... done ] [Loading ML file ssreflect_plugin.cmxs (using legacy method) ... done ] [Loading ML file ring_plugin.cmxs (using legacy method) ... done ] Serlib plugin: coq-elpi.elpi is not available: serlib support is missing.
Incremental checking for commands in this plugin will be impacted. [Loading ML file coq-elpi.elpi ... done ] [Loading ML file zify_plugin.cmxs (using legacy method) ... done ] [Loading ML file micromega_core_plugin.cmxs (using legacy method) ... done ] [Loading ML file micromega_plugin.cmxs (using legacy method) ... done ] [Loading ML file btauto_plugin.cmxs (using legacy method) ... done ] Notation "_ + _" was already used in scope nat_scope.
[notation-overridden,parsing,default]Notation "_ - _" was already used in scope nat_scope.
[notation-overridden,parsing,default]Notation "_ <= _" was already used in scope nat_scope.
[notation-overridden,parsing,default]Notation "_ < _" was already used in scope nat_scope.
[notation-overridden,parsing,default]Notation "_ >= _" was already used in scope nat_scope.
[notation-overridden,parsing,default]Notation "_ > _" was already used in scope nat_scope.
[notation-overridden,parsing,default]Notation "_ <= _ <= _" was already used in scope
nat_scope. [notation-overridden,parsing,default]Notation "_ < _ <= _" was already used in scope
nat_scope. [notation-overridden,parsing,default]Notation "_ <= _ < _" was already used in scope
nat_scope. [notation-overridden,parsing,default]Notation "_ < _ < _" was already used in scope
nat_scope. [notation-overridden,parsing,default]Notation "_ * _" was already used in scope nat_scope.
[notation-overridden,parsing,default]
Require Export prosa.analysis.facts.preemption.task.preemptive.
Require Export prosa.analysis.facts.preemption.rtc_threshold.preemptive.
Require Export prosa.analysis.facts.readiness.sequential.
Require Export prosa.model.task.preemption.fully_preemptive.
(** * RTA for Fully Preemptive FP Model *)
(** In this section we prove the RTA theorem for the fully preemptive FP model *)
(** ** Setup and Assumptions *)
Section RTAforFullyPreemptiveFPModelwithArrivalCurves .
(** We assume ideal uni-processor schedules. *)
#[local] Existing Instance ideal .processor_state.
(** 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}.
(** We assume that jobs and tasks are fully preemptive. *)
#[local] Existing Instance fully_preemptive_job_model .
#[local] Existing Instance fully_preemptive_task_model .
#[local] Existing Instance fully_preemptive_rtc_threshold .
(** Consider any arrival sequence with consistent, non-duplicate arrivals. *)
Variable arr_seq : arrival_sequence Job.
Hypothesis H_valid_arrival_sequence : valid_arrival_sequence arr_seq.
(** 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 the cost of a job cannot be larger than the task cost. *)
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.
(** Recall that we assume sequential readiness. *)
#[local] Instance sequential_readiness : JobReady _ _ :=
sequential_ready_instance arr_seq.
(** Next, consider any ideal uniprocessor schedule of this arrival sequence ... *)
Variable sched : schedule (ideal.processor_state Job).
Hypothesis H_sched_valid : valid_schedule sched arr_seq.
Hypothesis H_jobs_come_from_arrival_sequence :
jobs_come_from_arrival_sequence sched arr_seq.
(** 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.
(** Next, we assume that the schedule is a work-conserving schedule... *)
Hypothesis H_work_conserving : work_conserving arr_seq sched.
(** ... and the schedule respects the scheduling policy. *)
Hypothesis H_respects_policy : respects_FP_policy_at_preemption_point arr_seq sched FP.
(** ** Total Workload and Length of Busy Interval *)
(** We introduce the abbreviation [rbf] for the task request bound function,
which is defined as [task_cost(T) × max_arrivals(T,Δ)] for a task T. *)
Let rbf := task_request_bound_function.
(** Next, we introduce [task_rbf] as an abbreviation
for the task request bound function of task [tsk]. *)
Let task_rbf := rbf tsk.
(** Using the sum of individual request bound functions, we define
the request bound function of all tasks with higher priority
... *)
Let total_hep_rbf := total_hep_request_bound_function_FP ts tsk.
