Library rt.restructuring.analysis.fixed_priority.rta.nonpr_reg.response_time_bound
(* ----------------------------------[ coqtop ]---------------------------------
Welcome to Coq 8.10.1 (October 2019)
----------------------------------------------------------------------------- *)
From rt.util Require Import all.
From rt.restructuring.behavior Require Import all.
From rt.restructuring.analysis.basic_facts Require Import all.
From rt.restructuring.model Require Import job task workload processor.ideal readiness.basic.
From rt.restructuring.model.arrival Require Import arrival_curves.
From rt.restructuring.model.preemption Require Import valid_model
job.parameters task.parameters rtc_threshold.valid_rtct.
From rt.restructuring.model.schedule Require Import
work_conserving priority_based.priorities priority_based.preemption_aware.
From rt.restructuring.analysis.arrival Require Import workload_bound rbf.
From rt.restructuring.analysis.fixed_priority Require Export rta.response_time_bound.
From rt.restructuring.analysis.facts Require Import priority_inversion_is_bounded.
From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq path fintype bigop.
RTA for FP-schedulers with Bounded Non-Preemprive 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 uniprocessor 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 maping jobs
to theirs preemption points ...
... and assume that it defines a valid preemption
model with bounded nonpreemptive 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.
Variable higher_eq_priority : 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 thejob_preemptable
function (i.e., jobs have bounded nonpreemptive 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 | ~~ higher_eq_priority tsk_other tsk)
(task_max_nonpreemptive_segment tsk_other - ε).
\max_(tsk_other <- ts | ~~ higher_eq_priority 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 nonpreemtive 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 nonpreemptive 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 300)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 304)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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_is_bounded.max_length_of_priority_inversion.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 312)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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)
(job_task j))
(job_max_nonpreemptive_segment j_lp - ε) <=
\max_(tsk_other <- ts | ~~ higher_eq_priority tsk_other tsk)
(task_max_nonpreemptive_segment tsk_other - ε)
----------------------------------------------------------------------------- *)
apply leq_trans with
(\max_(j_lp <- arrivals_between arr_seq 0 t
| ~~ higher_eq_priority (job_task j_lp) tsk)
(task_max_nonpreemptive_segment (job_task j_lp) - ε)).
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 327)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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)
(job_task j))
(job_max_nonpreemptive_segment j_lp - ε) <=
\max_(j_lp <- arrivals_between arr_seq 0 t | ~~
higher_eq_priority
(job_task j_lp) tsk)
(task_max_nonpreemptive_segment (job_task j_lp) - ε)
subgoal 2 (ID 328) is:
\max_(j_lp <- arrivals_between arr_seq 0 t | ~~
higher_eq_priority
(job_task j_lp) tsk)
(task_max_nonpreemptive_segment (job_task j_lp) - ε) <=
\max_(tsk_other <- ts | ~~ higher_eq_priority tsk_other tsk)
(task_max_nonpreemptive_segment tsk_other - ε)
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 327)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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)
(job_task j))
(job_max_nonpreemptive_segment j_lp - ε) <=
\max_(j_lp <- arrivals_between arr_seq 0 t | ~~
higher_eq_priority
(job_task j_lp) tsk)
(task_max_nonpreemptive_segment (job_task j_lp) - ε)
----------------------------------------------------------------------------- *)
rewrite TSK.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 330)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 | ~~
higher_eq_priority
(job_task j_lp) tsk)
(task_max_nonpreemptive_segment (job_task j_lp) - ε)
----------------------------------------------------------------------------- *)
apply leq_big_max.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 331)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 ->
~~ higher_eq_priority (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 334)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 : ~~ higher_eq_priority (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 341)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 : ~~ higher_eq_priority (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 374)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 : ~~ higher_eq_priority (job_task j') tsk
============================
arrives_in arr_seq j'
----------------------------------------------------------------------------- *)
by eapply in_arrivals_implies_arrived; eauto 2.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 328)
subgoal 1 (ID 328) is:
\max_(j_lp <- arrivals_between arr_seq 0 t | ~~
higher_eq_priority
(job_task j_lp) tsk)
(task_max_nonpreemptive_segment (job_task j_lp) - ε) <=
\max_(tsk_other <- ts | ~~ higher_eq_priority tsk_other tsk)
(task_max_nonpreemptive_segment tsk_other - ε)
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 328)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 | ~~
higher_eq_priority
(job_task j_lp) tsk)
(task_max_nonpreemptive_segment (job_task j_lp) - ε) <=
\max_(tsk_other <- ts | ~~ higher_eq_priority tsk_other tsk)
(task_max_nonpreemptive_segment tsk_other - ε)
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 328)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 | ~~
higher_eq_priority
(job_task j_lp) tsk)
(task_max_nonpreemptive_segment (job_task j_lp) - ε) <=
\max_(tsk_other <- ts | ~~ higher_eq_priority tsk_other tsk)
(task_max_nonpreemptive_segment tsk_other - ε)
----------------------------------------------------------------------------- *)
apply /bigmax_leq_seqP.
