Library rt.restructuring.analysis.edf.rta.nonpr_reg.concrete_models.preemptive
(* ----------------------------------[ 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.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.rta Require Import nonpr_reg.response_time_bound.
Assume we have a fully preemptive model.
From rt.restructuring.model Require Import preemption.preemptive.
From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq path fintype bigop.
From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq path fintype bigop.
RTA for Fully Preemptive EDF Model
In this section we prove the RTA theorem for the fully preemptive EDF model
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}.
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.
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.
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 uniprocessor schedule of the 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.
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 nonpreemptive
segments).
Let's define some local names for clarity.
Let response_time_bounded_by :=
task_response_time_bound arr_seq sched.
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.
task_response_time_bound arr_seq sched.
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.
We introduce the abbreviation "rbf" for the task request bound function,
which is defined as [task_cost(T) × max_arrivals(T,Δ)] for a task T.
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 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 = task_rbf (A + ε) + bound_on_total_hep_workload A (A + F) ∧
F ≤ R.
Hypothesis H_R_is_maximum:
∀ (A : duration),
is_in_search_space A →
∃ (F : duration),
A + F = task_rbf (A + ε) + bound_on_total_hep_workload A (A + F) ∧
F ≤ R.
Now, we can leverage the results for the abstract model with bounded nonpreemptive segments
to establish a response-time bound for the more concrete model of fully preemptive scheduling.
Theorem uniprocessor_response_time_bound_fully_preemptive_edf:
response_time_bounded_by tsk R.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 271)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : 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
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H4 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_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_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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
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
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
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 =
task_rbf (A + ε) + bound_on_total_hep_workload A (A + F) /\
F <= R
============================
response_time_bounded_by tsk R
----------------------------------------------------------------------------- *)
Proof.
have BLOCK: blocking_bound ts tsk = 0.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 277)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : 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
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H4 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_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_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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
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
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
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 =
task_rbf (A + ε) + bound_on_total_hep_workload A (A + F) /\
F <= R
============================
blocking_bound ts tsk = 0
subgoal 2 (ID 279) is:
response_time_bounded_by tsk R
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 277)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : 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
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H4 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_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_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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
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
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
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 =
task_rbf (A + ε) + bound_on_total_hep_workload A (A + F) /\
F <= R
============================
blocking_bound ts tsk = 0
----------------------------------------------------------------------------- *)
by rewrite /blocking_bound /parameters.task_max_nonpreemptive_segment
/preemptive.fully_preemptive_model subnn big1_eq.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 279)
subgoal 1 (ID 279) is:
response_time_bounded_by tsk R
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 279)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : 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
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H4 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_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_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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
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
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
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 =
task_rbf (A + ε) + bound_on_total_hep_workload A (A + F) /\
F <= R
BLOCK : blocking_bound ts tsk = 0
============================
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 321)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : 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
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H4 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_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_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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
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
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
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 =
task_rbf (A + ε) + bound_on_total_hep_workload A (A + F) /\
F <= R
BLOCK : blocking_bound ts tsk = 0
============================
consistent_arrival_times arr_seq
subgoal 2 (ID 322) is:
arrival_sequence_uniq arr_seq
subgoal 3 (ID 323) is:
jobs_come_from_arrival_sequence sched arr_seq
subgoal 4 (ID 324) is:
jobs_must_arrive_to_execute sched
subgoal 5 (ID 325) is:
completed_jobs_dont_execute sched
subgoal 6 (ID 326) is:
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
subgoal 7 (ID 327) is:
sequential_tasks sched
subgoal 8 (ID 328) is:
work_conserving arr_seq sched
subgoal 9 (ID 329) is:
respects_policy_at_preemption_point arr_seq sched
subgoal 10 (ID 330) is:
all_jobs_from_taskset arr_seq ?ts0
subgoal 11 (ID 331) is:
cost_of_jobs_from_arrival_sequence_le_task_cost arr_seq
subgoal 12 (ID 332) is:
valid_taskset_arrival_curve ?ts0 max_arrivals
