Library rt.analysis.parallel.bertogna_fp_comp
Require Import rt.util.all.
Require Import rt.analysis.parallel.bertogna_fp_theory.
From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq fintype bigop div path.
Module ResponseTimeIterationFP.
Import ResponseTimeAnalysisFP.
(* In this section, we define the algorithm of Bertogna and Cirinei's
response-time analysis for FP scheduling with parallel jobs. *)
Section Analysis.
Context {sporadic_task: eqType}.
Variable task_cost: sporadic_task → time.
Variable task_period: sporadic_task → time.
Variable task_deadline: sporadic_task → time.
(* During the iterations of the algorithm, we pass around pairs
of tasks and computed response-time bounds. *)
Let task_with_response_time := (sporadic_task × time)%type.
Context {Job: eqType}.
Variable job_cost: Job → time.
Variable job_deadline: Job → time.
Variable job_task: Job → sporadic_task.
(* Consider a platform with num_cpus processors, ... *)
Variable num_cpus: nat.
(* ..., and priorities based on an FP policy. *)
Variable higher_priority: FP_policy sporadic_task.
(* Next we define the fixed-point iteration for computing
Bertogna's response-time bound of a task set. *)
(* First, given a sequence of pairs R_prev = <..., (tsk_hp, R_hp)> of
response-time bounds for the higher-priority tasks, we define an
iteration that computes the response-time bound of the current task:
R_tsk (0) = task_cost tsk
R_tsk (step + 1) = f (R step),
where f is the response-time recurrence, step is the number of iterations,
and R_tsk (0) is the initial state. *)
Definition per_task_rta (tsk: sporadic_task)
(R_prev: seq task_with_response_time) (step: nat) :=
iter step
(fun t ⇒ task_cost tsk +
div_floor
(total_interference_bound_fp task_cost task_period R_prev t)
num_cpus)
(task_cost tsk).
(* To ensure that the iteration converges, we will apply per_task_rta
a "sufficient" number of times: task_deadline tsk - task_cost tsk + 1.
This corresponds to the time complexity of the iteration. *)
Definition max_steps (tsk: sporadic_task) := task_deadline tsk - task_cost tsk + 1.
(* Next we compute the response-time bounds for the entire task set.
Since high-priority tasks may not be schedulable, we allow the
computation to fail.
Thus, given the response-time bound of previous tasks, we either
(a) append the computed response-time bound (tsk, R) of the current task
to the list of pairs, or,
(b) return None if the response-time analysis failed. *)
Definition fp_bound_of_task hp_pairs tsk :=
if hp_pairs is Some rt_bounds then
let R := per_task_rta tsk rt_bounds (max_steps tsk) in
if R ≤ task_deadline tsk then
Some (rcons rt_bounds (tsk, R))
else None
else None.
(* The response-time analysis for a given task set is defined
as a left-fold (reduce) based on the function above.
This either returns a list of task and response-time bounds, or None. *)
Definition fp_claimed_bounds (ts: seq sporadic_task) :=
foldl fp_bound_of_task (Some [::]) ts.
(* The schedulability test simply checks if we got a list of
response-time bounds (i.e., if the computation did not fail). *)
Definition fp_schedulable (ts: seq sporadic_task) :=
fp_claimed_bounds ts ≠ None.
(* In the following section, we prove several helper lemmas about the
list of response-time bounds. The results seem trivial, but must be proven
nonetheless since the list of response-time bounds is computed with
a specific algorithm and there are no lemmas in the library for that. *)
Section SimpleLemmas.
(* First, we show that the first component of the computed list is the set of tasks. *)
Lemma fp_claimed_bounds_unzip :
∀ ts hp_bounds,
fp_claimed_bounds ts = Some hp_bounds →
unzip1 hp_bounds = ts.
Proof.
unfold fp_claimed_bounds in *; intros ts.
induction ts using last_ind; first by destruct hp_bounds.
{
intros hp_bounds SOME.
destruct (lastP hp_bounds) as [| hp_bounds'].
{
rewrite -cats1 foldl_cat /= in SOME.
unfold fp_bound_of_task at 1 in SOME; simpl in *; desf.
by destruct l.
}
rewrite -cats1 foldl_cat /= in SOME.
unfold fp_bound_of_task at 1 in SOME; simpl in *; desf.
move: H0 ⇒ /eqP EQSEQ.
rewrite eqseq_rcons in EQSEQ.
move: EQSEQ ⇒ /andP [/eqP SUBST /eqP EQSEQ]; subst.
unfold unzip1; rewrite map_rcons; f_equal.
by apply IHts.
}
Qed.
(* Next, we show that some properties of the analysis are preserved for the
prefixes of the list: (a) the tasks do not change, (b) R <= deadline,
(c) R is computed using the response-time equation, ... *)
Lemma fp_claimed_bounds_rcons :
∀ ts' hp_bounds tsk1 tsk2 R,
(fp_claimed_bounds (rcons ts' tsk1) = Some (rcons hp_bounds (tsk2, R)) →
(fp_claimed_bounds ts' = Some hp_bounds ∧
tsk1 = tsk2 ∧
R = per_task_rta tsk1 hp_bounds (max_steps tsk1) ∧
R ≤ task_deadline tsk1)).
Proof.
intros ts hp_bounds tsk tsk' R.
rewrite -cats1.
unfold fp_claimed_bounds in ×.
rewrite foldl_cat /=.
unfold fp_bound_of_task at 1; simpl; desf.
intros EQ; inversion EQ; move: EQ H0 ⇒ _ /eqP EQ.
rewrite eqseq_rcons in EQ.
move: EQ ⇒ /andP [/eqP EQ /eqP RESP].
by inversion RESP; repeat split; subst.
Qed.
(* ..., which implies that any prefix of the computation is the computation
of the prefix. *)
Lemma fp_claimed_bounds_take :
∀ ts hp_bounds i,
fp_claimed_bounds ts = Some hp_bounds →
i ≤ size hp_bounds →
fp_claimed_bounds (take i ts) = Some (take i hp_bounds).
Proof.
intros ts hp_bounds i SOME LTi.
have UNZIP := fp_claimed_bounds_unzip ts hp_bounds SOME.
rewrite <- UNZIP in ×.
rewrite -[hp_bounds]take_size /unzip1 map_take in SOME.
fold (unzip1 hp_bounds) in *; clear UNZIP.
rewrite leq_eqVlt in LTi.
move: LTi ⇒ /orP [/eqP EQ | LTi]; first by subst.
remember (size hp_bounds) as len; apply eq_leq in Heqlen.
induction len; first by rewrite ltn0 in LTi.
{
assert (TAKElen: fp_claimed_bounds (take len (unzip1 (hp_bounds))) =
Some (take len (hp_bounds))).
{
assert (∃ p, p ∈ hp_bounds).
{
destruct hp_bounds; first by rewrite ltn0 in Heqlen.
by ∃ t; rewrite in_cons eq_refl orTb.
} destruct H as [[tsk R] _].
rewrite (take_nth tsk) in SOME; last by rewrite size_map.
rewrite (take_nth (tsk,R)) in SOME; last by done.
destruct (nth (tsk, R) hp_bounds len) as [tsk_len R_len].
by apply fp_claimed_bounds_rcons in SOME; des.
}
rewrite ltnS leq_eqVlt in LTi.
move: LTi ⇒ /orP [/eqP EQ | LESS]; first by subst.
apply ltnW in Heqlen.
by specialize (IHlen Heqlen TAKElen LESS).