(* ----------------------------------[ coqtop ]---------------------------------
1 focused subgoal
(shelved: 1) (ID 411)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 ->
~~ higher_eq_priority (job_task i) tsk ->
task_max_nonpreemptive_segment (job_task i) - ε <=
\max_(tsk_other <- ts | ~~ higher_eq_priority tsk_other tsk)
(task_max_nonpreemptive_segment tsk_other - ε)
----------------------------------------------------------------------------- *)
intros j' JINB NOTHEP.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 414)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 : ~~ higher_eq_priority (job_task j') tsk
============================
task_max_nonpreemptive_segment (job_task j') - ε <=
\max_(tsk_other <- ts | ~~ higher_eq_priority 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 423)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 : ~~ higher_eq_priority (job_task j') tsk
============================
job_task j' \in ts
----------------------------------------------------------------------------- *)
apply H_all_jobs_from_taskset.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 425)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 : ~~ higher_eq_priority (job_task j') tsk
============================
arrives_in arr_seq j'
----------------------------------------------------------------------------- *)
apply mem_bigcat_nat_exists in JINB.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 426)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 : ~~ higher_eq_priority (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 300)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 304)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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_is_bounded.max_length_of_priority_inversion.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 312)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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)
(job_task j))
(job_max_nonpreemptive_segment j_lp - ε) <=
\max_(tsk_other <- ts | ~~ higher_eq_priority tsk_other tsk)
(task_max_nonpreemptive_segment tsk_other - ε)
----------------------------------------------------------------------------- *)
apply leq_trans with
(\max_(j_lp <- arrivals_between arr_seq 0 t
| ~~ higher_eq_priority (job_task j_lp) tsk)
(task_max_nonpreemptive_segment (job_task j_lp) - ε)).
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 327)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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)
(job_task j))
(job_max_nonpreemptive_segment j_lp - ε) <=
\max_(j_lp <- arrivals_between arr_seq 0 t | ~~
higher_eq_priority
(job_task j_lp) tsk)
(task_max_nonpreemptive_segment (job_task j_lp) - ε)
subgoal 2 (ID 328) is:
\max_(j_lp <- arrivals_between arr_seq 0 t | ~~
higher_eq_priority
(job_task j_lp) tsk)
(task_max_nonpreemptive_segment (job_task j_lp) - ε) <=
\max_(tsk_other <- ts | ~~ higher_eq_priority tsk_other tsk)
(task_max_nonpreemptive_segment tsk_other - ε)
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 327)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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)
(job_task j))
(job_max_nonpreemptive_segment j_lp - ε) <=
\max_(j_lp <- arrivals_between arr_seq 0 t | ~~
higher_eq_priority
(job_task j_lp) tsk)
(task_max_nonpreemptive_segment (job_task j_lp) - ε)
----------------------------------------------------------------------------- *)
rewrite TSK.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 330)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 | ~~
higher_eq_priority
(job_task j_lp) tsk)
(task_max_nonpreemptive_segment (job_task j_lp) - ε)
----------------------------------------------------------------------------- *)
apply leq_big_max.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 331)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 ->
~~ higher_eq_priority (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 334)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 : ~~ higher_eq_priority (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 341)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 : ~~ higher_eq_priority (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 374)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 : ~~ higher_eq_priority (job_task j') tsk
============================
arrives_in arr_seq j'
----------------------------------------------------------------------------- *)
by eapply in_arrivals_implies_arrived; eauto 2.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 328)
subgoal 1 (ID 328) is:
\max_(j_lp <- arrivals_between arr_seq 0 t | ~~
higher_eq_priority
(job_task j_lp) tsk)
(task_max_nonpreemptive_segment (job_task j_lp) - ε) <=
\max_(tsk_other <- ts | ~~ higher_eq_priority tsk_other tsk)
(task_max_nonpreemptive_segment tsk_other - ε)
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 328)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 | ~~
