Newer
Older
Require Import prosa.classic.util.all.
Require Import prosa.classic.analysis.global.jitter.bertogna_edf_theory.
From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq fintype bigop div path.
Import ResponseTimeAnalysisEDFJitter.
(* In this section, we define the algorithm for Bertogna and Cirinei's
response-time analysis for EDF scheduling with release jitter. *)
Section Analysis.
Context {sporadic_task: eqType}.
Variable task_cost: sporadic_task -> time.
Variable task_period: sporadic_task -> time.
Variable task_deadline: sporadic_task -> time.
Variable task_jitter: sporadic_task -> time.
(* As input for each iteration of the algorithm, we consider pairs
of tasks and computed response-time bounds. *)
Let task_with_response_time := (sporadic_task * time)%type.
Variable job_arrival: Job -> time.
Variable job_cost: Job -> time.
Variable job_deadline: Job -> time.
Variable job_jitter: Job -> time.
(* Consider a platform with num_cpus processors. *)
(* First, recall the interference bound under EDF, ... *)
Let I (rt_bounds: seq task_with_response_time)
(tsk: sporadic_task) (delta: time) :=
total_interference_bound_edf task_cost task_period task_deadline task_jitter tsk rt_bounds delta.
(* ..., which yields the following response-time bound. *)
Definition edf_response_time_bound (rt_bounds: seq task_with_response_time)
(tsk: sporadic_task) (delta: time) :=
task_cost tsk + div_floor (I rt_bounds tsk delta) num_cpus.
(* Also note that a response-time R is only valid if R + jitter is no larger
Definition jitter_plus_R_le_deadline (pair: task_with_response_time) :=
task_jitter tsk + R <= task_deadline tsk.
(* Next we define the fixed-point iteration for computing
Bertogna's response-time bound of a task set. *)
(* Given a sequence 'rt_bounds' of task and response-time bounds
from the previous iteration, we compute the response-time
bound of a single task using the RTA for EDF. *)
Definition update_bound (rt_bounds: seq task_with_response_time)
(pair : task_with_response_time) :=
let (tsk, R) := pair in
(tsk, edf_response_time_bound rt_bounds tsk R).
(* To compute the response-time bounds of the entire task set,
We start the iteration with a sequence of tasks and costs:
<(task1, cost1), (task2, cost2), ...>. *)
Let initial_state (ts: seq sporadic_task) :=
map (fun t => (t, task_cost t)) ts.
(* Then, we successively update the the response-time bounds based
on the slack computed in the previous iteration. *)
Definition edf_rta_iteration (rt_bounds: seq task_with_response_time) :=
(* To ensure that the procedure converges, we run the iteration a
"sufficient" number of times: task_deadline tsk - task_cost tsk + 1.
This corresponds to the time complexity of the procedure. *)
\sum_(tsk <- ts) (task_deadline tsk - task_cost tsk) + 1.
(* This yields the following definition for the RTA. At the end of
the iteration, we check if all computed response-time bounds
are less than or equal to the deadline, in which case they are
valid. *)
Definition edf_claimed_bounds (ts: seq sporadic_task) :=
let R_values := iter (max_steps ts) edf_rta_iteration (initial_state ts) in
if (all jitter_plus_R_le_deadline R_values) then
(* The schedulability test simply checks if we got a list of
response-time bounds (i.e., if the computation did not fail). *)
Definition edf_schedulable (ts: seq sporadic_task) :=
(* In the following section, we prove several helper lemmas about the
list of tasks/response-time bounds. *)
Section SimpleLemmas.
(* Updating a single response-time bound does not modify the task. *)
Lemma edf_claimed_bounds_unzip1_update_bound :
forall l rt_bounds,
unzip1 (map (update_bound rt_bounds) l) = unzip1 l.
Proof.
induction l; first by done.
intros rt_bounds.
simpl; f_equal; last by done.
by unfold update_bound; desf.
Qed.
(* At any point of the iteration, the tasks are the same. *)
Lemma edf_claimed_bounds_unzip1_iteration :
forall l k,
unzip1 (iter k edf_rta_iteration (initial_state l)) = l.
Proof.
intros l k; clear -k.
induction k; simpl.
{
unfold initial_state.
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
by simpl; rewrite IHl.
}
{
unfold edf_rta_iteration.
by rewrite edf_claimed_bounds_unzip1_update_bound.
}
Qed.
(* The iteration preserves the size of the list. *)
Lemma edf_claimed_bounds_size :
forall l k,
size (iter k edf_rta_iteration (initial_state l)) = size l.
Proof.
intros l k; clear -k.
induction k; simpl; first by rewrite size_map.
by rewrite size_map.
Qed.
(* If the analysis succeeds, the computed response-time bounds are no smaller
than the task cost. *)
Lemma edf_claimed_bounds_ge_cost :
forall l k tsk R,
(tsk, R) \in (iter k edf_rta_iteration (initial_state l)) ->
R >= task_cost tsk.
Proof.
intros l k tsk R IN.
destruct k.
{
move: IN => /mapP IN; destruct IN as [x IN EQ]; inversion EQ.
by apply leqnn.
}
{
rewrite iterS in IN.
move: IN => /mapP IN; destruct IN as [x IN EQ].
unfold update_bound in EQ; destruct x; inversion EQ.
by unfold edf_response_time_bound; apply leq_addr.
