Library rt.restructuring.analysis.edf.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.analysis.basic_facts.preemption Require Import
rtc_threshold.job_preemptable.
From rt.restructuring.analysis.facts Require Import priority_inversion_is_bounded.
From rt.restructuring.model.schedule Require Import
work_conserving priority_based.priorities priority_based.edf priority_based.preemption_aware.
From rt.restructuring.analysis.arrival Require Import workload_bound rbf.
From rt.restructuring.analysis.edf Require Export rta.response_time_bound.
From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq path fintype bigop.
RTA for EDF-schedulers with Bounded Non-Preemprive Segments
Consider any type of tasks ...
Context {Task : TaskType}.
Context `{TaskCost Task}.
Context `{TaskDeadline Task}.
Context `{TaskRunToCompletionThreshold Task}.
Context `{TaskMaxNonpreemptiveSegment Task}.
Context `{TaskCost Task}.
Context `{TaskDeadline 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}.
For clarity, let's denote the relative deadline of a task as D.
Consider the EDF policy that indicates a higher-or-equal priority relation.
Note that we do not relate the EDF policy with the scheduler. However, we
define functions for Interference and Interfering Workload that actively use
the concept of priorities.
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.
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.
We introduce as an abbreviation "rbf" for the task request bound function,
which is defined as [task_cost(T) × max_arrivals(T,Δ)] for a task T.
Next, we introduce task_rbf as an abbreviation for the task
request bound function of task tsk.
Using the sum of individual request bound functions, we define the request bound
function of all tasks (total request bound function).
Next, we define an upper bound on interfering workload received from jobs
of other tasks with higher-than-or-equal priority.
Let bound_on_total_hep_workload A Δ :=
\sum_(tsk_o <- ts | tsk_o != tsk)
rbf tsk_o (minn ((A + ε) + D tsk - D tsk_o) Δ).
\sum_(tsk_o <- ts | tsk_o != tsk)
rbf tsk_o (minn ((A + ε) + D tsk - D tsk_o) Δ).
Let's define some local names for clarity.
Let max_length_of_priority_inversion :=
max_length_of_priority_inversion arr_seq EDF.
Let task_rbf_changes_at A := task_rbf_changes_at tsk A.
Let bound_on_total_hep_workload_changes_at :=
bound_on_total_hep_workload_changes_at ts tsk.
Let response_time_bounded_by := task_response_time_bound arr_seq sched.
max_length_of_priority_inversion arr_seq EDF.
Let task_rbf_changes_at A := task_rbf_changes_at tsk A.
Let bound_on_total_hep_workload_changes_at :=
bound_on_total_hep_workload_changes_at 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_o <- ts | (tsk_o != tsk) && (D tsk < D tsk_o))
(task_max_nonpreemptive_segment tsk_o - ε).
\max_(tsk_o <- ts | (tsk_o != tsk) && (D tsk < D tsk_o))
(task_max_nonpreemptive_segment tsk_o - 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 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 →
t ≤ job_arrival j →
max_length_of_priority_inversion j t ≤ blocking_bound.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 306)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
============================
forall (j : Job) (t : nat),
arrives_in arr_seq j ->
job_task j = tsk ->
t <= job_arrival j ->
max_length_of_priority_inversion j t <= blocking_bound
----------------------------------------------------------------------------- *)
Proof.
intros j t ARR TSK LE; unfold max_length_of_priority_inversion, blocking_bound.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 313)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
============================
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq EDF
j t <=
\max_(tsk_o <- ts | (tsk_o != tsk) && (D tsk < D tsk_o))
(task_max_nonpreemptive_segment tsk_o - ε)
----------------------------------------------------------------------------- *)
apply leq_trans with
(\max_(j_lp <- arrivals_between arr_seq 0 t | ~~ EDF j_lp j)
(task_max_nonpreemptive_segment (job_task j_lp) - ε)).
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 325)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
============================
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq EDF
j t <=
\max_(j_lp <- arrivals_between arr_seq 0 t | ~~ EDF j_lp j)
(task_max_nonpreemptive_segment (job_task j_lp) - ε)
subgoal 2 (ID 326) is:
\max_(j_lp <- arrivals_between arr_seq 0 t | ~~ EDF j_lp j)
(task_max_nonpreemptive_segment (job_task j_lp) - ε) <=
\max_(tsk_o <- ts | (tsk_o != tsk) && (D tsk < D tsk_o))
(task_max_nonpreemptive_segment tsk_o - ε)
----------------------------------------------------------------------------- *)
- apply leq_big_max.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 327)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
============================
forall i : Job,
i \in arrivals_between arr_seq 0 t ->
~~ EDF i j ->
job_max_nonpreemptive_segment i - ε <=
task_max_nonpreemptive_segment (job_task i) - ε
subgoal 2 (ID 326) is:
\max_(j_lp <- arrivals_between arr_seq 0 t | ~~ EDF j_lp j)
(task_max_nonpreemptive_segment (job_task j_lp) - ε) <=
\max_(tsk_o <- ts | (tsk_o != tsk) && (D tsk < D tsk_o))
(task_max_nonpreemptive_segment tsk_o - ε)
----------------------------------------------------------------------------- *)
intros j' JINB NOTHEP.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 330)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NOTHEP : ~~ EDF j' j
============================
job_max_nonpreemptive_segment j' - ε <=
task_max_nonpreemptive_segment (job_task j') - ε
subgoal 2 (ID 326) is:
\max_(j_lp <- arrivals_between arr_seq 0 t | ~~ EDF j_lp j)
(task_max_nonpreemptive_segment (job_task j_lp) - ε) <=
\max_(tsk_o <- ts | (tsk_o != tsk) && (D tsk < D tsk_o))
(task_max_nonpreemptive_segment tsk_o - ε)
----------------------------------------------------------------------------- *)
rewrite leq_sub2r //.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 337)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NOTHEP : ~~ EDF j' j
============================
job_max_nonpreemptive_segment j' <=
task_max_nonpreemptive_segment (job_task j')
subgoal 2 (ID 326) is:
\max_(j_lp <- arrivals_between arr_seq 0 t | ~~ EDF j_lp j)
(task_max_nonpreemptive_segment (job_task j_lp) - ε) <=
\max_(tsk_o <- ts | (tsk_o != tsk) && (D tsk < D tsk_o))
(task_max_nonpreemptive_segment tsk_o - ε)
----------------------------------------------------------------------------- *)
apply in_arrivals_implies_arrived in JINB.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 361)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : arrives_in arr_seq j'
NOTHEP : ~~ EDF j' j
============================
job_max_nonpreemptive_segment j' <=
task_max_nonpreemptive_segment (job_task j')
subgoal 2 (ID 326) is:
\max_(j_lp <- arrivals_between arr_seq 0 t | ~~ EDF j_lp j)
(task_max_nonpreemptive_segment (job_task j_lp) - ε) <=
\max_(tsk_o <- ts | (tsk_o != tsk) && (D tsk < D tsk_o))
(task_max_nonpreemptive_segment tsk_o - ε)
----------------------------------------------------------------------------- *)
by apply H_valid_model_with_bounded_nonpreemptive_segments.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 326)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
============================
\max_(j_lp <- arrivals_between arr_seq 0 t | ~~ EDF j_lp j)
(task_max_nonpreemptive_segment (job_task j_lp) - ε) <=
\max_(tsk_o <- ts | (tsk_o != tsk) && (D tsk < D tsk_o))
(task_max_nonpreemptive_segment tsk_o - ε)
----------------------------------------------------------------------------- *)
- apply /bigmax_leq_seqP.
