Library prosa.results.fixed_priority.rta.bounded_nps
(* ----------------------------------[ coqtop ]---------------------------------
Welcome to Coq 8.11.2 (June 2020)
----------------------------------------------------------------------------- *)
Require Export prosa.analysis.definitions.schedulability.
Require Export prosa.analysis.definitions.request_bound_function.
Require Export prosa.analysis.facts.model.sequential.
Require Export prosa.analysis.facts.busy_interval.priority_inversion.
Require Export prosa.results.fixed_priority.rta.bounded_pi.
Throughout this file, we assume ideal uni-processor schedules.
Throughout this file, we assume the basic (i.e., Liu & Layland) readiness model.
RTA for FP-schedulers with Bounded Non-Preemptive Segments
Consider any type of tasks ...
Context {Task : TaskType}.
Context `{TaskCost Task}.
Context `{TaskRunToCompletionThreshold Task}.
Context `{TaskMaxNonpreemptiveSegment Task}.
Context `{TaskCost Task}.
Context `{TaskRunToCompletionThreshold Task}.
Context `{TaskMaxNonpreemptiveSegment Task}.
... and any type of jobs associated with these tasks.
Context {Job : JobType}.
Context `{JobTask Job Task}.
Context `{JobArrival Job}.
Context `{JobCost Job}.
Context `{JobTask Job Task}.
Context `{JobArrival Job}.
Context `{JobCost Job}.
Consider any arrival sequence with consistent, non-duplicate arrivals.
Variable arr_seq : arrival_sequence Job.
Hypothesis H_arrival_times_are_consistent : consistent_arrival_times arr_seq.
Hypothesis H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq.
Hypothesis H_arrival_times_are_consistent : consistent_arrival_times arr_seq.
Hypothesis H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq.
Next, consider any ideal uni-processor schedule of this arrival sequence ...
Variable sched : schedule (ideal.processor_state Job).
Hypothesis H_jobs_come_from_arrival_sequence:
jobs_come_from_arrival_sequence sched arr_seq.
Hypothesis H_jobs_come_from_arrival_sequence:
jobs_come_from_arrival_sequence sched arr_seq.
... where jobs do not execute before their arrival or after completion.
Hypothesis H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched.
Hypothesis H_completed_jobs_dont_execute : completed_jobs_dont_execute sched.
Hypothesis H_completed_jobs_dont_execute : completed_jobs_dont_execute sched.
In addition, we assume the existence of a function mapping jobs
to theirs preemption points ...
... and assume that it defines a valid preemption
model with bounded non-preemptive segments.
Hypothesis H_valid_model_with_bounded_nonpreemptive_segments:
valid_model_with_bounded_nonpreemptive_segments arr_seq sched.
valid_model_with_bounded_nonpreemptive_segments arr_seq sched.
Consider an FP policy that indicates a higher-or-equal priority
relation, and assume that the relation is reflexive and
transitive.
Context `{FP_policy Task}.
Hypothesis H_priority_is_reflexive : reflexive_priorities.
Hypothesis H_priority_is_transitive : transitive_priorities.
Hypothesis H_priority_is_reflexive : reflexive_priorities.
Hypothesis H_priority_is_transitive : transitive_priorities.
Assume we have sequential tasks, i.e, jobs from the same task
execute in the order of their arrival.
Next, we assume that the schedule is a work-conserving schedule...
... and the schedule respects the policy defined by the [job_preemptable]
function (i.e., jobs have bounded non-preemptive segments).
Consider an arbitrary task set ts, ...
... assume that all jobs come from the task set, ...
... and the cost of a job cannot be larger than the task cost.
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.
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.
Consider a valid preemption model...
...and a valid task run-to-completion threshold function. That is,
[task_run_to_completion_threshold tsk] is (1) no bigger than [tsk]'s
cost, (2) for any job of task [tsk] job_run_to_completion_threshold
is bounded by task_run_to_completion_threshold.
Let's define some local names for clarity.
Let max_length_of_priority_inversion :=
max_length_of_priority_inversion arr_seq.
Let task_rbf := task_request_bound_function tsk.
Let total_hep_rbf := total_hep_request_bound_function_FP ts tsk.
Let total_ohep_rbf := total_ohep_request_bound_function_FP ts tsk.
Let response_time_bounded_by := task_response_time_bound arr_seq sched.
max_length_of_priority_inversion arr_seq.
Let task_rbf := task_request_bound_function tsk.
Let total_hep_rbf := total_hep_request_bound_function_FP ts tsk.
Let total_ohep_rbf := total_ohep_request_bound_function_FP ts tsk.
Let response_time_bounded_by := task_response_time_bound arr_seq sched.
We also define a bound for the priority inversion caused by jobs with lower priority.
Definition blocking_bound :=
\max_(tsk_other <- ts | ~~ hep_task tsk_other tsk)
(task_max_nonpreemptive_segment tsk_other - ε).
\max_(tsk_other <- ts | ~~ hep_task tsk_other tsk)
(task_max_nonpreemptive_segment tsk_other - ε).
Priority inversion is bounded
In this section, we prove that a priority inversion for task [tsk] is bounded by the maximum length of non-preemptive segments among the tasks with lower priority.
First, we prove that the maximum length of a priority inversion of a job j is
bounded by the maximum length of a non-preemptive section of a task with
lower-priority task (i.e., the blocking term).
Lemma priority_inversion_is_bounded_by_blocking:
∀ j t,
arrives_in arr_seq j →
job_task j = tsk →
max_length_of_priority_inversion j t ≤ blocking_bound.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1947)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
============================
forall (j : Job) (t : instant),
arrives_in arr_seq j ->
job_task j = tsk -> max_length_of_priority_inversion j t <= blocking_bound
----------------------------------------------------------------------------- *)
Proof.
intros j t ARR TSK.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1951)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : instant
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
============================
max_length_of_priority_inversion j t <= blocking_bound
----------------------------------------------------------------------------- *)
rewrite /max_length_of_priority_inversion /blocking_bound /FP_to_JLFP
/priority_inversion.max_length_of_priority_inversion.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1959)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : instant
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
============================
\max_(j_lp <- arrivals_before arr_seq t | ~~ hep_job j_lp j)
(job_max_nonpreemptive_segment j_lp - ε) <=
\max_(tsk_other <- ts | ~~ hep_task tsk_other tsk)
(task_max_nonpreemptive_segment tsk_other - ε)
----------------------------------------------------------------------------- *)
apply leq_trans with
(\max_(j_lp <- arrivals_between arr_seq 0 t
| ~~ hep_task (job_task j_lp) tsk)
(task_max_nonpreemptive_segment (job_task j_lp) - ε)).
