Library prosa.results.fixed_priority.rta.fully_preemptive
(* ----------------------------------[ coqtop ]---------------------------------
Welcome to Coq 8.11.2 (June 2020)
----------------------------------------------------------------------------- *)
Require Export prosa.results.fixed_priority.rta.bounded_nps.
Require Export prosa.analysis.facts.preemption.task.preemptive.
Require Export prosa.analysis.facts.preemption.rtc_threshold.preemptive.
From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq path fintype bigop.
RTA for Fully Preemptive FP Model
In this section we prove the RTA theorem for the fully preemptive FP model
Furthermore, we assume the fully preemptive task 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}.
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 the task set, ...
... and the cost of a job cannot be larger than the task cost.
Let max_arrivals be a family of valid arrival curves, i.e., for
any task [tsk] in ts [max_arrival tsk] is (1) an arrival bound of
[tsk], and (2) it is a monotonic function that equals 0 for the
empty interval [delta = 0].
Context `{MaxArrivals Task}.
Hypothesis H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals.
Hypothesis H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts.
Hypothesis H_valid_arrival_curve : valid_taskset_arrival_curve ts max_arrivals.
Hypothesis H_is_arrival_curve : taskset_respects_max_arrivals arr_seq ts.
Let [tsk] be any task in ts that is to be analyzed.
Next, consider any ideal uniprocessor schedule of this arrival sequence ...
Variable sched : schedule (ideal.processor_state Job).
Hypothesis H_jobs_come_from_arrival_sequence:
jobs_come_from_arrival_sequence sched arr_seq.
Hypothesis H_jobs_come_from_arrival_sequence:
jobs_come_from_arrival_sequence sched arr_seq.
... where jobs do not execute before their arrival or after completion.
Hypothesis H_jobs_must_arrive_to_execute : jobs_must_arrive_to_execute sched.
Hypothesis H_completed_jobs_dont_execute : completed_jobs_dont_execute sched.
Hypothesis H_completed_jobs_dont_execute : completed_jobs_dont_execute sched.
Consider an FP policy that indicates a higher-or-equal priority relation,
and assume that the relation is reflexive and transitive.
Context `{FP_policy Task}.
Hypothesis H_priority_is_reflexive : reflexive_priorities.
Hypothesis H_priority_is_transitive : transitive_priorities.
Hypothesis H_priority_is_reflexive : reflexive_priorities.
Hypothesis H_priority_is_transitive : transitive_priorities.
Assume we have sequential tasks, i.e, tasks 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 with higher priority
...
... and the request bound function of all tasks with higher
priority other than task [tsk].
Let L be any positive fixed point of the busy interval recurrence, determined by
the sum of blocking and higher-or-equal-priority workload.
Variable L : duration.
Hypothesis H_L_positive : L > 0.
Hypothesis H_fixed_point : L = total_hep_rbf L.
Hypothesis H_L_positive : L > 0.
Hypothesis H_fixed_point : L = total_hep_rbf L.
Response-Time Bound
Next, consider any value R, and assume that for any given
arrival A from 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) + total_ohep_rbf (A + F) ∧
F ≤ R.
Hypothesis H_R_is_maximum:
∀ (A : duration),
is_in_search_space A →
∃ (F : duration),
A + F = task_rbf (A + ε) + total_ohep_rbf (A + F) ∧
F ≤ 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 fully preemptive
scheduling.
Let response_time_bounded_by := task_response_time_bound arr_seq sched.
Theorem uniprocessor_response_time_bound_fully_preemptive_fp:
response_time_bounded_by tsk R.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1581)
Task : TaskType
H : TaskCost Task
Job : JobType
H0 : JobTask Job Task
H1 : JobArrival Job
H2 : 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
H3 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_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
H4 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks arr_seq 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_hep_rbf := total_hep_request_bound_function_FP ts tsk
: duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP ts tsk
: duration -> nat
L : duration
H_L_positive : 0 < L
H_fixed_point : L = total_hep_rbf L
is_in_search_space := bounded_pi.is_in_search_space tsk L : nat -> bool
R : duration
H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
A + F = task_rbf (A + ε) + total_ohep_rbf (A + F) /\
F <= R
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
============================
response_time_bounded_by tsk R
----------------------------------------------------------------------------- *)
Proof.