higher_eq_priority
(job_task j_lp) tsk)
(task_max_nonpreemptive_segment (job_task j_lp) - ε) <=
\max_(tsk_other <- ts | ~~ higher_eq_priority tsk_other tsk)
(task_max_nonpreemptive_segment tsk_other - ε)
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 328)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 | ~~
higher_eq_priority
(job_task j_lp) tsk)
(task_max_nonpreemptive_segment (job_task j_lp) - ε) <=
\max_(tsk_other <- ts | ~~ higher_eq_priority tsk_other tsk)
(task_max_nonpreemptive_segment tsk_other - ε)
----------------------------------------------------------------------------- *)
apply /bigmax_leq_seqP.
(* ----------------------------------[ coqtop ]---------------------------------
1 focused subgoal
(shelved: 1) (ID 411)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 ->
~~ higher_eq_priority (job_task i) tsk ->
task_max_nonpreemptive_segment (job_task i) - ε <=
\max_(tsk_other <- ts | ~~ higher_eq_priority tsk_other tsk)
(task_max_nonpreemptive_segment tsk_other - ε)
----------------------------------------------------------------------------- *)
intros j' JINB NOTHEP.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 414)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 : ~~ higher_eq_priority (job_task j') tsk
============================
task_max_nonpreemptive_segment (job_task j') - ε <=
\max_(tsk_other <- ts | ~~ higher_eq_priority 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 423)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 : ~~ higher_eq_priority (job_task j') tsk
============================
job_task j' \in ts
----------------------------------------------------------------------------- *)
apply H_all_jobs_from_taskset.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 425)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 : ~~ higher_eq_priority (job_task j') tsk
============================
arrives_in arr_seq j'
----------------------------------------------------------------------------- *)
apply mem_bigcat_nat_exists in JINB.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 426)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 : ~~ higher_eq_priority (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 311)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) tsk
blocking_bound
----------------------------------------------------------------------------- *)
Proof.
intros j ARR TSK POS t1 t2 PREF.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 320)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
============================
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 t2 <=
blocking_bound
----------------------------------------------------------------------------- *)
case NEQ: (t2 - t1 ≤ blocking_bound).
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 375)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
NEQ : (t2 - t1 <= blocking_bound) = true
============================
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 t2 <=
blocking_bound
subgoal 2 (ID 422) is:
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 t2 <=
blocking_bound
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 375)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
NEQ : (t2 - t1 <= blocking_bound) = true
============================
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 t2 <=
blocking_bound
----------------------------------------------------------------------------- *)
apply leq_trans with (t2 - t1); last by done.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 423)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
NEQ : (t2 - t1 <= blocking_bound) = true
============================
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 t2 <=
t2 - t1
----------------------------------------------------------------------------- *)
rewrite /cumulative_priority_inversion /is_priority_inversion.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 430)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
NEQ : (t2 - t1 <= blocking_bound) = true
============================
\sum_(t1 <= t < t2)
match sched t with
| Some jlp => ~~ FP_to_JLFP Job Task 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 467)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
NEQ : (t2 - t1 <= blocking_bound) = true
============================
forall i : nat,
true ->
match sched i with
| Some jlp => ~~ FP_to_JLFP Job Task jlp j
| None => false
end <= 1
----------------------------------------------------------------------------- *)
intros t _; case: (sched t); last by done.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 503)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
NEQ : (t2 - t1 <= blocking_bound) = true
t : nat
============================
forall a : Job, ~~ FP_to_JLFP Job Task a j <= 1
----------------------------------------------------------------------------- *)
by intros s; case: (FP_to_JLFP Job Task s j).