(* ----------------------------------[ coqtop ]---------------------------------
1 focused subgoal
(shelved: 1) (ID 404)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
============================
forall i : Job,
i \in arrivals_between arr_seq 0 t ->
~~ EDF i j ->
task_max_nonpreemptive_segment (job_task i) - ε <=
\max_(tsk_o <- ts | (tsk_o != tsk) && (D tsk < D tsk_o))
(task_max_nonpreemptive_segment tsk_o - ε)
----------------------------------------------------------------------------- *)
intros j' JINB NOTHEP.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 407)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NOTHEP : ~~ EDF j' j
============================
task_max_nonpreemptive_segment (job_task j') - ε <=
\max_(tsk_o <- ts | (tsk_o != tsk) && (D tsk < D tsk_o))
(task_max_nonpreemptive_segment tsk_o - ε)
----------------------------------------------------------------------------- *)
apply leq_bigmax_cond_seq with (i0 := (job_task j')) (F := fun tsk ⇒ task_max_nonpreemptive_segment tsk - 1).
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 416)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NOTHEP : ~~ EDF j' j
============================
job_task j' \in ts
subgoal 2 (ID 417) is:
(job_task j' != tsk) && (D tsk < D (job_task j'))
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 416)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NOTHEP : ~~ EDF j' j
============================
job_task j' \in ts
----------------------------------------------------------------------------- *)
apply H_all_jobs_from_taskset.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 418)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NOTHEP : ~~ EDF j' j
============================
arrives_in arr_seq j'
----------------------------------------------------------------------------- *)
apply mem_bigcat_nat_exists in JINB.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 419)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : exists i : nat, j' \in arrivals_at arr_seq i /\ 0 <= i < t
NOTHEP : ~~ EDF j' j
============================
arrives_in arr_seq j'
----------------------------------------------------------------------------- *)
by inversion JINB as [ta' [JIN' _]]; ∃ ta'.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 417)
subgoal 1 (ID 417) is:
(job_task j' != tsk) && (D tsk < D (job_task j'))
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 417)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NOTHEP : ~~ EDF j' j
============================
(job_task j' != tsk) && (D tsk < D (job_task j'))
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 417)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NOTHEP : ~~ EDF j' j
============================
(job_task j' != tsk) && (D tsk < D (job_task j'))
----------------------------------------------------------------------------- *)
have NINTSK: job_task j' != tsk.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 440)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NOTHEP : ~~ EDF j' j
============================
job_task j' != tsk
subgoal 2 (ID 442) is:
(job_task j' != tsk) && (D tsk < D (job_task j'))
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 440)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NOTHEP : ~~ EDF j' j
============================
job_task j' != tsk
----------------------------------------------------------------------------- *)
apply/eqP; intros TSKj'.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 482)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NOTHEP : ~~ EDF j' j
TSKj' : job_task j' = tsk
============================
False
----------------------------------------------------------------------------- *)
rewrite /EDF -ltnNge in NOTHEP.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 544)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
TSKj' : job_task j' = tsk
NOTHEP : job_deadline j < job_deadline j'
============================
False
----------------------------------------------------------------------------- *)
rewrite /job_deadline /job_deadline.job_deadline_from_task_deadline in NOTHEP.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 607)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
TSKj' : job_task j' = tsk
NOTHEP : job_arrival j + task_deadline (job_task j) <
job_arrival j' + task_deadline (job_task j')
============================
False
----------------------------------------------------------------------------- *)
rewrite TSKj' TSK ltn_add2r in NOTHEP.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 673)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
TSKj' : job_task j' = tsk
NOTHEP : job_arrival j < job_arrival j'
============================
False
----------------------------------------------------------------------------- *)
move: NOTHEP; rewrite ltnNge; move ⇒ /negP T; apply: T.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 711)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
TSKj' : job_task j' = tsk
============================
job_arrival j' <= job_arrival j
----------------------------------------------------------------------------- *)
apply leq_trans with t; last by done.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 712)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
TSKj' : job_task j' = tsk
============================
job_arrival j' <= t
----------------------------------------------------------------------------- *)
eapply in_arrivals_implies_arrived_between in JINB; last by eauto 2.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 716)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : arrived_between j' 0 t
TSKj' : job_task j' = tsk
============================
job_arrival j' <= t
----------------------------------------------------------------------------- *)
move: JINB; move ⇒ /andP [_ T].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 760)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
TSKj' : job_task j' = tsk
T : job_arrival j' < t
============================
job_arrival j' <= t
----------------------------------------------------------------------------- *)
by apply ltnW.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 442)
subgoal 1 (ID 442) is:
(job_task j' != tsk) && (D tsk < D (job_task j'))
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 442)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NOTHEP : ~~ EDF j' j
NINTSK : job_task j' != tsk
============================
(job_task j' != tsk) && (D tsk < D (job_task j'))
----------------------------------------------------------------------------- *)
apply/andP; split; first by done.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 788)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NOTHEP : ~~ EDF j' j
NINTSK : job_task j' != tsk
============================
D tsk < D (job_task j')
----------------------------------------------------------------------------- *)
rewrite /EDF -ltnNge in NOTHEP.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 850)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NINTSK : job_task j' != tsk
NOTHEP : job_deadline j < job_deadline j'
============================
D tsk < D (job_task j')
----------------------------------------------------------------------------- *)
rewrite -TSK.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 852)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NINTSK : job_task j' != tsk
NOTHEP : job_deadline j < job_deadline j'
============================
D (job_task j) < D (job_task j')
----------------------------------------------------------------------------- *)
have ARRLE: job_arrival j' < job_arrival j.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 857)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NINTSK : job_task j' != tsk
NOTHEP : job_deadline j < job_deadline j'
============================
job_arrival j' < job_arrival j
subgoal 2 (ID 859) is:
D (job_task j) < D (job_task j')
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 857)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NINTSK : job_task j' != tsk
NOTHEP : job_deadline j < job_deadline j'
============================
job_arrival j' < job_arrival j
----------------------------------------------------------------------------- *)
apply leq_trans with t; last by done.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 860)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NINTSK : job_task j' != tsk
NOTHEP : job_deadline j < job_deadline j'
============================
job_arrival j' < t
----------------------------------------------------------------------------- *)
eapply in_arrivals_implies_arrived_between in JINB; last by eauto 2.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 864)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : arrived_between j' 0 t
NINTSK : job_task j' != tsk
NOTHEP : job_deadline j < job_deadline j'
============================
job_arrival j' < t
----------------------------------------------------------------------------- *)
by move: JINB; move ⇒ /andP [_ T].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 859)
subgoal 1 (ID 859) is:
D (job_task j) < D (job_task j')
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 859)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NINTSK : job_task j' != tsk
NOTHEP : job_deadline j < job_deadline j'
ARRLE : job_arrival j' < job_arrival j
============================
D (job_task j) < D (job_task j')
----------------------------------------------------------------------------- *)
rewrite /job_deadline /job_deadline.job_deadline_from_task_deadline in NOTHEP.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 972)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NINTSK : job_task j' != tsk
ARRLE : job_arrival j' < job_arrival j
NOTHEP : job_arrival j + task_deadline (job_task j) <
job_arrival j' + task_deadline (job_task j')
============================
D (job_task j) < D (job_task j')
----------------------------------------------------------------------------- *)
rewrite /D; ssromega.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
∀ j t,
arrives_in arr_seq j →
job_task j = tsk →
t ≤ job_arrival j →
max_length_of_priority_inversion j t ≤ blocking_bound.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 306)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
============================
forall (j : Job) (t : nat),
arrives_in arr_seq j ->
job_task j = tsk ->
t <= job_arrival j ->
max_length_of_priority_inversion j t <= blocking_bound
----------------------------------------------------------------------------- *)
Proof.
intros j t ARR TSK LE; unfold max_length_of_priority_inversion, blocking_bound.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 313)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
============================
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq EDF
j t <=
\max_(tsk_o <- ts | (tsk_o != tsk) && (D tsk < D tsk_o))
(task_max_nonpreemptive_segment tsk_o - ε)
----------------------------------------------------------------------------- *)
apply leq_trans with
(\max_(j_lp <- arrivals_between arr_seq 0 t | ~~ EDF j_lp j)
(task_max_nonpreemptive_segment (job_task j_lp) - ε)).