(** ... and the request bound function of all tasks with higher
priority other than task [tsk]. *)
Let total_ohep_rbf := total_ohep_request_bound_function_FP ts tsk.
(** Let L be any positive fixed point of the busy interval recurrence, determined by
the sum of blocking and higher-or-equal-priority workload. *)
Variable L : duration.
Hypothesis H_L_positive : L > 0 .
Hypothesis H_fixed_point : L = total_hep_rbf L.
(** ** Response-Time Bound *)
(** To reduce the time complexity of the analysis, recall the notion of search space. *)
Let is_in_search_space := is_in_search_space tsk L.
(** Next, consider any value [R], and assume that for any given
arrival [A] from search space there is a solution of the
response-time bound recurrence which is bounded by [R]. *)
Variable R : duration.
Hypothesis H_R_is_maximum :
forall (A : duration),
is_in_search_space A ->
exists (F : duration),
A + F >= task_rbf (A + ε) + total_ohep_rbf (A + F) /\
R >= F.
(** Now, we can leverage the results for the abstract model with
bounded non-preemptive segments to establish a response-time
bound for the more concrete model of fully preemptive
scheduling. *)
Let response_time_bounded_by := task_response_time_bound arr_seq sched.
Theorem uniprocessor_response_time_bound_fully_preemptive_fp :
response_time_bounded_by tsk R.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H3 : 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 sched : schedule (ideal.processor_state Job) H_sched_valid : valid_schedule sched arr_seq H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : L = total_hep_rbf L is_in_search_space := bounded_pi.is_in_search_space tsk
L : nat -> bool R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
task_rbf (A + 1 ) +
total_ohep_rbf (A + F) <=
A + F /\
F <= Rresponse_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop
response_time_bounded_by tsk R
Proof .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H3 : 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 sched : schedule (ideal.processor_state Job) H_sched_valid : valid_schedule sched arr_seq H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : L = total_hep_rbf L is_in_search_space := bounded_pi.is_in_search_space tsk
L : nat -> bool R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
task_rbf (A + 1 ) +
total_ohep_rbf (A + F) <=
A + F /\
F <= Rresponse_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop
response_time_bounded_by tsk R
have BLOCK: blocking_bound ts tsk = 0 .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H3 : 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 sched : schedule (ideal.processor_state Job) H_sched_valid : valid_schedule sched arr_seq H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : L = total_hep_rbf L is_in_search_space := bounded_pi.is_in_search_space tsk
L : nat -> bool R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
task_rbf (A + 1 ) +
total_ohep_rbf (A + F) <=
A + F /\
F <= Rresponse_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop
blocking_bound ts tsk = 0
{ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H3 : 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 sched : schedule (ideal.processor_state Job) H_sched_valid : valid_schedule sched arr_seq H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : L = total_hep_rbf L is_in_search_space := bounded_pi.is_in_search_space tsk
L : nat -> bool R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
task_rbf (A + 1 ) +
total_ohep_rbf (A + F) <=
A + F /\
F <= Rresponse_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop
blocking_bound ts tsk = 0
by rewrite /blocking_bound /parameters.task_max_nonpreemptive_segment
/fully_preemptive_task_model subnn big1_eq. } Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H3 : 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 sched : schedule (ideal.processor_state Job) H_sched_valid : valid_schedule sched arr_seq H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : L = total_hep_rbf L is_in_search_space := bounded_pi.is_in_search_space tsk
L : nat -> bool R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
task_rbf (A + 1 ) +
total_ohep_rbf (A + F) <=
A + F /\
F <= Rresponse_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop BLOCK : blocking_bound ts tsk = 0
response_time_bounded_by tsk R
eapply uniprocessor_response_time_bound_fp_with_bounded_nonpreemptive_segments with (L:=L) => //
.Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H3 : 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 sched : schedule (ideal.processor_state Job) H_sched_valid : valid_schedule sched arr_seq H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : L = total_hep_rbf L is_in_search_space := bounded_pi.is_in_search_space tsk
L : nat -> bool R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
task_rbf (A + 1 ) +
total_ohep_rbf (A + F) <=
A + F /\
F <= Rresponse_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop BLOCK : blocking_bound ts tsk = 0
work_bearing_readiness arr_seq sched
- Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H3 : 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 sched : schedule (ideal.processor_state Job) H_sched_valid : valid_schedule sched arr_seq H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : L = total_hep_rbf L is_in_search_space := bounded_pi.is_in_search_space tsk
L : nat -> bool R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
task_rbf (A + 1 ) +
total_ohep_rbf (A + F) <=
A + F /\
F <= Rresponse_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop BLOCK : blocking_bound ts tsk = 0
work_bearing_readiness arr_seq sched
exact : sequential_readiness_implies_work_bearing_readiness.
- Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H3 : 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 sched : schedule (ideal.processor_state Job) H_sched_valid : valid_schedule sched arr_seq H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : L = total_hep_rbf L is_in_search_space := bounded_pi.is_in_search_space tsk
L : nat -> bool R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
task_rbf (A + 1 ) +
total_ohep_rbf (A + F) <=
A + F /\
F <= Rresponse_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop BLOCK : blocking_bound ts tsk = 0
sequential_tasks arr_seq sched
exact : sequential_readiness_implies_sequential_tasks.
- Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H3 : 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 sched : schedule (ideal.processor_state Job) H_sched_valid : valid_schedule sched arr_seq H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : L = total_hep_rbf L is_in_search_space := bounded_pi.is_in_search_space tsk
L : nat -> bool R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
task_rbf (A + 1 ) +
total_ohep_rbf (A + F) <=
A + F /\
F <= Rresponse_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop BLOCK : blocking_bound ts tsk = 0
L =
blocking_bound ts tsk +
total_hep_request_bound_function_FP ts tsk L
by rewrite BLOCK add0n.
- Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H3 : 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 sched : schedule (ideal.processor_state Job) H_sched_valid : valid_schedule sched arr_seq H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : L = total_hep_rbf L is_in_search_space := bounded_pi.is_in_search_space tsk
L : nat -> bool R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
task_rbf (A + 1 ) +
total_ohep_rbf (A + F) <=
A + F /\
F <= Rresponse_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop BLOCK : blocking_bound ts tsk = 0
forall A : duration,
bounded_pi.is_in_search_space tsk L A ->
exists F : duration,
blocking_bound ts tsk +
(task_request_bound_function tsk (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_request_bound_function_FP ts tsk (A + F) <=
A + F /\ F + (task_cost tsk - task_rtct tsk) <= R
move => A /andP [LT NEQ].Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H3 : 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 sched : schedule (ideal.processor_state Job) H_sched_valid : valid_schedule sched arr_seq H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : L = total_hep_rbf L is_in_search_space := bounded_pi.is_in_search_space tsk
L : nat -> bool R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
task_rbf (A + 1 ) +
total_ohep_rbf (A + F) <=
A + F /\
F <= Rresponse_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop BLOCK : blocking_bound ts tsk = 0 A : duration LT : A < L NEQ : task_request_bound_function tsk A
!= task_request_bound_function tsk (A + 1 )
exists F : duration,
blocking_bound ts tsk +
(task_request_bound_function tsk (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_request_bound_function_FP ts tsk (A + F) <=
A + F /\ F + (task_cost tsk - task_rtct tsk) <= R
edestruct H_R_is_maximum as [F [FIX BOUND]].Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H3 : 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 sched : schedule (ideal.processor_state Job) H_sched_valid : valid_schedule sched arr_seq H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : L = total_hep_rbf L is_in_search_space := bounded_pi.is_in_search_space tsk
L : nat -> bool R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
task_rbf (A + 1 ) +
total_ohep_rbf (A + F) <=
A + F /\
F <= Rresponse_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop BLOCK : blocking_bound ts tsk = 0 A : duration LT : A < L NEQ : task_request_bound_function tsk A
!= task_request_bound_function tsk (A + 1 )
is_in_search_space ?A
{ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H3 : 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 sched : schedule (ideal.processor_state Job) H_sched_valid : valid_schedule sched arr_seq H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : L = total_hep_rbf L is_in_search_space := bounded_pi.is_in_search_space tsk
L : nat -> bool R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
task_rbf (A + 1 ) +
total_ohep_rbf (A + F) <=
A + F /\
F <= Rresponse_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop BLOCK : blocking_bound ts tsk = 0 A : duration LT : A < L NEQ : task_request_bound_function tsk A
!= task_request_bound_function tsk (A + 1 )
is_in_search_space ?A