subgoal 13 (ID 333) is:
taskset_respects_max_arrivals arr_seq ?ts0
subgoal 14 (ID 334) is:
tsk \in ?ts0
subgoal 15 (ID 335) is:
valid_preemption_model arr_seq sched
subgoal 16 (ID 336) is:
valid_rtct.valid_task_run_to_completion_threshold arr_seq tsk
subgoal 17 (ID 337) is:
0 < L
subgoal 18 (ID 338) is:
L = total_request_bound_function ?ts0 L
subgoal 19 (ID 339) is:
forall A : duration,
(A < L) &&
(response_time_bound.task_rbf_changes_at tsk A
|| response_time_bound.bound_on_total_hep_workload_changes_at ?ts0 tsk A) ->
exists F : duration,
A + F =
blocking_bound ?ts0 tsk +
(task_request_bound_function tsk (A + ε) -
(task_cost tsk - parameters.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 - parameters.task_run_to_completion_threshold tsk) <= R
----------------------------------------------------------------------------- *)
all: eauto 2 with basic_facts.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 339)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : 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
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H4 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_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_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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
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
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
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 =
task_rbf (A + ε) + bound_on_total_hep_workload A (A + F) /\
F <= R
BLOCK : blocking_bound ts tsk = 0
============================
forall A : duration,
(A < L) &&
(response_time_bound.task_rbf_changes_at tsk A
|| response_time_bound.bound_on_total_hep_workload_changes_at ts tsk A) ->
exists F : duration,
A + F =
blocking_bound ts tsk +
(task_request_bound_function tsk (A + ε) -
(task_cost tsk - parameters.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 - parameters.task_run_to_completion_threshold tsk) <=
R
----------------------------------------------------------------------------- *)
- move ⇒ A /andP [LT NEQ].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 396)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : 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
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H4 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_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_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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
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
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
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 =
task_rbf (A + ε) + bound_on_total_hep_workload A (A + F) /\
F <= R
BLOCK : blocking_bound ts tsk = 0
A : duration
LT : A < L
NEQ : response_time_bound.task_rbf_changes_at tsk A
|| response_time_bound.bound_on_total_hep_workload_changes_at ts tsk
A
============================
exists F : duration,
A + F =
blocking_bound ts tsk +
(task_request_bound_function tsk (A + ε) -
(task_cost tsk - parameters.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 - parameters.task_run_to_completion_threshold tsk) <=
R
----------------------------------------------------------------------------- *)
specialize (H_R_is_maximum A); feed H_R_is_maximum.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 399)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : 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
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H4 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_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_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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
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
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
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, A : duration
H_R_is_maximum : is_in_search_space A ->
exists F : duration,
A + F =
task_rbf (A + ε) + bound_on_total_hep_workload A (A + F) /\
F <= R
BLOCK : blocking_bound ts tsk = 0
LT : A < L
NEQ : response_time_bound.task_rbf_changes_at tsk A
|| response_time_bound.bound_on_total_hep_workload_changes_at ts tsk
A
============================
is_in_search_space A
subgoal 2 (ID 404) is:
exists F : duration,
A + F =
blocking_bound ts tsk +
(task_request_bound_function tsk (A + ε) -
(task_cost tsk - parameters.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 - parameters.task_run_to_completion_threshold tsk) <= R
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 399)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : 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
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H4 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_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_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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
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
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
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, A : duration
H_R_is_maximum : is_in_search_space A ->
exists F : duration,
A + F =
task_rbf (A + ε) + bound_on_total_hep_workload A (A + F) /\
F <= R
BLOCK : blocking_bound ts tsk = 0
LT : A < L
NEQ : response_time_bound.task_rbf_changes_at tsk A
|| response_time_bound.bound_on_total_hep_workload_changes_at ts tsk
A
============================
is_in_search_space A
----------------------------------------------------------------------------- *)
by apply/andP; split.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 404)
subgoal 1 (ID 404) is:
exists F : duration,
A + F =
blocking_bound ts tsk +
(task_request_bound_function tsk (A + ε) -
(task_cost tsk - parameters.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 - parameters.task_run_to_completion_threshold tsk) <= R
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 404)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : 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
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H4 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_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_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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
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
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
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, A : duration
H_R_is_maximum : exists F : duration,
A + F =
task_rbf (A + ε) + bound_on_total_hep_workload A (A + F) /\
F <= R
BLOCK : blocking_bound ts tsk = 0
LT : A < L
NEQ : response_time_bound.task_rbf_changes_at tsk A
|| response_time_bound.bound_on_total_hep_workload_changes_at ts tsk
A
============================
exists F : duration,
A + F =
blocking_bound ts tsk +