have BLOCK: blocking_bound ts tsk = 0.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 1587)
Task : TaskType
H : TaskCost Task
Job : JobType
H0 : JobTask Job Task
H1 : JobArrival Job
H2 : 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
H3 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_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
H4 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks arr_seq 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_hep_rbf := total_hep_request_bound_function_FP ts tsk
: duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP ts tsk
: duration -> nat
L : duration
H_L_positive : 0 < L
H_fixed_point : L = total_hep_rbf L
is_in_search_space := bounded_pi.is_in_search_space tsk L : nat -> bool
R : duration
H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
A + F = task_rbf (A + ε) + total_ohep_rbf (A + F) /\
F <= R
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
============================
blocking_bound ts tsk = 0
subgoal 2 (ID 1589) is:
response_time_bounded_by tsk R
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1587)
Task : TaskType
H : TaskCost Task
Job : JobType
H0 : JobTask Job Task
H1 : JobArrival Job
H2 : 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
H3 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_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
H4 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks arr_seq 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_hep_rbf := total_hep_request_bound_function_FP ts tsk
: duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP ts tsk
: duration -> nat
L : duration
H_L_positive : 0 < L
H_fixed_point : L = total_hep_rbf L
is_in_search_space := bounded_pi.is_in_search_space tsk L : nat -> bool
R : duration
H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
A + F = task_rbf (A + ε) + total_ohep_rbf (A + F) /\
F <= R
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
============================
blocking_bound ts tsk = 0
----------------------------------------------------------------------------- *)
by rewrite /blocking_bound /parameters.task_max_nonpreemptive_segment
/fully_preemptive.fully_preemptive_model subnn big1_eq.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1589)
subgoal 1 (ID 1589) is:
response_time_bounded_by tsk R
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1589)
Task : TaskType
H : TaskCost Task
Job : JobType
H0 : JobTask Job Task
H1 : JobArrival Job
H2 : 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
H3 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_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
H4 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks arr_seq 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_hep_rbf := total_hep_request_bound_function_FP ts tsk
: duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP ts tsk
: duration -> nat
L : duration
H_L_positive : 0 < L
H_fixed_point : L = total_hep_rbf L
is_in_search_space := bounded_pi.is_in_search_space tsk L : nat -> bool
R : duration
H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
A + F = task_rbf (A + ε) + total_ohep_rbf (A + F) /\
F <= R
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
BLOCK : blocking_bound ts tsk = 0
============================
response_time_bounded_by tsk R
----------------------------------------------------------------------------- *)
eapply uniprocessor_response_time_bound_fp_with_bounded_nonpreemptive_segments.
(* ----------------------------------[ coqtop ]---------------------------------
21 focused subgoals
(shelved: 2) (ID 1632)
Task : TaskType
H : TaskCost Task
Job : JobType
H0 : JobTask Job Task
H1 : JobArrival Job
H2 : 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
H3 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_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
H4 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks arr_seq 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_hep_rbf := total_hep_request_bound_function_FP ts tsk
: duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP ts tsk
: duration -> nat
L : duration
H_L_positive : 0 < L
H_fixed_point : L = total_hep_rbf L
is_in_search_space := bounded_pi.is_in_search_space tsk L : nat -> bool
R : duration
H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
A + F = task_rbf (A + ε) + total_ohep_rbf (A + F) /\
F <= R
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
BLOCK : blocking_bound ts tsk = 0