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 325)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
============================
priority_inversion_is_bounded.max_length_of_priority_inversion arr_seq EDF
j t <=
\max_(j_lp <- arrivals_between arr_seq 0 t | ~~ EDF j_lp j)
(task_max_nonpreemptive_segment (job_task j_lp) - ε)
subgoal 2 (ID 326) is:
\max_(j_lp <- arrivals_between arr_seq 0 t | ~~ EDF j_lp j)
(task_max_nonpreemptive_segment (job_task j_lp) - ε) <=
\max_(tsk_o <- ts | (tsk_o != tsk) && (D tsk < D tsk_o))
(task_max_nonpreemptive_segment tsk_o - ε)
----------------------------------------------------------------------------- *)
- apply leq_big_max.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 327)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
============================
forall i : Job,
i \in arrivals_between arr_seq 0 t ->
~~ EDF i j ->
job_max_nonpreemptive_segment i - ε <=
task_max_nonpreemptive_segment (job_task i) - ε
subgoal 2 (ID 326) is:
\max_(j_lp <- arrivals_between arr_seq 0 t | ~~ EDF j_lp j)
(task_max_nonpreemptive_segment (job_task j_lp) - ε) <=
\max_(tsk_o <- ts | (tsk_o != tsk) && (D tsk < D tsk_o))
(task_max_nonpreemptive_segment tsk_o - ε)
----------------------------------------------------------------------------- *)
intros j' JINB NOTHEP.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 330)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NOTHEP : ~~ EDF j' j
============================
job_max_nonpreemptive_segment j' - ε <=
task_max_nonpreemptive_segment (job_task j') - ε
subgoal 2 (ID 326) is:
\max_(j_lp <- arrivals_between arr_seq 0 t | ~~ EDF j_lp j)
(task_max_nonpreemptive_segment (job_task j_lp) - ε) <=
\max_(tsk_o <- ts | (tsk_o != tsk) && (D tsk < D tsk_o))
(task_max_nonpreemptive_segment tsk_o - ε)
----------------------------------------------------------------------------- *)
rewrite leq_sub2r //.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 337)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NOTHEP : ~~ EDF j' j
============================
job_max_nonpreemptive_segment j' <=
task_max_nonpreemptive_segment (job_task j')
subgoal 2 (ID 326) is:
\max_(j_lp <- arrivals_between arr_seq 0 t | ~~ EDF j_lp j)
(task_max_nonpreemptive_segment (job_task j_lp) - ε) <=
\max_(tsk_o <- ts | (tsk_o != tsk) && (D tsk < D tsk_o))
(task_max_nonpreemptive_segment tsk_o - ε)
----------------------------------------------------------------------------- *)
apply in_arrivals_implies_arrived in JINB.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 361)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : arrives_in arr_seq j'
NOTHEP : ~~ EDF j' j
============================
job_max_nonpreemptive_segment j' <=
task_max_nonpreemptive_segment (job_task j')
subgoal 2 (ID 326) is:
\max_(j_lp <- arrivals_between arr_seq 0 t | ~~ EDF j_lp j)
(task_max_nonpreemptive_segment (job_task j_lp) - ε) <=
\max_(tsk_o <- ts | (tsk_o != tsk) && (D tsk < D tsk_o))
(task_max_nonpreemptive_segment tsk_o - ε)
----------------------------------------------------------------------------- *)
by apply H_valid_model_with_bounded_nonpreemptive_segments.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 326)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
============================
\max_(j_lp <- arrivals_between arr_seq 0 t | ~~ EDF j_lp j)
(task_max_nonpreemptive_segment (job_task j_lp) - ε) <=
\max_(tsk_o <- ts | (tsk_o != tsk) && (D tsk < D tsk_o))
(task_max_nonpreemptive_segment tsk_o - ε)
----------------------------------------------------------------------------- *)
- apply /bigmax_leq_seqP.
(* ----------------------------------[ coqtop ]---------------------------------
1 focused subgoal
(shelved: 1) (ID 404)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
============================
forall i : Job,
i \in arrivals_between arr_seq 0 t ->
~~ EDF i j ->
task_max_nonpreemptive_segment (job_task i) - ε <=
\max_(tsk_o <- ts | (tsk_o != tsk) && (D tsk < D tsk_o))
(task_max_nonpreemptive_segment tsk_o - ε)
----------------------------------------------------------------------------- *)
intros j' JINB NOTHEP.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 407)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NOTHEP : ~~ EDF j' j
============================
task_max_nonpreemptive_segment (job_task j') - ε <=
\max_(tsk_o <- ts | (tsk_o != tsk) && (D tsk < D tsk_o))
(task_max_nonpreemptive_segment tsk_o - ε)
----------------------------------------------------------------------------- *)
apply leq_bigmax_cond_seq with (i0 := (job_task j')) (F := fun tsk ⇒ task_max_nonpreemptive_segment tsk - 1).
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 416)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NOTHEP : ~~ EDF j' j
============================
job_task j' \in ts
subgoal 2 (ID 417) is:
(job_task j' != tsk) && (D tsk < D (job_task j'))
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 416)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NOTHEP : ~~ EDF j' j
============================
job_task j' \in ts
----------------------------------------------------------------------------- *)
apply H_all_jobs_from_taskset.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 418)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NOTHEP : ~~ EDF j' j
============================
arrives_in arr_seq j'
----------------------------------------------------------------------------- *)
apply mem_bigcat_nat_exists in JINB.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 419)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : exists i : nat, j' \in arrivals_at arr_seq i /\ 0 <= i < t
NOTHEP : ~~ EDF j' j
============================
arrives_in arr_seq j'
----------------------------------------------------------------------------- *)
by inversion JINB as [ta' [JIN' _]]; ∃ ta'.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 417)
subgoal 1 (ID 417) is:
(job_task j' != tsk) && (D tsk < D (job_task j'))
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 417)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NOTHEP : ~~ EDF j' j
============================
(job_task j' != tsk) && (D tsk < D (job_task j'))
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 417)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NOTHEP : ~~ EDF j' j
============================
(job_task j' != tsk) && (D tsk < D (job_task j'))
----------------------------------------------------------------------------- *)
have NINTSK: job_task j' != tsk.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 440)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NOTHEP : ~~ EDF j' j
============================
job_task j' != tsk
subgoal 2 (ID 442) is:
(job_task j' != tsk) && (D tsk < D (job_task j'))
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 440)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NOTHEP : ~~ EDF j' j
============================
job_task j' != tsk
----------------------------------------------------------------------------- *)
apply/eqP; intros TSKj'.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 482)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NOTHEP : ~~ EDF j' j
TSKj' : job_task j' = tsk
============================
False
----------------------------------------------------------------------------- *)
rewrite /EDF -ltnNge in NOTHEP.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 544)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
TSKj' : job_task j' = tsk
NOTHEP : job_deadline j < job_deadline j'
============================
False
----------------------------------------------------------------------------- *)
rewrite /job_deadline /job_deadline.job_deadline_from_task_deadline in NOTHEP.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 607)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
TSKj' : job_task j' = tsk
NOTHEP : job_arrival j + task_deadline (job_task j) <
job_arrival j' + task_deadline (job_task j')
============================
False
----------------------------------------------------------------------------- *)
rewrite TSKj' TSK ltn_add2r in NOTHEP.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 673)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
TSKj' : job_task j' = tsk
NOTHEP : job_arrival j < job_arrival j'
============================
False
----------------------------------------------------------------------------- *)
move: NOTHEP; rewrite ltnNge; move ⇒ /negP T; apply: T.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 711)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
TSKj' : job_task j' = tsk
============================
job_arrival j' <= job_arrival j
----------------------------------------------------------------------------- *)
apply leq_trans with t; last by done.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 712)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
TSKj' : job_task j' = tsk
============================
job_arrival j' <= t
----------------------------------------------------------------------------- *)
eapply in_arrivals_implies_arrived_between in JINB; last by eauto 2.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 716)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : arrived_between j' 0 t
TSKj' : job_task j' = tsk
============================
job_arrival j' <= t
----------------------------------------------------------------------------- *)
move: JINB; move ⇒ /andP [_ T].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 760)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
TSKj' : job_task j' = tsk
T : job_arrival j' < t
============================
job_arrival j' <= t
----------------------------------------------------------------------------- *)
by apply ltnW.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 442)
subgoal 1 (ID 442) is:
(job_task j' != tsk) && (D tsk < D (job_task j'))
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 442)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NOTHEP : ~~ EDF j' j
NINTSK : job_task j' != tsk
============================
