Library prosa.classic.implementation.global.basic.bertogna_fp_example
Require Import prosa.classic.util.all.
Require Import prosa.classic.model.arrival.basic.job prosa.classic.model.arrival.basic.task prosa.classic.model.priority.
Require Import prosa.classic.model.schedule.global.schedulability.
Require Import prosa.classic.model.schedule.global.basic.schedule prosa.classic.model.schedule.global.basic.platform.
Require Import prosa.classic.analysis.global.basic.workload_bound
prosa.classic.analysis.global.basic.interference_bound_fp
prosa.classic.analysis.global.basic.bertogna_fp_comp.
Require Import prosa.classic.implementation.job prosa.classic.implementation.task
prosa.classic.implementation.arrival_sequence.
Require Import prosa.classic.implementation.global.basic.schedule.
From mathcomp Require Import ssreflect ssrbool ssrnat eqtype seq bigop div.
Module ResponseTimeAnalysisFP.
Import Job Schedule SporadicTaskset Priority Schedulability Platform InterferenceBoundFP WorkloadBound ResponseTimeIterationFP.
Import ConcreteJob ConcreteTask ConcreteArrivalSequence ConcreteScheduler.
(* In this section, we instantiate a simple example to show that the theorems
contain no contradictory assumptions. *)
Section ExampleRTA.
Let tsk1 := {| task_id := 1; task_cost := 2; task_period := 5; task_deadline := 3|}.
Let tsk2 := {| task_id := 2; task_cost := 4; task_period := 6; task_deadline := 5|}.
Let tsk3 := {| task_id := 3; task_cost := 3; task_period := 12; task_deadline := 11|}.
(* Let ts be a task set containing these three tasks (sorted by rate-monotonic priority). *)
Program Let ts := Build_set [:: tsk1; tsk2; tsk3] _.
Section FactsAboutTaskset.
Fact ts_has_valid_parameters:
valid_sporadic_taskset task_cost task_period task_deadline ts.
Proof.
intros tsk IN.
repeat (move: IN ⇒ /orP [/eqP EQ | IN]; subst; compute); by done.
Qed.
Fact ts_has_constrained_deadlines:
∀ tsk,
tsk \in ts →
task_deadline tsk ≤ task_period tsk.
Proof.
intros tsk IN.
repeat (move: IN ⇒ /orP [/eqP EQ | IN]; subst; compute); by done.
Qed.
End FactsAboutTaskset.
(* Assume there are two processors. *)
Let num_cpus := 2.
(* Recall the FP RTA schedulability test. *)
Let schedulability_test :=
fp_schedulable task_cost task_period task_deadline num_cpus.
(* Now we show that the schedulability test returns true. *)
Fact schedulability_test_succeeds :
schedulability_test ts = true.
Proof.
unfold schedulability_test, fp_schedulable, fp_claimed_bounds; simpl.
unfold total_interference_bound_fp, div_floor.
rewrite big_nil div0n addn0 /=.
unfold div_floor; simpl.
set I2 := total_interference_bound_fp task_cost task_period tsk2
[:: (tsk1, 2)].
assert (H1: I2 4 = 1).
{
by unfold I2, total_interference_bound_fp; rewrite big_cons big_nil; compute.
}
rewrite H1.
have H2: 4 + 1 %/ num_cpus = 4 by compute.
rewrite H2 H1 H2.
have H3: 3 < 5 by compute.
rewrite H3.
unfold fp_bound_of_task.
have H4: per_task_rta task_cost task_period num_cpus tsk3 [:: (tsk1, 2); (tsk2, 4)]
(max_steps task_cost task_deadline tsk3) = 5.