(task_request_bound_function tsk (A + ε) -
(task_cost tsk - parameters.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 - parameters.task_run_to_completion_threshold tsk) <=
R
----------------------------------------------------------------------------- *)
move: H_R_is_maximum ⇒ [F [FIX BOUND]].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 453)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : 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
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H4 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_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_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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
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
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
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, A : duration
H_R_is_maximum : exists F : duration,
A + F =
task_rbf (A + ε) + bound_on_total_hep_workload A (A + F) /\
F <= R
BLOCK : blocking_bound ts tsk = 0
LT : A < L
NEQ : response_time_bound.task_rbf_changes_at tsk A
|| response_time_bound.bound_on_total_hep_workload_changes_at ts tsk
A
F : duration
FIX : A + F = task_rbf (A + ε) + bound_on_total_hep_workload A (A + F)
BOUND : F <= R
============================
exists F0 : duration,
A + F0 =
blocking_bound ts tsk +
(task_request_bound_function tsk (A + ε) -
(task_cost tsk - parameters.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 + F0)) /\
F0 + (task_cost tsk - parameters.task_run_to_completion_threshold tsk) <=
R
----------------------------------------------------------------------------- *)
∃ F; split.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 457)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : 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
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H4 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_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_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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
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
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
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, A : duration
H_R_is_maximum : exists F : duration,
A + F =
task_rbf (A + ε) + bound_on_total_hep_workload A (A + F) /\
F <= R
BLOCK : blocking_bound ts tsk = 0
LT : A < L
NEQ : response_time_bound.task_rbf_changes_at tsk A
|| response_time_bound.bound_on_total_hep_workload_changes_at ts tsk
A
F : duration
FIX : A + F = task_rbf (A + ε) + bound_on_total_hep_workload A (A + F)
BOUND : F <= R
============================
A + F =
blocking_bound ts tsk +
(task_request_bound_function tsk (A + ε) -
(task_cost tsk - parameters.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))
subgoal 2 (ID 458) is:
F + (task_cost tsk - parameters.task_run_to_completion_threshold tsk) <= R
----------------------------------------------------------------------------- *)
+ by rewrite BLOCK add0n subnn subn0.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 458)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : 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
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H4 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_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_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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
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
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
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, A : duration
H_R_is_maximum : exists F : duration,
A + F =
task_rbf (A + ε) + bound_on_total_hep_workload A (A + F) /\
F <= R
BLOCK : blocking_bound ts tsk = 0
LT : A < L
NEQ : response_time_bound.task_rbf_changes_at tsk A
|| response_time_bound.bound_on_total_hep_workload_changes_at ts tsk
A
F : duration
FIX : A + F = task_rbf (A + ε) + bound_on_total_hep_workload A (A + F)
BOUND : F <= R
============================
F + (task_cost tsk - parameters.task_run_to_completion_threshold tsk) <= R
----------------------------------------------------------------------------- *)
+ by rewrite subnn addn0.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
End RTAforFullyPreemptiveEDFModelwithArrivalCurves.
response_time_bounded_by tsk R.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 271)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : 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
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H4 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_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_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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
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
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
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 =
task_rbf (A + ε) + bound_on_total_hep_workload A (A + F) /\
F <= R
============================
response_time_bounded_by tsk R
----------------------------------------------------------------------------- *)
Proof.
have BLOCK: blocking_bound ts tsk = 0.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 277)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : 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
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H4 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_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_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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
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
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
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 =
task_rbf (A + ε) + bound_on_total_hep_workload A (A + F) /\
F <= R
============================
blocking_bound ts tsk = 0
subgoal 2 (ID 279) is:
response_time_bounded_by tsk R
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 277)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : 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
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H4 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_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_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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
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
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
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 =
task_rbf (A + ε) + bound_on_total_hep_workload A (A + F) /\
F <= R
============================
blocking_bound ts tsk = 0
----------------------------------------------------------------------------- *)
by rewrite /blocking_bound /parameters.task_max_nonpreemptive_segment