(job_task j' != tsk) && (D tsk < D (job_task j'))
----------------------------------------------------------------------------- *)
apply/andP; split; first by done.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 788)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NOTHEP : ~~ EDF j' j
NINTSK : job_task j' != tsk
============================
D tsk < D (job_task j')
----------------------------------------------------------------------------- *)
rewrite /EDF -ltnNge in NOTHEP.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 850)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NINTSK : job_task j' != tsk
NOTHEP : job_deadline j < job_deadline j'
============================
D tsk < D (job_task j')
----------------------------------------------------------------------------- *)
rewrite -TSK.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 852)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NINTSK : job_task j' != tsk
NOTHEP : job_deadline j < job_deadline j'
============================
D (job_task j) < D (job_task j')
----------------------------------------------------------------------------- *)
have ARRLE: job_arrival j' < job_arrival j.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 857)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NINTSK : job_task j' != tsk
NOTHEP : job_deadline j < job_deadline j'
============================
job_arrival j' < job_arrival j
subgoal 2 (ID 859) is:
D (job_task j) < D (job_task j')
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 857)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NINTSK : job_task j' != tsk
NOTHEP : job_deadline j < job_deadline j'
============================
job_arrival j' < job_arrival j
----------------------------------------------------------------------------- *)
apply leq_trans with t; last by done.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 860)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NINTSK : job_task j' != tsk
NOTHEP : job_deadline j < job_deadline j'
============================
job_arrival j' < t
----------------------------------------------------------------------------- *)
eapply in_arrivals_implies_arrived_between in JINB; last by eauto 2.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 864)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : arrived_between j' 0 t
NINTSK : job_task j' != tsk
NOTHEP : job_deadline j < job_deadline j'
============================
job_arrival j' < t
----------------------------------------------------------------------------- *)
by move: JINB; move ⇒ /andP [_ T].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 859)
subgoal 1 (ID 859) is:
D (job_task j) < D (job_task j')
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 859)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NINTSK : job_task j' != tsk
NOTHEP : job_deadline j < job_deadline j'
ARRLE : job_arrival j' < job_arrival j
============================
D (job_task j) < D (job_task j')
----------------------------------------------------------------------------- *)
rewrite /job_deadline /job_deadline.job_deadline_from_task_deadline in NOTHEP.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 972)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
j : Job
t : nat
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
LE : t <= job_arrival j
j' : Job
JINB : j' \in arrivals_between arr_seq 0 t
NINTSK : job_task j' != tsk
ARRLE : job_arrival j' < job_arrival j
NOTHEP : job_arrival j + task_deadline (job_task j) <
job_arrival j' + task_deadline (job_task j')
============================
D (job_task j) < D (job_task j')
----------------------------------------------------------------------------- *)
rewrite /D; ssromega.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
Using the lemma above, we prove that the priority inversion of the task is bounded by
the maximum length of a nonpreemptive section of lower-priority tasks.
Lemma priority_inversion_is_bounded:
priority_inversion_is_bounded_by arr_seq sched _ tsk blocking_bound.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 321)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
============================
priority_inversion_is_bounded_by arr_seq sched (edf.EDF Task Job) tsk
blocking_bound
----------------------------------------------------------------------------- *)
Proof.
move ⇒ j ARR TSK POS t1 t2 PREF; move: (PREF) ⇒ [_ [_ [_ /andP [T _]]]].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 401)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
============================
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 t2 <=
blocking_bound
----------------------------------------------------------------------------- *)
destruct (leqP (t2 - t1) blocking_bound) as [NEQ|NEQ].
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 424)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : t2 - t1 <= blocking_bound
============================
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 t2 <=
blocking_bound
subgoal 2 (ID 425) is:
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 t2 <=
blocking_bound
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 424)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : t2 - t1 <= blocking_bound
============================
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 t2 <=
blocking_bound
----------------------------------------------------------------------------- *)
apply leq_trans with (t2 - t1); last by done.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 426)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : t2 - t1 <= blocking_bound
============================
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 t2 <= t2 - t1
----------------------------------------------------------------------------- *)
rewrite /cumulative_priority_inversion /is_priority_inversion.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 433)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : t2 - t1 <= blocking_bound
============================
\sum_(t1 <= t < t2)
match sched t with
| Some jlp => ~~ edf.EDF Task Job jlp j
| None => false
end <= t2 - t1
----------------------------------------------------------------------------- *)
rewrite -[X in _ ≤ X]addn0 -[t2 - t1]mul1n -iter_addn -big_const_nat.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 461)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : t2 - t1 <= blocking_bound
============================
\sum_(t1 <= t < t2)
match sched t with
| Some jlp => ~~ edf.EDF Task Job jlp j
| None => false
end <= \sum_(t1 <= i < t2) 1
----------------------------------------------------------------------------- *)
rewrite leq_sum //.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 470)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : t2 - t1 <= blocking_bound
============================
forall i : nat,
true ->
match sched i with
| Some jlp => ~~ edf.EDF Task Job jlp j
| None => false
end <= 1
----------------------------------------------------------------------------- *)
intros t _; case: (sched t); last by done.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 506)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : t2 - t1 <= blocking_bound
t : nat
============================
forall a : Job, ~~ edf.EDF Task Job a j <= 1
----------------------------------------------------------------------------- *)
by intros s; case: edf.EDF.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 425)
subgoal 1 (ID 425) is:
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 t2 <=
blocking_bound
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 425)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : blocking_bound < t2 - t1
============================
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 t2 <=
blocking_bound
----------------------------------------------------------------------------- *)
edestruct @preemption_time_exists as [ppt [PPT NEQ2]]; eauto 2 with basic_facts.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 589)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
NEQ2 : t1 <= ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq (edf.EDF Task Job) j t1
============================
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 t2 <=
blocking_bound
----------------------------------------------------------------------------- *)
move: NEQ2 ⇒ /andP [GE LE].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 651)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) j t1
============================
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 t2 <=
blocking_bound
----------------------------------------------------------------------------- *)
apply leq_trans with (cumulative_priority_inversion sched _ j t1 ppt);
last apply leq_trans with (ppt - t1).
(* ----------------------------------[ coqtop ]---------------------------------
3 subgoals (ID 662)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) j t1
============================
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 t2 <=
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 ppt
subgoal 2 (ID 664) is:
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 ppt <= ppt - t1
subgoal 3 (ID 665) is:
ppt - t1 <= blocking_bound
----------------------------------------------------------------------------- *)
- rewrite /cumulative_priority_inversion /is_priority_inversion.
(* ----------------------------------[ coqtop ]---------------------------------
3 subgoals (ID 671)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) j t1
============================
\sum_(t1 <= t < t2)
match sched t with
| Some jlp => ~~ edf.EDF Task Job jlp j
| None => false
end <=
\sum_(t1 <= t < ppt)
match sched t with
| Some jlp => ~~ edf.EDF Task Job jlp j
| None => false
end
subgoal 2 (ID 664) is:
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 ppt <= ppt - t1
subgoal 3 (ID 665) is:
ppt - t1 <= blocking_bound
----------------------------------------------------------------------------- *)
rewrite (@big_cat_nat _ _ _ ppt) //=; last first.