}
Qed.
(* If the analysis suceeds, the computed response-time bounds are no larger
than the deadline. *)
Lemma edf_claimed_bounds_le_deadline :
forall ts rt_bounds tsk R,
edf_claimed_bounds ts = Some rt_bounds ->
(tsk, R) \in rt_bounds ->
task_jitter tsk + R <= task_deadline tsk.
Proof.
intros ts rt_bounds tsk R SOME PAIR; unfold edf_claimed_bounds in SOME.
destruct (all jitter_plus_R_le_deadline (iter (max_steps ts)
edf_rta_iteration (initial_state ts))) eqn:DEADLINE;
last by done.
move: DEADLINE => /allP DEADLINE.
inversion SOME as [EQ]; rewrite -EQ in PAIR.
by specialize (DEADLINE (tsk, R) PAIR).
Qed.
(* The list contains a response-time bound for every task in the task set. *)
Lemma edf_claimed_bounds_has_R_for_every_task :
forall ts rt_bounds tsk,
edf_claimed_bounds ts = Some rt_bounds ->
tsk \in ts ->
exists R,
(tsk, R) \in rt_bounds.
Proof.
intros ts rt_bounds tsk SOME IN.
unfold edf_claimed_bounds in SOME.
destruct (all jitter_plus_R_le_deadline (iter (max_steps ts) edf_rta_iteration (initial_state ts)));
last by done.
inversion SOME as [EQ]; clear SOME EQ.
generalize dependent tsk.
induction (max_steps ts) as [| step]; simpl in *.
{
intros tsk IN; unfold initial_state.
exists (task_cost tsk).
by apply/mapP; exists tsk.
}
{
intros tsk IN.
set prev_state := iter step edf_rta_iteration (initial_state ts).
fold prev_state in IN, IHstep.
specialize (IHstep tsk IN); des.
exists (edf_response_time_bound prev_state tsk R).
by apply/mapP; exists (tsk, R); [by done | by f_equal].
}
Qed.
End SimpleLemmas.
(* As required by the proof of convergence, we show that the
interference bound is monotonically increasing with both
the size of the interval and the value of the previous
response-time bounds.
TODO: move to bertogna_edf_theory.v? *)
Section MonotonicityOfInterferenceBound.
Variable tsk tsk_other: sporadic_task.
Hypothesis H_period_positive: task_period tsk_other > 0.
Variable delta delta' R R': time.
Hypothesis H_delta_monotonic: delta <= delta'.
Hypothesis H_response_time_monotonic: R <= R'.
Hypothesis H_cost_le_rt_bound: task_cost tsk_other <= R.
Lemma interference_bound_edf_monotonic :
interference_bound_edf task_cost task_period task_deadline task_jitter tsk delta (tsk_other, R) <=
interference_bound_edf task_cost task_period task_deadline task_jitter tsk delta' (tsk_other, R').
Proof.
rename H_response_time_monotonic into LEr, H_delta_monotonic into LEx,
H_cost_le_rt_bound into LEcost, H_period_positive into GEperiod.
unfold interference_bound_edf, interference_bound_generic.
rewrite leq_min; apply/andP; split.
{
rewrite leq_min; apply/andP; split.
apply leq_trans with (n := (minn (W_jitter task_cost task_period task_jitter (fst (tsk_other, R))
(snd (tsk_other, R)) delta) (delta - task_cost tsk + 1)));
first by apply geq_minl.
apply leq_trans with (n := W_jitter task_cost task_period task_jitter (fst (tsk_other, R))
(snd (tsk_other, R)) delta);
[by apply geq_minl | by apply W_monotonic].
apply leq_trans with (n := minn (W_jitter task_cost task_period task_jitter (fst (tsk_other, R)) (snd (tsk_other, R)) delta) (delta - task_cost tsk + 1));
first by apply geq_minl.
apply leq_trans with (n := delta - task_cost tsk + 1);
first by apply geq_minr.
by rewrite leq_add2r leq_sub2r.
}
{
apply leq_trans with (n := edf_specific_interference_bound task_cost task_period
task_deadline task_jitter tsk tsk_other R);
first by apply geq_minr.
unfold edf_specific_interference_bound; simpl.
rewrite leq_add2l leq_min; apply/andP; split; first by apply geq_minl.
apply leq_trans with (n := task_deadline tsk %% task_period tsk_other -
(task_deadline tsk_other - R - task_jitter tsk_other));
[by apply geq_minr | by rewrite 2?leq_sub2l 2?leq_sub2r // leq_sub2l].
}
Qed.
End MonotonicityOfInterferenceBound.
(* In this section, we prove the convergence of the RTA procedure.
Since we define the RTA procedure as the application of a function
a fixed number of times, this translates into proving that the value
of the iteration at (max_steps ts) is equal to the value at (max_steps ts) + 1. *)
(* Consider any list of tasks with valid parameters. *)
Variable ts: list sporadic_task.
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
Hypothesis H_valid_task_parameters:
valid_sporadic_taskset task_cost task_period task_deadline ts.
(* To simplify, let f denote the RTA procedure. *)
Let f (k: nat) := iter k edf_rta_iteration (initial_state ts).