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 422)
subgoal 1 (ID 422) is:
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 t2 <=
blocking_bound
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 422)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
NEQ : (t2 - t1 <= blocking_bound) = false
============================
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 t2 <=
blocking_bound
----------------------------------------------------------------------------- *)
move: NEQ ⇒ /negP /negP; rewrite -ltnNge; move ⇒ BOUND.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 597)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
============================
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 t2 <=
blocking_bound
----------------------------------------------------------------------------- *)
edestruct (@preemption_time_exists) as [ppt [PPT NEQ]]; eauto 2; move: NEQ ⇒ /andP [GE LE].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 714)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) j t1
============================
cumulative_priority_inversion sched (FP_to_JLFP Job Task) 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 724)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) j t1
============================
ppt - t1 <= blocking_bound
subgoal 2 (ID 723) is:
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 ppt <=
ppt - t1
subgoal 3 (ID 721) is:
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 t2 <=
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 ppt
----------------------------------------------------------------------------- *)
- rewrite leq_subLR.
(* ----------------------------------[ coqtop ]---------------------------------
3 subgoals (ID 729)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) j t1
============================
ppt <= t1 + blocking_bound
subgoal 2 (ID 723) is:
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 ppt <=
ppt - t1
subgoal 3 (ID 721) is:
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 t2 <=
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 ppt
----------------------------------------------------------------------------- *)
apply leq_trans with (t1 + max_length_of_priority_inversion j t1); first by done.
(* ----------------------------------[ coqtop ]---------------------------------
3 subgoals (ID 731)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) j t1
============================
t1 + max_length_of_priority_inversion j t1 <= t1 + blocking_bound
subgoal 2 (ID 723) is:
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 ppt <=
ppt - t1
subgoal 3 (ID 721) is:
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 t2 <=
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 ppt
----------------------------------------------------------------------------- *)
by rewrite leq_add2l; eapply priority_inversion_is_bounded_by_blocking; eauto 2.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 723)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) j t1
============================
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 ppt <=
ppt - t1
subgoal 2 (ID 721) is:
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 t2 <=
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 ppt
----------------------------------------------------------------------------- *)
- rewrite /cumulative_priority_inversion /is_priority_inversion.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 747)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) j t1
============================
\sum_(t1 <= t < ppt)
match sched t with
| Some jlp => ~~ FP_to_JLFP Job Task jlp j
| None => false
end <= ppt - t1
subgoal 2 (ID 721) is:
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 t2 <=
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 ppt
----------------------------------------------------------------------------- *)
rewrite -[X in _ ≤ X]addn0 -[ppt - t1]mul1n -iter_addn -big_const_nat.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 775)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) j t1
============================
\sum_(t1 <= t < ppt)
match sched t with
| Some jlp => ~~ FP_to_JLFP Job Task jlp j
| None => false
end <= \sum_(t1 <= i < ppt) 1
subgoal 2 (ID 721) is:
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 t2 <=
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 ppt
----------------------------------------------------------------------------- *)
rewrite leq_sum //; intros t _; case: (sched t); last by done.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 820)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) j t1
t : nat
============================
forall a : Job, ~~ FP_to_JLFP Job Task a j <= 1
subgoal 2 (ID 721) is:
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 t2 <=
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 ppt
----------------------------------------------------------------------------- *)
by intros s; case: (FP_to_JLFP Job Task s j).