(* ----------------------------------[ coqtop ]---------------------------------
4 subgoals (ID 718)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) j t1
============================
ppt <= t2
subgoal 2 (ID 694) is:
\sum_(t1 <= i < ppt)
match sched i with
| Some jlp => ~~ edf.EDF Task Job jlp j
| None => false
end +
\sum_(ppt <= i < t2)
match sched i with
| Some jlp => ~~ edf.EDF Task Job jlp j
| None => false
end <=
\sum_(t1 <= t < ppt)
match sched t with
| Some jlp => ~~ edf.EDF Task Job jlp j
| None => false
end
subgoal 3 (ID 664) is:
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 ppt <= ppt - t1
subgoal 4 (ID 665) is:
ppt - t1 <= blocking_bound
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 718)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) j t1
============================
ppt <= t2
----------------------------------------------------------------------------- *)
rewrite ltn_subRL in NEQ.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 806)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
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
(edf.EDF Task Job) j t1
NEQ : t1 + blocking_bound < t2
============================
ppt <= t2
----------------------------------------------------------------------------- *)
apply leq_trans with (t1 + blocking_bound); last by apply ltnW.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 807)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
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
(edf.EDF Task Job) j t1
NEQ : 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 811)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
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
(edf.EDF Task Job) j t1
NEQ : t1 + blocking_bound < t2
============================
t1 + max_length_of_priority_inversion j t1 <= t1 + blocking_bound
----------------------------------------------------------------------------- *)
by rewrite leq_add2l; eapply priority_inversion_is_bounded_by_blocking; eauto 2.
(* ----------------------------------[ coqtop ]---------------------------------
3 subgoals (ID 694)
subgoal 1 (ID 694) is:
\sum_(t1 <= i < ppt)
match sched i with
| Some jlp => ~~ edf.EDF Task Job jlp j
| None => false
end +
\sum_(ppt <= i < t2)
match sched i with
| Some jlp => ~~ edf.EDF Task Job jlp j
| None => false
end <=
\sum_(t1 <= t < ppt)
match sched t with
| Some jlp => ~~ edf.EDF Task Job jlp j
| None => false
end
subgoal 2 (ID 664) is:
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 ppt <= ppt - t1
subgoal 3 (ID 665) is:
ppt - t1 <= blocking_bound
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
3 subgoals (ID 694)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) j t1
============================
\sum_(t1 <= i < ppt)
match sched i with
| Some jlp => ~~ edf.EDF Task Job jlp j
| None => false
end +
\sum_(ppt <= i < t2)
match sched i with
| Some jlp => ~~ edf.EDF Task Job jlp j
| None => false
end <=
\sum_(t1 <= t < ppt)
match sched t with
| Some jlp => ~~ edf.EDF Task Job jlp j
| None => false
end
subgoal 2 (ID 664) is:
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 ppt <= ppt - t1
subgoal 3 (ID 665) is:
ppt - t1 <= blocking_bound
----------------------------------------------------------------------------- *)
rewrite -[X in _ ≤ X]addn0 leq_add2l leqn0.
(* ----------------------------------[ coqtop ]---------------------------------
3 subgoals (ID 843)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) j t1
============================
\sum_(ppt <= i < t2)
match sched i with
| Some jlp => ~~ edf.EDF Task Job jlp j
| None => false
end == 0
subgoal 2 (ID 664) is:
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 ppt <= ppt - t1
subgoal 3 (ID 665) is:
ppt - t1 <= blocking_bound
----------------------------------------------------------------------------- *)
rewrite big_nat_cond big1 //; move ⇒ t /andP [/andP [GEt LTt] _ ].
(* ----------------------------------[ coqtop ]---------------------------------
3 subgoals (ID 968)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) j t1
t : nat
GEt : ppt <= t
LTt : t < t2
============================
match sched t with
| Some jlp => ~~ edf.EDF Task Job jlp j
| None => false
end = 0
subgoal 2 (ID 664) is:
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 ppt <= ppt - t1
subgoal 3 (ID 665) is:
ppt - t1 <= blocking_bound
----------------------------------------------------------------------------- *)
case SCHED: (sched t) ⇒ [s | ]; last by done.
(* ----------------------------------[ coqtop ]---------------------------------
3 subgoals (ID 1038)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) j t1
t : nat
GEt : ppt <= t
LTt : t < t2
s : Job
SCHED : sched t = Some s
============================
~~ edf.EDF Task Job s j = 0
subgoal 2 (ID 664) is:
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 ppt <= ppt - t1
subgoal 3 (ID 665) is:
ppt - t1 <= blocking_bound
----------------------------------------------------------------------------- *)
edestruct @not_quiet_implies_exists_scheduled_hp_job
with (K := ppt - t1) (t := t) as [j_hp [ARRB [HP SCHEDHP]]]; eauto 2 with basic_facts.
(* ----------------------------------[ coqtop ]---------------------------------
5 subgoals (ID 1150)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) 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 1151) is:
t1 + (ppt - t1) <= t < t2
subgoal 3 (ID 1165) is:
~~ edf.EDF Task Job s j = 0
subgoal 4 (ID 664) is:
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 ppt <= ppt - t1
subgoal 5 (ID 665) is:
ppt - t1 <= blocking_bound
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1150)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) 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)
----------------------------------------------------------------------------- *)
∃ ppt; split.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 1191)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) j t1
t : nat
GEt : ppt <= t
LTt : t < t2
s : Job
SCHED : sched t = Some s
============================
preemption_time.preemption_time sched ppt
subgoal 2 (ID 1192) is:
t1 <= ppt <= t1 + (ppt - t1)
----------------------------------------------------------------------------- *)
by done.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1192)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) j t1
t : nat
GEt : ppt <= t
LTt : t < t2
s : Job
SCHED : sched t = Some s
============================
t1 <= ppt <= t1 + (ppt - t1)
----------------------------------------------------------------------------- *)
by rewrite subnKC //; apply/andP; split.
(* ----------------------------------[ coqtop ]---------------------------------
4 subgoals (ID 1151)
subgoal 1 (ID 1151) is:
t1 + (ppt - t1) <= t < t2
subgoal 2 (ID 1165) is:
~~ edf.EDF Task Job s j = 0
subgoal 3 (ID 664) is:
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 ppt <= ppt - t1
subgoal 4 (ID 665) is:
ppt - t1 <= blocking_bound
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
4 subgoals (ID 1151)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) 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 1165) is:
~~ edf.EDF Task Job s j = 0
subgoal 3 (ID 664) is:
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 ppt <= ppt - t1
subgoal 4 (ID 665) is:
ppt - t1 <= blocking_bound
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1151)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) 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 ]---------------------------------
3 subgoals (ID 1165)
subgoal 1 (ID 1165) is:
~~ edf.EDF Task Job s j = 0
subgoal 2 (ID 664) is:
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 ppt <= ppt - t1
subgoal 3 (ID 665) is:
ppt - t1 <= blocking_bound
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
3 subgoals (ID 1165)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) 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 : edf.EDF Task Job j_hp j
SCHEDHP : scheduled_at sched j_hp t
============================
~~ edf.EDF Task Job s j = 0
subgoal 2 (ID 664) is:
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 ppt <= ppt - t1
subgoal 3 (ID 665) is:
ppt - t1 <= blocking_bound
----------------------------------------------------------------------------- *)
apply/eqP; rewrite eqb0 Bool.negb_involutive.
(* ----------------------------------[ coqtop ]---------------------------------
3 subgoals (ID 1363)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) 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 : edf.EDF Task Job j_hp j
SCHEDHP : scheduled_at sched j_hp t
============================
edf.EDF Task Job s j
subgoal 2 (ID 664) is:
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 ppt <= ppt - t1
subgoal 3 (ID 665) is:
ppt - t1 <= blocking_bound
----------------------------------------------------------------------------- *)
enough (EQ : s = j_hp); first by subst.