/preemptive.fully_preemptive_model subnn big1_eq.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 279)
subgoal 1 (ID 279) is:
response_time_bounded_by tsk R
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 279)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : 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
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H4 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_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_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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
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
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
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 =
task_rbf (A + ε) + bound_on_total_hep_workload A (A + F) /\
F <= R
BLOCK : blocking_bound ts tsk = 0
============================
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 321)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : 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
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H4 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_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_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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
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
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
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 =
task_rbf (A + ε) + bound_on_total_hep_workload A (A + F) /\
F <= R
BLOCK : blocking_bound ts tsk = 0
============================
consistent_arrival_times arr_seq
subgoal 2 (ID 322) is:
arrival_sequence_uniq arr_seq
subgoal 3 (ID 323) is:
jobs_come_from_arrival_sequence sched arr_seq
subgoal 4 (ID 324) is:
jobs_must_arrive_to_execute sched
subgoal 5 (ID 325) is:
completed_jobs_dont_execute sched
subgoal 6 (ID 326) is:
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
subgoal 7 (ID 327) is:
sequential_tasks sched
subgoal 8 (ID 328) is:
work_conserving arr_seq sched
subgoal 9 (ID 329) is:
respects_policy_at_preemption_point arr_seq sched
subgoal 10 (ID 330) is:
all_jobs_from_taskset arr_seq ?ts0
subgoal 11 (ID 331) is:
cost_of_jobs_from_arrival_sequence_le_task_cost arr_seq
subgoal 12 (ID 332) is:
valid_taskset_arrival_curve ?ts0 max_arrivals
subgoal 13 (ID 333) is:
taskset_respects_max_arrivals arr_seq ?ts0
subgoal 14 (ID 334) is:
tsk \in ?ts0
subgoal 15 (ID 335) is:
valid_preemption_model arr_seq sched
subgoal 16 (ID 336) is:
valid_rtct.valid_task_run_to_completion_threshold arr_seq tsk
subgoal 17 (ID 337) is:
0 < L
subgoal 18 (ID 338) is:
L = total_request_bound_function ?ts0 L
subgoal 19 (ID 339) is:
forall A : duration,
(A < L) &&
(response_time_bound.task_rbf_changes_at tsk A
|| response_time_bound.bound_on_total_hep_workload_changes_at ?ts0 tsk A) ->
exists F : duration,
A + F =
blocking_bound ?ts0 tsk +
(task_request_bound_function tsk (A + ε) -
(task_cost tsk - parameters.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 - parameters.task_run_to_completion_threshold tsk) <= R
----------------------------------------------------------------------------- *)
all: eauto 2 with basic_facts.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 339)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : 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
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H4 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_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_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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
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
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
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 =
task_rbf (A + ε) + bound_on_total_hep_workload A (A + F) /\
F <= R
BLOCK : blocking_bound ts tsk = 0
============================
forall A : duration,
(A < L) &&
(response_time_bound.task_rbf_changes_at tsk A
|| response_time_bound.bound_on_total_hep_workload_changes_at ts tsk A) ->
exists F : duration,
A + F =
blocking_bound ts tsk +
(task_request_bound_function tsk (A + ε) -
(task_cost tsk - parameters.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 - parameters.task_run_to_completion_threshold tsk) <=
R
----------------------------------------------------------------------------- *)
- move ⇒ A /andP [LT NEQ].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 396)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : 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
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H4 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_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_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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
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
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
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 =
task_rbf (A + ε) + bound_on_total_hep_workload A (A + F) /\
F <= R
BLOCK : blocking_bound ts tsk = 0
A : duration
LT : A < L
NEQ : response_time_bound.task_rbf_changes_at tsk A
|| response_time_bound.bound_on_total_hep_workload_changes_at ts tsk
A
============================
exists F : duration,
A + F =
blocking_bound ts tsk +
(task_request_bound_function tsk (A + ε) -
(task_cost tsk - parameters.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 - parameters.task_run_to_completion_threshold tsk) <=
R
----------------------------------------------------------------------------- *)
specialize (H_R_is_maximum A); feed H_R_is_maximum.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 399)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : 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
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H4 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_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_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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
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
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
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, A : duration
H_R_is_maximum : is_in_search_space A ->
exists F : duration,
A + F =
task_rbf (A + ε) + bound_on_total_hep_workload A (A + F) /\
F <= R
BLOCK : blocking_bound ts tsk = 0
LT : A < L
NEQ : response_time_bound.task_rbf_changes_at tsk A
|| response_time_bound.bound_on_total_hep_workload_changes_at ts tsk
A
============================
is_in_search_space A
subgoal 2 (ID 404) is:
exists F : duration,
A + F =
blocking_bound ts tsk +
(task_request_bound_function tsk (A + ε) -
(task_cost tsk - parameters.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 - parameters.task_run_to_completion_threshold tsk) <= R