}
Qed.
(* If the analysis suceeds, the computed response-time bounds are no larger
than the deadline... *)
Lemma fp_claimed_bounds_le_deadline :
∀ ts' rt_bounds tsk R,
fp_claimed_bounds ts' = Some rt_bounds →
(tsk, R) ∈ rt_bounds →
R ≤ task_deadline tsk.
Proof.
intros ts; induction ts as [| ts' tsk_lst] using last_ind.
{
intros rt_bounds tsk R SOME IN.
by inversion SOME; subst; rewrite in_nil in IN.
}
{
intros rt_bounds tsk_i R SOME IN.
destruct (lastP rt_bounds) as [|rt_bounds (tsk_lst', R_lst)];
first by rewrite in_nil in IN.
rewrite mem_rcons in_cons in IN; move: IN ⇒ /orP IN.
destruct IN as [LAST | FRONT].
{
move: LAST ⇒ /eqP LAST.
rewrite -cats1 in SOME.
unfold fp_claimed_bounds in ×.
rewrite foldl_cat /= in SOME.
unfold fp_bound_of_task in SOME.
desf; rename H0 into EQ.
move: EQ ⇒ /eqP EQ.
rewrite eqseq_rcons in EQ.
move: EQ ⇒ /andP [_ /eqP EQ].
inversion EQ; subst.
by apply Heq0.
}
{
apply IHts with (rt_bounds := rt_bounds); last by ins.
by apply fp_claimed_bounds_rcons in SOME; des.
}
}
Qed.
(* ... and the computed response-time bounds are no smaller than
the task costs. *)
Lemma fp_claimed_bounds_ge_cost :
∀ ts' rt_bounds tsk R,
fp_claimed_bounds ts' = Some rt_bounds →
(tsk, R) ∈ rt_bounds →
R ≥ task_cost tsk.
Proof.
intros ts; induction ts as [| ts' tsk_lst] using last_ind.
{
intros rt_bounds tsk R SOME IN.
by inversion SOME; subst; rewrite in_nil in IN.
}
{
intros rt_bounds tsk_i R SOME IN.
destruct (lastP rt_bounds) as [|rt_bounds (tsk_lst', R_lst)];
first by rewrite in_nil in IN.
rewrite mem_rcons in_cons in IN; move: IN ⇒ /orP IN.
destruct IN as [LAST | FRONT].
{
move: LAST ⇒ /eqP LAST.
rewrite -cats1 in SOME.
unfold fp_claimed_bounds in ×.
rewrite foldl_cat /= in SOME.
unfold fp_bound_of_task in SOME.
desf; rename H0 into EQ.
move: EQ ⇒ /eqP EQ.
rewrite eqseq_rcons in EQ.
move: EQ ⇒ /andP [_ /eqP EQ].
inversion EQ; subst.
by destruct (max_steps tsk_lst');
[by apply leqnn | by apply leq_addr].
}
{
apply IHts with (rt_bounds := rt_bounds); last by ins.
by apply fp_claimed_bounds_rcons in SOME; des.
}
}
Qed.
(* Short lemma about unfolding the iteration one step. *)
Lemma per_task_rta_fold :
∀ tsk rt_bounds,
task_cost tsk +
div_floor (total_interference_bound_fp task_cost task_period rt_bounds
(per_task_rta tsk rt_bounds (max_steps tsk))) num_cpus
= per_task_rta tsk rt_bounds (max_steps tsk).+1.
Proof.
by done.
Qed.
End SimpleLemmas.
(* In this section, we prove that if the task set is sorted by priority,
the tasks in fp_claimed_bounds are interfering tasks. *)
Section HighPriorityTasks.
(* Consider a list of previous tasks and a task tsk to be analyzed. *)
Variable ts: taskset_of sporadic_task.
(* Assume that the task set is sorted by unique priorities, ... *)
Hypothesis H_task_set_is_sorted: sorted higher_priority ts.
Hypothesis H_task_set_has_unique_priorities:
FP_is_antisymmetric_over_task_set higher_priority ts.
(* ...the priority order is transitive, ...*)
Hypothesis H_priority_transitive: FP_is_transitive higher_priority.
(* ... and that the response-time analysis succeeds. *)
Variable hp_bounds: seq task_with_response_time.
Variable R: time.
Hypothesis H_analysis_succeeds: fp_claimed_bounds ts = Some hp_bounds.
(* Let's refer to tasks by index. *)
Variable elem: sporadic_task.
Let TASK := nth elem ts.
(* We prove that higher-priority tasks have smaller index. *)
Lemma fp_claimed_bounds_hp_tasks_have_smaller_index :
∀ hp_idx idx,
hp_idx < size ts →
idx < size ts →
hp_idx ≠ idx →
higher_priority (TASK hp_idx) (TASK idx) →
hp_idx < idx.
Proof.
unfold TASK; clear TASK.
rename ts into ts'; destruct ts' as [ts UNIQ]; simpl in ×.
intros hp_idx idx LThp LT NEQ HP.
rewrite ltn_neqAle; apply/andP; split; first by done.
by apply sorted_rel_implies_le_idx with (leT := higher_priority) (s := ts) (x0 := elem).
Qed.
End HighPriorityTasks.
(* In this section, we show that the fixed-point iteration converges. *)
Section Convergence.
(* Consider any set of higher-priority tasks. *)
Variable ts_hp: seq sporadic_task.
(* Assume that the response-time analysis succeeds for the higher-priority tasks. *)
Variable rt_bounds: seq task_with_response_time.
Hypothesis H_test_succeeds: fp_claimed_bounds ts_hp = Some rt_bounds.
(* Consider any task tsk to be analyzed, ... *)
Variable tsk: sporadic_task.
(* ... and assume all tasks have valid parameters. *)
Hypothesis H_valid_task_parameters:
valid_sporadic_taskset task_cost task_period task_deadline (rcons ts_hp tsk).
(* To simplify, let f denote the fixed-point iteration. *)
Let f := per_task_rta tsk rt_bounds.
(* Assume that f (max_steps tsk) is no larger than the deadline. *)
Hypothesis H_no_larger_than_deadline: f (max_steps tsk) ≤ task_deadline tsk.
(* First, we show that f is monotonically increasing. *)
Lemma bertogna_fp_comp_f_monotonic :
∀ x1 x2, x1 ≤ x2 → f x1 ≤ f x2.
Proof.
unfold valid_sporadic_taskset, is_valid_sporadic_task in ×.
rename H_test_succeeds into SOME,
H_valid_task_parameters into VALID.
intros x1 x2 LEx; unfold f, per_task_rta.
apply fun_mon_iter_mon; [by ins | by ins; apply leq_addr |].
clear LEx x1 x2; intros x1 x2 LEx.
rewrite leq_add2l leq_div2r //.
unfold total_interference_bound_fp.
rewrite big_seq_cond.
rewrite [\sum_(_ <- _ | true) _]big_seq_cond.
apply leq_sum; move ⇒ i /andP [IN _].
destruct i as [i R].
have GE_COST := fp_claimed_bounds_ge_cost ts_hp rt_bounds i R SOME IN.
have UNZIP := fp_claimed_bounds_unzip ts_hp rt_bounds SOME.
unfold interference_bound_generic; simpl.
apply W_monotonic; try (by done).
have INts: i ∈ ts_hp by rewrite -UNZIP; apply/mapP; ∃ (i, R).
by exploit (VALID i);
[by rewrite mem_rcons in_cons INts orbT | by ins; des].