(* ----------------------------------[ coqtop ]---------------------------------
3 subgoals (ID 1367)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) 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 : edf.EDF Task Job j_hp j
SCHEDHP : scheduled_at sched j_hp t
============================
s = j_hp
subgoal 2 (ID 664) is:
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 ppt <= ppt - t1
subgoal 3 (ID 665) is:
ppt - t1 <= blocking_bound
----------------------------------------------------------------------------- *)
move: SCHED ⇒ /eqP SCHED; rewrite -scheduled_at_def in SCHED.
(* ----------------------------------[ coqtop ]---------------------------------
3 subgoals (ID 1530)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) j t1
t : nat
GEt : ppt <= t
LTt : t < t2
s, j_hp : Job
ARRB : arrived_between j_hp t1 (succn t)
HP : edf.EDF Task Job j_hp j
SCHEDHP : scheduled_at sched j_hp t
SCHED : scheduled_at sched s t
============================
s = j_hp
subgoal 2 (ID 664) is:
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 ppt <= ppt - t1
subgoal 3 (ID 665) is:
ppt - t1 <= blocking_bound
----------------------------------------------------------------------------- *)
by eapply ideal_proc_model_is_a_uniprocessor_model; [exact SCHED | exact SCHEDHP].
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 664)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) j t1
============================
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 ppt <= ppt - t1
subgoal 2 (ID 665) is:
ppt - t1 <= blocking_bound
----------------------------------------------------------------------------- *)
- rewrite /cumulative_priority_inversion /is_priority_inversion.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 1541)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) j t1
============================
\sum_(t1 <= t < ppt)
match sched t with
| Some jlp => ~~ edf.EDF Task Job jlp j
| None => false
end <= ppt - t1
subgoal 2 (ID 665) is:
ppt - t1 <= blocking_bound
----------------------------------------------------------------------------- *)
rewrite -[X in _ ≤ X]addn0 -[ppt - t1]mul1n -iter_addn -big_const_nat.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 1569)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) j t1
============================
\sum_(t1 <= t < ppt)
match sched t with
| Some jlp => ~~ edf.EDF Task Job jlp j
| None => false
end <= \sum_(t1 <= i < ppt) 1
subgoal 2 (ID 665) is:
ppt - t1 <= blocking_bound
----------------------------------------------------------------------------- *)
rewrite leq_sum //.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 1578)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) j t1
============================
forall i : nat,
true ->
match sched i with
| Some jlp => ~~ edf.EDF Task Job jlp j
| None => false
end <= 1
subgoal 2 (ID 665) is:
ppt - t1 <= blocking_bound
----------------------------------------------------------------------------- *)
intros t _; case: (sched t); last by done.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 1614)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) j t1
t : nat
============================
forall a : Job, ~~ edf.EDF Task Job a j <= 1
subgoal 2 (ID 665) is:
ppt - t1 <= blocking_bound
----------------------------------------------------------------------------- *)
by intros s; case: edf.EDF.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 665)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) j t1
============================
ppt - t1 <= blocking_bound
----------------------------------------------------------------------------- *)
- rewrite leq_subLR.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1642)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) j t1
============================
ppt <= t1 + blocking_bound
----------------------------------------------------------------------------- *)
apply leq_trans with (t1 + max_length_of_priority_inversion j t1); first by done.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1644)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) j t1
============================
t1 + max_length_of_priority_inversion j t1 <= t1 + blocking_bound
----------------------------------------------------------------------------- *)
rewrite leq_add2l; eapply priority_inversion_is_bounded_by_blocking; eauto 2.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
End PriorityInversionIsBounded.
priority_inversion_is_bounded_by arr_seq sched _ tsk blocking_bound.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 321)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
============================
priority_inversion_is_bounded_by arr_seq sched (edf.EDF Task Job) tsk
blocking_bound
----------------------------------------------------------------------------- *)
Proof.
move ⇒ j ARR TSK POS t1 t2 PREF; move: (PREF) ⇒ [_ [_ [_ /andP [T _]]]].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 401)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
============================
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 t2 <=
blocking_bound
----------------------------------------------------------------------------- *)
destruct (leqP (t2 - t1) blocking_bound) as [NEQ|NEQ].
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 424)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : t2 - t1 <= blocking_bound
============================
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 t2 <=
blocking_bound
subgoal 2 (ID 425) is:
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 t2 <=
blocking_bound
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 424)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : t2 - t1 <= blocking_bound
============================
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 t2 <=
blocking_bound
----------------------------------------------------------------------------- *)
apply leq_trans with (t2 - t1); last by done.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 426)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : t2 - t1 <= blocking_bound
============================
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 t2 <= t2 - t1
----------------------------------------------------------------------------- *)
rewrite /cumulative_priority_inversion /is_priority_inversion.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 433)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : t2 - t1 <= blocking_bound
============================
\sum_(t1 <= t < t2)
match sched t with
| Some jlp => ~~ edf.EDF Task Job jlp j
| None => false
end <= t2 - t1
----------------------------------------------------------------------------- *)
rewrite -[X in _ ≤ X]addn0 -[t2 - t1]mul1n -iter_addn -big_const_nat.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 461)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : t2 - t1 <= blocking_bound
============================
\sum_(t1 <= t < t2)
match sched t with
| Some jlp => ~~ edf.EDF Task Job jlp j
| None => false
end <= \sum_(t1 <= i < t2) 1
----------------------------------------------------------------------------- *)
rewrite leq_sum //.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 470)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : t2 - t1 <= blocking_bound
============================
forall i : nat,
true ->
match sched i with
| Some jlp => ~~ edf.EDF Task Job jlp j
| None => false
end <= 1
----------------------------------------------------------------------------- *)
intros t _; case: (sched t); last by done.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 506)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : t2 - t1 <= blocking_bound
t : nat
============================
forall a : Job, ~~ edf.EDF Task Job a j <= 1
----------------------------------------------------------------------------- *)
by intros s; case: edf.EDF.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 425)
subgoal 1 (ID 425) is:
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 t2 <=
blocking_bound
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 425)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : blocking_bound < t2 - t1
============================
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 t2 <=
blocking_bound
----------------------------------------------------------------------------- *)
edestruct @preemption_time_exists as [ppt [PPT NEQ2]]; eauto 2 with basic_facts.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 589)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : blocking_bound < t2 - t1
ppt : instant
PPT : preemption_time.preemption_time sched ppt
NEQ2 : t1 <= ppt <=
t1 +
priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq (edf.EDF Task Job) j t1
============================
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 t2 <=
blocking_bound
----------------------------------------------------------------------------- *)
move: NEQ2 ⇒ /andP [GE LE].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 651)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) j t1
============================
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 t2 <=
blocking_bound
----------------------------------------------------------------------------- *)
apply leq_trans with (cumulative_priority_inversion sched _ j t1 ppt);
last apply leq_trans with (ppt - t1).
(* ----------------------------------[ coqtop ]---------------------------------
3 subgoals (ID 662)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) j t1
============================
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 t2 <=
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 ppt
subgoal 2 (ID 664) is:
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 ppt <= ppt - t1
subgoal 3 (ID 665) is:
ppt - t1 <= blocking_bound
----------------------------------------------------------------------------- *)
- rewrite /cumulative_priority_inversion /is_priority_inversion.
(* ----------------------------------[ coqtop ]---------------------------------
3 subgoals (ID 671)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) j t1
============================
\sum_(t1 <= t < t2)
match sched t with
| Some jlp => ~~ edf.EDF Task Job jlp j
| None => false
end <=
\sum_(t1 <= t < ppt)
match sched t with
| Some jlp => ~~ edf.EDF Task Job jlp j
| None => false
end
subgoal 2 (ID 664) is:
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 ppt <= ppt - t1
subgoal 3 (ID 665) is:
ppt - t1 <= blocking_bound
----------------------------------------------------------------------------- *)
rewrite (@big_cat_nat _ _ _ ppt) //=; last first.