(* Since the iteration is applied directly to a list of tasks and response-times,
we define a corresponding relation "<=" over those lists. *)
(* Let 'all_le' be a binary relation over lists of tasks/response-time bounds.
It states that every element of list l1 has a response-time bound R that is less
than or equal to the corresponding response-time bound R' in list l2 (point-wise).
In addition, the relation states that the tasks of both lists are unchanged. *)
Let all_le := fun (l1 l2: list task_with_response_time) =>
(unzip1 l1 == unzip1 l2) &&
all (fun p => (snd (fst p)) <= (snd (snd p))) (zip l1 l2).
(* Similarly, we define a strict version of 'all_le' called 'one_lt', which states that
there exists at least one element whose response-time bound increases. *)
Let one_lt := fun (l1 l2: list task_with_response_time) =>
(unzip1 l1 == unzip1 l2) &&
has (fun p => (snd (fst p)) < (snd (snd p))) (zip l1 l2).
(* Next, we prove some basic properties about the relation all_le. *)
Section RelationProperties.
(* The relation is reflexive, ... *)
Lemma all_le_reflexive : reflexive all_le.
Proof.
intros l; unfold all_le; rewrite eq_refl andTb.
destruct l; first by done.
by apply/(zipP t (fun x y => snd x <= snd y)).
(* ... and transitive. *)
Lemma all_le_transitive: transitive all_le.
unfold transitive, all_le.
move => y x z /andP [/eqP ZIPxy LExy] /andP [/eqP ZIPyz LEyz].
apply/andP; split; first by rewrite ZIPxy -ZIPyz.
move: LExy => /(zipP _ (fun x y => snd x <= snd y)) LExy.
move: LEyz => /(zipP _ (fun x y => snd x <= snd y)) LEyz.
assert (SIZExy: size (unzip1 x) = size (unzip1 y)).
by rewrite ZIPxy.
assert (SIZEyz: size (unzip1 y) = size (unzip1 z)).
by rewrite ZIPyz.
rewrite 2!size_map in SIZExy; rewrite 2!size_map in SIZEyz.
destruct y.
apply size0nil in SIZExy; symmetry in SIZEyz.
by apply size0nil in SIZEyz; subst.
specialize (LExy t); specialize (LEyz t).
apply/(zipP t (fun x y => snd x <= snd y));
first by rewrite SIZExy -SIZEyz.
intros i LTi.
exploit LExy; first by rewrite SIZExy.
rewrite size_zip -SIZEyz -SIZExy minnn in LTi.
by rewrite size_zip -SIZExy minnn; apply LTi.
exploit LEyz; first by apply SIZEyz.
{
rewrite size_zip SIZExy SIZEyz minnn in LTi.
by rewrite size_zip SIZEyz minnn; apply LTi.
}
(* At any step of the iteration, the corresponding list
is larger than or equal to the initial state. *)
Lemma bertogna_edf_comp_iteration_preserves_minimum :
forall step, all_le (initial_state ts) (f step).
unfold f.
intros step; destruct step; first by apply all_le_reflexive.
apply/andP; split.
{
assert (UNZIP0 := edf_claimed_bounds_unzip1_iteration ts 0).
by simpl in UNZIP0; rewrite UNZIP0 edf_claimed_bounds_unzip1_iteration.
destruct ts as [| tsk0 ts'].
{
clear -step; induction step; first by done.
by rewrite iterSr IHstep.
}
apply/(zipP (tsk0,0) (fun x y => snd x <= snd y));
first by rewrite edf_claimed_bounds_size size_map.
intros i LTi; rewrite iterS; unfold edf_rta_iteration at 1.
have MAP := @nth_map _ (tsk0,0) _ (tsk0,0).
rewrite size_zip edf_claimed_bounds_size size_map minnn in LTi.
rewrite MAP; clear MAP; last by rewrite edf_claimed_bounds_size.
destruct (nth (tsk0, 0) (initial_state (tsk0 :: ts')) i) as [tsk_i R_i] eqn:SUBST.
rewrite SUBST; unfold update_bound.
unfold initial_state in SUBST.
have MAP := @nth_map _ tsk0 _ (tsk0, 0).
rewrite ?MAP // in SUBST; inversion SUBST; clear MAP.
assert (EQtsk: tsk_i = fst (nth (tsk0, 0) (iter step edf_rta_iteration
(initial_state (tsk0 :: ts'))) i)).
{
have MAP := @nth_map _ (tsk0,0) _ tsk0 (fun x => fst x).
rewrite -MAP; clear MAP; last by rewrite edf_claimed_bounds_size.
have UNZIP := edf_claimed_bounds_unzip1_iteration; unfold unzip1 in UNZIP.
by rewrite UNZIP; symmetry.
}
destruct (nth (tsk0, 0) (iter step edf_rta_iteration (initial_state (tsk0 :: ts')))) as [tsk_i' R_i'].
by simpl in EQtsk; rewrite -EQtsk; subst; apply leq_addr.
(* The application of the function is inductive. *)
Lemma bertogna_edf_comp_iteration_inductive (P : seq task_with_response_time -> Type) :
P (initial_state ts) ->
(forall k, P (f k) -> P (f (k.+1))) ->
P (f (max_steps ts)).