by apply /andP; split ; eauto 2 . } Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H3 : 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 sched : schedule (ideal.processor_state Job) H_sched_valid : valid_schedule sched arr_seq H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : L = total_hep_rbf L is_in_search_space := bounded_pi.is_in_search_space tsk
L : nat -> bool R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
task_rbf (A + 1 ) +
total_ohep_rbf (A + F) <=
A + F /\
F <= Rresponse_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop BLOCK : blocking_bound ts tsk = 0 A : duration LT : A < L NEQ : task_request_bound_function tsk A
!= task_request_bound_function tsk (A + 1 ) F : duration FIX : task_rbf (A + 1 ) + total_ohep_rbf (A + F) <=
A + F BOUND : F <= R
exists F : duration,
blocking_bound ts tsk +
(task_request_bound_function tsk (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_request_bound_function_FP ts tsk (A + F) <=
A + F /\ F + (task_cost tsk - task_rtct tsk) <= R
exists F ; split .Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H3 : 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 sched : schedule (ideal.processor_state Job) H_sched_valid : valid_schedule sched arr_seq H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : L = total_hep_rbf L is_in_search_space := bounded_pi.is_in_search_space tsk
L : nat -> bool R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
task_rbf (A + 1 ) +
total_ohep_rbf (A + F) <=
A + F /\
F <= Rresponse_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop BLOCK : blocking_bound ts tsk = 0 A : duration LT : A < L NEQ : task_request_bound_function tsk A
!= task_request_bound_function tsk (A + 1 ) F : duration FIX : task_rbf (A + 1 ) + total_ohep_rbf (A + F) <=
A + F BOUND : F <= R
blocking_bound ts tsk +
(task_request_bound_function tsk (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_request_bound_function_FP ts tsk (A + F) <=
A + F
+ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H3 : 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 sched : schedule (ideal.processor_state Job) H_sched_valid : valid_schedule sched arr_seq H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : L = total_hep_rbf L is_in_search_space := bounded_pi.is_in_search_space tsk
L : nat -> bool R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
task_rbf (A + 1 ) +
total_ohep_rbf (A + F) <=
A + F /\
F <= Rresponse_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop BLOCK : blocking_bound ts tsk = 0 A : duration LT : A < L NEQ : task_request_bound_function tsk A
!= task_request_bound_function tsk (A + 1 ) F : duration FIX : task_rbf (A + 1 ) + total_ohep_rbf (A + F) <=
A + F BOUND : F <= R
blocking_bound ts tsk +
(task_request_bound_function tsk (A + 1 ) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_request_bound_function_FP ts tsk (A + F) <=
A + F
by rewrite BLOCK add0n subnn subn0.
+ Task : TaskType H : TaskCost Task Job : JobType H0 : JobTask Job Task H1 : JobArrival Job H2 : JobCost Job arr_seq : arrival_sequence Job H_valid_arrival_sequence : valid_arrival_sequence
arr_seq 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 H3 : 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 sched : schedule (ideal.processor_state Job) H_sched_valid : valid_schedule sched arr_seq H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence
sched arr_seq FP : FP_policy Task H_priority_is_reflexive : reflexive_task_priorities FP H_priority_is_transitive : transitive_task_priorities
FP H_work_conserving : work_conserving arr_seq sched H_respects_policy : respects_FP_policy_at_preemption_point
arr_seq sched FP rbf := task_request_bound_function : Task -> duration -> nat task_rbf := rbf tsk : duration -> nat total_hep_rbf := total_hep_request_bound_function_FP ts
tsk : duration -> nat total_ohep_rbf := total_ohep_request_bound_function_FP
ts tsk : duration -> nat L : duration H_L_positive : 0 < LH_fixed_point : L = total_hep_rbf L is_in_search_space := bounded_pi.is_in_search_space tsk
L : nat -> bool R : duration H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
task_rbf (A + 1 ) +
total_ohep_rbf (A + F) <=
A + F /\
F <= Rresponse_time_bounded_by := task_response_time_bound
arr_seq sched : Task -> duration -> Prop BLOCK : blocking_bound ts tsk = 0 A : duration LT : A < L NEQ : task_request_bound_function tsk A
!= task_request_bound_function tsk (A + 1 ) F : duration FIX : task_rbf (A + 1 ) + total_ohep_rbf (A + F) <=
A + F BOUND : F <= R
F + (task_cost tsk - task_rtct tsk) <= R
by rewrite subnn addn0.
Qed .
End RTAforFullyPreemptiveFPModelwithArrivalCurves .