{
unfold per_task_rta; simpl.
unfold total_interference_bound_fp at 9.
rewrite !big_cons big_nil /=.
unfold total_interference_bound_fp at 8.
rewrite !big_cons big_nil /=.
unfold total_interference_bound_fp at 7.
rewrite !big_cons big_nil /=.
unfold total_interference_bound_fp at 6.
rewrite !big_cons big_nil /=.
unfold total_interference_bound_fp at 5.
rewrite !big_cons big_nil /=.
unfold total_interference_bound_fp at 4.
rewrite !big_cons big_nil /=.
unfold total_interference_bound_fp at 3.
rewrite !big_cons big_nil /=.
unfold total_interference_bound_fp at 2.
rewrite !big_cons big_nil /=.
unfold total_interference_bound_fp at 1.
by rewrite !big_cons big_nil /=; compute.
}
by rewrite H4.
Qed.
(* Let arr_seq be the periodic arrival sequence from ts. *)
Let arr_seq := periodic_arrival_sequence ts.
(* Assume rate-monotonic priorities. *)
Let higher_priority := FP_to_JLDP job_task (RM task_period).
Section FactsAboutPriorityOrder.
Lemma ts_has_unique_priorities :
FP_is_antisymmetric_over_task_set (RM task_period) ts.
Proof.
unfold RM; intros tsk tsk' IN IN' HP HP'.
have EQ: task_period tsk = task_period tsk' by apply/eqP; rewrite eqn_leq HP HP'.
clear HP HP'.
rewrite !in_cons 2!in_nil 2!orbF in IN IN'; des; rewrite IN IN'; try (by done);
subst tsk tsk'; simpl in *; by done.
Qed.
Lemma priority_is_total :
FP_is_total_over_task_set (RM task_period) ts.
Proof.
unfold RM; intros tsk tsk' IN IN'.
destruct (leqP (task_period tsk) (task_period tsk'));
[by left | by right; apply ltnW].
Qed.
End FactsAboutPriorityOrder.
(* Let sched be the work-conserving RM scheduler. *)
Let sched := scheduler job_arrival job_cost num_cpus arr_seq higher_priority.
(* Recall the definition of deadline miss. *)
Let no_deadline_missed_by :=
task_misses_no_deadline job_arrival job_cost job_deadline job_task arr_seq sched.
(* Next, we prove that ts is schedulable with the result of the test. *)
Corollary ts_is_schedulable:
∀ tsk,
tsk \in ts →
no_deadline_missed_by tsk.
Proof.
intros tsk IN.
have VALID := periodic_arrivals_valid_job_parameters ts ts_has_valid_parameters.
have TSVALID := ts_has_valid_parameters.
unfold valid_sporadic_job, valid_realtime_job in *; des.
try ( apply taskset_schedulable_by_fp_rta with (task_cost := task_cost)
(task_period := task_period) (task_deadline := task_deadline)
(ts0 := ts) (higher_priority0 := RM task_period); try (by done) ) ||
apply taskset_schedulable_by_fp_rta with (task_cost := task_cost)
(task_period := task_period) (task_deadline := task_deadline)
(ts := ts) (higher_priority := RM task_period); try (by done).
- by apply ts_has_constrained_deadlines.
- by apply ts_has_unique_priorities.
- by apply priority_is_total.
- by apply RM_is_transitive.
- by apply periodic_arrivals_all_jobs_from_taskset.
- by apply periodic_arrivals_are_sporadic.
- by apply scheduler_jobs_come_from_arrival_sequence.
- by apply scheduler_jobs_must_arrive_to_execute.
- apply scheduler_completed_jobs_dont_execute.
-- by apply periodic_arrivals_are_consistent.
-- by apply periodic_arrivals_is_a_set.
- apply scheduler_sequential_jobs.
-- by apply periodic_arrivals_are_consistent.
-- by apply periodic_arrivals_is_a_set.
- by apply scheduler_work_conserving, periodic_arrivals_are_consistent.
- apply scheduler_respects_policy.
-- by apply periodic_arrivals_are_consistent.
-- by intros t; apply RM_is_transitive.
-- by intros t x y; apply leq_total.
- by apply schedulability_test_succeeds.
Qed.