Proof.
by intros P0 Pn; induction (max_steps ts); last by apply Pn.
Qed.
(* As a last step, we show that edf_rta_iteration preserves order, i.e., for any
list l1 no smaller than the initial state, and list l2 such that
l1 <= l2, we have (edf_rta_iteration l1) <= (edf_rta_iteration l2). *)
Lemma bertogna_edf_comp_iteration_preserves_order :
forall l1 l2,
all_le (initial_state ts) l1 ->
all_le l1 l2 ->
all_le (edf_rta_iteration l1) (edf_rta_iteration l2).
rename H_valid_task_parameters into VALID.
intros x1 x2 LEinit LE.
move: LE => /andP [/eqP ZIP LE]; unfold all_le.
assert (UNZIP': unzip1 (edf_rta_iteration x1) = unzip1 (edf_rta_iteration x2)).
by rewrite 2!edf_claimed_bounds_unzip1_update_bound.
}
apply/andP; split; first by rewrite UNZIP'.
apply f_equal with (B := nat) (f := fun x => size x) in UNZIP'.
rename UNZIP' into SIZE.
rewrite size_map [size (unzip1 _)]size_map in SIZE.
move: LE => /(zipP _ (fun x y => snd x <= snd y)) LE.
destruct x1 as [| p0 x1'], x2 as [| p0' x2']; try (by ins).
apply/(zipP p0 (fun x y => snd x <= snd y)); first by done.
intros i LTi.
exploit LE; first by rewrite 2!size_map in SIZE.
{
by rewrite size_zip 2!size_map -size_zip in LTi; apply LTi.
}
rewrite 2!size_map in SIZE.
instantiate (1 := p0); intro LEi.
rewrite (nth_map p0);
last by rewrite size_zip 2!size_map -SIZE minnn in LTi.
rewrite (nth_map p0);
last by rewrite size_zip 2!size_map SIZE minnn in LTi.
unfold update_bound, edf_response_time_bound; desf; simpl.
rename s into tsk_i, s0 into tsk_i', t into R_i, t0 into R_i', Heq into EQ, Heq0 into EQ'.
assert (EQtsk: tsk_i = tsk_i').
{
destruct p0 as [tsk0 R0], p0' as [tsk0' R0']; simpl in H2; subst.
have MAP := @nth_map _ (tsk0',R0) _ tsk0' (fun x => fst x) i ((tsk0', R0) :: x1').
have MAP' := @nth_map _ (tsk0',R0) _ tsk0' (fun x => fst x) i ((tsk0', R0') :: x2').
assert (FSTeq: fst (nth (tsk0', R0)((tsk0', R0) :: x1') i) =
fst (nth (tsk0',R0) ((tsk0', R0') :: x2') i)).
rewrite -MAP;
last by simpl; rewrite size_zip 2!size_map /= -H0 minnn in LTi.
rewrite -MAP';
last by simpl; rewrite size_zip 2!size_map /= H0 minnn in LTi.
by f_equal; simpl; f_equal.
apply f_equal with (B := sporadic_task) (f := fun x => fst x) in EQ.
apply f_equal with (B := sporadic_task) (f := fun x => fst x) in EQ'.
by rewrite FSTeq EQ' /= in EQ; rewrite EQ.
}
subst tsk_i'; rewrite leq_add2l.
unfold I, total_interference_bound_edf; apply leq_div2r.
rewrite 2!big_cons.
destruct p0 as [tsk0 R0], p0' as [tsk0' R0'].
simpl in H2; subst tsk0'.
rename R_i into delta, R_i' into delta'.
rewrite EQ EQ' in LEi; simpl in LEi.
rename H0 into SIZE, H1 into UNZIP; clear EQ EQ'.
assert (SUBST: forall l delta,
\sum_(j <- l | let '(tsk_other, _) := j in
interference_bound_edf task_cost task_period task_deadline task_jitter tsk_i delta
\sum_(j <- l | different_task tsk_i (fst j))
interference_bound_edf task_cost task_period task_deadline task_jitter tsk_i delta j).
{
intros l x; clear -l.
induction l; first by rewrite 2!big_nil.
by rewrite 2!big_cons; rewrite IHl; desf; rewrite /= Heq in Heq0.
} rewrite 2!SUBST; clear SUBST.
assert (VALID': valid_sporadic_taskset task_cost task_period task_deadline
(unzip1 ((tsk0, R0) :: x1'))).
{
move: LEinit => /andP [/eqP EQinit _].
rewrite -EQinit; unfold valid_sporadic_taskset.
move => tsk /mapP IN. destruct IN as [p INinit EQ]; subst.
by move: INinit => /mapP INinit; destruct INinit as [tsk INtsk]; subst; apply VALID.
}
assert (GE_COST: all (fun p => task_cost (fst p) <= snd p) ((tsk0, R0) :: x1')).