Qed.
(* If the iteration converged at an earlier step, then it remains stable. *)
Lemma bertogna_fp_comp_f_converges_early :
(∃ k, k ≤ max_steps tsk ∧ f k = f k.+1) →
f (max_steps tsk) = f (max_steps tsk).+1.
Proof.
by intros EX; des; apply iter_fix with (k := k).
Qed.
(* Else, we derive a contradiction. *)
Section DerivingContradiction.
(* Assume instead that the iteration continued to diverge. *)
Hypothesis H_keeps_diverging:
∀ k,
k ≤ max_steps tsk → f k ≠ f k.+1.
(* By monotonicity, it follows that the value always increases. *)
Lemma bertogna_fp_comp_f_increases :
∀ k,
k ≤ max_steps tsk →
f k < f k.+1.
Proof.
intros k LT.
rewrite ltn_neqAle; apply/andP; split.
by apply H_keeps_diverging.
by apply bertogna_fp_comp_f_monotonic, leqnSn.
Qed.
(* In the end, the response-time bound must exceed the deadline. Contradiction! *)
Lemma bertogna_fp_comp_rt_grows_too_much :
∀ k,
k ≤ max_steps tsk →
f k > k + task_cost tsk - 1.
Proof.
have INC := bertogna_fp_comp_f_increases.
rename H_valid_task_parameters into TASK_PARAMS.
unfold valid_sporadic_taskset, is_valid_sporadic_task in *; des.
exploit (TASK_PARAMS tsk);
[by rewrite mem_rcons in_cons eq_refl orTb | intro PARAMS; des].
induction k.
{
intros _; rewrite add0n -addn1 subh1;
first by rewrite -addnBA // subnn addn0 /= leqnn.
by apply PARAMS.
}
{
intros LT.
specialize (IHk (ltnW LT)).
apply leq_ltn_trans with (n := f k); last by apply INC, ltnW.
rewrite -addn1 -addnA [1 + _]addnC addnA -addnBA // subnn addn0.
rewrite -(ltn_add2r 1) in IHk.
rewrite subh1 in IHk;
last by apply leq_trans with (n := task_cost tsk);
[by apply PARAMS | by apply leq_addl].
by rewrite -addnBA // subnn addn0 addn1 ltnS in IHk.
}
Qed.
End DerivingContradiction.
(* Using the lemmas above, we prove the convergence of the iteration after max_steps. *)
Lemma per_task_rta_converges:
f (max_steps tsk) = f (max_steps tsk).+1.
Proof.
have TOOMUCH := bertogna_fp_comp_rt_grows_too_much.
have INC := bertogna_fp_comp_f_increases.
rename H_no_larger_than_deadline into LE,
H_valid_task_parameters into TASK_PARAMS.
unfold valid_sporadic_taskset, is_valid_sporadic_task in *; des.
(* Either f converges by the deadline or not. *)
destruct ([∃ k in 'I_(max_steps tsk).+1, f k = f k.+1]) eqn:EX.
{
move: EX ⇒ /exists_inP EX; destruct EX as [k _ ITERk].
apply bertogna_fp_comp_f_converges_early.
by ∃ k; split; [by rewrite -ltnS; apply ltn_ord | by apply/eqP].
}
(* If not, then we reach a contradiction *)
apply negbT in EX; rewrite negb_exists_in in EX.
move: EX ⇒ /forall_inP EX.
rewrite leqNgt in LE; move: LE ⇒ /negP LE.
exfalso; apply LE.
assert (DIFF: ∀ k : nat, k ≤ max_steps tsk → f k ≠ f k.+1).
{
intros k LEk; rewrite -ltnS in LEk.
by exploit (EX (Ordinal LEk)); [by done | intro DIFF; apply DIFF].
}
exploit TOOMUCH; [by apply DIFF | by apply leq_addr |].
exploit (TASK_PARAMS tsk);
[by rewrite mem_rcons in_cons eq_refl orTb | intro PARAMS; des].
rewrite subh1; last by apply PARAMS2.
rewrite -addnBA // subnn addn0 subn1 prednK //.
intros LT; apply (leq_ltn_trans LT).
by rewrite /max_steps [_ - _ + 1]addn1; apply INC, leq_addr.
Qed.
End Convergence.
Section MainProof.
(* Consider a task set ts. *)
Variable ts: taskset_of sporadic_task.
(* Assume that all tasks have valid parameters, ... *)
Hypothesis H_valid_task_parameters:
valid_sporadic_taskset task_cost task_period task_deadline ts.
(* ...and constrained deadlines.*)
Hypothesis H_constrained_deadlines:
∀ tsk, tsk ∈ ts → task_deadline tsk ≤ task_period tsk.
(* Assume that the task set is totally ordered by unique priorities,
and that the priority order is transitive. *)
Hypothesis H_task_set_is_sorted: sorted higher_priority ts.
Hypothesis H_task_set_has_unique_priorities:
FP_is_antisymmetric_over_task_set higher_priority ts.
Hypothesis H_priority_is_total:
FP_is_total_over_task_set higher_priority ts.
Hypothesis H_priority_transitive: FP_is_transitive higher_priority.
(* Next, consider any arrival sequence such that...*)
Context {arr_seq: arrival_sequence Job}.
(* ...all jobs come from task set ts, ...*)
Hypothesis H_all_jobs_from_taskset:
∀ (j: JobIn arr_seq), job_task j ∈ ts.
(* ...they have valid parameters,...*)
Hypothesis H_valid_job_parameters:
∀ (j: JobIn arr_seq),
valid_sporadic_job task_cost task_deadline job_cost job_deadline job_task j.
(* ... and satisfy the sporadic task model.*)
Hypothesis H_sporadic_tasks:
sporadic_task_model task_period arr_seq job_task.
(* Then, consider any platform with at least one CPU such that...*)
Variable sched: schedule num_cpus arr_seq.
Hypothesis H_at_least_one_cpu: num_cpus > 0.
(* ...jobs only execute after they arrived and no longer
than their execution costs,... *)
Hypothesis H_jobs_must_arrive_to_execute:
jobs_must_arrive_to_execute sched.
Hypothesis H_completed_jobs_dont_execute:
completed_jobs_dont_execute job_cost sched.
(* Assume that the scheduler is work-conserving and enforces the FP policy. *)
Hypothesis H_work_conserving: work_conserving job_cost sched.
Hypothesis H_enforces_FP_policy:
enforces_FP_policy job_cost job_task sched higher_priority.
Let no_deadline_missed_by_task (tsk: sporadic_task) :=
task_misses_no_deadline job_cost job_deadline job_task sched tsk.
Let no_deadline_missed_by_job :=
job_misses_no_deadline job_cost job_deadline sched.
Let response_time_bounded_by (tsk: sporadic_task) :=
is_response_time_bound_of_task job_cost job_task tsk sched.
(* In the following theorem, we prove that any response-time bound contained
in fp_claimed_bounds is safe. The proof follows by induction on the task set:
Induction hypothesis: all higher-priority tasks have safe response-time bounds.
Inductive step: We prove that the response-time bound of the current task is safe.
Note that the inductive step is a direct application of the main Theorem from
bertogna_fp_theory.v. *)
Theorem fp_analysis_yields_response_time_bounds :
∀ tsk R,
(tsk, R) ∈ fp_claimed_bounds ts →
response_time_bounded_by tsk R.