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 1976)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : instant
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
============================
\max_(j_lp <- arrivals_before arr_seq t | ~~ hep_job j_lp j)
(job_max_nonpreemptive_segment j_lp - ε) <=
\max_(j_lp <- arrivals_between arr_seq 0 t | ~~
hep_task (job_task j_lp) tsk)
(task_max_nonpreemptive_segment (job_task j_lp) - ε)
subgoal 2 (ID 1977) is:
\max_(j_lp <- arrivals_between arr_seq 0 t | ~~ hep_task (job_task j_lp) tsk)
(task_max_nonpreemptive_segment (job_task j_lp) - ε) <=
\max_(tsk_other <- ts | ~~ hep_task tsk_other tsk)
(task_max_nonpreemptive_segment tsk_other - ε)
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1976)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : instant
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
============================
\max_(j_lp <- arrivals_before arr_seq t | ~~ hep_job j_lp j)
(job_max_nonpreemptive_segment j_lp - ε) <=
\max_(j_lp <- arrivals_between arr_seq 0 t | ~~
hep_task (job_task j_lp) tsk)
(task_max_nonpreemptive_segment (job_task j_lp) - ε)
----------------------------------------------------------------------------- *)
rewrite /hep_job TSK.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1984)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : instant
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
============================
\max_(j_lp <- arrivals_before arr_seq t | ~~ hep_task (job_task j_lp) tsk)
(job_max_nonpreemptive_segment j_lp - ε) <=
\max_(j_lp <- arrivals_between arr_seq 0 t | ~~
hep_task (job_task j_lp) tsk)
(task_max_nonpreemptive_segment (job_task j_lp) - ε)
----------------------------------------------------------------------------- *)
apply leq_big_max.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1985)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : instant
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
============================
forall i : Job,
i \in arrivals_between arr_seq 0 t ->
~~ hep_task (job_task i) tsk ->
job_max_nonpreemptive_segment i - ε <=
task_max_nonpreemptive_segment (job_task i) - ε
----------------------------------------------------------------------------- *)
intros j' JINB NOTHEP.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1988)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : instant
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NOTHEP : ~~ hep_task (job_task j') tsk
============================
job_max_nonpreemptive_segment j' - ε <=
task_max_nonpreemptive_segment (job_task j') - ε
----------------------------------------------------------------------------- *)
rewrite leq_sub2r //.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1995)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : instant
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NOTHEP : ~~ hep_task (job_task j') tsk
============================
job_max_nonpreemptive_segment j' <=
task_max_nonpreemptive_segment (job_task j')
----------------------------------------------------------------------------- *)
apply H_valid_model_with_bounded_nonpreemptive_segments.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2028)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : instant
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NOTHEP : ~~ hep_task (job_task j') tsk
============================
arrives_in arr_seq j'
----------------------------------------------------------------------------- *)
by eapply in_arrivals_implies_arrived; eauto 2.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1977)
subgoal 1 (ID 1977) is:
\max_(j_lp <- arrivals_between arr_seq 0 t | ~~ hep_task (job_task j_lp) tsk)
(task_max_nonpreemptive_segment (job_task j_lp) - ε) <=
\max_(tsk_other <- ts | ~~ hep_task tsk_other tsk)
(task_max_nonpreemptive_segment tsk_other - ε)
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1977)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : instant
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
============================
\max_(j_lp <- arrivals_between arr_seq 0 t | ~~
hep_task (job_task j_lp) tsk)
(task_max_nonpreemptive_segment (job_task j_lp) - ε) <=
\max_(tsk_other <- ts | ~~ hep_task tsk_other tsk)
(task_max_nonpreemptive_segment tsk_other - ε)
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1977)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : instant
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
============================
\max_(j_lp <- arrivals_between arr_seq 0 t | ~~
hep_task (job_task j_lp) tsk)
(task_max_nonpreemptive_segment (job_task j_lp) - ε) <=
\max_(tsk_other <- ts | ~~ hep_task tsk_other tsk)
(task_max_nonpreemptive_segment tsk_other - ε)
----------------------------------------------------------------------------- *)
apply /bigmax_leq_seqP.
(* ----------------------------------[ coqtop ]---------------------------------
1 focused subgoal
(shelved: 1) (ID 2065)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : instant
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
============================
forall i : Job,
i \in arrivals_between arr_seq 0 t ->
~~ hep_task (job_task i) tsk ->
task_max_nonpreemptive_segment (job_task i) - ε <=
\max_(tsk_other <- ts | ~~ hep_task tsk_other tsk)
(task_max_nonpreemptive_segment tsk_other - ε)
----------------------------------------------------------------------------- *)
intros j' JINB NOTHEP.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2068)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : instant
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NOTHEP : ~~ hep_task (job_task j') tsk
============================
task_max_nonpreemptive_segment (job_task j') - ε <=
\max_(tsk_other <- ts | ~~ hep_task tsk_other tsk)
(task_max_nonpreemptive_segment tsk_other - ε)
----------------------------------------------------------------------------- *)
apply leq_bigmax_cond_seq with
(i0 := (job_task j')) (F := fun tsk ⇒ task_max_nonpreemptive_segment tsk - 1); last by done.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2077)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : instant
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NOTHEP : ~~ hep_task (job_task j') tsk
============================
job_task j' \in ts
----------------------------------------------------------------------------- *)
apply H_all_jobs_from_taskset.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2079)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : instant
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NOTHEP : ~~ hep_task (job_task j') tsk
============================
arrives_in arr_seq j'
----------------------------------------------------------------------------- *)
apply mem_bigcat_nat_exists in JINB.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2080)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : instant
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
j' : Job
JINB : exists i : nat, j' \in arrivals_at arr_seq i /\ 0 <= i < t
NOTHEP : ~~ hep_task (job_task j') tsk
============================
arrives_in arr_seq j'
----------------------------------------------------------------------------- *)
by inversion JINB as [ta' [JIN' _]]; ∃ ta'.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
∀ j t,
arrives_in arr_seq j →
job_task j = tsk →
max_length_of_priority_inversion j t ≤ blocking_bound.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1947)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
============================
forall (j : Job) (t : instant),
arrives_in arr_seq j ->
job_task j = tsk -> max_length_of_priority_inversion j t <= blocking_bound
----------------------------------------------------------------------------- *)
Proof.