{
clear LE; move: LEinit => /andP [/eqP UNZIP' LE].
move: LE => /(zipP _ (fun x y => snd x <= snd y)) LE.
specialize (LE (tsk0, R0)).
apply/(all_nthP (tsk0,R0)).
intros j LTj; generalize UNZIP'; simpl; intro SIZE'.
have F := @f_equal _ _ size (unzip1 (initial_state ts)).
apply F in SIZE'; clear F; rewrite /= 3!size_map in SIZE'.
exploit LE; [by rewrite size_map /= | |].
rewrite size_zip size_map /= SIZE' minnn.
by simpl in LTj; apply LTj.
clear LE; intro LE.
unfold initial_state in LE.
have MAP := @nth_map _ tsk0 _ (tsk0,R0).
rewrite MAP /= in LE;
[clear MAP | by rewrite SIZE'; simpl in LTj].
apply leq_trans with (n := task_cost (nth tsk0 ts j));
[apply eq_leq; f_equal | by done].
have MAP := @nth_map _ (tsk0, R0) _ tsk0 (fun x => fst x).
rewrite -MAP; [clear MAP | by done].
unfold unzip1 in UNZIP'; rewrite -UNZIP'; f_equal.
clear -ts; induction ts; [by done | by simpl; f_equal].
assert (LESUM: \sum_(j <- x1' | different_task tsk_i (fst j))
interference_bound_edf task_cost task_period task_deadline task_jitter tsk_i delta j <=
\sum_(j <- x2' | different_task tsk_i (fst j))
interference_bound_edf task_cost task_period task_deadline task_jitter tsk_i delta' j).
set elem := (tsk0, R0); rewrite 2!(big_nth elem).
rewrite -SIZE.
rewrite big_mkcond [\sum_(_ <- _ | different_task _ _)_]big_mkcond.
rewrite big_seq_cond [\sum_(_ <- _ | true) _]big_seq_cond.
apply leq_sum; intros j; rewrite andbT; intros INj.
rewrite mem_iota add0n subn0 in INj; move: INj => /andP [_ INj].
assert (FSTeq: fst (nth elem x1' j) = fst (nth elem x2' j)).
{
have MAP := @nth_map _ elem _ tsk0 (fun x => fst x).
by rewrite -2?MAP -?SIZE //; f_equal.
} rewrite -FSTeq.
destruct (different_task tsk_i (fst (nth elem x1' j))) eqn:INTERF;
last by done.
{
exploit (LE elem); [by rewrite /= SIZE | | intro LEj].
{
rewrite size_zip 2!size_map /= -SIZE minnn in LTi.
by rewrite size_zip /= -SIZE minnn; apply (leq_ltn_trans INj).
}
simpl in LEj.
exploit (VALID' (fst (nth elem x1' j))); last intro VALIDj.
{
apply/mapP; exists (nth elem x1' j); last by done.
by rewrite in_cons; apply/orP; right; rewrite mem_nth.
}
exploit (GE_COST (nth elem x1' j)); last intro GE_COSTj.
{
by rewrite in_cons; apply/orP; right; rewrite mem_nth.
}
unfold is_valid_sporadic_task in *.
destruct (nth elem x1' j) as [tsk_j R_j] eqn:SIMPL1,
(nth elem x2' j) as [tsk_j' R_j'] eqn:SIMPL2.
rewrite SIMPL1 SIMPL2 in LEj.
simpl in FSTeq; rewrite -FSTeq; simpl in LEj; simpl in VALIDj; des.
by apply interference_bound_edf_monotonic.
}
}
destruct (different_task tsk_i tsk0) eqn:INTERFtsk0; last by done.
apply leq_add; last by done.
{
exploit (LE (tsk0, R0)); [by rewrite /= SIZE | | intro LEj];
first by instantiate (1 := 0); rewrite size_zip /= -SIZE minnn.
exploit (VALID' tsk0); first by rewrite in_cons; apply/orP; left.
exploit (GE_COST (tsk0, R0)); first by rewrite in_cons eq_refl orTb.
unfold is_valid_sporadic_task; intros GE_COST0 VALID0; des; simpl in LEj.
by apply interference_bound_edf_monotonic.
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
(* It follows from the properties above that the iteration is monotonically increasing. *)
Lemma bertogna_edf_comp_iteration_monotonic: forall k, all_le (f k) (f k.+1).
Proof.
unfold f; intros k.
apply fun_mon_iter_mon_generic with (x1 := k) (x2 := k.+1);
try (by done);
[ by apply all_le_reflexive
| by apply all_le_transitive
| by apply bertogna_edf_comp_iteration_preserves_order
| by apply bertogna_edf_comp_iteration_preserves_minimum].
Qed.
End RelationProperties.
(* Knowing that the iteration is monotonically increasing (with respect to all_le),
we show that the RTA procedure converges to a fixed point. *)
(* First, note that when there are no tasks, the iteration trivially converges. *)
Lemma bertogna_edf_comp_f_converges_with_no_tasks :
size ts = 0 ->
f (max_steps ts) = f (max_steps ts).+1.
Proof.
intro SIZE; destruct ts; last by inversion SIZE.
unfold max_steps; rewrite big_nil /=.
by unfold edf_rta_iteration.
Qed.
(* Otherwise, if the iteration reached a fixed point before (max_steps ts), then
the value at (max_steps ts) is still at a fixed point. *)
Lemma bertogna_edf_comp_f_converges_early :
(exists k, k <= max_steps ts /\ f k = f k.+1) ->
f (max_steps ts) = f (max_steps ts).+1.
Proof.
by intros EX; des; apply fixedpoint.iter_fix with (k := k).