End ExampleRTA.
End ResponseTimeAnalysisFP.
Require Import prosa.classic.model.arrival.basic.job prosa.classic.model.arrival.basic.task prosa.classic.model.priority.
Require Import prosa.classic.model.schedule.global.schedulability.
Require Import prosa.classic.model.schedule.global.basic.schedule prosa.classic.model.schedule.global.basic.platform.
Require Import prosa.classic.analysis.global.basic.workload_bound
prosa.classic.analysis.global.basic.interference_bound_fp
prosa.classic.analysis.global.basic.bertogna_fp_comp.
Require Import prosa.classic.implementation.job prosa.classic.implementation.task
prosa.classic.implementation.arrival_sequence.
Require Import prosa.classic.implementation.global.basic.schedule.
From mathcomp Require Import ssreflect ssrbool ssrnat eqtype seq bigop div.
Module ResponseTimeAnalysisFP.
Import Job Schedule SporadicTaskset Priority Schedulability Platform InterferenceBoundFP WorkloadBound ResponseTimeIterationFP.
Import ConcreteJob ConcreteTask ConcreteArrivalSequence ConcreteScheduler.
(* In this section, we instantiate a simple example to show that the theorems
contain no contradictory assumptions. *)
Section ExampleRTA.
Let tsk1 := {| task_id := 1; task_cost := 2; task_period := 5; task_deadline := 3|}.
Let tsk2 := {| task_id := 2; task_cost := 4; task_period := 6; task_deadline := 5|}.
Let tsk3 := {| task_id := 3; task_cost := 3; task_period := 12; task_deadline := 11|}.
(* Let ts be a task set containing these three tasks (sorted by rate-monotonic priority). *)
Program Let ts := Build_set [:: tsk1; tsk2; tsk3] _.
Section FactsAboutTaskset.
Fact ts_has_valid_parameters:
valid_sporadic_taskset task_cost task_period task_deadline ts.
Proof.
intros tsk IN.
repeat (move: IN ⇒ /orP [/eqP EQ | IN]; subst; compute); by done.
Qed.
Fact ts_has_constrained_deadlines:
∀ tsk,
tsk \in ts →
task_deadline tsk ≤ task_period tsk.
Proof.
intros tsk IN.
repeat (move: IN ⇒ /orP [/eqP EQ | IN]; subst; compute); by done.
Qed.
End FactsAboutTaskset.
(* Assume there are two processors. *)
Let num_cpus := 2.
(* Recall the FP RTA schedulability test. *)
Let schedulability_test :=
fp_schedulable task_cost task_period task_deadline num_cpus.
(* Now we show that the schedulability test returns true. *)
Fact schedulability_test_succeeds :
schedulability_test ts = true.
Proof.
unfold schedulability_test, fp_schedulable, fp_claimed_bounds; simpl.
unfold total_interference_bound_fp, div_floor.
rewrite big_nil div0n addn0 /=.
unfold div_floor; simpl.
set I2 := total_interference_bound_fp task_cost task_period tsk2
[:: (tsk1, 2)].
assert (H1: I2 4 = 1).
{
by unfold I2, total_interference_bound_fp; rewrite big_cons big_nil; compute.
}
rewrite H1.
have H2: 4 + 1 %/ num_cpus = 4 by compute.
rewrite H2 H1 H2.
have H3: 3 < 5 by compute.
rewrite H3.
unfold fp_bound_of_task.
have H4: per_task_rta task_cost task_period num_cpus tsk3 [:: (tsk1, 2); (tsk2, 4)]
(max_steps task_cost task_deadline tsk3) = 5.