intros j t ARR TSK.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1951)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : instant
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
============================
max_length_of_priority_inversion j t <= blocking_bound
----------------------------------------------------------------------------- *)
rewrite /max_length_of_priority_inversion /blocking_bound /FP_to_JLFP
/priority_inversion.max_length_of_priority_inversion.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1959)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : instant
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
============================
\max_(j_lp <- arrivals_before arr_seq t | ~~ hep_job j_lp j)
(job_max_nonpreemptive_segment j_lp - ε) <=
\max_(tsk_other <- ts | ~~ hep_task tsk_other tsk)
(task_max_nonpreemptive_segment tsk_other - ε)
----------------------------------------------------------------------------- *)
apply leq_trans with
(\max_(j_lp <- arrivals_between arr_seq 0 t
| ~~ hep_task (job_task j_lp) tsk)
(task_max_nonpreemptive_segment (job_task j_lp) - ε)).
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 1976)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : instant
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
============================
\max_(j_lp <- arrivals_before arr_seq t | ~~ hep_job j_lp j)
(job_max_nonpreemptive_segment j_lp - ε) <=
\max_(j_lp <- arrivals_between arr_seq 0 t | ~~
hep_task (job_task j_lp) tsk)
(task_max_nonpreemptive_segment (job_task j_lp) - ε)
subgoal 2 (ID 1977) is:
\max_(j_lp <- arrivals_between arr_seq 0 t | ~~ hep_task (job_task j_lp) tsk)
(task_max_nonpreemptive_segment (job_task j_lp) - ε) <=
\max_(tsk_other <- ts | ~~ hep_task tsk_other tsk)
(task_max_nonpreemptive_segment tsk_other - ε)
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1976)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : instant
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
============================
\max_(j_lp <- arrivals_before arr_seq t | ~~ hep_job j_lp j)
(job_max_nonpreemptive_segment j_lp - ε) <=
\max_(j_lp <- arrivals_between arr_seq 0 t | ~~
hep_task (job_task j_lp) tsk)
(task_max_nonpreemptive_segment (job_task j_lp) - ε)
----------------------------------------------------------------------------- *)
rewrite /hep_job TSK.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1984)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : instant
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
============================
\max_(j_lp <- arrivals_before arr_seq t | ~~ hep_task (job_task j_lp) tsk)
(job_max_nonpreemptive_segment j_lp - ε) <=
\max_(j_lp <- arrivals_between arr_seq 0 t | ~~
hep_task (job_task j_lp) tsk)
(task_max_nonpreemptive_segment (job_task j_lp) - ε)
----------------------------------------------------------------------------- *)
apply leq_big_max.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1985)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : instant
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
============================
forall i : Job,
i \in arrivals_between arr_seq 0 t ->
~~ hep_task (job_task i) tsk ->
job_max_nonpreemptive_segment i - ε <=
task_max_nonpreemptive_segment (job_task i) - ε
----------------------------------------------------------------------------- *)
intros j' JINB NOTHEP.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1988)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : instant
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NOTHEP : ~~ hep_task (job_task j') tsk
============================
job_max_nonpreemptive_segment j' - ε <=
task_max_nonpreemptive_segment (job_task j') - ε
----------------------------------------------------------------------------- *)
rewrite leq_sub2r //.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1995)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : instant
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NOTHEP : ~~ hep_task (job_task j') tsk
============================
job_max_nonpreemptive_segment j' <=
task_max_nonpreemptive_segment (job_task j')
----------------------------------------------------------------------------- *)
apply H_valid_model_with_bounded_nonpreemptive_segments.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2028)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : instant
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NOTHEP : ~~ hep_task (job_task j') tsk
============================
arrives_in arr_seq j'
----------------------------------------------------------------------------- *)
by eapply in_arrivals_implies_arrived; eauto 2.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1977)
subgoal 1 (ID 1977) is:
\max_(j_lp <- arrivals_between arr_seq 0 t | ~~ hep_task (job_task j_lp) tsk)
(task_max_nonpreemptive_segment (job_task j_lp) - ε) <=
\max_(tsk_other <- ts | ~~ hep_task tsk_other tsk)
(task_max_nonpreemptive_segment tsk_other - ε)
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1977)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : instant
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
============================
\max_(j_lp <- arrivals_between arr_seq 0 t | ~~
hep_task (job_task j_lp) tsk)
(task_max_nonpreemptive_segment (job_task j_lp) - ε) <=
\max_(tsk_other <- ts | ~~ hep_task tsk_other tsk)
(task_max_nonpreemptive_segment tsk_other - ε)
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1977)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : instant
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
============================
\max_(j_lp <- arrivals_between arr_seq 0 t | ~~
hep_task (job_task j_lp) tsk)
(task_max_nonpreemptive_segment (job_task j_lp) - ε) <=
\max_(tsk_other <- ts | ~~ hep_task tsk_other tsk)
(task_max_nonpreemptive_segment tsk_other - ε)
----------------------------------------------------------------------------- *)
apply /bigmax_leq_seqP.
(* ----------------------------------[ coqtop ]---------------------------------
1 focused subgoal
(shelved: 1) (ID 2065)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : instant
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
============================
forall i : Job,
i \in arrivals_between arr_seq 0 t ->
~~ hep_task (job_task i) tsk ->
task_max_nonpreemptive_segment (job_task i) - ε <=
\max_(tsk_other <- ts | ~~ hep_task tsk_other tsk)
(task_max_nonpreemptive_segment tsk_other - ε)
----------------------------------------------------------------------------- *)
intros j' JINB NOTHEP.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2068)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : instant
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NOTHEP : ~~ hep_task (job_task j') tsk
============================
task_max_nonpreemptive_segment (job_task j') - ε <=
\max_(tsk_other <- ts | ~~ hep_task tsk_other tsk)
(task_max_nonpreemptive_segment tsk_other - ε)
----------------------------------------------------------------------------- *)
apply leq_bigmax_cond_seq with
(i0 := (job_task j')) (F := fun tsk ⇒ task_max_nonpreemptive_segment tsk - 1); last by done.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2077)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : instant
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NOTHEP : ~~ hep_task (job_task j') tsk
============================
job_task j' \in ts
----------------------------------------------------------------------------- *)
apply H_all_jobs_from_taskset.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2079)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : instant
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NOTHEP : ~~ hep_task (job_task j') tsk
============================
arrives_in arr_seq j'
----------------------------------------------------------------------------- *)
apply mem_bigcat_nat_exists in JINB.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2080)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : instant
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
j' : Job
JINB : exists i : nat, j' \in arrivals_at arr_seq i /\ 0 <= i < t
NOTHEP : ~~ hep_task (job_task j') tsk
============================
arrives_in arr_seq j'
----------------------------------------------------------------------------- *)
by inversion JINB as [ta' [JIN' _]]; ∃ ta'.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
Using the above lemma, we prove that the priority inversion of the task is bounded by blocking_bound.
