Library prosa.results.edf.rta.limited_preemptive
(* ----------------------------------[ coqtop ]---------------------------------
Welcome to Coq 8.11.2 (June 2020)
----------------------------------------------------------------------------- *)
Require Export prosa.results.edf.rta.bounded_nps.
Require Export prosa.analysis.facts.preemption.rtc_threshold.limited.
From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq path fintype bigop.
RTA for EDF with Fixed Preemption Points
In this module we prove the RTA theorem for EDF-schedulers with fixed preemption points.
Require Import prosa.model.priority.edf.
Require Import prosa.model.processor.ideal.
Require Import prosa.model.readiness.basic.
Require Import prosa.model.processor.ideal.
Require Import prosa.model.readiness.basic.
Furthermore, we assume the task model with fixed preemption points.
Require Import prosa.model.preemption.limited_preemptive.
Require Import prosa.model.task.preemption.limited_preemptive.
Require Import prosa.model.task.preemption.limited_preemptive.
Consider any type of tasks ...
... and any type of jobs associated with these tasks.
Context {Job : JobType}.
Context `{JobTask Job Task}.
Context `{JobArrival Job}.
Context `{JobCost Job}.
Context `{JobTask Job Task}.
Context `{JobArrival Job}.
Context `{JobCost Job}.
Consider any arrival sequence with consistent, non-duplicate arrivals.
Variable arr_seq : arrival_sequence Job.
Hypothesis H_arrival_times_are_consistent : consistent_arrival_times arr_seq.
Hypothesis H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq.
Hypothesis H_arrival_times_are_consistent : consistent_arrival_times arr_seq.
Hypothesis H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq.
Consider an arbitrary task set ts, ...
... assume that all jobs come from this task set, ...
... and the cost of a job cannot be larger than the task cost.
Next, we assume we have the model with fixed preemption points.
I.e., each task is divided into a number of non-preemptive segments
by inserting statically predefined preemption points.
Context `{JobPreemptionPoints Job}.
Context `{TaskPreemptionPoints Task}.
Hypothesis H_valid_model_with_fixed_preemption_points:
valid_fixed_preemption_points_model arr_seq ts.
Context `{TaskPreemptionPoints Task}.
Hypothesis H_valid_model_with_fixed_preemption_points:
valid_fixed_preemption_points_model arr_seq ts.
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.
Next, consider any ideal uni-processor schedule with limited
preemptions 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_schedule_with_limited_preemptions:
schedule_respects_preemption_model arr_seq sched.
Hypothesis H_jobs_come_from_arrival_sequence:
jobs_come_from_arrival_sequence sched arr_seq.
Hypothesis H_schedule_with_limited_preemptions:
schedule_respects_preemption_model arr_seq sched.
... 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.
Assume we have sequential tasks, i.e, jobs from the
same task execute in the order of their arrival.
Next, we assume that the schedule is a work-conserving schedule...
... and the schedule respects the policy defined by the
[job_preemptable] function (i.e., jobs have bounded non-preemptive
segments).
Total Workload and Length of Busy Interval
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).
We define a bound for the priority inversion caused by jobs with lower priority.
Let blocking_bound :=
\max_(tsk_other <- ts | (tsk_other != tsk) && (task_deadline tsk_other > task_deadline tsk))
(task_max_nonpreemptive_segment tsk_other - ε).
\max_(tsk_other <- ts | (tsk_other != tsk) && (task_deadline tsk_other > task_deadline tsk))
(task_max_nonpreemptive_segment tsk_other - ε).
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) + task_deadline tsk - task_deadline tsk_o) Δ).
\sum_(tsk_o <- ts | tsk_o != tsk)
rbf tsk_o (minn ((A + ε) + task_deadline tsk - task_deadline tsk_o) Δ).
Let L be any positive fixed point of the busy interval recurrence.
Response-Time Bound
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_last_nonpr_segment tsk - ε))
+ bound_on_total_hep_workload A (A + F) ∧
F + (task_last_nonpr_segment tsk - ε) ≤ R.
Hypothesis H_R_is_maximum:
∀ (A : duration),
is_in_search_space A →
∃ (F : duration),
A + F = blocking_bound
+ (task_rbf (A + ε) - (task_last_nonpr_segment tsk - ε))
+ bound_on_total_hep_workload A (A + F) ∧
F + (task_last_nonpr_segment tsk - ε) ≤ R.
Now, we can leverage the results for the abstract model with bounded non-preemptive segments
to establish a response-time bound for the more concrete model of fixed preemption points.
Let response_time_bounded_by := task_response_time_bound arr_seq sched.
Theorem uniprocessor_response_time_bound_edf_with_fixed_preemption_points:
response_time_bounded_by tsk R.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2205)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H4 : JobPreemptionPoints Job
H5 : TaskPreemptionPoints Task
H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts
H6 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute 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
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
blocking_bound := \max_(tsk_other <- ts | (tsk_other != tsk) &&
(task_deadline tsk <
task_deadline tsk_other))
(task_max_nonpreemptive_segment tsk_other - ε) : nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + task_deadline tsk -
task_deadline tsk_o) Δ)
: nat -> nat -> nat
L : duration
H_L_positive : 0 < L
H_fixed_point : L = total_rbf L
is_in_search_space := bounded_pi.is_in_search_space ts tsk L
: duration -> bool
R : duration
H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
A + F =
blocking_bound +
(task_rbf (A + ε) - (task_last_nonpr_segment tsk - ε)) +
bound_on_total_hep_workload A (A + F) /\
F + (task_last_nonpr_segment tsk - ε) <= R
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
============================
response_time_bounded_by tsk R
----------------------------------------------------------------------------- *)
Proof.