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 721)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) j t1
============================
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 t2 <=
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 ppt
----------------------------------------------------------------------------- *)
- rewrite /cumulative_priority_inversion /is_priority_inversion.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 837)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) j t1
============================
\sum_(t1 <= t < t2)
match sched t with
| Some jlp => ~~ FP_to_JLFP Job Task jlp j
| None => false
end <=
\sum_(t1 <= t < ppt)
match sched t with
| Some jlp => ~~ FP_to_JLFP Job Task jlp j
| None => false
end
----------------------------------------------------------------------------- *)
rewrite (@big_cat_nat _ _ _ ppt) //=; last first.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 884)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) j t1
============================
ppt <= t2
subgoal 2 (ID 860) is:
\sum_(t1 <= i < ppt)
match sched i with
| Some jlp => ~~ FP_to_JLFP Job Task jlp j
| None => false
end +
\sum_(ppt <= i < t2)
match sched i with
| Some jlp => ~~ FP_to_JLFP Job Task jlp j
| None => false
end <=
\sum_(t1 <= t < ppt)
match sched t with
| Some jlp => ~~ FP_to_JLFP Job Task jlp j
| None => false
end
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 884)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) j t1
============================
ppt <= t2
----------------------------------------------------------------------------- *)
rewrite ltn_subRL in BOUND.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 968)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) j t1
BOUND : t1 + blocking_bound < t2
============================
ppt <= t2
----------------------------------------------------------------------------- *)
apply leq_trans with (t1 + blocking_bound); last by apply ltnW.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 969)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) 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 973)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) 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 860)
subgoal 1 (ID 860) is:
\sum_(t1 <= i < ppt)
match sched i with
| Some jlp => ~~ FP_to_JLFP Job Task jlp j
| None => false
end +
\sum_(ppt <= i < t2)
match sched i with
| Some jlp => ~~ FP_to_JLFP Job Task jlp j
| None => false
end <=
\sum_(t1 <= t < ppt)
match sched t with
| Some jlp => ~~ FP_to_JLFP Job Task jlp j
| None => false
end
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 860)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) j t1
============================
\sum_(t1 <= i < ppt)
match sched i with
| Some jlp => ~~ FP_to_JLFP Job Task jlp j
| None => false
end +
\sum_(ppt <= i < t2)
match sched i with
| Some jlp => ~~ FP_to_JLFP Job Task jlp j
| None => false
end <=
\sum_(t1 <= t < ppt)
match sched t with
| Some jlp => ~~ FP_to_JLFP Job Task jlp j
| None => false
end
----------------------------------------------------------------------------- *)
rewrite -[X in _ ≤ X]addn0 leq_add2l leqn0.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1001)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) j t1
============================
\sum_(ppt <= i < t2)
match sched i with
| Some jlp => ~~ FP_to_JLFP Job Task jlp j
| None => false
end == 0
----------------------------------------------------------------------------- *)
rewrite big_nat_cond big1 //; move ⇒ t /andP [/andP [GEt LTt] _ ].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1126)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) j t1
t : nat
GEt : ppt <= t
LTt : t < t2
============================
match sched t with
| Some jlp => ~~ FP_to_JLFP Job Task jlp j
| None => false
end = 0
----------------------------------------------------------------------------- *)
case SCHED: (sched t) ⇒ [s | ]; last by done.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1192)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) j t1
t : nat
GEt : ppt <= t
LTt : t < t2
s : Job
SCHED : sched t = Some s
============================
~~ FP_to_JLFP Job Task 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 1296)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) j t1
t : nat
GEt : ppt <= t
LTt : t < t2
s : Job
SCHED : sched t = Some s
============================
exists pr_t : instant,
preemption_time.preemption_time sched pr_t /\
t1 <= pr_t <= t1 + (ppt - t1)
subgoal 2 (ID 1297) is:
t1 + (ppt - t1) <= t < t2
subgoal 3 (ID 1311) is:
~~ FP_to_JLFP Job Task s j = 0
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1296)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) j t1
t : nat
GEt : ppt <= t
LTt : t < t2
s : Job
SCHED : sched t = Some s
============================
exists pr_t : instant,
preemption_time.preemption_time sched pr_t /\
t1 <= pr_t <= t1 + (ppt - t1)
----------------------------------------------------------------------------- *)
by ∃ ppt; split; [done | rewrite subnKC //; apply/andP].