Lemma priority_inversion_is_bounded:
priority_inversion_is_bounded_by
arr_seq sched tsk blocking_bound.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1957)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
============================
priority_inversion_is_bounded_by arr_seq sched tsk blocking_bound
----------------------------------------------------------------------------- *)
Proof.
intros j ARR TSK POS t1 t2 PREF.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1966)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
============================
cumulative_priority_inversion sched j t1 t2 <= blocking_bound
----------------------------------------------------------------------------- *)
case NEQ: (t2 - t1 ≤ blocking_bound).
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 2021)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
NEQ : (t2 - t1 <= blocking_bound) = true
============================
cumulative_priority_inversion sched j t1 t2 <= blocking_bound
subgoal 2 (ID 2068) is:
cumulative_priority_inversion sched j t1 t2 <= blocking_bound
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2021)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
NEQ : (t2 - t1 <= blocking_bound) = true
============================
cumulative_priority_inversion sched j t1 t2 <= blocking_bound
----------------------------------------------------------------------------- *)
apply leq_trans with (t2 - t1); last by done.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2069)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
NEQ : (t2 - t1 <= blocking_bound) = true
============================
cumulative_priority_inversion sched j t1 t2 <= t2 - t1
----------------------------------------------------------------------------- *)
rewrite /cumulative_priority_inversion /is_priority_inversion.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2076)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
NEQ : (t2 - t1 <= blocking_bound) = true
============================
\sum_(t1 <= t < t2)
match sched t with
| Some jlp => ~~ hep_job jlp j
| None => false
end <= t2 - t1
----------------------------------------------------------------------------- *)
rewrite -[X in _ ≤ X]addn0 -[t2 - t1]mul1n -iter_addn -big_const_nat leq_sum //.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2113)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
NEQ : (t2 - t1 <= blocking_bound) = true
============================
forall i : nat,
true ->
match sched i with
| Some jlp => ~~ hep_job jlp j
| None => false
end <= 1
----------------------------------------------------------------------------- *)
intros t _; case: (sched t); last by done.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2149)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
NEQ : (t2 - t1 <= blocking_bound) = true
t : nat
============================
forall a : Job, ~~ hep_job a j <= 1
----------------------------------------------------------------------------- *)
by intros s; case: (hep_job s j).
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2068)
subgoal 1 (ID 2068) is:
cumulative_priority_inversion sched j t1 t2 <= blocking_bound
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2068)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
NEQ : (t2 - t1 <= blocking_bound) = false
============================
cumulative_priority_inversion sched j t1 t2 <= blocking_bound
----------------------------------------------------------------------------- *)
move: NEQ ⇒ /negP /negP; rewrite -ltnNge; move ⇒ BOUND.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2243)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
============================
cumulative_priority_inversion sched j t1 t2 <= blocking_bound
----------------------------------------------------------------------------- *)
edestruct (@preemption_time_exists) as [ppt [PPT NEQ]]; eauto 2; move: NEQ ⇒ /andP [GE LE].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2360)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
============================
cumulative_priority_inversion sched j t1 t2 <= blocking_bound
----------------------------------------------------------------------------- *)
apply leq_trans with (cumulative_priority_inversion sched j t1 ppt);
last apply leq_trans with (ppt - t1); first last.
(* ----------------------------------[ coqtop ]---------------------------------
3 subgoals (ID 2369)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
============================
ppt - t1 <= blocking_bound
subgoal 2 (ID 2368) is:
cumulative_priority_inversion sched j t1 ppt <= ppt - t1
subgoal 3 (ID 2366) is:
cumulative_priority_inversion sched j t1 t2 <=
cumulative_priority_inversion sched j t1 ppt
----------------------------------------------------------------------------- *)
- rewrite leq_subLR.
(* ----------------------------------[ coqtop ]---------------------------------
3 subgoals (ID 2374)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
============================
ppt <= t1 + blocking_bound
subgoal 2 (ID 2368) is:
cumulative_priority_inversion sched j t1 ppt <= ppt - t1
subgoal 3 (ID 2366) is:
cumulative_priority_inversion sched j t1 t2 <=
cumulative_priority_inversion sched j t1 ppt
----------------------------------------------------------------------------- *)
apply leq_trans with (t1 + max_length_of_priority_inversion j t1); first by done.
(* ----------------------------------[ coqtop ]---------------------------------
3 subgoals (ID 2376)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
============================
t1 + max_length_of_priority_inversion j t1 <= t1 + blocking_bound
subgoal 2 (ID 2368) is:
cumulative_priority_inversion sched j t1 ppt <= ppt - t1
subgoal 3 (ID 2366) is:
cumulative_priority_inversion sched j t1 t2 <=
cumulative_priority_inversion sched j t1 ppt
----------------------------------------------------------------------------- *)
by rewrite leq_add2l; eapply priority_inversion_is_bounded_by_blocking; eauto 2.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 2368)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
============================
cumulative_priority_inversion sched j t1 ppt <= ppt - t1
subgoal 2 (ID 2366) is:
cumulative_priority_inversion sched j t1 t2 <=
cumulative_priority_inversion sched j t1 ppt
----------------------------------------------------------------------------- *)
- rewrite /cumulative_priority_inversion /is_priority_inversion.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 2392)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
============================
\sum_(t1 <= t < ppt)
match sched t with
| Some jlp => ~~ hep_job jlp j
| None => false
end <= ppt - t1
subgoal 2 (ID 2366) is:
cumulative_priority_inversion sched j t1 t2 <=
cumulative_priority_inversion sched j t1 ppt
----------------------------------------------------------------------------- *)
rewrite -[X in _ ≤ X]addn0 -[ppt - t1]mul1n -iter_addn -big_const_nat.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 2420)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
============================
\sum_(t1 <= t < ppt)
match sched t with
| Some jlp => ~~ hep_job jlp j
| None => false
end <= \sum_(t1 <= i < ppt) 1
subgoal 2 (ID 2366) is:
cumulative_priority_inversion sched j t1 t2 <=
cumulative_priority_inversion sched j t1 ppt
----------------------------------------------------------------------------- *)
rewrite leq_sum //; intros t _; case: (sched t); last by done.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 2465)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
t : nat
============================
forall a : Job, ~~ hep_job a j <= 1
subgoal 2 (ID 2366) is:
cumulative_priority_inversion sched j t1 t2 <=
cumulative_priority_inversion sched j t1 ppt
----------------------------------------------------------------------------- *)
by intros s; case: (hep_job s j).