============================
consistent_arrival_times arr_seq
subgoal 2 (ID 1633) is:
arrival_sequence_uniq arr_seq
subgoal 3 (ID 1634) is:
jobs_come_from_arrival_sequence sched arr_seq
subgoal 4 (ID 1635) is:
jobs_must_arrive_to_execute sched
subgoal 5 (ID 1636) is:
completed_jobs_dont_execute sched
subgoal 6 (ID 1637) is:
valid_model_with_bounded_nonpreemptive_segments arr_seq sched
subgoal 7 (ID 1638) is:
reflexive_priorities
subgoal 8 (ID 1639) is:
transitive_priorities
subgoal 9 (ID 1640) is:
sequential_tasks arr_seq sched
subgoal 10 (ID 1641) is:
work_conserving arr_seq sched
subgoal 11 (ID 1642) is:
respects_policy_at_preemption_point arr_seq sched
subgoal 12 (ID 1643) is:
all_jobs_from_taskset arr_seq ?ts
subgoal 13 (ID 1644) is:
arrivals_have_valid_job_costs arr_seq
subgoal 14 (ID 1645) is:
valid_taskset_arrival_curve ?ts max_arrivals
subgoal 15 (ID 1646) is:
taskset_respects_max_arrivals arr_seq ?ts
subgoal 16 (ID 1647) is:
tsk \in ?ts
subgoal 17 (ID 1648) is:
valid_preemption_model arr_seq sched
subgoal 18 (ID 1649) is:
valid_task_run_to_completion_threshold arr_seq tsk
subgoal 19 (ID 1650) is:
0 < ?L
subgoal 20 (ID 1651) is:
?L = blocking_bound ?ts tsk + total_hep_request_bound_function_FP ?ts tsk ?L
subgoal 21 (ID 1652) is:
forall A : duration,
bounded_pi.is_in_search_space tsk ?L A ->
exists F : duration,
A + F =
blocking_bound ?ts tsk +
(task_request_bound_function tsk (A + ε) - (task_cost tsk - task_rtct tsk)) +
total_ohep_request_bound_function_FP ?ts tsk (A + F) /\
F + (task_cost tsk - task_rtct tsk) <= R
----------------------------------------------------------------------------- *)
all: eauto 2 with basic_facts.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 1651)
Task : TaskType
H : TaskCost Task
Job : JobType
H0 : JobTask Job Task
H1 : JobArrival Job
H2 : 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
H3 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_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
H4 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks arr_seq 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_hep_rbf := total_hep_request_bound_function_FP ts tsk
: duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP ts tsk
: duration -> nat
L : duration
H_L_positive : 0 < L
H_fixed_point : L = total_hep_rbf L
is_in_search_space := bounded_pi.is_in_search_space tsk L : nat -> bool
R : duration
H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
A + F = task_rbf (A + ε) + total_ohep_rbf (A + F) /\
F <= R
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
BLOCK : blocking_bound ts tsk = 0
============================
L = blocking_bound ts tsk + total_hep_request_bound_function_FP ts tsk L
subgoal 2 (ID 1652) is:
forall A : duration,
bounded_pi.is_in_search_space tsk L A ->
exists F : duration,
A + F =
blocking_bound ts tsk +
(task_request_bound_function tsk (A + ε) - (task_cost tsk - task_rtct tsk)) +
total_ohep_request_bound_function_FP ts tsk (A + F) /\
F + (task_cost tsk - task_rtct tsk) <= R
----------------------------------------------------------------------------- *)
- by rewrite BLOCK add0n.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1652)
Task : TaskType
H : TaskCost Task
Job : JobType
H0 : JobTask Job Task
H1 : JobArrival Job
H2 : 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
H3 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_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
H4 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks arr_seq 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_hep_rbf := total_hep_request_bound_function_FP ts tsk
: duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP ts tsk
: duration -> nat
L : duration
H_L_positive : 0 < L
H_fixed_point : L = total_hep_rbf L
is_in_search_space := bounded_pi.is_in_search_space tsk L : nat -> bool
R : duration
H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
A + F = task_rbf (A + ε) + total_ohep_rbf (A + F) /\
F <= R
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
BLOCK : blocking_bound ts tsk = 0
============================
forall A : duration,
bounded_pi.is_in_search_space tsk L A ->
exists F : duration,
A + F =
blocking_bound ts tsk +
(task_request_bound_function tsk (A + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_request_bound_function_FP ts tsk (A + F) /\
F + (task_cost tsk - task_rtct tsk) <= R
----------------------------------------------------------------------------- *)
- move ⇒ A /andP [LT NEQ].