Proof.
rename H_valid_job_parameters into JOBPARAMS, H_valid_task_parameters into TASKPARAMS.
unfold valid_sporadic_taskset in ×.
intros tsk R MATCH.
assert (SOME: ∃ hp_bounds, fp_claimed_bounds ts = Some hp_bounds ∧
(tsk, R) ∈ hp_bounds).
{
destruct (fp_claimed_bounds ts); last by done.
by ∃ l; split.
} clear MATCH; des; rename SOME0 into IN.
have UNZIP := fp_claimed_bounds_unzip ts hp_bounds SOME.
set elem := (tsk,R).
move: IN ⇒ /(nthP elem) [idx LTidx EQ].
set NTH := fun k ⇒ nth elem hp_bounds k.
set TASK := fun k ⇒ (NTH k).1.
set RESP := fun k ⇒ (NTH k).2.
cut (response_time_bounded_by (TASK idx) (RESP idx));
first by unfold TASK, RESP, NTH; rewrite EQ.
clear EQ.
assert (PAIR: ∀ idx, (TASK idx, RESP idx) = NTH idx).
{
by intros i; unfold TASK, RESP; destruct (NTH i).
}
assert (SUBST: ∀ i, i < size hp_bounds → TASK i = nth tsk ts i).
{
by intros i LTi; rewrite /TASK /NTH -UNZIP (nth_map elem) //.
}
assert (SIZE: size hp_bounds = size ts).
{
by rewrite -UNZIP size_map.
}
induction idx as [idx IH'] using strong_ind.
assert (IH: ∀ tsk_hp R_hp, (tsk_hp, R_hp) ∈ take idx hp_bounds → response_time_bounded_by tsk_hp R_hp).
{
intros tsk_hp R_hp INhp.
move: INhp ⇒ /(nthP elem) [k LTk EQ].
rewrite size_take LTidx in LTk.
rewrite nth_take in EQ; last by done.
cut (response_time_bounded_by (TASK k) (RESP k));
first by unfold TASK, RESP, NTH; rewrite EQ.
by apply IH'; try (by done); apply (ltn_trans LTk).
} clear IH'.
unfold response_time_bounded_by in ×.
exploit (fp_claimed_bounds_rcons (take idx ts) (take idx hp_bounds) (TASK idx) (TASK idx) (RESP idx)).
{
by rewrite PAIR SUBST // -2?take_nth -?SIZE // (fp_claimed_bounds_take _ hp_bounds).
}
intros [_ [_ [REC DL]]].
apply bertogna_cirinei_response_time_bound_fp with
(task_cost0 := task_cost) (task_period0 := task_period)
(task_deadline0 := task_deadline) (job_deadline0 := job_deadline) (tsk0 := (TASK idx))
(job_task0 := job_task) (ts0 := ts) (hp_bounds0 := take idx hp_bounds)
(higher_eq_priority := higher_priority); try (by done).
{
cut (NTH idx ∈ hp_bounds = true);
[intros IN | by apply mem_nth].
by rewrite set_mem -UNZIP; apply/mapP; ∃ (TASK idx, RESP idx); rewrite PAIR.
}
{
intros hp_tsk IN INTERF.
∃ (RESP (index hp_tsk ts)).
move: (IN) ⇒ INDEX; apply nth_index with (x0 := tsk) in INDEX.
rewrite -{1}[hp_tsk]INDEX -SUBST; last by rewrite SIZE index_mem.
assert (UNIQ: uniq hp_bounds).
{
apply map_uniq with (f := fst); unfold unzip1 in *; rewrite UNZIP.
by destruct ts.
}
rewrite -filter_idx_lt_take //.
{
rewrite PAIR mem_filter; apply/andP; split;
last by apply mem_nth; rewrite SIZE index_mem.
{
rewrite /NTH index_uniq; [| by rewrite SIZE index_mem | by done ].
{
move: INTERF ⇒ /andP [HP NEQ].
apply fp_claimed_bounds_hp_tasks_have_smaller_index with
(ts := ts) (elem := tsk) (hp_bounds := hp_bounds);
try (by done);
[by rewrite index_mem | by rewrite -SIZE | | by rewrite INDEX -SUBST].
apply/eqP; intro BUG; subst idx.
rewrite SUBST -{1}INDEX in NEQ;
first by rewrite eq_refl in NEQ.
by rewrite SIZE index_mem INDEX.
}
}
}
}
{
rewrite REC per_task_rta_fold.
apply per_task_rta_converges with (ts_hp := take idx ts);
[by apply fp_claimed_bounds_take; try (by apply ltnW) | | by rewrite -REC ].
rewrite SUBST // -take_nth -?SIZE //.
by intros i IN; eapply TASKPARAMS, mem_take, IN.
}
Qed.
(* Therefore, if the schedulability test suceeds, ...*)
Hypothesis H_test_succeeds: fp_schedulable ts.
(*..., no task misses its deadline. *)
Theorem taskset_schedulable_by_fp_rta :
∀ tsk, tsk ∈ ts → no_deadline_missed_by_task tsk.
Proof.
have RLIST := (fp_analysis_yields_response_time_bounds).
have UNZIP := (fp_claimed_bounds_unzip ts).
have DL := (fp_claimed_bounds_le_deadline ts).
unfold no_deadline_missed_by_task, task_misses_no_deadline,
job_misses_no_deadline, completed,
fp_schedulable, valid_sporadic_job in ×.
rename H_valid_job_parameters into JOBPARAMS.
move ⇒ tsk INtsk j JOBtsk.
destruct (fp_claimed_bounds ts) as [rt_bounds |]; last by ins.
feed (UNZIP rt_bounds); first by done.
assert (EX: ∃ R, (tsk, R) ∈ rt_bounds).
{
rewrite set_mem -UNZIP in INtsk; move: INtsk ⇒ /mapP EX.
by destruct EX as [p]; destruct p as [tsk' R]; simpl in *; subst tsk'; ∃ R.
} des.
exploit (RLIST tsk R); [by ins | by apply JOBtsk | intro COMPLETED].
exploit (DL rt_bounds tsk R); [by ins | by ins | clear DL; intro DL].
rewrite eqn_leq; apply/andP; split; first by apply cumulative_service_le_job_cost.
apply leq_trans with (n := service sched j (job_arrival j + R)); last first.
{
unfold valid_sporadic_taskset, is_valid_sporadic_task in ×.
apply extend_sum; rewrite // leq_add2l.
specialize (JOBPARAMS j); des; rewrite JOBPARAMS1.
by rewrite JOBtsk.
}
rewrite leq_eqVlt; apply/orP; left; rewrite eq_sym.
by apply COMPLETED.
Qed.
(* For completeness, since all jobs of the arrival sequence
are spawned by the task set, we also conclude that no job in
the schedule misses its deadline. *)
Theorem jobs_schedulable_by_fp_rta :
∀ (j: JobIn arr_seq), no_deadline_missed_by_job j.
Proof.
intros j.
have SCHED := taskset_schedulable_by_fp_rta.
unfold no_deadline_missed_by_task, task_misses_no_deadline in ×.
apply SCHED with (tsk := job_task j); last by done.
by apply H_all_jobs_from_taskset.
Qed.
End MainProof.
End Analysis.
End ResponseTimeIterationFP.