move: (H_valid_model_with_fixed_preemption_points) ⇒ [MLP [BEG [END [INCR [HYP1 [HYP2 HYP3]]]]]].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2267)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H4 : JobPreemptionPoints Job
H5 : TaskPreemptionPoints Task
H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts
H6 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute 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
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
blocking_bound := \max_(tsk_other <- ts | (tsk_other != tsk) &&
(task_deadline tsk <
task_deadline tsk_other))
(task_max_nonpreemptive_segment tsk_other - ε) : nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + task_deadline tsk -
task_deadline tsk_o) Δ)
: nat -> nat -> nat
L : duration
H_L_positive : 0 < L
H_fixed_point : L = total_rbf L
is_in_search_space := bounded_pi.is_in_search_space ts tsk L
: duration -> bool
R : duration
H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
A + F =
blocking_bound +
(task_rbf (A + ε) - (task_last_nonpr_segment tsk - ε)) +
bound_on_total_hep_workload A (A + F) /\
F + (task_last_nonpr_segment tsk - ε) <= R
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
MLP : valid_limited_preemptions_job_model arr_seq
BEG : task_beginning_of_execution_in_preemption_points ts
END : task_end_of_execution_in_preemption_points ts
INCR : nondecreasing_task_preemption_points ts
HYP1 : consistent_job_segment_count arr_seq
HYP2 : job_respects_segment_lengths arr_seq
HYP3 : task_segments_are_nonempty ts
============================
response_time_bounded_by tsk R
----------------------------------------------------------------------------- *)
move: (MLP) ⇒ [BEGj [ENDj _]].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2290)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H4 : JobPreemptionPoints Job
H5 : TaskPreemptionPoints Task
H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts
H6 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute 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
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
blocking_bound := \max_(tsk_other <- ts | (tsk_other != tsk) &&
(task_deadline tsk <
task_deadline tsk_other))
(task_max_nonpreemptive_segment tsk_other - ε) : nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + task_deadline tsk -
task_deadline tsk_o) Δ)
: nat -> nat -> nat
L : duration
H_L_positive : 0 < L
H_fixed_point : L = total_rbf L
is_in_search_space := bounded_pi.is_in_search_space ts tsk L
: duration -> bool
R : duration
H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
A + F =
blocking_bound +
(task_rbf (A + ε) - (task_last_nonpr_segment tsk - ε)) +
bound_on_total_hep_workload A (A + F) /\
F + (task_last_nonpr_segment tsk - ε) <= R
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
MLP : valid_limited_preemptions_job_model arr_seq
BEG : task_beginning_of_execution_in_preemption_points ts
END : task_end_of_execution_in_preemption_points ts
INCR : nondecreasing_task_preemption_points ts
HYP1 : consistent_job_segment_count arr_seq
HYP2 : job_respects_segment_lengths arr_seq
HYP3 : task_segments_are_nonempty ts
BEGj : beginning_of_execution_in_preemption_points arr_seq
ENDj : end_of_execution_in_preemption_points arr_seq
============================
response_time_bounded_by tsk R
----------------------------------------------------------------------------- *)
case: (posnP (task_cost tsk)) ⇒ [ZERO|POSt].
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 2313)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H4 : JobPreemptionPoints Job
H5 : TaskPreemptionPoints Task
H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts
H6 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute 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
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
blocking_bound := \max_(tsk_other <- ts | (tsk_other != tsk) &&
(task_deadline tsk <
task_deadline tsk_other))
(task_max_nonpreemptive_segment tsk_other - ε) : nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + task_deadline tsk -
task_deadline tsk_o) Δ)
: nat -> nat -> nat
L : duration
H_L_positive : 0 < L
H_fixed_point : L = total_rbf L
is_in_search_space := bounded_pi.is_in_search_space ts tsk L
: duration -> bool
R : duration
H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
A + F =
blocking_bound +
(task_rbf (A + ε) - (task_last_nonpr_segment tsk - ε)) +
bound_on_total_hep_workload A (A + F) /\
F + (task_last_nonpr_segment tsk - ε) <= R
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
MLP : valid_limited_preemptions_job_model arr_seq
BEG : task_beginning_of_execution_in_preemption_points ts
END : task_end_of_execution_in_preemption_points ts
INCR : nondecreasing_task_preemption_points ts
HYP1 : consistent_job_segment_count arr_seq
HYP2 : job_respects_segment_lengths arr_seq
HYP3 : task_segments_are_nonempty ts
BEGj : beginning_of_execution_in_preemption_points arr_seq
ENDj : end_of_execution_in_preemption_points arr_seq
ZERO : task_cost tsk = 0
============================
response_time_bounded_by tsk R
subgoal 2 (ID 2314) is:
response_time_bounded_by tsk R
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2313)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H4 : JobPreemptionPoints Job
H5 : TaskPreemptionPoints Task
H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts
H6 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute 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
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
blocking_bound := \max_(tsk_other <- ts | (tsk_other != tsk) &&
(task_deadline tsk <
task_deadline tsk_other))
(task_max_nonpreemptive_segment tsk_other - ε) : nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + task_deadline tsk -
task_deadline tsk_o) Δ)
: nat -> nat -> nat
L : duration
H_L_positive : 0 < L
H_fixed_point : L = total_rbf L
is_in_search_space := bounded_pi.is_in_search_space ts tsk L
: duration -> bool
R : duration
H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
A + F =
blocking_bound +
(task_rbf (A + ε) - (task_last_nonpr_segment tsk - ε)) +
bound_on_total_hep_workload A (A + F) /\
F + (task_last_nonpr_segment tsk - ε) <= R
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
MLP : valid_limited_preemptions_job_model arr_seq
BEG : task_beginning_of_execution_in_preemption_points ts
END : task_end_of_execution_in_preemption_points ts
INCR : nondecreasing_task_preemption_points ts
HYP1 : consistent_job_segment_count arr_seq
HYP2 : job_respects_segment_lengths arr_seq
HYP3 : task_segments_are_nonempty ts
BEGj : beginning_of_execution_in_preemption_points arr_seq
ENDj : end_of_execution_in_preemption_points arr_seq
ZERO : task_cost tsk = 0
============================
response_time_bounded_by tsk R
----------------------------------------------------------------------------- *)
intros j ARR TSK.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2318)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H4 : JobPreemptionPoints Job
H5 : TaskPreemptionPoints Task
H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts
H6 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute 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