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1727)
Task : TaskType
H : TaskCost Task
Job : JobType
H0 : JobTask Job Task
H1 : JobArrival Job
H2 : 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
H3 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_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
H4 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks arr_seq 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_hep_rbf := total_hep_request_bound_function_FP ts tsk
: duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP ts tsk
: duration -> nat
L : duration
H_L_positive : 0 < L
H_fixed_point : L = total_hep_rbf L
is_in_search_space := bounded_pi.is_in_search_space tsk L : nat -> bool
R : duration
H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
A + F = task_rbf (A + ε) + total_ohep_rbf (A + F) /\
F <= R
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
BLOCK : blocking_bound ts tsk = 0
A : duration
LT : A < L
NEQ : task_request_bound_function tsk A
!= task_request_bound_function tsk (A + ε)
============================
exists F : duration,
A + F =
blocking_bound ts tsk +
(task_request_bound_function tsk (A + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_request_bound_function_FP ts tsk (A + F) /\
F + (task_cost tsk - task_rtct tsk) <= R
----------------------------------------------------------------------------- *)
edestruct H_R_is_maximum as [F [FIX BOUND]].
(* ----------------------------------[ coqtop ]---------------------------------
2 focused subgoals
(shelved: 1) (ID 1731)
Task : TaskType
H : TaskCost Task
Job : JobType
H0 : JobTask Job Task
H1 : JobArrival Job
H2 : 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
H3 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_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
H4 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks arr_seq 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_hep_rbf := total_hep_request_bound_function_FP ts tsk
: duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP ts tsk
: duration -> nat
L : duration
H_L_positive : 0 < L
H_fixed_point : L = total_hep_rbf L
is_in_search_space := bounded_pi.is_in_search_space tsk L : nat -> bool
R : duration
H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
A + F = task_rbf (A + ε) + total_ohep_rbf (A + F) /\
F <= R
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
BLOCK : blocking_bound ts tsk = 0
A : duration
LT : A < L
NEQ : task_request_bound_function tsk A
!= task_request_bound_function tsk (A + ε)
============================
is_in_search_space ?A
subgoal 2 (ID 1741) is:
exists F0 : duration,
A + F0 =
blocking_bound ts tsk +
(task_request_bound_function tsk (A + ε) - (task_cost tsk - task_rtct tsk)) +
total_ohep_request_bound_function_FP ts tsk (A + F0) /\
F0 + (task_cost tsk - task_rtct tsk) <= R
----------------------------------------------------------------------------- *)
{
(* ----------------------------------[ coqtop ]---------------------------------
1 focused subgoal
(shelved: 1) (ID 1731)
Task : TaskType
H : TaskCost Task
Job : JobType
H0 : JobTask Job Task
H1 : JobArrival Job
H2 : 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
H3 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_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
H4 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks arr_seq 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_hep_rbf := total_hep_request_bound_function_FP ts tsk
: duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP ts tsk
: duration -> nat
L : duration
H_L_positive : 0 < L
H_fixed_point : L = total_hep_rbf L
is_in_search_space := bounded_pi.is_in_search_space tsk L : nat -> bool
R : duration
H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
A + F = task_rbf (A + ε) + total_ohep_rbf (A + F) /\
F <= R
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
BLOCK : blocking_bound ts tsk = 0
A : duration
LT : A < L
NEQ : task_request_bound_function tsk A
!= task_request_bound_function tsk (A + ε)
============================
is_in_search_space ?A
----------------------------------------------------------------------------- *)
by apply/andP; split; eauto 2.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1741)
subgoal 1 (ID 1741) is:
exists F0 : duration,
A + F0 =
blocking_bound ts tsk +
(task_request_bound_function tsk (A + ε) - (task_cost tsk - task_rtct tsk)) +
total_ohep_request_bound_function_FP ts tsk (A + F0) /\
F0 + (task_cost tsk - task_rtct tsk) <= R
----------------------------------------------------------------------------- *)
}
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1741)
Task : TaskType
H : TaskCost Task
Job : JobType
H0 : JobTask Job Task
H1 : JobArrival Job
H2 : 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
H3 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_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
H4 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks arr_seq 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_hep_rbf := total_hep_request_bound_function_FP ts tsk
: duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP ts tsk
: duration -> nat
L : duration
H_L_positive : 0 < L
H_fixed_point : L = total_hep_rbf L
is_in_search_space := bounded_pi.is_in_search_space tsk L : nat -> bool
R : duration
H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
A + F = task_rbf (A + ε) + total_ohep_rbf (A + F) /\
F <= R
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
BLOCK : blocking_bound ts tsk = 0
A : duration
LT : A < L
NEQ : task_request_bound_function tsk A
!= task_request_bound_function tsk (A + ε)
F : duration
FIX : A + F = task_rbf (A + ε) + total_ohep_rbf (A + F)
BOUND : F <= R
============================
exists F0 : duration,
A + F0 =
blocking_bound ts tsk +
(task_request_bound_function tsk (A + ε) -
(task_cost tsk - task_rtct tsk)) +
total_ohep_request_bound_function_FP ts tsk (A + F0) /\
F0 + (task_cost tsk - task_rtct tsk) <= R
----------------------------------------------------------------------------- *)
∃ F; split.