Require Import rt.analysis.parallel.bertogna_fp_theory.
From mathcomp Require Import ssreflect ssrbool ssrfun eqtype ssrnat seq fintype bigop div path.
Module ResponseTimeIterationFP.
Import ResponseTimeAnalysisFP.
(* In this section, we define the algorithm of Bertogna and Cirinei's
response-time analysis for FP scheduling with parallel jobs. *)
Section Analysis.
Context {sporadic_task: eqType}.
Variable task_cost: sporadic_task → time.
Variable task_period: sporadic_task → time.
Variable task_deadline: sporadic_task → time.
(* During the iterations of the algorithm, we pass around pairs
of tasks and computed response-time bounds. *)
Let task_with_response_time := (sporadic_task × time)%type.
Context {Job: eqType}.
Variable job_cost: Job → time.
Variable job_deadline: Job → time.
Variable job_task: Job → sporadic_task.
(* Consider a platform with num_cpus processors, ... *)
Variable num_cpus: nat.
(* ..., and priorities based on an FP policy. *)
Variable higher_priority: FP_policy sporadic_task.
(* Next we define the fixed-point iteration for computing
Bertogna's response-time bound of a task set. *)
(* First, given a sequence of pairs R_prev = <..., (tsk_hp, R_hp)> of
response-time bounds for the higher-priority tasks, we define an
iteration that computes the response-time bound of the current task:
R_tsk (0) = task_cost tsk
R_tsk (step + 1) = f (R step),
where f is the response-time recurrence, step is the number of iterations,
and R_tsk (0) is the initial state. *)
Definition per_task_rta (tsk: sporadic_task)
(R_prev: seq task_with_response_time) (step: nat) :=
iter step
(fun t ⇒ task_cost tsk +
div_floor
(total_interference_bound_fp task_cost task_period R_prev t)
num_cpus)
(task_cost tsk).
(* To ensure that the iteration converges, we will apply per_task_rta
a "sufficient" number of times: task_deadline tsk - task_cost tsk + 1.
This corresponds to the time complexity of the iteration. *)
Definition max_steps (tsk: sporadic_task) := task_deadline tsk - task_cost tsk + 1.
(* Next we compute the response-time bounds for the entire task set.
Since high-priority tasks may not be schedulable, we allow the
computation to fail.
Thus, given the response-time bound of previous tasks, we either
(a) append the computed response-time bound (tsk, R) of the current task
to the list of pairs, or,
(b) return None if the response-time analysis failed. *)
Definition fp_bound_of_task hp_pairs tsk :=
if hp_pairs is Some rt_bounds then
let R := per_task_rta tsk rt_bounds (max_steps tsk) in
if R ≤ task_deadline tsk then
Some (rcons rt_bounds (tsk, R))
else None
else None.
(* The response-time analysis for a given task set is defined
as a left-fold (reduce) based on the function above.
This either returns a list of task and response-time bounds, or None. *)
Definition fp_claimed_bounds (ts: seq sporadic_task) :=
foldl fp_bound_of_task (Some [::]) ts.
(* The schedulability test simply checks if we got a list of
response-time bounds (i.e., if the computation did not fail). *)
Definition fp_schedulable (ts: seq sporadic_task) :=
fp_claimed_bounds ts ≠ None.
(* In the following section, we prove several helper lemmas about the
list of response-time bounds. The results seem trivial, but must be proven
nonetheless since the list of response-time bounds is computed with
a specific algorithm and there are no lemmas in the library for that. *)
Section SimpleLemmas.
(* First, we show that the first component of the computed list is the set of tasks. *)
Lemma fp_claimed_bounds_unzip :
∀ ts hp_bounds,
fp_claimed_bounds ts = Some hp_bounds →
unzip1 hp_bounds = ts.
Proof.
unfold fp_claimed_bounds in *; intros ts.
induction ts using last_ind; first by destruct hp_bounds.
{
intros hp_bounds SOME.
destruct (lastP hp_bounds) as [| hp_bounds'].
{
rewrite -cats1 foldl_cat /= in SOME.
unfold fp_bound_of_task at 1 in SOME; simpl in *; desf.
by destruct l.
}
rewrite -cats1 foldl_cat /= in SOME.
unfold fp_bound_of_task at 1 in SOME; simpl in *; desf.
move: H0 ⇒ /eqP EQSEQ.
rewrite eqseq_rcons in EQSEQ.
move: EQSEQ ⇒ /andP [/eqP SUBST /eqP EQSEQ]; subst.
unfold unzip1; rewrite map_rcons; f_equal.
by apply IHts.
}
Qed.
(* Next, we show that some properties of the analysis are preserved for the
prefixes of the list: (a) the tasks do not change, (b) R <= deadline,
(c) R is computed using the response-time equation, ... *)
Lemma fp_claimed_bounds_rcons :
∀ ts' hp_bounds tsk1 tsk2 R,
(fp_claimed_bounds (rcons ts' tsk1) = Some (rcons hp_bounds (tsk2, R)) →
(fp_claimed_bounds ts' = Some hp_bounds ∧
tsk1 = tsk2 ∧
R = per_task_rta tsk1 hp_bounds (max_steps tsk1) ∧
R ≤ task_deadline tsk1)).
Proof.
intros ts hp_bounds tsk tsk' R.
rewrite -cats1.
unfold fp_claimed_bounds in ×.
rewrite foldl_cat /=.
unfold fp_bound_of_task at 1; simpl; desf.
intros EQ; inversion EQ; move: EQ H0 ⇒ _ /eqP EQ.
rewrite eqseq_rcons in EQ.
move: EQ ⇒ /andP [/eqP EQ /eqP RESP].
by inversion RESP; repeat split; subst.
Qed.
(* ..., which implies that any prefix of the computation is the computation
of the prefix. *)
Lemma fp_claimed_bounds_take :
∀ ts hp_bounds i,
fp_claimed_bounds ts = Some hp_bounds →
i ≤ size hp_bounds →
fp_claimed_bounds (take i ts) = Some (take i hp_bounds).
Proof.
intros ts hp_bounds i SOME LTi.
have UNZIP := fp_claimed_bounds_unzip ts hp_bounds SOME.
rewrite <- UNZIP in ×.
rewrite -[hp_bounds]take_size /unzip1 map_take in SOME.
fold (unzip1 hp_bounds) in *; clear UNZIP.
rewrite leq_eqVlt in LTi.
move: LTi ⇒ /orP [/eqP EQ | LTi]; first by subst.
remember (size hp_bounds) as len; apply eq_leq in Heqlen.
induction len; first by rewrite ltn0 in LTi.
{
assert (TAKElen: fp_claimed_bounds (take len (unzip1 (hp_bounds))) =
Some (take len (hp_bounds))).
{
assert (∃ p, p ∈ hp_bounds).
{
destruct hp_bounds; first by rewrite ltn0 in Heqlen.
by ∃ t; rewrite in_cons eq_refl orTb.
} destruct H as [[tsk R] _].
rewrite (take_nth tsk) in SOME; last by rewrite size_map.
rewrite (take_nth (tsk,R)) in SOME; last by done.
destruct (nth (tsk, R) hp_bounds len) as [tsk_len R_len].
by apply fp_claimed_bounds_rcons in SOME; des.
}
rewrite ltnS leq_eqVlt in LTi.
move: LTi ⇒ /orP [/eqP EQ | LESS]; first by subst.
apply ltnW in Heqlen.
by specialize (IHlen Heqlen TAKElen LESS).