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
blocking_bound := \max_(tsk_other <- ts | (tsk_other != tsk) &&
(task_deadline tsk <
task_deadline tsk_other))
(task_max_nonpreemptive_segment tsk_other - ε) : nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + task_deadline tsk -
task_deadline tsk_o) Δ)
: nat -> nat -> nat
L : duration
H_L_positive : 0 < L
H_fixed_point : L = total_rbf L
is_in_search_space := bounded_pi.is_in_search_space ts tsk L
: duration -> bool
R : duration
H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
A + F =
blocking_bound +
(task_rbf (A + ε) - (task_last_nonpr_segment tsk - ε)) +
bound_on_total_hep_workload A (A + F) /\
F + (task_last_nonpr_segment tsk - ε) <= R
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
MLP : valid_limited_preemptions_job_model arr_seq
BEG : task_beginning_of_execution_in_preemption_points ts
END : task_end_of_execution_in_preemption_points ts
INCR : nondecreasing_task_preemption_points ts
HYP1 : consistent_job_segment_count arr_seq
HYP2 : job_respects_segment_lengths arr_seq
HYP3 : task_segments_are_nonempty ts
BEGj : beginning_of_execution_in_preemption_points arr_seq
ENDj : end_of_execution_in_preemption_points arr_seq
ZERO : task_cost tsk = 0
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
============================
job_response_time_bound sched j R
----------------------------------------------------------------------------- *)
move: (H_valid_job_cost _ ARR) ⇒ POSt.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2322)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H4 : JobPreemptionPoints Job
H5 : TaskPreemptionPoints Task
H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts
H6 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute 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
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
blocking_bound := \max_(tsk_other <- ts | (tsk_other != tsk) &&
(task_deadline tsk <
task_deadline tsk_other))
(task_max_nonpreemptive_segment tsk_other - ε) : nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + task_deadline tsk -
task_deadline tsk_o) Δ)
: nat -> nat -> nat
L : duration
H_L_positive : 0 < L
H_fixed_point : L = total_rbf L
is_in_search_space := bounded_pi.is_in_search_space ts tsk L
: duration -> bool
R : duration
H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
A + F =
blocking_bound +
(task_rbf (A + ε) - (task_last_nonpr_segment tsk - ε)) +
bound_on_total_hep_workload A (A + F) /\
F + (task_last_nonpr_segment tsk - ε) <= R
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
MLP : valid_limited_preemptions_job_model arr_seq
BEG : task_beginning_of_execution_in_preemption_points ts
END : task_end_of_execution_in_preemption_points ts
INCR : nondecreasing_task_preemption_points ts
HYP1 : consistent_job_segment_count arr_seq
HYP2 : job_respects_segment_lengths arr_seq
HYP3 : task_segments_are_nonempty ts
BEGj : beginning_of_execution_in_preemption_points arr_seq
ENDj : end_of_execution_in_preemption_points arr_seq
ZERO : task_cost tsk = 0
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
POSt : valid_job_cost j
============================
job_response_time_bound sched j R
----------------------------------------------------------------------------- *)
move: POSt; rewrite /valid_job_cost TSK ZERO leqn0; move ⇒ /eqP Z.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2376)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H4 : JobPreemptionPoints Job
H5 : TaskPreemptionPoints Task
H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts
H6 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute 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
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
blocking_bound := \max_(tsk_other <- ts | (tsk_other != tsk) &&
(task_deadline tsk <
task_deadline tsk_other))
(task_max_nonpreemptive_segment tsk_other - ε) : nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + task_deadline tsk -
task_deadline tsk_o) Δ)
: nat -> nat -> nat
L : duration
H_L_positive : 0 < L
H_fixed_point : L = total_rbf L
is_in_search_space := bounded_pi.is_in_search_space ts tsk L
: duration -> bool
R : duration
H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
A + F =
blocking_bound +
(task_rbf (A + ε) - (task_last_nonpr_segment tsk - ε)) +
bound_on_total_hep_workload A (A + F) /\
F + (task_last_nonpr_segment tsk - ε) <= R
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
MLP : valid_limited_preemptions_job_model arr_seq
BEG : task_beginning_of_execution_in_preemption_points ts
END : task_end_of_execution_in_preemption_points ts
INCR : nondecreasing_task_preemption_points ts
HYP1 : consistent_job_segment_count arr_seq
HYP2 : job_respects_segment_lengths arr_seq
HYP3 : task_segments_are_nonempty ts
BEGj : beginning_of_execution_in_preemption_points arr_seq
ENDj : end_of_execution_in_preemption_points arr_seq
ZERO : task_cost tsk = 0
j : Job
ARR : arrives_in arr_seq j
TSK : job_task j = tsk
Z : job_cost j = 0
============================
job_response_time_bound sched j R
----------------------------------------------------------------------------- *)
by rewrite /job_response_time_bound /completed_by Z.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2314)
subgoal 1 (ID 2314) is:
response_time_bounded_by tsk R
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2314)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H4 : JobPreemptionPoints Job
H5 : TaskPreemptionPoints Task
H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts
H6 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute 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
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
blocking_bound := \max_(tsk_other <- ts | (tsk_other != tsk) &&
(task_deadline tsk <
task_deadline tsk_other))
(task_max_nonpreemptive_segment tsk_other - ε) : nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + task_deadline tsk -
task_deadline tsk_o) Δ)
: nat -> nat -> nat
L : duration
H_L_positive : 0 < L
H_fixed_point : L = total_rbf L
is_in_search_space := bounded_pi.is_in_search_space ts tsk L
: duration -> bool
R : duration
H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
A + F =
blocking_bound +
(task_rbf (A + ε) - (task_last_nonpr_segment tsk - ε)) +
bound_on_total_hep_workload A (A + F) /\
F + (task_last_nonpr_segment tsk - ε) <= R
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
MLP : valid_limited_preemptions_job_model arr_seq
BEG : task_beginning_of_execution_in_preemption_points ts
END : task_end_of_execution_in_preemption_points ts
INCR : nondecreasing_task_preemption_points ts
HYP1 : consistent_job_segment_count arr_seq
HYP2 : job_respects_segment_lengths arr_seq
HYP3 : task_segments_are_nonempty ts
BEGj : beginning_of_execution_in_preemption_points arr_seq
ENDj : end_of_execution_in_preemption_points arr_seq
POSt : 0 < task_cost tsk
============================
response_time_bounded_by tsk R
----------------------------------------------------------------------------- *)
eapply uniprocessor_response_time_bound_edf_with_bounded_nonpreemptive_segments with (L0 := L).