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 1297)
subgoal 1 (ID 1297) is:
t1 + (ppt - t1) <= t < t2
subgoal 2 (ID 1311) is:
~~ FP_to_JLFP Job Task s j = 0
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 1297)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) 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 1311) is:
~~ FP_to_JLFP Job Task s j = 0
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1297)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) 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 1311)
subgoal 1 (ID 1311) is:
~~ FP_to_JLFP Job Task s j = 0
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1311)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) 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 : FP_to_JLFP Job Task j_hp j
SCHEDHP : scheduled_at sched j_hp t
============================
~~ FP_to_JLFP Job Task s j = 0
----------------------------------------------------------------------------- *)
apply/eqP; rewrite eqb0 Bool.negb_involutive.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1529)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) 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 : FP_to_JLFP Job Task j_hp j
SCHEDHP : scheduled_at sched j_hp t
============================
FP_to_JLFP Job Task s j
----------------------------------------------------------------------------- *)
enough (EQef : s = j_hp); first by subst;auto.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1533)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) 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 : FP_to_JLFP Job Task 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 1559)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) 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 : FP_to_JLFP Job Task 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 311)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) tsk
blocking_bound
----------------------------------------------------------------------------- *)
Proof.
intros j ARR TSK POS t1 t2 PREF.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 320)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
============================
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 t2 <=
blocking_bound
----------------------------------------------------------------------------- *)
case NEQ: (t2 - t1 ≤ blocking_bound).
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 375)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
NEQ : (t2 - t1 <= blocking_bound) = true
============================
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 t2 <=
blocking_bound
subgoal 2 (ID 422) is:
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 t2 <=
blocking_bound
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 375)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
NEQ : (t2 - t1 <= blocking_bound) = true
============================
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 t2 <=
blocking_bound
----------------------------------------------------------------------------- *)
apply leq_trans with (t2 - t1); last by done.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 423)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
NEQ : (t2 - t1 <= blocking_bound) = true
============================
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 t2 <=
t2 - t1
----------------------------------------------------------------------------- *)
rewrite /cumulative_priority_inversion /is_priority_inversion.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 430)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
NEQ : (t2 - t1 <= blocking_bound) = true
============================
\sum_(t1 <= t < t2)
match sched t with
| Some jlp => ~~ FP_to_JLFP Job Task 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 467)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
NEQ : (t2 - t1 <= blocking_bound) = true
============================
forall i : nat,
true ->
match sched i with
| Some jlp => ~~ FP_to_JLFP Job Task jlp j
| None => false
end <= 1
----------------------------------------------------------------------------- *)
intros t _; case: (sched t); last by done.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 503)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
NEQ : (t2 - t1 <= blocking_bound) = true
t : nat
============================
forall a : Job, ~~ FP_to_JLFP Job Task a j <= 1
----------------------------------------------------------------------------- *)
by intros s; case: (FP_to_JLFP Job Task s j).
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 422)
subgoal 1 (ID 422) is:
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 t2 <=
blocking_bound
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 422)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
NEQ : (t2 - t1 <= blocking_bound) = false
============================
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 t2 <=
blocking_bound
----------------------------------------------------------------------------- *)
move: NEQ ⇒ /negP /negP; rewrite -ltnNge; move ⇒ BOUND.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 597)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
============================
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 t2 <=
blocking_bound
----------------------------------------------------------------------------- *)
edestruct (@preemption_time_exists) as [ppt [PPT NEQ]]; eauto 2; move: NEQ ⇒ /andP [GE LE].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 714)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) j t1
============================
cumulative_priority_inversion sched (FP_to_JLFP Job Task) 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 724)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) j t1
============================
ppt - t1 <= blocking_bound
subgoal 2 (ID 723) is:
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 ppt <=
ppt - t1
subgoal 3 (ID 721) is:
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 t2 <=
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 ppt
----------------------------------------------------------------------------- *)
- rewrite leq_subLR.
(* ----------------------------------[ coqtop ]---------------------------------
3 subgoals (ID 729)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) j t1
============================
ppt <= t1 + blocking_bound
subgoal 2 (ID 723) is:
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 ppt <=
ppt - t1
subgoal 3 (ID 721) is:
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 t2 <=
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 ppt
----------------------------------------------------------------------------- *)
apply leq_trans with (t1 + max_length_of_priority_inversion j t1); first by done.