Qed.
(* Else, we derive a contradiction. *)
Section DerivingContradiction.
(* Assume that there are tasks. *)
Hypothesis H_at_least_one_task: size ts > 0.
(* Assume that the iteration continued to diverge. *)
Hypothesis H_keeps_diverging:
forall k,
k <= max_steps ts -> f k != f k.+1.
(* Since the iteration is monotonically increasing, it must be
strictly increasing. *)
Lemma bertogna_edf_comp_f_increases :
forall k,
k <= max_steps ts -> one_lt (f k) (f k.+1).
rename H_at_least_one_task into NONEMPTY.
intros step LEstep; unfold one_lt; apply/andP; split;
first by rewrite 2!edf_claimed_bounds_unzip1_iteration.
rewrite -[has _ _]negbK; apply/negP; unfold not; intro ALL.
rewrite -all_predC in ALL.
move: ALL => /allP ALL.
exploit (H_keeps_diverging step); [by done | intro DIFF].
assert (DUMMY: exists tsk: sporadic_task, True).
destruct ts as [|tsk0]; first by rewrite ltnn in NONEMPTY.
by exists tsk0.
des; clear DUMMY.
move: DIFF => /eqP DIFF; apply DIFF.
apply eq_from_nth with (x0 := (tsk, 0));
first by simpl; rewrite size_map.
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
intros i LTi.
remember (nth (tsk, 0)(f step) i) as p_i;rewrite -Heqp_i.
remember (nth (tsk, 0)(f step.+1) i) as p_i';rewrite -Heqp_i'.
rename Heqp_i into EQ, Heqp_i' into EQ'.
exploit (ALL (p_i, p_i')).
{
rewrite EQ EQ'.
rewrite -nth_zip; last by unfold f; rewrite iterS size_map.
apply mem_nth; rewrite size_zip.
unfold f; rewrite iterS size_map.
by rewrite minnn.
}
unfold predC; simpl; rewrite -ltnNge; intro LTp.
have GROWS := bertogna_edf_comp_iteration_monotonic step.
move: GROWS => /andP [_ /allP GROWS].
exploit (GROWS (p_i, p_i')).
{
rewrite EQ EQ'.
rewrite -nth_zip; last by unfold f; rewrite iterS size_map.
apply mem_nth; rewrite size_zip.
unfold f; rewrite iterS size_map.
by rewrite minnn.
}
simpl; intros LE.
destruct p_i as [tsk_i R_i], p_i' as [tsk_i' R_i'].
simpl in *.
assert (EQtsk: tsk_i = tsk_i').
{
unfold edf_rta_iteration in EQ'.
rewrite (nth_map (tsk, 0)) in EQ'; last by done.
by unfold update_bound in EQ'; desf.
}
rewrite EQtsk; f_equal.
by apply/eqP; rewrite eqn_leq; apply/andP; split.
}
(* In the end, each response-time bound is so high that the sum
of all response-time bounds exceeds the sum of all deadlines.
Contradiction! *)
Lemma bertogna_edf_comp_rt_grows_too_much :
forall k,
k <= max_steps ts ->
\sum_((tsk, R) <- f k) (R - task_cost tsk) + 1 > k.
have LT := bertogna_edf_comp_f_increases.
have MONO := bertogna_edf_comp_iteration_monotonic.
rename H_at_least_one_task into NONEMPTY.
unfold valid_sporadic_taskset, is_valid_sporadic_task in *.
rename H_valid_task_parameters into VALID.
intros step LE.
assert (DUMMY: exists tsk: sporadic_task, True).
destruct ts as [|tsk0]; first by rewrite ltnn in NONEMPTY.
by exists tsk0.
} destruct DUMMY as [elem _].
induction step.
{
by rewrite addn1.
rewrite -addn1 ltn_add2r.
apply leq_ltn_trans with (n := \sum_(i <- f step) (let '(tsk, R) := i in R - task_cost tsk)).
{
rewrite -ltnS; rewrite addn1 in IHstep.
by apply IHstep, ltnW.
}
rewrite (eq_bigr (fun x => snd x - task_cost (fst x)));
last by ins; destruct i.
rewrite [\sum_(_ <- f step.+1)_](eq_bigr (fun x => snd x - task_cost (fst x)));
last by ins; destruct i.
unfold f at 2; rewrite iterS.
rewrite big_map; fold (f step).
rewrite -(ltn_add2r (\sum_(i <- f step) task_cost (fst i))).
rewrite -2!big_split /=.
rewrite big_seq_cond [\sum_(_ <- _ | true)_]big_seq_cond.
rewrite (eq_bigr (fun i => snd i)); last first.
{
intro i; rewrite andbT; intro IN;
rewrite addnBAC; first by rewrite -addnBA // subnn addn0.
have GE_COST := edf_claimed_bounds_ge_cost ts step.
by destruct i; apply GE_COST.
}
rewrite [\sum_(_ <- _ | _)(_ - _ + _)](eq_bigr (fun i => snd (update_bound (f step) i))); last first.