(* ----------------------------------[ coqtop ]---------------------------------
19 focused subgoals
(shelved: 1) (ID 2409)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H4 : JobPreemptionPoints Job
H5 : TaskPreemptionPoints Task
H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts
H6 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute 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
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
blocking_bound := \max_(tsk_other <- ts | (tsk_other != tsk) &&
(task_deadline tsk <
task_deadline tsk_other))
(task_max_nonpreemptive_segment tsk_other - ε) : nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + task_deadline tsk -
task_deadline tsk_o) Δ)
: nat -> nat -> nat
L : duration
H_L_positive : 0 < L
H_fixed_point : L = total_rbf L
is_in_search_space := bounded_pi.is_in_search_space ts tsk L
: duration -> bool
R : duration
H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
A + F =
blocking_bound +
(task_rbf (A + ε) - (task_last_nonpr_segment tsk - ε)) +
bound_on_total_hep_workload A (A + F) /\
F + (task_last_nonpr_segment tsk - ε) <= R
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
MLP : valid_limited_preemptions_job_model arr_seq
BEG : task_beginning_of_execution_in_preemption_points ts
END : task_end_of_execution_in_preemption_points ts
INCR : nondecreasing_task_preemption_points ts
HYP1 : consistent_job_segment_count arr_seq
HYP2 : job_respects_segment_lengths arr_seq
HYP3 : task_segments_are_nonempty ts
BEGj : beginning_of_execution_in_preemption_points arr_seq
ENDj : end_of_execution_in_preemption_points arr_seq
POSt : 0 < task_cost tsk
============================
consistent_arrival_times arr_seq
subgoal 2 (ID 2410) is:
arrival_sequence_uniq arr_seq
subgoal 3 (ID 2411) is:
jobs_come_from_arrival_sequence sched arr_seq
subgoal 4 (ID 2412) is:
jobs_must_arrive_to_execute sched
subgoal 5 (ID 2413) is:
completed_jobs_dont_execute sched
subgoal 6 (ID 2414) is:
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
subgoal 7 (ID 2415) is:
sequential_tasks sched
subgoal 8 (ID 2416) is:
work_conserving arr_seq sched
subgoal 9 (ID 2417) is:
respects_policy_at_preemption_point arr_seq sched
subgoal 10 (ID 2418) is:
all_jobs_from_taskset arr_seq ?ts0
subgoal 11 (ID 2419) is:
arrivals_have_valid_job_costs arr_seq
subgoal 12 (ID 2420) is:
valid_taskset_arrival_curve ?ts0 max_arrivals
subgoal 13 (ID 2421) is:
taskset_respects_max_arrivals arr_seq ?ts0
subgoal 14 (ID 2422) is:
tsk \in ?ts0
subgoal 15 (ID 2423) is:
valid_preemption_model arr_seq sched
subgoal 16 (ID 2424) is:
valid_task_run_to_completion_threshold arr_seq tsk
subgoal 17 (ID 2425) is:
0 < L
subgoal 18 (ID 2426) is:
L = total_request_bound_function ?ts0 L
subgoal 19 (ID 2427) is:
forall A : duration,
bounded_pi.is_in_search_space ?ts0 tsk L A ->
exists F : duration,
A + F =
bounded_nps.blocking_bound ?ts0 tsk +
(task_request_bound_function tsk (A + ε) -
(task_cost tsk - task_run_to_completion_threshold tsk)) +
\sum_(tsk_o <- ?ts0 | tsk_o != tsk)
task_request_bound_function tsk_o
(minn (A + ε + task_deadline tsk - task_deadline tsk_o) (A + F)) /\
F + (task_cost tsk - task_run_to_completion_threshold tsk) <= R
----------------------------------------------------------------------------- *)
all: eauto 2 with basic_facts.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2427)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H4 : JobPreemptionPoints Job
H5 : TaskPreemptionPoints Task
H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts
H6 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute 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
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
blocking_bound := \max_(tsk_other <- ts | (tsk_other != tsk) &&
(task_deadline tsk <
task_deadline tsk_other))
(task_max_nonpreemptive_segment tsk_other - ε) : nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + task_deadline tsk -
task_deadline tsk_o) Δ)
: nat -> nat -> nat
L : duration
H_L_positive : 0 < L
H_fixed_point : L = total_rbf L
is_in_search_space := bounded_pi.is_in_search_space ts tsk L
: duration -> bool
R : duration
H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
A + F =
blocking_bound +
(task_rbf (A + ε) - (task_last_nonpr_segment tsk - ε)) +
bound_on_total_hep_workload A (A + F) /\
F + (task_last_nonpr_segment tsk - ε) <= R
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
MLP : valid_limited_preemptions_job_model arr_seq
BEG : task_beginning_of_execution_in_preemption_points ts
END : task_end_of_execution_in_preemption_points ts
INCR : nondecreasing_task_preemption_points ts
HYP1 : consistent_job_segment_count arr_seq
HYP2 : job_respects_segment_lengths arr_seq
HYP3 : task_segments_are_nonempty ts
BEGj : beginning_of_execution_in_preemption_points arr_seq
ENDj : end_of_execution_in_preemption_points arr_seq
POSt : 0 < task_cost tsk
============================
forall A : duration,
bounded_pi.is_in_search_space ts tsk L A ->
exists F : duration,
A + F =
bounded_nps.blocking_bound ts tsk +
(task_request_bound_function tsk (A + ε) -
(task_cost tsk - task_run_to_completion_threshold tsk)) +
\sum_(tsk_o <- ts | tsk_o != tsk)
task_request_bound_function tsk_o
(minn (A + ε + task_deadline tsk - task_deadline tsk_o) (A + F)) /\
F + (task_cost tsk - task_run_to_completion_threshold tsk) <= R
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2427)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H4 : JobPreemptionPoints Job
H5 : TaskPreemptionPoints Task
H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts
H6 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute 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
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
blocking_bound := \max_(tsk_other <- ts | (tsk_other != tsk) &&
(task_deadline tsk <
task_deadline tsk_other))
(task_max_nonpreemptive_segment tsk_other - ε) : nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + task_deadline tsk -
task_deadline tsk_o) Δ)
: nat -> nat -> nat
L : duration
H_L_positive : 0 < L
H_fixed_point : L = total_rbf L
is_in_search_space := bounded_pi.is_in_search_space ts tsk L
: duration -> bool
R : duration
H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
A + F =
blocking_bound +
(task_rbf (A + ε) - (task_last_nonpr_segment tsk - ε)) +
bound_on_total_hep_workload A (A + F) /\
F + (task_last_nonpr_segment tsk - ε) <= R