}
Qed.
(* If the analysis suceeds, the computed response-time bounds are no larger
than the deadline... *)
Lemma fp_claimed_bounds_le_deadline :
∀ ts' rt_bounds tsk R,
fp_claimed_bounds ts' = Some rt_bounds →
(tsk, R) ∈ rt_bounds →
R ≤ task_deadline tsk.
Proof.
intros ts; induction ts as [| ts' tsk_lst] using last_ind.
{
intros rt_bounds tsk R SOME IN.
by inversion SOME; subst; rewrite in_nil in IN.
}
{
intros rt_bounds tsk_i R SOME IN.
destruct (lastP rt_bounds) as [|rt_bounds (tsk_lst', R_lst)];
first by rewrite in_nil in IN.
rewrite mem_rcons in_cons in IN; move: IN ⇒ /orP IN.
destruct IN as [LAST | FRONT].
{
move: LAST ⇒ /eqP LAST.
rewrite -cats1 in SOME.
unfold fp_claimed_bounds in ×.
rewrite foldl_cat /= in SOME.
unfold fp_bound_of_task in SOME.
desf; rename H0 into EQ.
move: EQ ⇒ /eqP EQ.
rewrite eqseq_rcons in EQ.
move: EQ ⇒ /andP [_ /eqP EQ].
inversion EQ; subst.
by apply Heq0.
}
{
apply IHts with (rt_bounds := rt_bounds); last by ins.
by apply fp_claimed_bounds_rcons in SOME; des.
}
}
Qed.
(* ... and the computed response-time bounds are no smaller than
the task costs. *)
Lemma fp_claimed_bounds_ge_cost :
∀ ts' rt_bounds tsk R,
fp_claimed_bounds ts' = Some rt_bounds →
(tsk, R) ∈ rt_bounds →
R ≥ task_cost tsk.
Proof.
intros ts; induction ts as [| ts' tsk_lst] using last_ind.
{
intros rt_bounds tsk R SOME IN.
by inversion SOME; subst; rewrite in_nil in IN.
}
{
intros rt_bounds tsk_i R SOME IN.
destruct (lastP rt_bounds) as [|rt_bounds (tsk_lst', R_lst)];
first by rewrite in_nil in IN.
rewrite mem_rcons in_cons in IN; move: IN ⇒ /orP IN.
destruct IN as [LAST | FRONT].
{
move: LAST ⇒ /eqP LAST.
rewrite -cats1 in SOME.
unfold fp_claimed_bounds in ×.
rewrite foldl_cat /= in SOME.
unfold fp_bound_of_task in SOME.
desf; rename H0 into EQ.
move: EQ ⇒ /eqP EQ.
rewrite eqseq_rcons in EQ.
move: EQ ⇒ /andP [_ /eqP EQ].
inversion EQ; subst.
by destruct (max_steps tsk_lst');
[by apply leqnn | by apply leq_addr].
}
{
apply IHts with (rt_bounds := rt_bounds); last by ins.
by apply fp_claimed_bounds_rcons in SOME; des.
}
}
Qed.
(* Short lemma about unfolding the iteration one step. *)
Lemma per_task_rta_fold :
∀ tsk rt_bounds,
task_cost tsk +
div_floor (total_interference_bound_fp task_cost task_period rt_bounds
(per_task_rta tsk rt_bounds (max_steps tsk))) num_cpus
= per_task_rta tsk rt_bounds (max_steps tsk).+1.
Proof.
by done.
Qed.
End SimpleLemmas.
(* In this section, we prove that if the task set is sorted by priority,
the tasks in fp_claimed_bounds are interfering tasks. *)
Section HighPriorityTasks.
(* Consider a list of previous tasks and a task tsk to be analyzed. *)
Variable ts: taskset_of sporadic_task.
(* Assume that the task set is sorted by unique priorities, ... *)
Hypothesis H_task_set_is_sorted: sorted higher_priority ts.
Hypothesis H_task_set_has_unique_priorities:
FP_is_antisymmetric_over_task_set higher_priority ts.
(* ...the priority order is transitive, ...*)
Hypothesis H_priority_transitive: FP_is_transitive higher_priority.
(* ... and that the response-time analysis succeeds. *)
Variable hp_bounds: seq task_with_response_time.
Variable R: time.
Hypothesis H_analysis_succeeds: fp_claimed_bounds ts = Some hp_bounds.
(* Let's refer to tasks by index. *)
Variable elem: sporadic_task.
Let TASK := nth elem ts.
(* We prove that higher-priority tasks have smaller index. *)
Lemma fp_claimed_bounds_hp_tasks_have_smaller_index :
∀ hp_idx idx,
hp_idx < size ts →
idx < size ts →
hp_idx ≠ idx →
higher_priority (TASK hp_idx) (TASK idx) →
hp_idx < idx.
Proof.
unfold TASK; clear TASK.
rename ts into ts'; destruct ts' as [ts UNIQ]; simpl in ×.
intros hp_idx idx LThp LT NEQ HP.
rewrite ltn_neqAle; apply/andP; split; first by done.
by apply sorted_rel_implies_le_idx with (leT := higher_priority) (s := ts) (x0 := elem).
Qed.
End HighPriorityTasks.
(* In this section, we show that the fixed-point iteration converges. *)
Section Convergence.
(* Consider any set of higher-priority tasks. *)
Variable ts_hp: seq sporadic_task.
(* Assume that the response-time analysis succeeds for the higher-priority tasks. *)
Variable rt_bounds: seq task_with_response_time.
Hypothesis H_test_succeeds: fp_claimed_bounds ts_hp = Some rt_bounds.
(* Consider any task tsk to be analyzed, ... *)
Variable tsk: sporadic_task.
(* ... and assume all tasks have valid parameters. *)
Hypothesis H_valid_task_parameters:
valid_sporadic_taskset task_cost task_period task_deadline (rcons ts_hp tsk).
(* To simplify, let f denote the fixed-point iteration. *)
Let f := per_task_rta tsk rt_bounds.
(* Assume that f (max_steps tsk) is no larger than the deadline. *)
Hypothesis H_no_larger_than_deadline: f (max_steps tsk) ≤ task_deadline tsk.
(* First, we show that f is monotonically increasing. *)
Lemma bertogna_fp_comp_f_monotonic :
∀ x1 x2, x1 ≤ x2 → f x1 ≤ f x2.
Proof.
unfold valid_sporadic_taskset, is_valid_sporadic_task in ×.
rename H_test_succeeds into SOME,
H_valid_task_parameters into VALID.
intros x1 x2 LEx; unfold f, per_task_rta.
apply fun_mon_iter_mon; [by ins | by ins; apply leq_addr |].
clear LEx x1 x2; intros x1 x2 LEx.
rewrite leq_add2l leq_div2r //.
unfold total_interference_bound_fp.
rewrite big_seq_cond.
rewrite [\sum_(_ <- _ | true) _]big_seq_cond.
apply leq_sum; move ⇒ i /andP [IN _].
destruct i as [i R].
have GE_COST := fp_claimed_bounds_ge_cost ts_hp rt_bounds i R SOME IN.
have UNZIP := fp_claimed_bounds_unzip ts_hp rt_bounds SOME.
unfold interference_bound_generic; simpl.
apply W_monotonic; try (by done).
have INts: i ∈ ts_hp by rewrite -UNZIP; apply/mapP; ∃ (i, R).
by exploit (VALID i);
[by rewrite mem_rcons in_cons INts orbT | by ins; des].