{
intro i; rewrite andbT; intro IN.
unfold update_bound; destruct i; simpl.
rewrite addnBAC; first by rewrite -addnBA // subnn addn0.
apply (edf_claimed_bounds_ge_cost ts step.+1).
by rewrite iterS; apply/mapP; exists (s, t).
specialize (LT step (ltnW LE)).
specialize (MONO step).
move: LT => /andP [_ LT]; move: LT => /hasP LT.
destruct LT as [[x1 x2] INzip LT]; simpl in *.
move: MONO => /andP [_ /(zipP _ (fun x y => snd x <= snd y)) MONO].
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
rewrite 2!(big_nth (elem, 0)).
apply mem_zip_exists with (elem := (elem, 0)) (elem' := (elem, 0)) in INzip; des;
last by rewrite size_map.
rewrite -> big_cat_nat with (m := 0) (n := idx) (p := size (f step));
[simpl | by done | by apply ltnW].
rewrite -> big_cat_nat with (m := idx) (n := idx.+1) (p := size (f step));
[simpl | by done | by done].
rewrite big_nat_recr /=; last by done.
rewrite -> big_cat_nat with (m := 0) (n := idx) (p := size (f step));
[simpl | by done | by apply ltnW].
rewrite -> big_cat_nat with (m := idx) (n := idx.+1) (p := size (f step));
[simpl | by done | by done].
rewrite big_nat_recr /=; last by done.
rewrite [\sum_(idx <= i < idx) _]big_geq // add0n.
rewrite [\sum_(idx <= i < idx) _]big_geq // add0n.
rewrite -addn1 -addnA; apply leq_add.
{
rewrite big_nat_cond [\sum_(_ <= _ < _ | true) _]big_nat_cond.
apply leq_sum; move => i /andP [/andP [LT1 LT2] _].
exploit (MONO (elem,0)); [by rewrite size_map | | intro LEi].
{
rewrite size_zip; apply (ltn_trans LT2).
by apply leq_trans with (n := size (f step));
[by done | by rewrite size_map minnn].
}
unfold edf_rta_iteration in LEi.
by rewrite -> nth_map with (x1 := (elem, 0)) in LEi;
last by apply (ltn_trans LT2).
}
rewrite -addnA [_ + 1]addnC addnA; apply leq_add.
{
unfold edf_rta_iteration in INzip2; rewrite addn1.
rewrite -> nth_map with (x1 := (elem, 0)) in INzip2; last by done.
by rewrite -INzip2 -INzip1.
}
{
rewrite big_nat_cond [\sum_(_ <= _ < _ | true) _]big_nat_cond.
apply leq_sum; move => i /andP [/andP [LT1 LT2] _].
exploit (MONO (elem,0));
[ by rewrite size_map
| by rewrite size_zip; apply (leq_trans LT2); rewrite size_map minnn | intro LEi ].
unfold edf_rta_iteration in LEi.
by rewrite -> nth_map with (x1 := (elem, 0)) in LEi; last by done.
}
(* Using the lemmas above, we prove that edf_rta_iteration reaches
a fixed point after (max_steps ts) step, ... *)
Lemma edf_claimed_bounds_finds_fixed_point_of_list :
forall rt_bounds,
edf_claimed_bounds ts = Some rt_bounds ->
valid_sporadic_taskset task_cost task_period task_deadline ts ->
f (max_steps ts) = edf_rta_iteration (f (max_steps ts)).
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
Proof.
intros rt_bounds SOME VALID.
unfold valid_sporadic_taskset, is_valid_sporadic_task in *.
unfold edf_claimed_bounds in SOME; desf.
rename Heq into LE.
fold (f (max_steps ts)) in *; fold (f (max_steps ts).+1).
(* Either the task set is empty or not. *)
destruct (size ts == 0) eqn:EMPTY;
first by apply bertogna_edf_comp_f_converges_with_no_tasks; apply/eqP.
apply negbT in EMPTY; rewrite -lt0n in EMPTY.
(* Either f converges by the deadline or not. *)
destruct ([exists k in 'I_((max_steps ts).+1), f k == f k.+1]) eqn:EX.
{
move: EX => /exists_inP EX; destruct EX as [k _ ITERk].
destruct k as [k LTk]; simpl in ITERk.
apply bertogna_edf_comp_f_converges_early.
exists k; split; [by apply LTk | 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.
assert (SAMESUM: \sum_(tsk <- ts) task_cost tsk = \sum_(p <- f (max_steps ts)) task_cost (fst p)).
{
have MAP := @big_map _ 0 addn _ _ (fun x => fst x) (f (max_steps ts))
(fun x => true) (fun x => task_cost x).
have UNZIP := edf_claimed_bounds_unzip1_iteration ts (max_steps ts).
fold (f (max_steps ts)) in UNZIP; unfold unzip1 in UNZIP.
by rewrite UNZIP in MAP; rewrite MAP.
}
(* Show that the sum is less than the sum of all deadlines. *)
assert (SUM: \sum_(p <- f (max_steps ts)) (snd p - task_cost (fst p)) + 1 <= max_steps ts).
{
unfold max_steps at 2; rewrite leq_add2r.
rewrite -(leq_add2r (\sum_(tsk <- ts) task_cost tsk)).
rewrite {1}SAMESUM -2!big_split /=.
rewrite big_seq_cond [\sum_(_ <- _ | true)_]big_seq_cond.
rewrite (eq_bigr (fun x => snd x)); last first.