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
MLP : valid_limited_preemptions_job_model arr_seq
BEG : task_beginning_of_execution_in_preemption_points ts
END : task_end_of_execution_in_preemption_points ts
INCR : nondecreasing_task_preemption_points ts
HYP1 : consistent_job_segment_count arr_seq
HYP2 : job_respects_segment_lengths arr_seq
HYP3 : task_segments_are_nonempty ts
BEGj : beginning_of_execution_in_preemption_points arr_seq
ENDj : end_of_execution_in_preemption_points arr_seq
POSt : 0 < task_cost tsk
============================
forall A : duration,
bounded_pi.is_in_search_space ts tsk L A ->
exists F : duration,
A + F =
bounded_nps.blocking_bound ts tsk +
(task_request_bound_function tsk (A + ε) -
(task_cost tsk - task_run_to_completion_threshold tsk)) +
\sum_(tsk_o <- ts | tsk_o != tsk)
task_request_bound_function tsk_o
(minn (A + ε + task_deadline tsk - task_deadline tsk_o) (A + F)) /\
F + (task_cost tsk - task_run_to_completion_threshold tsk) <= R
----------------------------------------------------------------------------- *)
rewrite subKn; first by done.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2469)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H4 : JobPreemptionPoints Job
H5 : TaskPreemptionPoints Task
H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts
H6 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute 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
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
blocking_bound := \max_(tsk_other <- ts | (tsk_other != tsk) &&
(task_deadline tsk <
task_deadline tsk_other))
(task_max_nonpreemptive_segment tsk_other - ε) : nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + task_deadline tsk -
task_deadline tsk_o) Δ)
: nat -> nat -> nat
L : duration
H_L_positive : 0 < L
H_fixed_point : L = total_rbf L
is_in_search_space := bounded_pi.is_in_search_space ts tsk L
: duration -> bool
R : duration
H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
A + F =
blocking_bound +
(task_rbf (A + ε) - (task_last_nonpr_segment tsk - ε)) +
bound_on_total_hep_workload A (A + F) /\
F + (task_last_nonpr_segment tsk - ε) <= R
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
MLP : valid_limited_preemptions_job_model arr_seq
BEG : task_beginning_of_execution_in_preemption_points ts
END : task_end_of_execution_in_preemption_points ts
INCR : nondecreasing_task_preemption_points ts
HYP1 : consistent_job_segment_count arr_seq
HYP2 : job_respects_segment_lengths arr_seq
HYP3 : task_segments_are_nonempty ts
BEGj : beginning_of_execution_in_preemption_points arr_seq
ENDj : end_of_execution_in_preemption_points arr_seq
POSt : 0 < task_cost tsk
============================
task_last_nonpr_segment tsk - ε <= task_cost tsk
----------------------------------------------------------------------------- *)
rewrite /task_last_nonpr_segment -(leq_add2r 1) subn1 !addn1 prednK; last first.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 2493)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H4 : JobPreemptionPoints Job
H5 : TaskPreemptionPoints Task
H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts
H6 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute 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
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
blocking_bound := \max_(tsk_other <- ts | (tsk_other != tsk) &&
(task_deadline tsk <
task_deadline tsk_other))
(task_max_nonpreemptive_segment tsk_other - ε) : nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + task_deadline tsk -
task_deadline tsk_o) Δ)
: nat -> nat -> nat
L : duration
H_L_positive : 0 < L
H_fixed_point : L = total_rbf L
is_in_search_space := bounded_pi.is_in_search_space ts tsk L
: duration -> bool
R : duration
H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
A + F =
blocking_bound +
(task_rbf (A + ε) - (task_last_nonpr_segment tsk - ε)) +
bound_on_total_hep_workload A (A + F) /\
F + (task_last_nonpr_segment tsk - ε) <= R
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
MLP : valid_limited_preemptions_job_model arr_seq
BEG : task_beginning_of_execution_in_preemption_points ts
END : task_end_of_execution_in_preemption_points ts
INCR : nondecreasing_task_preemption_points ts
HYP1 : consistent_job_segment_count arr_seq
HYP2 : job_respects_segment_lengths arr_seq
HYP3 : task_segments_are_nonempty ts
BEGj : beginning_of_execution_in_preemption_points arr_seq
ENDj : end_of_execution_in_preemption_points arr_seq
POSt : 0 < task_cost tsk
============================
0 < last0 (distances (task_preemption_points tsk))
subgoal 2 (ID 2492) is:
last0 (distances (task_preemption_points tsk)) <= succn (task_cost tsk)
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2493)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H4 : JobPreemptionPoints Job
H5 : TaskPreemptionPoints Task
H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts
H6 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute 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
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
blocking_bound := \max_(tsk_other <- ts | (tsk_other != tsk) &&
(task_deadline tsk <
task_deadline tsk_other))
(task_max_nonpreemptive_segment tsk_other - ε) : nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + task_deadline tsk -
task_deadline tsk_o) Δ)
: nat -> nat -> nat
L : duration
H_L_positive : 0 < L
H_fixed_point : L = total_rbf L
is_in_search_space := bounded_pi.is_in_search_space ts tsk L
: duration -> bool
R : duration
H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
A + F =
blocking_bound +
(task_rbf (A + ε) - (task_last_nonpr_segment tsk - ε)) +
bound_on_total_hep_workload A (A + F) /\
F + (task_last_nonpr_segment tsk - ε) <= R
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
MLP : valid_limited_preemptions_job_model arr_seq
BEG : task_beginning_of_execution_in_preemption_points ts
END : task_end_of_execution_in_preemption_points ts
INCR : nondecreasing_task_preemption_points ts
HYP1 : consistent_job_segment_count arr_seq
HYP2 : job_respects_segment_lengths arr_seq
HYP3 : task_segments_are_nonempty ts
BEGj : beginning_of_execution_in_preemption_points arr_seq
ENDj : end_of_execution_in_preemption_points arr_seq
POSt : 0 < task_cost tsk
============================
0 < last0 (distances (task_preemption_points tsk))
----------------------------------------------------------------------------- *)