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2366)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
============================
cumulative_priority_inversion sched j t1 t2 <=
cumulative_priority_inversion sched j t1 ppt
----------------------------------------------------------------------------- *)
- rewrite /cumulative_priority_inversion /is_priority_inversion.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2482)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
============================
\sum_(t1 <= t < t2)
match sched t with
| Some jlp => ~~ hep_job jlp j
| None => false
end <=
\sum_(t1 <= t < ppt)
match sched t with
| Some jlp => ~~ hep_job jlp j
| None => false
end
----------------------------------------------------------------------------- *)
rewrite (@big_cat_nat _ _ _ ppt) //=; last first.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 2529)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
============================
ppt <= t2
subgoal 2 (ID 2505) is:
\sum_(t1 <= i < ppt)
match sched i with
| Some jlp => ~~ hep_job jlp j
| None => false
end +
\sum_(ppt <= i < t2)
match sched i with
| Some jlp => ~~ hep_job jlp j
| None => false
end <=
\sum_(t1 <= t < ppt)
match sched t with
| Some jlp => ~~ hep_job jlp j
| None => false
end
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2529)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
============================
ppt <= t2
----------------------------------------------------------------------------- *)
rewrite ltn_subRL in BOUND.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2613)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
BOUND : t1 + blocking_bound < t2
============================
ppt <= t2
----------------------------------------------------------------------------- *)
apply leq_trans with (t1 + blocking_bound); last by apply ltnW.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2614)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
BOUND : t1 + blocking_bound < t2
============================
ppt <= t1 + blocking_bound
----------------------------------------------------------------------------- *)
apply leq_trans with (t1 + max_length_of_priority_inversion j t1); first by done.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2618)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
BOUND : t1 + blocking_bound < t2
============================
t1 + max_length_of_priority_inversion j t1 <= t1 + blocking_bound
----------------------------------------------------------------------------- *)
rewrite leq_add2l; eapply priority_inversion_is_bounded_by_blocking; eauto 2.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2505)
subgoal 1 (ID 2505) is:
\sum_(t1 <= i < ppt)
match sched i with
| Some jlp => ~~ hep_job jlp j
| None => false
end +
\sum_(ppt <= i < t2)
match sched i with
| Some jlp => ~~ hep_job jlp j
| None => false
end <=
\sum_(t1 <= t < ppt)
match sched t with
| Some jlp => ~~ hep_job jlp j
| None => false
end
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2505)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
============================
\sum_(t1 <= i < ppt)
match sched i with
| Some jlp => ~~ hep_job jlp j
| None => false
end +
\sum_(ppt <= i < t2)
match sched i with
| Some jlp => ~~ hep_job jlp j
| None => false
end <=
\sum_(t1 <= t < ppt)
match sched t with
| Some jlp => ~~ hep_job jlp j
| None => false
end
----------------------------------------------------------------------------- *)
rewrite -[X in _ ≤ X]addn0 leq_add2l leqn0.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2646)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
============================
\sum_(ppt <= i < t2)
match sched i with
| Some jlp => ~~ hep_job jlp j
| None => false
end == 0
----------------------------------------------------------------------------- *)
rewrite big_nat_cond big1 //; move ⇒ t /andP [/andP [GEt LTt] _ ].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2771)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
t : nat
GEt : ppt <= t
LTt : t < t2
============================
match sched t with
| Some jlp => ~~ hep_job jlp j
| None => false
end = 0
----------------------------------------------------------------------------- *)
case SCHED: (sched t) ⇒ [s | ]; last by done.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2837)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
t : nat
GEt : ppt <= t
LTt : t < t2
s : Job
SCHED : sched t = Some s
============================
~~ hep_job s j = 0
----------------------------------------------------------------------------- *)
edestruct (@not_quiet_implies_exists_scheduled_hp_job)
with (K := ppt - t1) (t1 := t1) (t2 := t2) (t := t) as [j_hp [ARRB [HP SCHEDHP]]]; eauto 2.
(* ----------------------------------[ coqtop ]---------------------------------
3 subgoals (ID 2941)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
t : nat
GEt : ppt <= t
LTt : t < t2
s : Job
SCHED : sched t = Some s
============================
exists pr_t : instant,
preemption_time sched pr_t /\ t1 <= pr_t <= t1 + (ppt - t1)
subgoal 2 (ID 2942) is:
t1 + (ppt - t1) <= t < t2
subgoal 3 (ID 2956) is:
~~ hep_job s j = 0
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2941)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
t : nat
GEt : ppt <= t
LTt : t < t2
s : Job
SCHED : sched t = Some s
============================
exists pr_t : instant,
preemption_time sched pr_t /\ t1 <= pr_t <= t1 + (ppt - t1)
----------------------------------------------------------------------------- *)
by ∃ ppt; split; [done | rewrite subnKC //; apply/andP].