Qed.
(* If the iteration converged at an earlier step, then it remains stable. *)
Lemma bertogna_fp_comp_f_converges_early :
(∃ k, k ≤ max_steps tsk ∧ f k = f k.+1) →
f (max_steps tsk) = f (max_steps tsk).+1.
Proof.
by intros EX; des; apply iter_fix with (k := k).
Qed.
(* Else, we derive a contradiction. *)
Section DerivingContradiction.
(* Assume instead that the iteration continued to diverge. *)
Hypothesis H_keeps_diverging:
∀ k,
k ≤ max_steps tsk → f k ≠ f k.+1.
(* By monotonicity, it follows that the value always increases. *)
Lemma bertogna_fp_comp_f_increases :
∀ k,
k ≤ max_steps tsk →
f k < f k.+1.
Proof.
intros k LT.
rewrite ltn_neqAle; apply/andP; split.
by apply H_keeps_diverging.
by apply bertogna_fp_comp_f_monotonic, leqnSn.
Qed.
(* In the end, the response-time bound must exceed the deadline. Contradiction! *)
Lemma bertogna_fp_comp_rt_grows_too_much :
∀ k,
k ≤ max_steps tsk →
f k > k + task_cost tsk - 1.
Proof.
have INC := bertogna_fp_comp_f_increases.
rename H_valid_task_parameters into TASK_PARAMS.
unfold valid_sporadic_taskset, is_valid_sporadic_task in *; des.
exploit (TASK_PARAMS tsk);
[by rewrite mem_rcons in_cons eq_refl orTb | intro PARAMS; des].
induction k.
{
intros _; rewrite add0n -addn1 subh1;
first by rewrite -addnBA // subnn addn0 /= leqnn.
by apply PARAMS.
}
{
intros LT.
specialize (IHk (ltnW LT)).
apply leq_ltn_trans with (n := f k); last by apply INC, ltnW.
rewrite -addn1 -addnA [1 + _]addnC addnA -addnBA // subnn addn0.
rewrite -(ltn_add2r 1) in IHk.
rewrite subh1 in IHk;
last by apply leq_trans with (n := task_cost tsk);
[by apply PARAMS | by apply leq_addl].
by rewrite -addnBA // subnn addn0 addn1 ltnS in IHk.
}
Qed.
End DerivingContradiction.
(* Using the lemmas above, we prove the convergence of the iteration after max_steps. *)
Lemma per_task_rta_converges:
f (max_steps tsk) = f (max_steps tsk).+1.
Proof.
have TOOMUCH := bertogna_fp_comp_rt_grows_too_much.
have INC := bertogna_fp_comp_f_increases.
rename H_no_larger_than_deadline into LE,
H_valid_task_parameters into TASK_PARAMS.
unfold valid_sporadic_taskset, is_valid_sporadic_task in *; des.
(* Either f converges by the deadline or not. *)
destruct ([∃ k in 'I_(max_steps tsk).+1, f k = f k.+1]) eqn:EX.
{
move: EX ⇒ /exists_inP EX; destruct EX as [k _ ITERk].
apply bertogna_fp_comp_f_converges_early.
by ∃ k; split; [by rewrite -ltnS; apply ltn_ord | by apply/eqP].
}
(* If not, then we reach a contradiction *)
apply negbT in EX; rewrite negb_exists_in in EX.
move: EX ⇒ /forall_inP EX.
rewrite leqNgt in LE; move: LE ⇒ /negP LE.
exfalso; apply LE.
assert (DIFF: ∀ k : nat, k ≤ max_steps tsk → f k ≠ f k.+1).
{
intros k LEk; rewrite -ltnS in LEk.
by exploit (EX (Ordinal LEk)); [by done | intro DIFF; apply DIFF].
}
exploit TOOMUCH; [by apply DIFF | by apply leq_addr |].
exploit (TASK_PARAMS tsk);
[by rewrite mem_rcons in_cons eq_refl orTb | intro PARAMS; des].
rewrite subh1; last by apply PARAMS2.
rewrite -addnBA // subnn addn0 subn1 prednK //.
intros LT; apply (leq_ltn_trans LT).
by rewrite /max_steps [_ - _ + 1]addn1; apply INC, leq_addr.
Qed.
End Convergence.
Section MainProof.
(* Consider a task set ts. *)
Variable ts: taskset_of sporadic_task.
(* Assume that all tasks have valid parameters, ... *)
Hypothesis H_valid_task_parameters:
valid_sporadic_taskset task_cost task_period task_deadline ts.
(* ...and constrained deadlines.*)
Hypothesis H_constrained_deadlines:
∀ tsk, tsk ∈ ts → task_deadline tsk ≤ task_period tsk.
(* Assume that the task set is totally ordered by unique priorities,
and that the priority order is transitive. *)
Hypothesis H_task_set_is_sorted: sorted higher_priority ts.
Hypothesis H_task_set_has_unique_priorities:
FP_is_antisymmetric_over_task_set higher_priority ts.
Hypothesis H_priority_is_total:
FP_is_total_over_task_set higher_priority ts.
Hypothesis H_priority_transitive: FP_is_transitive higher_priority.
(* Next, consider any arrival sequence such that...*)
Context {arr_seq: arrival_sequence Job}.
(* ...all jobs come from task set ts, ...*)
Hypothesis H_all_jobs_from_taskset:
∀ (j: JobIn arr_seq), job_task j ∈ ts.
(* ...they have valid parameters,...*)
Hypothesis H_valid_job_parameters:
∀ (j: JobIn arr_seq),
valid_sporadic_job task_cost task_deadline job_cost job_deadline job_task j.
(* ... and satisfy the sporadic task model.*)
Hypothesis H_sporadic_tasks:
sporadic_task_model task_period arr_seq job_task.
(* Then, consider any platform with at least one CPU such that...*)
Variable sched: schedule num_cpus arr_seq.
Hypothesis H_at_least_one_cpu: num_cpus > 0.
(* ...jobs only execute after they arrived and no longer
than their execution costs,... *)
Hypothesis H_jobs_must_arrive_to_execute:
jobs_must_arrive_to_execute sched.
Hypothesis H_completed_jobs_dont_execute:
completed_jobs_dont_execute job_cost sched.
(* Assume that the scheduler is work-conserving and enforces the FP policy. *)
Hypothesis H_work_conserving: work_conserving job_cost sched.
Hypothesis H_enforces_FP_policy:
enforces_FP_policy job_cost job_task sched higher_priority.
Let no_deadline_missed_by_task (tsk: sporadic_task) :=
task_misses_no_deadline job_cost job_deadline job_task sched tsk.
Let no_deadline_missed_by_job :=
job_misses_no_deadline job_cost job_deadline sched.
Let response_time_bounded_by (tsk: sporadic_task) :=
is_response_time_bound_of_task job_cost job_task tsk sched.
(* In the following theorem, we prove that any response-time bound contained
in fp_claimed_bounds is safe. The proof follows by induction on the task set:
Induction hypothesis: all higher-priority tasks have safe response-time bounds.
Inductive step: We prove that the response-time bound of the current task is safe.
Note that the inductive step is a direct application of the main Theorem from
bertogna_fp_theory.v. *)
Theorem fp_analysis_yields_response_time_bounds :
∀ tsk R,
(tsk, R) ∈ fp_claimed_bounds ts →
response_time_bounded_by tsk R.