rewrite /last0 -nth_last.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2499)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H4 : JobPreemptionPoints Job
H5 : TaskPreemptionPoints Task
H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts
H6 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute 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
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
blocking_bound := \max_(tsk_other <- ts | (tsk_other != tsk) &&
(task_deadline tsk <
task_deadline tsk_other))
(task_max_nonpreemptive_segment tsk_other - ε) : nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + task_deadline tsk -
task_deadline tsk_o) Δ)
: nat -> nat -> nat
L : duration
H_L_positive : 0 < L
H_fixed_point : L = total_rbf L
is_in_search_space := bounded_pi.is_in_search_space ts tsk L
: duration -> bool
R : duration
H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
A + F =
blocking_bound +
(task_rbf (A + ε) - (task_last_nonpr_segment tsk - ε)) +
bound_on_total_hep_workload A (A + F) /\
F + (task_last_nonpr_segment tsk - ε) <= R
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
MLP : valid_limited_preemptions_job_model arr_seq
BEG : task_beginning_of_execution_in_preemption_points ts
END : task_end_of_execution_in_preemption_points ts
INCR : nondecreasing_task_preemption_points ts
HYP1 : consistent_job_segment_count arr_seq
HYP2 : job_respects_segment_lengths arr_seq
HYP3 : task_segments_are_nonempty ts
BEGj : beginning_of_execution_in_preemption_points arr_seq
ENDj : end_of_execution_in_preemption_points arr_seq
POSt : 0 < task_cost tsk
============================
0 <
nth 0 (distances (task_preemption_points tsk))
(predn (size (distances (task_preemption_points tsk))))
----------------------------------------------------------------------------- *)
apply HYP3; try by done.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2501)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H4 : JobPreemptionPoints Job
H5 : TaskPreemptionPoints Task
H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts
H6 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute 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
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
blocking_bound := \max_(tsk_other <- ts | (tsk_other != tsk) &&
(task_deadline tsk <
task_deadline tsk_other))
(task_max_nonpreemptive_segment tsk_other - ε) : nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + task_deadline tsk -
task_deadline tsk_o) Δ)
: nat -> nat -> nat
L : duration
H_L_positive : 0 < L
H_fixed_point : L = total_rbf L
is_in_search_space := bounded_pi.is_in_search_space ts tsk L
: duration -> bool
R : duration
H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
A + F =
blocking_bound +
(task_rbf (A + ε) - (task_last_nonpr_segment tsk - ε)) +
bound_on_total_hep_workload A (A + F) /\
F + (task_last_nonpr_segment tsk - ε) <= R
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
MLP : valid_limited_preemptions_job_model arr_seq
BEG : task_beginning_of_execution_in_preemption_points ts
END : task_end_of_execution_in_preemption_points ts
INCR : nondecreasing_task_preemption_points ts
HYP1 : consistent_job_segment_count arr_seq
HYP2 : job_respects_segment_lengths arr_seq
HYP3 : task_segments_are_nonempty ts
BEGj : beginning_of_execution_in_preemption_points arr_seq
ENDj : end_of_execution_in_preemption_points arr_seq
POSt : 0 < task_cost tsk
============================
predn (size (distances (task_preemption_points tsk))) <
size (distances (task_preemption_points tsk))
----------------------------------------------------------------------------- *)
rewrite -(ltn_add2r 1) !addn1 prednK //.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2539)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H4 : JobPreemptionPoints Job
H5 : TaskPreemptionPoints Task
H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts
H6 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute 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
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
blocking_bound := \max_(tsk_other <- ts | (tsk_other != tsk) &&
(task_deadline tsk <
task_deadline tsk_other))
(task_max_nonpreemptive_segment tsk_other - ε) : nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + task_deadline tsk -
task_deadline tsk_o) Δ)
: nat -> nat -> nat
L : duration
H_L_positive : 0 < L
H_fixed_point : L = total_rbf L
is_in_search_space := bounded_pi.is_in_search_space ts tsk L
: duration -> bool
R : duration
H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
A + F =
blocking_bound +
(task_rbf (A + ε) - (task_last_nonpr_segment tsk - ε)) +
bound_on_total_hep_workload A (A + F) /\
F + (task_last_nonpr_segment tsk - ε) <= R
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
MLP : valid_limited_preemptions_job_model arr_seq
BEG : task_beginning_of_execution_in_preemption_points ts
END : task_end_of_execution_in_preemption_points ts
INCR : nondecreasing_task_preemption_points ts
HYP1 : consistent_job_segment_count arr_seq
HYP2 : job_respects_segment_lengths arr_seq
HYP3 : task_segments_are_nonempty ts
BEGj : beginning_of_execution_in_preemption_points arr_seq
ENDj : end_of_execution_in_preemption_points arr_seq
POSt : 0 < task_cost tsk
============================
0 < size (distances (task_preemption_points tsk))
----------------------------------------------------------------------------- *)
move: (number_of_preemption_points_in_task_at_least_two
_ _ H_valid_model_with_fixed_preemption_points _ H_tsk_in_ts POSt) ⇒ Fact2.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2576)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H4 : JobPreemptionPoints Job
H5 : TaskPreemptionPoints Task
H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts
H6 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute 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
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
blocking_bound := \max_(tsk_other <- ts | (tsk_other != tsk) &&
(task_deadline tsk <
task_deadline tsk_other))
(task_max_nonpreemptive_segment tsk_other - ε) : nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + task_deadline tsk -
task_deadline tsk_o) Δ)
: nat -> nat -> nat
L : duration
H_L_positive : 0 < L
H_fixed_point : L = total_rbf L
is_in_search_space := bounded_pi.is_in_search_space ts tsk L
: duration -> bool
R : duration
H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
A + F =