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 2942)
subgoal 1 (ID 2942) is:
t1 + (ppt - t1) <= t < t2
subgoal 2 (ID 2956) is:
~~ hep_job s j = 0
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 2942)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
t : nat
GEt : ppt <= t
LTt : t < t2
s : Job
SCHED : sched t = Some s
============================
t1 + (ppt - t1) <= t < t2
subgoal 2 (ID 2956) is:
~~ hep_job s j = 0
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2942)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
t : nat
GEt : ppt <= t
LTt : t < t2
s : Job
SCHED : sched t = Some s
============================
t1 + (ppt - t1) <= t < t2
----------------------------------------------------------------------------- *)
by rewrite subnKC //; apply/andP; split.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2956)
subgoal 1 (ID 2956) is:
~~ hep_job s j = 0
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2956)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
t : nat
GEt : ppt <= t
LTt : t < t2
s : Job
SCHED : sched t = Some s
j_hp : Job
ARRB : arrived_between j_hp t1 (succn t)
HP : hep_job j_hp j
SCHEDHP : scheduled_at sched j_hp t
============================
~~ hep_job s j = 0
----------------------------------------------------------------------------- *)
apply/eqP; rewrite eqb0 Bool.negb_involutive.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 3174)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
t : nat
GEt : ppt <= t
LTt : t < t2
s : Job
SCHED : sched t = Some s
j_hp : Job
ARRB : arrived_between j_hp t1 (succn t)
HP : hep_job j_hp j
SCHEDHP : scheduled_at sched j_hp t
============================
hep_job s j
----------------------------------------------------------------------------- *)
enough (EQef : s = j_hp); first by subst;auto.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 3178)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
t : nat
GEt : ppt <= t
LTt : t < t2
s : Job
SCHED : sched t = Some s
j_hp : Job
ARRB : arrived_between j_hp t1 (succn t)
HP : hep_job j_hp j
SCHEDHP : scheduled_at sched j_hp t
============================
s = j_hp
----------------------------------------------------------------------------- *)
eapply ideal_proc_model_is_a_uniprocessor_model; eauto 2.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 3204)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
t : nat
GEt : ppt <= t
LTt : t < t2
s : Job
SCHED : sched t = Some s
j_hp : Job
ARRB : arrived_between j_hp t1 (succn t)
HP : hep_job j_hp j
SCHEDHP : scheduled_at sched j_hp t
============================
scheduled_at sched s t
----------------------------------------------------------------------------- *)
by rewrite scheduled_at_def SCHED.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
End PriorityInversionIsBounded.
priority_inversion_is_bounded_by
arr_seq sched tsk blocking_bound.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1957)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
============================
priority_inversion_is_bounded_by arr_seq sched tsk blocking_bound
----------------------------------------------------------------------------- *)
Proof.
intros j ARR TSK POS t1 t2 PREF.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1966)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
============================
cumulative_priority_inversion sched j t1 t2 <= blocking_bound
----------------------------------------------------------------------------- *)
case NEQ: (t2 - t1 ≤ blocking_bound).
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 2021)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
NEQ : (t2 - t1 <= blocking_bound) = true
============================
cumulative_priority_inversion sched j t1 t2 <= blocking_bound
subgoal 2 (ID 2068) is:
cumulative_priority_inversion sched j t1 t2 <= blocking_bound
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2021)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
NEQ : (t2 - t1 <= blocking_bound) = true
============================
cumulative_priority_inversion sched j t1 t2 <= blocking_bound
----------------------------------------------------------------------------- *)
apply leq_trans with (t2 - t1); last by done.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2069)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
NEQ : (t2 - t1 <= blocking_bound) = true
============================
cumulative_priority_inversion sched j t1 t2 <= t2 - t1
----------------------------------------------------------------------------- *)
rewrite /cumulative_priority_inversion /is_priority_inversion.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2076)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
NEQ : (t2 - t1 <= blocking_bound) = true
============================
\sum_(t1 <= t < t2)
match sched t with
| Some jlp => ~~ hep_job jlp j
| None => false
end <= t2 - t1
----------------------------------------------------------------------------- *)
rewrite -[X in _ ≤ X]addn0 -[t2 - t1]mul1n -iter_addn -big_const_nat leq_sum //.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2113)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
NEQ : (t2 - t1 <= blocking_bound) = true
============================
forall i : nat,
true ->
match sched i with
| Some jlp => ~~ hep_job jlp j
| None => false
end <= 1
----------------------------------------------------------------------------- *)
intros t _; case: (sched t); last by done.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2149)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
NEQ : (t2 - t1 <= blocking_bound) = true
t : nat
============================
forall a : Job, ~~ hep_job a j <= 1
----------------------------------------------------------------------------- *)
by intros s; case: (hep_job s j).
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2068)
subgoal 1 (ID 2068) is:
cumulative_priority_inversion sched j t1 t2 <= blocking_bound
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2068)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
NEQ : (t2 - t1 <= blocking_bound) = false
============================
cumulative_priority_inversion sched j t1 t2 <= blocking_bound
----------------------------------------------------------------------------- *)
move: NEQ ⇒ /negP /negP; rewrite -ltnNge; move ⇒ BOUND.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2243)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
============================
cumulative_priority_inversion sched j t1 t2 <= blocking_bound
----------------------------------------------------------------------------- *)
edestruct (@preemption_time_exists) as [ppt [PPT NEQ]]; eauto 2; move: NEQ ⇒ /andP [GE LE].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2360)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
============================
cumulative_priority_inversion sched j t1 t2 <= blocking_bound
----------------------------------------------------------------------------- *)
apply leq_trans with (cumulative_priority_inversion sched j t1 ppt);
last apply leq_trans with (ppt - t1); first last.
(* ----------------------------------[ coqtop ]---------------------------------
3 subgoals (ID 2369)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
============================
ppt - t1 <= blocking_bound
subgoal 2 (ID 2368) is:
cumulative_priority_inversion sched j t1 ppt <= ppt - t1
subgoal 3 (ID 2366) is:
cumulative_priority_inversion sched j t1 t2 <=
cumulative_priority_inversion sched j t1 ppt
----------------------------------------------------------------------------- *)
- rewrite leq_subLR.
(* ----------------------------------[ coqtop ]---------------------------------
3 subgoals (ID 2374)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
============================
ppt <= t1 + blocking_bound
subgoal 2 (ID 2368) is:
cumulative_priority_inversion sched j t1 ppt <= ppt - t1
subgoal 3 (ID 2366) is:
cumulative_priority_inversion sched j t1 t2 <=
cumulative_priority_inversion sched j t1 ppt
----------------------------------------------------------------------------- *)
apply leq_trans with (t1 + max_length_of_priority_inversion j t1); first by done.