Proof.
rename H_valid_job_parameters into JOBPARAMS, H_valid_task_parameters into TASKPARAMS.
unfold valid_sporadic_taskset in ×.
intros tsk R MATCH.
assert (SOME: ∃ hp_bounds, fp_claimed_bounds ts = Some hp_bounds ∧
(tsk, R) ∈ hp_bounds).
{
destruct (fp_claimed_bounds ts); last by done.
by ∃ l; split.
} clear MATCH; des; rename SOME0 into IN.
have UNZIP := fp_claimed_bounds_unzip ts hp_bounds SOME.
set elem := (tsk,R).
move: IN ⇒ /(nthP elem) [idx LTidx EQ].
set NTH := fun k ⇒ nth elem hp_bounds k.
set TASK := fun k ⇒ (NTH k).1.
set RESP := fun k ⇒ (NTH k).2.
cut (response_time_bounded_by (TASK idx) (RESP idx));
first by unfold TASK, RESP, NTH; rewrite EQ.
clear EQ.
assert (PAIR: ∀ idx, (TASK idx, RESP idx) = NTH idx).
{
by intros i; unfold TASK, RESP; destruct (NTH i).
}
assert (SUBST: ∀ i, i < size hp_bounds → TASK i = nth tsk ts i).
{
by intros i LTi; rewrite /TASK /NTH -UNZIP (nth_map elem) //.
}
assert (SIZE: size hp_bounds = size ts).
{
by rewrite -UNZIP size_map.
}
induction idx as [idx IH'] using strong_ind.
assert (IH: ∀ tsk_hp R_hp, (tsk_hp, R_hp) ∈ take idx hp_bounds → response_time_bounded_by tsk_hp R_hp).
{
intros tsk_hp R_hp INhp.
move: INhp ⇒ /(nthP elem) [k LTk EQ].
rewrite size_take LTidx in LTk.
rewrite nth_take in EQ; last by done.
cut (response_time_bounded_by (TASK k) (RESP k));
first by unfold TASK, RESP, NTH; rewrite EQ.
by apply IH'; try (by done); apply (ltn_trans LTk).
} clear IH'.
unfold response_time_bounded_by in ×.
exploit (fp_claimed_bounds_rcons (take idx ts) (take idx hp_bounds) (TASK idx) (TASK idx) (RESP idx)).
{
by rewrite PAIR SUBST // -2?take_nth -?SIZE // (fp_claimed_bounds_take _ hp_bounds).
}
intros [_ [_ [REC DL]]].
apply bertogna_cirinei_response_time_bound_fp with
(task_cost0 := task_cost) (task_period0 := task_period)
(task_deadline0 := task_deadline) (job_deadline0 := job_deadline) (tsk0 := (TASK idx))
(job_task0 := job_task) (ts0 := ts) (hp_bounds0 := take idx hp_bounds)
(higher_eq_priority := higher_priority); try (by done).
{
cut (NTH idx ∈ hp_bounds = true);
[intros IN | by apply mem_nth].
by rewrite set_mem -UNZIP; apply/mapP; ∃ (TASK idx, RESP idx); rewrite PAIR.
}
{
intros hp_tsk IN INTERF.
∃ (RESP (index hp_tsk ts)).
move: (IN) ⇒ INDEX; apply nth_index with (x0 := tsk) in INDEX.
rewrite -{1}[hp_tsk]INDEX -SUBST; last by rewrite SIZE index_mem.
assert (UNIQ: uniq hp_bounds).
{
apply map_uniq with (f := fst); unfold unzip1 in *; rewrite UNZIP.
by destruct ts.
}
rewrite -filter_idx_lt_take //.
{
rewrite PAIR mem_filter; apply/andP; split;
last by apply mem_nth; rewrite SIZE index_mem.
{
rewrite /NTH index_uniq; [| by rewrite SIZE index_mem | by done ].
{
move: INTERF ⇒ /andP [HP NEQ].
apply fp_claimed_bounds_hp_tasks_have_smaller_index with
(ts := ts) (elem := tsk) (hp_bounds := hp_bounds);
try (by done);
[by rewrite index_mem | by rewrite -SIZE | | by rewrite INDEX -SUBST].
apply/eqP; intro BUG; subst idx.
rewrite SUBST -{1}INDEX in NEQ;
first by rewrite eq_refl in NEQ.
by rewrite SIZE index_mem INDEX.
}
}
}
}
{
rewrite REC per_task_rta_fold.
apply per_task_rta_converges with (ts_hp := take idx ts);
[by apply fp_claimed_bounds_take; try (by apply ltnW) | | by rewrite -REC ].
rewrite SUBST // -take_nth -?SIZE //.
by intros i IN; eapply TASKPARAMS, mem_take, IN.
}
Qed.
(* Therefore, if the schedulability test suceeds, ...*)
Hypothesis H_test_succeeds: fp_schedulable ts.
(*..., no task misses its deadline. *)
Theorem taskset_schedulable_by_fp_rta :
∀ tsk, tsk ∈ ts → no_deadline_missed_by_task tsk.
Proof.
have RLIST := (fp_analysis_yields_response_time_bounds).
have UNZIP := (fp_claimed_bounds_unzip ts).
have DL := (fp_claimed_bounds_le_deadline ts).
unfold no_deadline_missed_by_task, task_misses_no_deadline,
job_misses_no_deadline, completed,
fp_schedulable, valid_sporadic_job in ×.
rename H_valid_job_parameters into JOBPARAMS.
move ⇒ tsk INtsk j JOBtsk.
destruct (fp_claimed_bounds ts) as [rt_bounds |]; last by ins.
feed (UNZIP rt_bounds); first by done.
assert (EX: ∃ R, (tsk, R) ∈ rt_bounds).
{
rewrite set_mem -UNZIP in INtsk; move: INtsk ⇒ /mapP EX.
by destruct EX as [p]; destruct p as [tsk' R]; simpl in *; subst tsk'; ∃ R.
} des.
exploit (RLIST tsk R); [by ins | by apply JOBtsk | intro COMPLETED].
exploit (DL rt_bounds tsk R); [by ins | by ins | clear DL; intro DL].
rewrite eqn_leq; apply/andP; split; first by apply cumulative_service_le_job_cost.
apply leq_trans with (n := service sched j (job_arrival j + R)); last first.
{
unfold valid_sporadic_taskset, is_valid_sporadic_task in ×.
apply extend_sum; rewrite // leq_add2l.
specialize (JOBPARAMS j); des; rewrite JOBPARAMS1.
by rewrite JOBtsk.
}
rewrite leq_eqVlt; apply/orP; left; rewrite eq_sym.
by apply COMPLETED.
Qed.
(* For completeness, since all jobs of the arrival sequence
are spawned by the task set, we also conclude that no job in
the schedule misses its deadline. *)
Theorem jobs_schedulable_by_fp_rta :
∀ (j: JobIn arr_seq), no_deadline_missed_by_job j.
Proof.
intros j.
have SCHED := taskset_schedulable_by_fp_rta.
unfold no_deadline_missed_by_task, task_misses_no_deadline in ×.
apply SCHED with (tsk := job_task j); last by done.
by apply H_all_jobs_from_taskset.
Qed.
End MainProof.
End Analysis.
End ResponseTimeIterationFP.