blocking_bound +
(task_rbf (A + ε) - (task_last_nonpr_segment tsk - ε)) +
bound_on_total_hep_workload A (A + F) /\
F + (task_last_nonpr_segment tsk - ε) <= R
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
MLP : valid_limited_preemptions_job_model arr_seq
BEG : task_beginning_of_execution_in_preemption_points ts
END : task_end_of_execution_in_preemption_points ts
INCR : nondecreasing_task_preemption_points ts
HYP1 : consistent_job_segment_count arr_seq
HYP2 : job_respects_segment_lengths arr_seq
HYP3 : task_segments_are_nonempty ts
BEGj : beginning_of_execution_in_preemption_points arr_seq
ENDj : end_of_execution_in_preemption_points arr_seq
POSt : 0 < task_cost tsk
Fact2 : 1 < size (task_preemption_points tsk)
============================
0 < size (distances (task_preemption_points tsk))
----------------------------------------------------------------------------- *)
move: (Fact2) ⇒ Fact3.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2578)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H4 : JobPreemptionPoints Job
H5 : TaskPreemptionPoints Task
H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts
H6 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute 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
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
blocking_bound := \max_(tsk_other <- ts | (tsk_other != tsk) &&
(task_deadline tsk <
task_deadline tsk_other))
(task_max_nonpreemptive_segment tsk_other - ε) : nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + task_deadline tsk -
task_deadline tsk_o) Δ)
: nat -> nat -> nat
L : duration
H_L_positive : 0 < L
H_fixed_point : L = total_rbf L
is_in_search_space := bounded_pi.is_in_search_space ts tsk L
: duration -> bool
R : duration
H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
A + F =
blocking_bound +
(task_rbf (A + ε) - (task_last_nonpr_segment tsk - ε)) +
bound_on_total_hep_workload A (A + F) /\
F + (task_last_nonpr_segment tsk - ε) <= R
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
MLP : valid_limited_preemptions_job_model arr_seq
BEG : task_beginning_of_execution_in_preemption_points ts
END : task_end_of_execution_in_preemption_points ts
INCR : nondecreasing_task_preemption_points ts
HYP1 : consistent_job_segment_count arr_seq
HYP2 : job_respects_segment_lengths arr_seq
HYP3 : task_segments_are_nonempty ts
BEGj : beginning_of_execution_in_preemption_points arr_seq
ENDj : end_of_execution_in_preemption_points arr_seq
POSt : 0 < task_cost tsk
Fact2, Fact3 : 1 < size (task_preemption_points tsk)
============================
0 < size (distances (task_preemption_points tsk))
----------------------------------------------------------------------------- *)
by rewrite size_of_seq_of_distances // addn1 ltnS // in Fact2.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2492)
subgoal 1 (ID 2492) is:
last0 (distances (task_preemption_points tsk)) <= succn (task_cost tsk)
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2492)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H4 : JobPreemptionPoints Job
H5 : TaskPreemptionPoints Task
H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts
H6 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute 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
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
blocking_bound := \max_(tsk_other <- ts | (tsk_other != tsk) &&
(task_deadline tsk <
task_deadline tsk_other))
(task_max_nonpreemptive_segment tsk_other - ε) : nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + task_deadline tsk -
task_deadline tsk_o) Δ)
: nat -> nat -> nat
L : duration
H_L_positive : 0 < L
H_fixed_point : L = total_rbf L
is_in_search_space := bounded_pi.is_in_search_space ts tsk L
: duration -> bool
R : duration
H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
A + F =
blocking_bound +
(task_rbf (A + ε) - (task_last_nonpr_segment tsk - ε)) +
bound_on_total_hep_workload A (A + F) /\
F + (task_last_nonpr_segment tsk - ε) <= R
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
MLP : valid_limited_preemptions_job_model arr_seq
BEG : task_beginning_of_execution_in_preemption_points ts
END : task_end_of_execution_in_preemption_points ts
INCR : nondecreasing_task_preemption_points ts
HYP1 : consistent_job_segment_count arr_seq
HYP2 : job_respects_segment_lengths arr_seq
HYP3 : task_segments_are_nonempty ts
BEGj : beginning_of_execution_in_preemption_points arr_seq
ENDj : end_of_execution_in_preemption_points arr_seq
POSt : 0 < task_cost tsk
============================
last0 (distances (task_preemption_points tsk)) <= succn (task_cost tsk)
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2492)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H4 : JobPreemptionPoints Job
H5 : TaskPreemptionPoints Task
H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts
H6 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute 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
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
blocking_bound := \max_(tsk_other <- ts | (tsk_other != tsk) &&
(task_deadline tsk <
task_deadline tsk_other))
(task_max_nonpreemptive_segment tsk_other - ε) : nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + task_deadline tsk -
task_deadline tsk_o) Δ)
: nat -> nat -> nat
L : duration
H_L_positive : 0 < L
H_fixed_point : L = total_rbf L
is_in_search_space := bounded_pi.is_in_search_space ts tsk L
: duration -> bool
R : duration
H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
A + F =
blocking_bound +
(task_rbf (A + ε) - (task_last_nonpr_segment tsk - ε)) +
bound_on_total_hep_workload A (A + F) /\
F + (task_last_nonpr_segment tsk - ε) <= R
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
MLP : valid_limited_preemptions_job_model arr_seq
BEG : task_beginning_of_execution_in_preemption_points ts
END : task_end_of_execution_in_preemption_points ts
INCR : nondecreasing_task_preemption_points ts
HYP1 : consistent_job_segment_count arr_seq
HYP2 : job_respects_segment_lengths arr_seq
HYP3 : task_segments_are_nonempty ts
BEGj : beginning_of_execution_in_preemption_points arr_seq
ENDj : end_of_execution_in_preemption_points arr_seq
POSt : 0 < task_cost tsk
============================
last0 (distances (task_preemption_points tsk)) <= succn (task_cost tsk)
----------------------------------------------------------------------------- *)
apply leq_trans with (task_max_nonpreemptive_segment tsk).