(* ----------------------------------[ coqtop ]---------------------------------
3 subgoals (ID 2376)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
============================
t1 + max_length_of_priority_inversion j t1 <= t1 + blocking_bound
subgoal 2 (ID 2368) is:
cumulative_priority_inversion sched j t1 ppt <= ppt - t1
subgoal 3 (ID 2366) is:
cumulative_priority_inversion sched j t1 t2 <=
cumulative_priority_inversion sched j t1 ppt
----------------------------------------------------------------------------- *)
by rewrite leq_add2l; eapply priority_inversion_is_bounded_by_blocking; eauto 2.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 2368)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
============================
cumulative_priority_inversion sched j t1 ppt <= ppt - t1
subgoal 2 (ID 2366) is:
cumulative_priority_inversion sched j t1 t2 <=
cumulative_priority_inversion sched j t1 ppt
----------------------------------------------------------------------------- *)
- rewrite /cumulative_priority_inversion /is_priority_inversion.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 2392)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
============================
\sum_(t1 <= t < ppt)
match sched t with
| Some jlp => ~~ hep_job jlp j
| None => false
end <= ppt - t1
subgoal 2 (ID 2366) is:
cumulative_priority_inversion sched j t1 t2 <=
cumulative_priority_inversion sched j t1 ppt
----------------------------------------------------------------------------- *)
rewrite -[X in _ ≤ X]addn0 -[ppt - t1]mul1n -iter_addn -big_const_nat.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 2420)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
============================
\sum_(t1 <= t < ppt)
match sched t with
| Some jlp => ~~ hep_job jlp j
| None => false
end <= \sum_(t1 <= i < ppt) 1
subgoal 2 (ID 2366) is:
cumulative_priority_inversion sched j t1 t2 <=
cumulative_priority_inversion sched j t1 ppt
----------------------------------------------------------------------------- *)
rewrite leq_sum //; intros t _; case: (sched t); last by done.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 2465)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
t : nat
============================
forall a : Job, ~~ hep_job a j <= 1
subgoal 2 (ID 2366) is:
cumulative_priority_inversion sched j t1 t2 <=
cumulative_priority_inversion sched j t1 ppt
----------------------------------------------------------------------------- *)
by intros s; case: (hep_job s j).
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2366)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
============================
cumulative_priority_inversion sched j t1 t2 <=
cumulative_priority_inversion sched j t1 ppt
----------------------------------------------------------------------------- *)
- rewrite /cumulative_priority_inversion /is_priority_inversion.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2482)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
============================
\sum_(t1 <= t < t2)
match sched t with
| Some jlp => ~~ hep_job jlp j
| None => false
end <=
\sum_(t1 <= t < ppt)
match sched t with
| Some jlp => ~~ hep_job jlp j
| None => false
end
----------------------------------------------------------------------------- *)
rewrite (@big_cat_nat _ _ _ ppt) //=; last first.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 2529)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
============================
ppt <= t2
subgoal 2 (ID 2505) is:
\sum_(t1 <= i < ppt)
match sched i with
| Some jlp => ~~ hep_job jlp j
| None => false
end +
\sum_(ppt <= i < t2)
match sched i with
| Some jlp => ~~ hep_job jlp j
| None => false
end <=
\sum_(t1 <= t < ppt)
match sched t with
| Some jlp => ~~ hep_job jlp j
| None => false
end
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2529)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
============================
ppt <= t2
----------------------------------------------------------------------------- *)
rewrite ltn_subRL in BOUND.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2613)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
BOUND : t1 + blocking_bound < t2
============================
ppt <= t2
----------------------------------------------------------------------------- *)
apply leq_trans with (t1 + blocking_bound); last by apply ltnW.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2614)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
BOUND : t1 + blocking_bound < t2
============================
ppt <= t1 + blocking_bound
----------------------------------------------------------------------------- *)
apply leq_trans with (t1 + max_length_of_priority_inversion j t1); first by done.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2618)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
BOUND : t1 + blocking_bound < t2
============================
t1 + max_length_of_priority_inversion j t1 <= t1 + blocking_bound
----------------------------------------------------------------------------- *)
rewrite leq_add2l; eapply priority_inversion_is_bounded_by_blocking; eauto 2.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2505)
subgoal 1 (ID 2505) is:
\sum_(t1 <= i < ppt)
match sched i with
| Some jlp => ~~ hep_job jlp j
| None => false
end +
\sum_(ppt <= i < t2)
match sched i with
| Some jlp => ~~ hep_job jlp j
| None => false
end <=
\sum_(t1 <= t < ppt)
match sched t with
| Some jlp => ~~ hep_job jlp j
| None => false
end
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2505)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
============================
\sum_(t1 <= i < ppt)
match sched i with
| Some jlp => ~~ hep_job jlp j
| None => false
end +
\sum_(ppt <= i < t2)
match sched i with
| Some jlp => ~~ hep_job jlp j
| None => false
end <=
\sum_(t1 <= t < ppt)
match sched t with
| Some jlp => ~~ hep_job jlp j
| None => false
end
----------------------------------------------------------------------------- *)
rewrite -[X in _ ≤ X]addn0 leq_add2l leqn0.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2646)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
============================
\sum_(ppt <= i < t2)
match sched i with
| Some jlp => ~~ hep_job jlp j
| None => false
end == 0
----------------------------------------------------------------------------- *)
rewrite big_nat_cond big1 //; move ⇒ t /andP [/andP [GEt LTt] _ ].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2771)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
t : nat
GEt : ppt <= t
LTt : t < t2
============================
match sched t with
| Some jlp => ~~ hep_job jlp j
| None => false
end = 0
----------------------------------------------------------------------------- *)
case SCHED: (sched t) ⇒ [s | ]; last by done.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2837)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
t : nat
GEt : ppt <= t
LTt : t < t2
s : Job
SCHED : sched t = Some s
============================
~~ hep_job s j = 0
----------------------------------------------------------------------------- *)
edestruct (@not_quiet_implies_exists_scheduled_hp_job)
with (K := ppt - t1) (t1 := t1) (t2 := t2) (t := t) as [j_hp [ARRB [HP SCHEDHP]]]; eauto 2.
(* ----------------------------------[ coqtop ]---------------------------------
3 subgoals (ID 2941)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
t : nat
GEt : ppt <= t
LTt : t < t2
s : Job
SCHED : sched t = Some s
============================
exists pr_t : instant,
preemption_time sched pr_t /\ t1 <= pr_t <= t1 + (ppt - t1)
subgoal 2 (ID 2942) is:
t1 + (ppt - t1) <= t < t2
subgoal 3 (ID 2956) is:
~~ hep_job s j = 0
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2941)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
t : nat
GEt : ppt <= t
LTt : t < t2
s : Job
SCHED : sched t = Some s
============================
exists pr_t : instant,
preemption_time sched pr_t /\ t1 <= pr_t <= t1 + (ppt - t1)
----------------------------------------------------------------------------- *)
by ∃ ppt; split; [done | rewrite subnKC //; apply/andP].