(* ----------------------------------[ coqtop ]---------------------------------
4 subgoals (ID 718)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) j t1
============================
ppt <= t2
subgoal 2 (ID 694) is:
\sum_(t1 <= i < ppt)
match sched i with
| Some jlp => ~~ edf.EDF Task Job jlp j
| None => false
end +
\sum_(ppt <= i < t2)
match sched i with
| Some jlp => ~~ edf.EDF Task Job jlp j
| None => false
end <=
\sum_(t1 <= t < ppt)
match sched t with
| Some jlp => ~~ edf.EDF Task Job jlp j
| None => false
end
subgoal 3 (ID 664) is:
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 ppt <= ppt - t1
subgoal 4 (ID 665) is:
ppt - t1 <= blocking_bound
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 718)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) j t1
============================
ppt <= t2
----------------------------------------------------------------------------- *)
rewrite ltn_subRL in NEQ.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 806)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
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
(edf.EDF Task Job) j t1
NEQ : t1 + blocking_bound < t2
============================
ppt <= t2
----------------------------------------------------------------------------- *)
apply leq_trans with (t1 + blocking_bound); last by apply ltnW.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 807)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
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
(edf.EDF Task Job) j t1
NEQ : 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 811)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
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
(edf.EDF Task Job) j t1
NEQ : t1 + blocking_bound < t2
============================
t1 + max_length_of_priority_inversion j t1 <= t1 + blocking_bound
----------------------------------------------------------------------------- *)
by rewrite leq_add2l; eapply priority_inversion_is_bounded_by_blocking; eauto 2.
(* ----------------------------------[ coqtop ]---------------------------------
3 subgoals (ID 694)
subgoal 1 (ID 694) is:
\sum_(t1 <= i < ppt)
match sched i with
| Some jlp => ~~ edf.EDF Task Job jlp j
| None => false
end +
\sum_(ppt <= i < t2)
match sched i with
| Some jlp => ~~ edf.EDF Task Job jlp j
| None => false
end <=
\sum_(t1 <= t < ppt)
match sched t with
| Some jlp => ~~ edf.EDF Task Job jlp j
| None => false
end
subgoal 2 (ID 664) is:
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 ppt <= ppt - t1
subgoal 3 (ID 665) is:
ppt - t1 <= blocking_bound
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
3 subgoals (ID 694)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) j t1
============================
\sum_(t1 <= i < ppt)
match sched i with
| Some jlp => ~~ edf.EDF Task Job jlp j
| None => false
end +
\sum_(ppt <= i < t2)
match sched i with
| Some jlp => ~~ edf.EDF Task Job jlp j
| None => false
end <=
\sum_(t1 <= t < ppt)
match sched t with
| Some jlp => ~~ edf.EDF Task Job jlp j
| None => false
end
subgoal 2 (ID 664) is:
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 ppt <= ppt - t1
subgoal 3 (ID 665) is:
ppt - t1 <= blocking_bound
----------------------------------------------------------------------------- *)
rewrite -[X in _ ≤ X]addn0 leq_add2l leqn0.
(* ----------------------------------[ coqtop ]---------------------------------
3 subgoals (ID 843)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) j t1
============================
\sum_(ppt <= i < t2)
match sched i with
| Some jlp => ~~ edf.EDF Task Job jlp j
| None => false
end == 0
subgoal 2 (ID 664) is:
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 ppt <= ppt - t1
subgoal 3 (ID 665) is:
ppt - t1 <= blocking_bound
----------------------------------------------------------------------------- *)
rewrite big_nat_cond big1 //; move ⇒ t /andP [/andP [GEt LTt] _ ].
(* ----------------------------------[ coqtop ]---------------------------------
3 subgoals (ID 968)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) j t1
t : nat
GEt : ppt <= t
LTt : t < t2
============================
match sched t with
| Some jlp => ~~ edf.EDF Task Job jlp j
| None => false
end = 0
subgoal 2 (ID 664) is:
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 ppt <= ppt - t1
subgoal 3 (ID 665) is:
ppt - t1 <= blocking_bound
----------------------------------------------------------------------------- *)
case SCHED: (sched t) ⇒ [s | ]; last by done.
(* ----------------------------------[ coqtop ]---------------------------------
3 subgoals (ID 1038)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) j t1
t : nat
GEt : ppt <= t
LTt : t < t2
s : Job
SCHED : sched t = Some s
============================
~~ edf.EDF Task Job s j = 0
subgoal 2 (ID 664) is:
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 ppt <= ppt - t1
subgoal 3 (ID 665) is:
ppt - t1 <= blocking_bound
----------------------------------------------------------------------------- *)
edestruct @not_quiet_implies_exists_scheduled_hp_job
with (K := ppt - t1) (t := t) as [j_hp [ARRB [HP SCHEDHP]]]; eauto 2 with basic_facts.
(* ----------------------------------[ coqtop ]---------------------------------
5 subgoals (ID 1150)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) 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 1151) is:
t1 + (ppt - t1) <= t < t2
subgoal 3 (ID 1165) is:
~~ edf.EDF Task Job s j = 0
subgoal 4 (ID 664) is:
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 ppt <= ppt - t1
subgoal 5 (ID 665) is:
ppt - t1 <= blocking_bound
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1150)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) 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)
----------------------------------------------------------------------------- *)
∃ ppt; split.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 1191)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) j t1
t : nat
GEt : ppt <= t
LTt : t < t2
s : Job
SCHED : sched t = Some s
============================
preemption_time.preemption_time sched ppt
subgoal 2 (ID 1192) is:
t1 <= ppt <= t1 + (ppt - t1)
----------------------------------------------------------------------------- *)
by done.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1192)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) j t1
t : nat
GEt : ppt <= t
LTt : t < t2
s : Job
SCHED : sched t = Some s
============================
t1 <= ppt <= t1 + (ppt - t1)
----------------------------------------------------------------------------- *)
by rewrite subnKC //; apply/andP; split.
(* ----------------------------------[ coqtop ]---------------------------------
4 subgoals (ID 1151)
subgoal 1 (ID 1151) is:
t1 + (ppt - t1) <= t < t2
subgoal 2 (ID 1165) is:
~~ edf.EDF Task Job s j = 0
subgoal 3 (ID 664) is:
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 ppt <= ppt - t1
subgoal 4 (ID 665) is:
ppt - t1 <= blocking_bound
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
4 subgoals (ID 1151)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) 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 1165) is:
~~ edf.EDF Task Job s j = 0
subgoal 3 (ID 664) is:
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 ppt <= ppt - t1
subgoal 4 (ID 665) is:
ppt - t1 <= blocking_bound
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1151)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) 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 ]---------------------------------
3 subgoals (ID 1165)
subgoal 1 (ID 1165) is:
~~ edf.EDF Task Job s j = 0
subgoal 2 (ID 664) is:
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 ppt <= ppt - t1
subgoal 3 (ID 665) is:
ppt - t1 <= blocking_bound
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
3 subgoals (ID 1165)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) 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 : edf.EDF Task Job j_hp j
SCHEDHP : scheduled_at sched j_hp t
============================
~~ edf.EDF Task Job s j = 0
subgoal 2 (ID 664) is:
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 ppt <= ppt - t1
subgoal 3 (ID 665) is:
ppt - t1 <= blocking_bound
----------------------------------------------------------------------------- *)
apply/eqP; rewrite eqb0 Bool.negb_involutive.
(* ----------------------------------[ coqtop ]---------------------------------
3 subgoals (ID 1363)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) 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 : edf.EDF Task Job j_hp j
SCHEDHP : scheduled_at sched j_hp t
============================
edf.EDF Task Job s j
subgoal 2 (ID 664) is:
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 ppt <= ppt - t1
subgoal 3 (ID 665) is:
ppt - t1 <= blocking_bound
----------------------------------------------------------------------------- *)
enough (EQ : s = j_hp); first by subst.
(* ----------------------------------[ coqtop ]---------------------------------
3 subgoals (ID 1367)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) 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 : edf.EDF Task Job j_hp j
SCHEDHP : scheduled_at sched j_hp t
============================
s = j_hp
subgoal 2 (ID 664) is:
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 ppt <= ppt - t1
subgoal 3 (ID 665) is:
ppt - t1 <= blocking_bound
----------------------------------------------------------------------------- *)
move: SCHED ⇒ /eqP SCHED; rewrite -scheduled_at_def in SCHED.