(* ----------------------------------[ coqtop ]---------------------------------
3 subgoals (ID 731)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) j t1
============================
t1 + max_length_of_priority_inversion j t1 <= t1 + blocking_bound
subgoal 2 (ID 723) is:
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 ppt <=
ppt - t1
subgoal 3 (ID 721) is:
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 t2 <=
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 ppt
----------------------------------------------------------------------------- *)
by rewrite leq_add2l; eapply priority_inversion_is_bounded_by_blocking; eauto 2.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 723)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) j t1
============================
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 ppt <=
ppt - t1
subgoal 2 (ID 721) is:
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 t2 <=
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 ppt
----------------------------------------------------------------------------- *)
- rewrite /cumulative_priority_inversion /is_priority_inversion.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 747)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) j t1
============================
\sum_(t1 <= t < ppt)
match sched t with
| Some jlp => ~~ FP_to_JLFP Job Task jlp j
| None => false
end <= ppt - t1
subgoal 2 (ID 721) is:
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 t2 <=
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 ppt
----------------------------------------------------------------------------- *)
rewrite -[X in _ ≤ X]addn0 -[ppt - t1]mul1n -iter_addn -big_const_nat.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 775)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) j t1
============================
\sum_(t1 <= t < ppt)
match sched t with
| Some jlp => ~~ FP_to_JLFP Job Task jlp j
| None => false
end <= \sum_(t1 <= i < ppt) 1
subgoal 2 (ID 721) is:
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 t2 <=
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 ppt
----------------------------------------------------------------------------- *)
rewrite leq_sum //; intros t _; case: (sched t); last by done.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 820)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) j t1
t : nat
============================
forall a : Job, ~~ FP_to_JLFP Job Task a j <= 1
subgoal 2 (ID 721) is:
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 t2 <=
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 ppt
----------------------------------------------------------------------------- *)
by intros s; case: (FP_to_JLFP Job Task s j).
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 721)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) j t1
============================
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 t2 <=
cumulative_priority_inversion sched (FP_to_JLFP Job Task) j t1 ppt
----------------------------------------------------------------------------- *)
- rewrite /cumulative_priority_inversion /is_priority_inversion.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 837)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) j t1
============================
\sum_(t1 <= t < t2)
match sched t with
| Some jlp => ~~ FP_to_JLFP Job Task jlp j
| None => false
end <=
\sum_(t1 <= t < ppt)
match sched t with
| Some jlp => ~~ FP_to_JLFP Job Task jlp j
| None => false
end
----------------------------------------------------------------------------- *)
rewrite (@big_cat_nat _ _ _ ppt) //=; last first.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 884)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) j t1
============================
ppt <= t2
subgoal 2 (ID 860) is:
\sum_(t1 <= i < ppt)
match sched i with
| Some jlp => ~~ FP_to_JLFP Job Task jlp j
| None => false
end +
\sum_(ppt <= i < t2)
match sched i with
| Some jlp => ~~ FP_to_JLFP Job Task jlp j
| None => false
end <=
\sum_(t1 <= t < ppt)
match sched t with
| Some jlp => ~~ FP_to_JLFP Job Task jlp j
| None => false
end
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 884)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) j t1
============================
ppt <= t2
----------------------------------------------------------------------------- *)
rewrite ltn_subRL in BOUND.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 968)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) j t1
BOUND : t1 + blocking_bound < t2
============================
ppt <= t2
----------------------------------------------------------------------------- *)
apply leq_trans with (t1 + blocking_bound); last by apply ltnW.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 969)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) 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 973)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) 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 860)
subgoal 1 (ID 860) is:
\sum_(t1 <= i < ppt)
match sched i with
| Some jlp => ~~ FP_to_JLFP Job Task jlp j
| None => false
end +
\sum_(ppt <= i < t2)
match sched i with
| Some jlp => ~~ FP_to_JLFP Job Task jlp j
| None => false
end <=
\sum_(t1 <= t < ppt)
match sched t with
| Some jlp => ~~ FP_to_JLFP Job Task jlp j
| None => false
end
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 860)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) j t1