{
unfold per_task_rta; simpl.
unfold total_interference_bound_fp at 9.
rewrite !big_cons big_nil /=.
unfold total_interference_bound_fp at 8.
rewrite !big_cons big_nil /=.
unfold total_interference_bound_fp at 7.
rewrite !big_cons big_nil /=.
unfold total_interference_bound_fp at 6.
rewrite !big_cons big_nil /=.
unfold total_interference_bound_fp at 5.
rewrite !big_cons big_nil /=.
unfold total_interference_bound_fp at 4.
rewrite !big_cons big_nil /=.
unfold total_interference_bound_fp at 3.
rewrite !big_cons big_nil /=.
unfold total_interference_bound_fp at 2.
rewrite !big_cons big_nil /=.
unfold total_interference_bound_fp at 1.
by rewrite !big_cons big_nil /=; compute.
}
by rewrite H4.
Qed.
(* Let arr_seq be the periodic arrival sequence from ts. *)
Let arr_seq := periodic_arrival_sequence ts.
(* Assume rate-monotonic priorities. *)
Let higher_priority := FP_to_JLDP job_task (RM task_period).
Section FactsAboutPriorityOrder.
Lemma ts_has_unique_priorities :
FP_is_antisymmetric_over_task_set (RM task_period) ts.
Proof.
unfold RM; intros tsk tsk' IN IN' HP HP'.
have EQ: task_period tsk = task_period tsk' by apply/eqP; rewrite eqn_leq HP HP'.
clear HP HP'.
rewrite !in_cons 2!in_nil 2!orbF in IN IN'; des; rewrite IN IN'; try (by done);
subst tsk tsk'; simpl in *; by done.
Qed.
Lemma priority_is_total :
FP_is_total_over_task_set (RM task_period) ts.
Proof.
unfold RM; intros tsk tsk' IN IN'.
destruct (leqP (task_period tsk) (task_period tsk'));
[by left | by right; apply ltnW].
Qed.
End FactsAboutPriorityOrder.
(* Let sched be the work-conserving RM scheduler. *)
Let sched := scheduler job_arrival job_cost num_cpus arr_seq higher_priority.
(* Recall the definition of deadline miss. *)
Let no_deadline_missed_by :=
task_misses_no_deadline job_arrival job_cost job_deadline job_task arr_seq sched.
(* Next, we prove that ts is schedulable with the result of the test. *)
Corollary ts_is_schedulable:
∀ tsk,
tsk \in ts →
no_deadline_missed_by tsk.
Proof.
intros tsk IN.
have VALID := periodic_arrivals_valid_job_parameters ts ts_has_valid_parameters.
have TSVALID := ts_has_valid_parameters.
unfold valid_sporadic_job, valid_realtime_job in *; des.
try ( apply taskset_schedulable_by_fp_rta with (task_cost := task_cost)
(task_period := task_period) (task_deadline := task_deadline)
(ts0 := ts) (higher_priority0 := RM task_period); try (by done) ) ||
apply taskset_schedulable_by_fp_rta with (task_cost := task_cost)
(task_period := task_period) (task_deadline := task_deadline)
(ts := ts) (higher_priority := RM task_period); try (by done).
- by apply ts_has_constrained_deadlines.
- by apply ts_has_unique_priorities.
- by apply priority_is_total.
- by apply RM_is_transitive.
- by apply periodic_arrivals_all_jobs_from_taskset.
- by apply periodic_arrivals_are_sporadic.
- by apply scheduler_jobs_come_from_arrival_sequence.
- by apply scheduler_jobs_must_arrive_to_execute.
- apply scheduler_completed_jobs_dont_execute.
-- by apply periodic_arrivals_are_consistent.
-- by apply periodic_arrivals_is_a_set.
- apply scheduler_sequential_jobs.
-- by apply periodic_arrivals_are_consistent.
-- by apply periodic_arrivals_is_a_set.
- by apply scheduler_work_conserving, periodic_arrivals_are_consistent.
- apply scheduler_respects_policy.
-- by apply periodic_arrivals_are_consistent.
-- by intros t; apply RM_is_transitive.
-- by intros t x y; apply leq_total.
- by apply schedulability_test_succeeds.
Qed.
End ExampleRTA.
End ResponseTimeAnalysisFP.