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 2623)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H4 : JobPreemptionPoints Job
H5 : TaskPreemptionPoints Task
H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts
H6 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute 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
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
blocking_bound := \max_(tsk_other <- ts | (tsk_other != tsk) &&
(task_deadline tsk <
task_deadline tsk_other))
(task_max_nonpreemptive_segment tsk_other - ε) : nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + task_deadline tsk -
task_deadline tsk_o) Δ)
: nat -> nat -> nat
L : duration
H_L_positive : 0 < L
H_fixed_point : L = total_rbf L
is_in_search_space := bounded_pi.is_in_search_space ts tsk L
: duration -> bool
R : duration
H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
A + F =
blocking_bound +
(task_rbf (A + ε) - (task_last_nonpr_segment tsk - ε)) +
bound_on_total_hep_workload A (A + F) /\
F + (task_last_nonpr_segment tsk - ε) <= R
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
MLP : valid_limited_preemptions_job_model arr_seq
BEG : task_beginning_of_execution_in_preemption_points ts
END : task_end_of_execution_in_preemption_points ts
INCR : nondecreasing_task_preemption_points ts
HYP1 : consistent_job_segment_count arr_seq
HYP2 : job_respects_segment_lengths arr_seq
HYP3 : task_segments_are_nonempty ts
BEGj : beginning_of_execution_in_preemption_points arr_seq
ENDj : end_of_execution_in_preemption_points arr_seq
POSt : 0 < task_cost tsk
============================
last0 (distances (task_preemption_points tsk)) <=
task_max_nonpreemptive_segment tsk
subgoal 2 (ID 2624) is:
task_max_nonpreemptive_segment tsk <= succn (task_cost tsk)
----------------------------------------------------------------------------- *)
- by apply last_of_seq_le_max_of_seq.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2624)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H4 : JobPreemptionPoints Job
H5 : TaskPreemptionPoints Task
H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts
H6 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute 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
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
blocking_bound := \max_(tsk_other <- ts | (tsk_other != tsk) &&
(task_deadline tsk <
task_deadline tsk_other))
(task_max_nonpreemptive_segment tsk_other - ε) : nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + task_deadline tsk -
task_deadline tsk_o) Δ)
: nat -> nat -> nat
L : duration
H_L_positive : 0 < L
H_fixed_point : L = total_rbf L
is_in_search_space := bounded_pi.is_in_search_space ts tsk L
: duration -> bool
R : duration
H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
A + F =
blocking_bound +
(task_rbf (A + ε) - (task_last_nonpr_segment tsk - ε)) +
bound_on_total_hep_workload A (A + F) /\
F + (task_last_nonpr_segment tsk - ε) <= R
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
MLP : valid_limited_preemptions_job_model arr_seq
BEG : task_beginning_of_execution_in_preemption_points ts
END : task_end_of_execution_in_preemption_points ts
INCR : nondecreasing_task_preemption_points ts
HYP1 : consistent_job_segment_count arr_seq
HYP2 : job_respects_segment_lengths arr_seq
HYP3 : task_segments_are_nonempty ts
BEGj : beginning_of_execution_in_preemption_points arr_seq
ENDj : end_of_execution_in_preemption_points arr_seq
POSt : 0 < task_cost tsk
============================
task_max_nonpreemptive_segment tsk <= succn (task_cost tsk)
----------------------------------------------------------------------------- *)
- rewrite -END; last by done.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2630)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H4 : JobPreemptionPoints Job
H5 : TaskPreemptionPoints Task
H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts
H6 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute 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
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
blocking_bound := \max_(tsk_other <- ts | (tsk_other != tsk) &&
(task_deadline tsk <
task_deadline tsk_other))
(task_max_nonpreemptive_segment tsk_other - ε) : nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + task_deadline tsk -
task_deadline tsk_o) Δ)
: nat -> nat -> nat
L : duration
H_L_positive : 0 < L
H_fixed_point : L = total_rbf L
is_in_search_space := bounded_pi.is_in_search_space ts tsk L
: duration -> bool
R : duration
H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
A + F =
blocking_bound +
(task_rbf (A + ε) - (task_last_nonpr_segment tsk - ε)) +
bound_on_total_hep_workload A (A + F) /\
F + (task_last_nonpr_segment tsk - ε) <= R
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
MLP : valid_limited_preemptions_job_model arr_seq
BEG : task_beginning_of_execution_in_preemption_points ts
END : task_end_of_execution_in_preemption_points ts
INCR : nondecreasing_task_preemption_points ts
HYP1 : consistent_job_segment_count arr_seq
HYP2 : job_respects_segment_lengths arr_seq
HYP3 : task_segments_are_nonempty ts
BEGj : beginning_of_execution_in_preemption_points arr_seq
ENDj : end_of_execution_in_preemption_points arr_seq
POSt : 0 < task_cost tsk
============================
task_max_nonpreemptive_segment tsk <=
succn (last0 (task_preemption_points tsk))
----------------------------------------------------------------------------- *)
apply ltnW; rewrite ltnS; try done.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 2636)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : JobCost Job
arr_seq : arrival_sequence Job
H_arrival_times_are_consistent : consistent_arrival_times arr_seq
H_arr_seq_is_a_set : arrival_sequence_uniq arr_seq
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_valid_job_cost : arrivals_have_valid_job_costs arr_seq
H4 : JobPreemptionPoints Job
H5 : TaskPreemptionPoints Task
H_valid_model_with_fixed_preemption_points : valid_fixed_preemption_points_model
arr_seq ts
H6 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals
H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts
tsk : Task
H_tsk_in_ts : tsk \in ts
sched : schedule (processor_state Job)
H_jobs_come_from_arrival_sequence : jobs_come_from_arrival_sequence sched
arr_seq
H_schedule_with_limited_preemptions : schedule_respects_preemption_model
arr_seq sched
H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched
H_completed_jobs_dont_execute : completed_jobs_dont_execute 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
rbf := task_request_bound_function : Task -> duration -> nat
task_rbf := rbf tsk : duration -> nat
total_rbf := total_request_bound_function ts : duration -> nat
blocking_bound := \max_(tsk_other <- ts | (tsk_other != tsk) &&
(task_deadline tsk <
task_deadline tsk_other))
(task_max_nonpreemptive_segment tsk_other - ε) : nat
bound_on_total_hep_workload := fun A Δ : nat =>
\sum_(tsk_o <- ts |
tsk_o != tsk)
rbf tsk_o
(minn
(A + ε + task_deadline tsk -
task_deadline tsk_o) Δ)
: nat -> nat -> nat
L : duration
H_L_positive : 0 < L
H_fixed_point : L = total_rbf L
is_in_search_space := bounded_pi.is_in_search_space ts tsk L
: duration -> bool
R : duration
H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
A + F =
blocking_bound +
(task_rbf (A + ε) - (task_last_nonpr_segment tsk - ε)) +
bound_on_total_hep_workload A (A + F) /\
F + (task_last_nonpr_segment tsk - ε) <= R
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
MLP : valid_limited_preemptions_job_model arr_seq
BEG : task_beginning_of_execution_in_preemption_points ts
END : task_end_of_execution_in_preemption_points ts
INCR : nondecreasing_task_preemption_points ts
HYP1 : consistent_job_segment_count arr_seq
HYP2 : job_respects_segment_lengths arr_seq
HYP3 : task_segments_are_nonempty ts
BEGj : beginning_of_execution_in_preemption_points arr_seq
ENDj : end_of_execution_in_preemption_points arr_seq
POSt : 0 < task_cost tsk
============================
task_max_nonpreemptive_segment tsk <= last0 (task_preemption_points tsk)
----------------------------------------------------------------------------- *)
by apply max_distance_in_seq_le_last_element_of_seq; eauto 2.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
End RTAforFixedPreemptionPointsModelwithArrivalCurves.