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 399)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : 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
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H4 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_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_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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
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
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
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, A : duration
H_R_is_maximum : is_in_search_space A ->
exists F : duration,
A + F =
task_rbf (A + ε) + bound_on_total_hep_workload A (A + F) /\
F <= R
BLOCK : blocking_bound ts tsk = 0
LT : A < L
NEQ : response_time_bound.task_rbf_changes_at tsk A
|| response_time_bound.bound_on_total_hep_workload_changes_at ts tsk
A
============================
is_in_search_space A
----------------------------------------------------------------------------- *)
by apply/andP; split.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 404)
subgoal 1 (ID 404) is:
exists F : duration,
A + F =
blocking_bound ts tsk +
(task_request_bound_function tsk (A + ε) -
(task_cost tsk - parameters.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 - parameters.task_run_to_completion_threshold tsk) <= R
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 404)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : 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
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H4 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_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_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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
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
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
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, A : duration
H_R_is_maximum : exists F : duration,
A + F =
task_rbf (A + ε) + bound_on_total_hep_workload A (A + F) /\
F <= R
BLOCK : blocking_bound ts tsk = 0
LT : A < L
NEQ : response_time_bound.task_rbf_changes_at tsk A
|| response_time_bound.bound_on_total_hep_workload_changes_at ts tsk
A
============================
exists F : duration,
A + F =
blocking_bound ts tsk +
(task_request_bound_function tsk (A + ε) -
(task_cost tsk - parameters.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 - parameters.task_run_to_completion_threshold tsk) <=
R
----------------------------------------------------------------------------- *)
move: H_R_is_maximum ⇒ [F [FIX BOUND]].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 453)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : 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
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H4 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_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_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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
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
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
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, A : duration
H_R_is_maximum : exists F : duration,
A + F =
task_rbf (A + ε) + bound_on_total_hep_workload A (A + F) /\
F <= R
BLOCK : blocking_bound ts tsk = 0
LT : A < L
NEQ : response_time_bound.task_rbf_changes_at tsk A
|| response_time_bound.bound_on_total_hep_workload_changes_at ts tsk
A
F : duration
FIX : A + F = task_rbf (A + ε) + bound_on_total_hep_workload A (A + F)
BOUND : F <= R
============================
exists F0 : duration,
A + F0 =
blocking_bound ts tsk +
(task_request_bound_function tsk (A + ε) -
(task_cost tsk - parameters.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 + F0)) /\
F0 + (task_cost tsk - parameters.task_run_to_completion_threshold tsk) <=
R
----------------------------------------------------------------------------- *)
∃ F; split.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 457)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : 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
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H4 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_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_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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
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
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
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, A : duration
H_R_is_maximum : exists F : duration,
A + F =
task_rbf (A + ε) + bound_on_total_hep_workload A (A + F) /\
F <= R
BLOCK : blocking_bound ts tsk = 0
LT : A < L
NEQ : response_time_bound.task_rbf_changes_at tsk A
|| response_time_bound.bound_on_total_hep_workload_changes_at ts tsk
A
F : duration
FIX : A + F = task_rbf (A + ε) + bound_on_total_hep_workload A (A + F)
BOUND : F <= R
============================
A + F =
blocking_bound ts tsk +
(task_request_bound_function tsk (A + ε) -
(task_cost tsk - parameters.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))
subgoal 2 (ID 458) is:
F + (task_cost tsk - parameters.task_run_to_completion_threshold tsk) <= R
----------------------------------------------------------------------------- *)
+ by rewrite BLOCK add0n subnn subn0.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 458)
Task : TaskType
H : TaskCost Task
H0 : TaskDeadline Task
Job : JobType
H1 : JobTask Job Task
H2 : JobArrival Job
H3 : 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
ts : seq Task
H_all_jobs_from_taskset : all_jobs_from_taskset arr_seq ts
H_job_cost_le_task_cost : cost_of_jobs_from_arrival_sequence_le_task_cost
arr_seq
H4 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_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_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
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
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
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
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, A : duration
H_R_is_maximum : exists F : duration,
A + F =
task_rbf (A + ε) + bound_on_total_hep_workload A (A + F) /\
F <= R
BLOCK : blocking_bound ts tsk = 0
LT : A < L
NEQ : response_time_bound.task_rbf_changes_at tsk A
|| response_time_bound.bound_on_total_hep_workload_changes_at ts tsk
A
F : duration
FIX : A + F = task_rbf (A + ε) + bound_on_total_hep_workload A (A + F)
BOUND : F <= R
============================
F + (task_cost tsk - parameters.task_run_to_completion_threshold tsk) <= R
----------------------------------------------------------------------------- *)
+ by rewrite subnn addn0.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
End RTAforFullyPreemptiveEDFModelwithArrivalCurves.