(* ----------------------------------[ coqtop ]---------------------------------
2 subgoals (ID 1782)
Task : TaskType
H : TaskCost Task
Job : JobType
H0 : JobTask Job Task
H1 : JobArrival Job
H2 : 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
H3 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_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
H4 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks arr_seq 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_hep_rbf := total_hep_request_bound_function_FP ts tsk
: duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP ts tsk
: duration -> nat
L : duration
H_L_positive : 0 < L
H_fixed_point : L = total_hep_rbf L
is_in_search_space := bounded_pi.is_in_search_space tsk L : nat -> bool
R : duration
H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
A + F = task_rbf (A + ε) + total_ohep_rbf (A + F) /\
F <= R
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
BLOCK : blocking_bound ts tsk = 0
A : duration
LT : A < L
NEQ : task_request_bound_function tsk A
!= task_request_bound_function tsk (A + ε)
F : duration
FIX : A + F = task_rbf (A + ε) + total_ohep_rbf (A + F)
BOUND : F <= R
============================
A + F =
blocking_bound ts tsk +
(task_request_bound_function tsk (A + ε) - (task_cost tsk - task_rtct tsk)) +
total_ohep_request_bound_function_FP ts tsk (A + F)
subgoal 2 (ID 1783) is:
F + (task_cost tsk - task_rtct tsk) <= R
----------------------------------------------------------------------------- *)
+ by rewrite BLOCK add0n subnn subn0.
(* ----------------------------------[ coqtop ]---------------------------------
1 subgoal (ID 1783)
Task : TaskType
H : TaskCost Task
Job : JobType
H0 : JobTask Job Task
H1 : JobArrival Job
H2 : 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
H3 : MaxArrivals Task
H_valid_arrival_curve : valid_taskset_arrival_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
H4 : FP_policy Task
H_priority_is_reflexive : reflexive_priorities
H_priority_is_transitive : transitive_priorities
H_sequential_tasks : sequential_tasks arr_seq 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_hep_rbf := total_hep_request_bound_function_FP ts tsk
: duration -> nat
total_ohep_rbf := total_ohep_request_bound_function_FP ts tsk
: duration -> nat
L : duration
H_L_positive : 0 < L
H_fixed_point : L = total_hep_rbf L
is_in_search_space := bounded_pi.is_in_search_space tsk L : nat -> bool
R : duration
H_R_is_maximum : forall A : duration,
is_in_search_space A ->
exists F : duration,
A + F = task_rbf (A + ε) + total_ohep_rbf (A + F) /\
F <= R
response_time_bounded_by := task_response_time_bound arr_seq sched
: Task -> duration -> Prop
BLOCK : blocking_bound ts tsk = 0
A : duration
LT : A < L
NEQ : task_request_bound_function tsk A
!= task_request_bound_function tsk (A + ε)
F : duration
FIX : A + F = task_rbf (A + ε) + total_ohep_rbf (A + F)
BOUND : F <= R
============================
F + (task_cost tsk - task_rtct tsk) <= R
----------------------------------------------------------------------------- *)
+ by rewrite subnn addn0.
(* ----------------------------------[ coqtop ]---------------------------------
No more subgoals.
----------------------------------------------------------------------------- *)
Qed.
End RTAforFullyPreemptiveFPModelwithArrivalCurves.