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 2942)
subgoal 1 (ID 2942) is:
t1 + (ppt - t1) <= t < t2
subgoal 2 (ID 2956) is:
~~ hep_job s j = 0
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 2942)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
t : nat
GEt : ppt <= t
LTt : t < t2
s : Job
SCHED : sched t = Some s
============================
t1 + (ppt - t1) <= t < t2
subgoal 2 (ID 2956) is:
~~ hep_job s j = 0
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2942)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
t : nat
GEt : ppt <= t
LTt : t < t2
s : Job
SCHED : sched t = Some s
============================
t1 + (ppt - t1) <= t < t2
----------------------------------------------------------------------------- *)
by rewrite subnKC //; apply/andP; split.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2956)
subgoal 1 (ID 2956) is:
~~ hep_job s j = 0
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2956)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
t : nat
GEt : ppt <= t
LTt : t < t2
s : Job
SCHED : sched t = Some s
j_hp : Job
ARRB : arrived_between j_hp t1 (succn t)
HP : hep_job j_hp j
SCHEDHP : scheduled_at sched j_hp t
============================
~~ hep_job s j = 0
----------------------------------------------------------------------------- *)
apply/eqP; rewrite eqb0 Bool.negb_involutive.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 3174)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
t : nat
GEt : ppt <= t
LTt : t < t2
s : Job
SCHED : sched t = Some s
j_hp : Job
ARRB : arrived_between j_hp t1 (succn t)
HP : hep_job j_hp j
SCHEDHP : scheduled_at sched j_hp t
============================
hep_job s j
----------------------------------------------------------------------------- *)
enough (EQef : s = j_hp); first by subst;auto.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 3178)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
t : nat
GEt : ppt <= t
LTt : t < t2
s : Job
SCHED : sched t = Some s
j_hp : Job
ARRB : arrived_between j_hp t1 (succn t)
HP : hep_job j_hp j
SCHEDHP : scheduled_at sched j_hp t
============================
s = j_hp
----------------------------------------------------------------------------- *)
eapply ideal_proc_model_is_a_uniprocessor_model; eauto 2.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 3204)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POS : 0 < job_cost j
t1, t2 : instant
PREF : busy_interval_prefix arr_seq sched j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 + priority_inversion.max_length_of_priority_inversion arr_seq j t1
t : nat
GEt : ppt <= t
LTt : t < t2
s : Job
SCHED : sched t = Some s
j_hp : Job
ARRB : arrived_between j_hp t1 (succn t)
HP : hep_job j_hp j
SCHEDHP : scheduled_at sched j_hp t
============================
scheduled_at sched s t
----------------------------------------------------------------------------- *)
by rewrite scheduled_at_def SCHED.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
End PriorityInversionIsBounded.
Response-Time Bound
In this section, we prove that the maximum among the solutions of the response-time bound recurrence is a response-time bound for [tsk].
Let L be any positive fixed point of the busy interval recurrence.
Variable L : duration.
Hypothesis H_L_positive : L > 0.
Hypothesis H_fixed_point : L = blocking_bound + total_hep_rbf L.
Hypothesis H_L_positive : L > 0.
Hypothesis H_fixed_point : L = blocking_bound + total_hep_rbf L.
To reduce the time complexity of the analysis, recall the notion of search space.
Next, consider any value R, and assume that for any given arrival offset A from the search
space there is a solution of the response-time bound recurrence that is bounded by R.
Variable R : duration.
Hypothesis H_R_is_maximum:
∀ (A : duration),
is_in_search_space A →
∃ (F : duration),
A + F = blocking_bound
+ (task_rbf (A + ε) - (task_cost tsk - task_run_to_completion_threshold tsk))
+ total_ohep_rbf (A + F) ∧
F + (task_cost tsk - task_run_to_completion_threshold tsk) ≤ R.
Hypothesis H_R_is_maximum:
∀ (A : duration),
is_in_search_space A →
∃ (F : duration),
A + F = blocking_bound
+ (task_rbf (A + ε) - (task_cost tsk - task_run_to_completion_threshold tsk))
+ total_ohep_rbf (A + F) ∧
F + (task_cost tsk - task_run_to_completion_threshold tsk) ≤ R.
Then, using the results for the general RTA for FP-schedulers, we establish a
response-time bound for the more concrete model of bounded nonpreemptive segments.
Note that in case of the general RTA for FP-schedulers, we just _assume_ that
the priority inversion is bounded. In this module we provide the preemption model
with bounded nonpreemptive segments and _prove_ that the priority inversion is
bounded.
Theorem uniprocessor_response_time_bound_fp_with_bounded_nonpreemptive_segments:
response_time_bounded_by tsk R.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1955)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
L : duration
H_L_positive : 0 < L
H_fixed_point : L = blocking_bound + 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,
A + F =
blocking_bound +
(task_rbf (A + ε) -
(task_cost tsk - task_run_to_completion_threshold tsk)) +
total_ohep_rbf (A + F) /\
F +
(task_cost tsk - task_run_to_completion_threshold tsk) <=
R
============================
response_time_bounded_by tsk R
----------------------------------------------------------------------------- *)
Proof.
eapply uniprocessor_response_time_bound_fp;
eauto using priority_inversion_is_bounded.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
End ResponseTimeBound.
End RTAforFPwithBoundedNonpreemptiveSegmentsWithArrivalCurves.
response_time_bounded_by tsk R.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1955)
Task : TaskType
H : TaskCost Task
H0 : TaskRunToCompletionThreshold Task
H1 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H2 : JobTask Job Task
H3 : JobArrival Job
H4 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute sched
H5 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
H6 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks sched
H_work_conserving : work_conserving arr_seq sched
H_respects_policy : respects_policy_at_preemption_point arr_seq sched
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H7 : 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
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
max_length_of_priority_inversion := priority_inversion.max_length_of_priority_inversion
arr_seq :
Job -> instant -> nat
task_rbf := task_request_bound_function 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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
L : duration
H_L_positive : 0 < L
H_fixed_point : L = blocking_bound + 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,
A + F =
blocking_bound +
(task_rbf (A + ε) -
(task_cost tsk - task_run_to_completion_threshold tsk)) +
total_ohep_rbf (A + F) /\
F +
(task_cost tsk - task_run_to_completion_threshold tsk) <=
R
============================
response_time_bounded_by tsk R
----------------------------------------------------------------------------- *)
Proof.
eapply uniprocessor_response_time_bound_fp;
eauto using priority_inversion_is_bounded.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
End ResponseTimeBound.
End RTAforFPwithBoundedNonpreemptiveSegmentsWithArrivalCurves.