(* ----------------------------------[ coqtop ]---------------------------------
3 subgoals (ID 1530)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) j t1
t : nat
GEt : ppt <= t
LTt : t < t2
s, j_hp : Job
ARRB : arrived_between j_hp t1 (succn t)
HP : edf.EDF Task Job j_hp j
SCHEDHP : scheduled_at sched j_hp t
SCHED : scheduled_at sched s t
============================
s = j_hp
subgoal 2 (ID 664) is:
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 ppt <= ppt - t1
subgoal 3 (ID 665) is:
ppt - t1 <= blocking_bound
----------------------------------------------------------------------------- *)
by eapply ideal_proc_model_is_a_uniprocessor_model; [exact SCHED | exact SCHEDHP].
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 664)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) j t1
============================
cumulative_priority_inversion sched (edf.EDF Task Job) j t1 ppt <= ppt - t1
subgoal 2 (ID 665) is:
ppt - t1 <= blocking_bound
----------------------------------------------------------------------------- *)
- rewrite /cumulative_priority_inversion /is_priority_inversion.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 1541)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) j t1
============================
\sum_(t1 <= t < ppt)
match sched t with
| Some jlp => ~~ edf.EDF Task Job jlp j
| None => false
end <= ppt - t1
subgoal 2 (ID 665) is:
ppt - t1 <= blocking_bound
----------------------------------------------------------------------------- *)
rewrite -[X in _ ≤ X]addn0 -[ppt - t1]mul1n -iter_addn -big_const_nat.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 1569)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) j t1
============================
\sum_(t1 <= t < ppt)
match sched t with
| Some jlp => ~~ edf.EDF Task Job jlp j
| None => false
end <= \sum_(t1 <= i < ppt) 1
subgoal 2 (ID 665) is:
ppt - t1 <= blocking_bound
----------------------------------------------------------------------------- *)
rewrite leq_sum //.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 1578)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) j t1
============================
forall i : nat,
true ->
match sched i with
| Some jlp => ~~ edf.EDF Task Job jlp j
| None => false
end <= 1
subgoal 2 (ID 665) is:
ppt - t1 <= blocking_bound
----------------------------------------------------------------------------- *)
intros t _; case: (sched t); last by done.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 1614)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) j t1
t : nat
============================
forall a : Job, ~~ edf.EDF Task Job a j <= 1
subgoal 2 (ID 665) is:
ppt - t1 <= blocking_bound
----------------------------------------------------------------------------- *)
by intros s; case: edf.EDF.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 665)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) j t1
============================
ppt - t1 <= blocking_bound
----------------------------------------------------------------------------- *)
- rewrite leq_subLR.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1642)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) j t1
============================
ppt <= t1 + blocking_bound
----------------------------------------------------------------------------- *)
apply leq_trans with (t1 + max_length_of_priority_inversion j t1); first by done.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1644)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 (edf.EDF Task Job) j t1 t2
T : t1 <= job_arrival j
NEQ : 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
(edf.EDF Task Job) j t1
============================
t1 + max_length_of_priority_inversion j t1 <= t1 + blocking_bound
----------------------------------------------------------------------------- *)
rewrite leq_add2l; eapply priority_inversion_is_bounded_by_blocking; eauto 2.
(* ----------------------------------[ 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.
To reduce the time complexity of the analysis, recall the notion of search space.
Let is_in_search_space A :=
(A < L) && (task_rbf_changes_at A || bound_on_total_hep_workload_changes_at A).
(A < L) && (task_rbf_changes_at A || bound_on_total_hep_workload_changes_at A).
Consider any value R, and assume that for any given arrival offset A in the search space,
there is a solution of the response-time bound recurrence which is bounded by R.
Variable R : duration.
Hypothesis H_R_is_maximum:
∀ (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))
+ bound_on_total_hep_workload A (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))
+ bound_on_total_hep_workload A (A + F) ∧
F + (task_cost tsk - task_run_to_completion_threshold tsk) ≤ R.
Then, using the results for the general RTA for EDF-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 EDF-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_edf_with_bounded_nonpreemptive_segments:
response_time_bounded_by tsk R.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 310)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 = total_rbf L
is_in_search_space := fun A : nat =>
(A < L) &&
(task_rbf_changes_at A
|| bound_on_total_hep_workload_changes_at A)
: nat -> bool
R : duration
H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
A + F =
blocking_bound +
(task_rbf (A + ε) -
(task_cost tsk - task_run_to_completion_threshold tsk)) +
bound_on_total_hep_workload A (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_edf; eauto 2.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 338)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 = total_rbf L
is_in_search_space := fun A : nat =>
(A < L) &&
(task_rbf_changes_at A
|| bound_on_total_hep_workload_changes_at A)
: nat -> bool
R : duration
H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
A + F =
blocking_bound +
(task_rbf (A + ε) -
(task_cost tsk - task_run_to_completion_threshold tsk)) +
bound_on_total_hep_workload A (A + F) /\
F +
(task_cost tsk - task_run_to_completion_threshold tsk) <=
R
============================
priority_inversion_is_bounded_by arr_seq sched (edf.EDF Task Job) tsk
blocking_bound
----------------------------------------------------------------------------- *)
by apply priority_inversion_is_bounded.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
End ResponseTimeBound.
End RTAforEDFwithBoundedNonpreemptiveSegmentsWithArrivalCurves.
response_time_bounded_by tsk R.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 310)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 = total_rbf L
is_in_search_space := fun A : nat =>
(A < L) &&
(task_rbf_changes_at A
|| bound_on_total_hep_workload_changes_at A)
: nat -> bool
R : duration
H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
A + F =
blocking_bound +
(task_rbf (A + ε) -
(task_cost tsk - task_run_to_completion_threshold tsk)) +
bound_on_total_hep_workload A (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_edf; eauto 2.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 338)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
H1 : TaskRunToCompletionThreshold Task
H2 : TaskMaxNonpreemptiveSegment Task
Job : JobType
H3 : JobTask Job Task
H4 : JobArrival Job
H5 : JobCost Job
D := [eta task_deadline] : Task -> duration
EDF := edf.EDF Task Job : JLFP_policy 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
H6 : JobPreemptable Job
H_valid_model_with_bounded_nonpreemptive_segments :
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
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
H7 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
H_valid_preemption_model : valid_preemption_model arr_seq sched
H_valid_run_to_completion_threshold : valid_task_run_to_completion_threshold
arr_seq tsk
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn (A + ε + D tsk - D tsk_o) Δ)
: nat -> nat -> nat
max_length_of_priority_inversion := priority_inversion_is_bounded.max_length_of_priority_inversion
arr_seq EDF :
Job -> instant -> nat
task_rbf_changes_at := [eta response_time_bound.task_rbf_changes_at tsk]
: duration -> bool
bound_on_total_hep_workload_changes_at := response_time_bound.bound_on_total_hep_workload_changes_at
ts tsk :
nat -> bool
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 = total_rbf L
is_in_search_space := fun A : nat =>
(A < L) &&
(task_rbf_changes_at A
|| bound_on_total_hep_workload_changes_at A)
: nat -> bool
R : duration
H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
A + F =
blocking_bound +
(task_rbf (A + ε) -
(task_cost tsk - task_run_to_completion_threshold tsk)) +
bound_on_total_hep_workload A (A + F) /\
F +
(task_cost tsk - task_run_to_completion_threshold tsk) <=
R
============================
priority_inversion_is_bounded_by arr_seq sched (edf.EDF Task Job) tsk
blocking_bound
----------------------------------------------------------------------------- *)
by apply priority_inversion_is_bounded.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
End ResponseTimeBound.
End RTAforEDFwithBoundedNonpreemptiveSegmentsWithArrivalCurves.