============================
\sum_(t1 <= i < ppt)
match sched i with
| Some jlp => ~~ FP_to_JLFP Job Task jlp j
| None => false
end +
\sum_(ppt <= i < t2)
match sched i with
| Some jlp => ~~ FP_to_JLFP Job Task jlp j
| None => false
end <=
\sum_(t1 <= t < ppt)
match sched t with
| Some jlp => ~~ FP_to_JLFP Job Task jlp j
| None => false
end
----------------------------------------------------------------------------- *)
rewrite -[X in _ ≤ X]addn0 leq_add2l leqn0.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1001)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) j t1
============================
\sum_(ppt <= i < t2)
match sched i with
| Some jlp => ~~ FP_to_JLFP Job Task jlp j
| None => false
end == 0
----------------------------------------------------------------------------- *)
rewrite big_nat_cond big1 //; move ⇒ t /andP [/andP [GEt LTt] _ ].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1126)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) j t1
t : nat
GEt : ppt <= t
LTt : t < t2
============================
match sched t with
| Some jlp => ~~ FP_to_JLFP Job Task jlp j
| None => false
end = 0
----------------------------------------------------------------------------- *)
case SCHED: (sched t) ⇒ [s | ]; last by done.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1192)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) j t1
t : nat
GEt : ppt <= t
LTt : t < t2
s : Job
SCHED : sched t = Some s
============================
~~ FP_to_JLFP Job Task 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 1296)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) j t1
t : nat
GEt : ppt <= t
LTt : t < t2
s : Job
SCHED : sched t = Some s
============================
exists pr_t : instant,
preemption_time.preemption_time sched pr_t /\
t1 <= pr_t <= t1 + (ppt - t1)
subgoal 2 (ID 1297) is:
t1 + (ppt - t1) <= t < t2
subgoal 3 (ID 1311) is:
~~ FP_to_JLFP Job Task s j = 0
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1296)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) j t1
t : nat
GEt : ppt <= t
LTt : t < t2
s : Job
SCHED : sched t = Some s
============================
exists pr_t : instant,
preemption_time.preemption_time sched pr_t /\
t1 <= pr_t <= t1 + (ppt - t1)
----------------------------------------------------------------------------- *)
by ∃ ppt; split; [done | rewrite subnKC //; apply/andP].
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 1297)
subgoal 1 (ID 1297) is:
t1 + (ppt - t1) <= t < t2
subgoal 2 (ID 1311) is:
~~ FP_to_JLFP Job Task s j = 0
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 1297)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) 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 1311) is:
~~ FP_to_JLFP Job Task s j = 0
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1297)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) 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 1311)
subgoal 1 (ID 1311) is:
~~ FP_to_JLFP Job Task s j = 0
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1311)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) 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 : FP_to_JLFP Job Task j_hp j
SCHEDHP : scheduled_at sched j_hp t
============================
~~ FP_to_JLFP Job Task s j = 0
----------------------------------------------------------------------------- *)
apply/eqP; rewrite eqb0 Bool.negb_involutive.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1529)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) 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 : FP_to_JLFP Job Task j_hp j
SCHEDHP : scheduled_at sched j_hp t
============================
FP_to_JLFP Job Task s j
----------------------------------------------------------------------------- *)
enough (EQef : s = j_hp); first by subst;auto.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1533)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) 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 : FP_to_JLFP Job Task 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 1559)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 (FP_to_JLFP Job Task) j t1 t2
BOUND : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
GE : t1 <= ppt
LE : ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq
(FP_to_JLFP Job Task) 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 : FP_to_JLFP Job Task 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 306)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 := fun A : duration =>
(A < L) && (task_rbf A != task_rbf (A + ε))
: duration -> 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 306)
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
higher_eq_priority : 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_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H6 : 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_is_bounded.max_length_of_priority_inversion
arr_seq (FP_to_JLFP Job Task)
: Job -> instant -> nat
task_rbf := task_request_bound_function tsk : duration -> nat
total_hep_rbf := total_hep_request_bound_function_FP higher_eq_priority ts
tsk : duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP higher_eq_priority
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 := fun A : duration =>
(A < L) && (task_rbf A != task_rbf (A + ε))
: duration -> 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.