{
intro i; rewrite andbT; intro IN.
rewrite addnBAC; first by rewrite -addnBA // subnn addn0.
have GE_COST := edf_claimed_bounds_ge_cost ts (max_steps ts).
fold (f (max_steps ts)) in GE_COST.
by destruct i; apply GE_COST.
}
rewrite (eq_bigr (fun x => task_deadline x)); last first.
{
intro i; rewrite andbT; intro IN.
rewrite addnBAC; first by rewrite -addnBA // subnn addn0.
by specialize (VALID i IN); des.
}
rewrite -2!big_seq_cond.
have MAP := @big_map _ 0 addn _ _ (fun x => fst x) (f (max_steps ts))
(fun x => true) (fun x => task_deadline x).
have UNZIP := edf_claimed_bounds_unzip1_iteration ts (max_steps ts).
fold (f (max_steps ts)) in UNZIP; unfold unzip1 in UNZIP.
rewrite UNZIP in MAP; rewrite MAP.
rewrite big_seq_cond [\sum_(_ <- _|true)_]big_seq_cond.
apply leq_sum; intro i; rewrite andbT; intro IN.
move: LE => /allP LE; unfold jitter_plus_R_le_deadline in LE.
specialize (LE i IN); destruct i; simpl.
by apply leq_trans with (n := task_jitter s + s0); first by apply leq_addl.
}
have TOOMUCH :=
bertogna_edf_comp_rt_grows_too_much EMPTY _ (max_steps ts) (leqnn (max_steps ts)).
exploit TOOMUCH; [| intro BUG].
{
intros k LEk; rewrite -ltnS in LEk.
by exploit (EX (Ordinal LEk)); [by done | by ins].
}
rewrite (eq_bigr (fun i => snd i - task_cost (fst i))) in BUG;
last by ins; destruct i.
by apply (leq_ltn_trans SUM) in BUG; rewrite ltnn in BUG.
Qed.
(* ...and since there cannot be a vector of response-time bounds with values less than
the task costs, this solution is also the least fixed point. *)
Lemma edf_claimed_bounds_finds_least_fixed_point :
forall v,
all_le (initial_state ts) v ->
v = edf_rta_iteration v ->
all_le (f (max_steps ts)) v.
Proof.
intros v GE0 EQ.
apply bertogna_edf_comp_iteration_inductive; first by done.
intros k GEk.
rewrite EQ.
apply bertogna_edf_comp_iteration_preserves_order; last by done.
by apply bertogna_edf_comp_iteration_preserves_minimum.
Qed.
(* Therefore, with regard to the response-time bound recurrence, ...*)
(* ..., the individual response-time bounds (elements of the list) are also fixed points. *)
Theorem edf_claimed_bounds_finds_fixed_point_for_each_bound :
forall tsk R rt_bounds,
edf_claimed_bounds ts = Some rt_bounds ->
(tsk, R) \in rt_bounds ->
R = edf_response_time_bound rt_bounds tsk R.
Proof.
intros tsk R rt_bounds SOME IN.
have CONV := edf_claimed_bounds_finds_fixed_point_of_list rt_bounds.
rewrite -iterS in CONV; fold (f (max_steps ts).+1) in CONV.
unfold edf_claimed_bounds in *; desf.
exploit (CONV); [by done | by done | intro ITER; clear CONV].
unfold f in ITER.
cut (update_bound (iter (max_steps ts)
edf_rta_iteration (initial_state ts)) (tsk,R) = (tsk, R)).
{
intros EQ.
have F := @f_equal _ _ (fun x => snd x) _ (tsk, R).
by apply F in EQ; simpl in EQ.
}
set s := iter (max_steps ts) edf_rta_iteration (initial_state ts).
fold s in ITER, IN.
move: IN => /(nthP (tsk,0)) IN; destruct IN as [i LT EQ].
generalize EQ; rewrite ITER iterS in EQ; intro EQ'.
fold s in EQ.
unfold edf_rta_iteration in EQ.
have MAP := @nth_map _ (tsk,0) _ _ (update_bound s).
by rewrite MAP // EQ' in EQ; rewrite EQ.
Qed.
(* ..., which in turn implies that the response-time bound is the fixed
point from Bertogna and Cirinei's equation. *)
Lemma edf_claimed_bounds_converges :
forall tsk R rt_bounds,
edf_claimed_bounds ts = Some rt_bounds ->
(tsk, R) \in rt_bounds ->
R = task_cost tsk + div_floor (I rt_bounds tsk R) num_cpus.
Proof.
intros tsk R rt_bounds SOME IN.
have CONV := edf_claimed_bounds_finds_fixed_point_of_list rt_bounds.
rewrite -iterS in CONV; fold (f (max_steps ts).+1) in CONV.
unfold edf_claimed_bounds in *; desf.
exploit (CONV); [by done | by done | intro ITER; clear CONV].
cut (update_bound (iter (max_steps ts)
edf_rta_iteration (initial_state ts)) (tsk,R) = (tsk, R)).
{
intros EQ.
have F := @f_equal _ _ (fun x => snd x) _ (tsk, R).
by apply F in EQ; simpl in EQ.
}
set s := iter (max_steps ts) edf_rta_iteration (initial_state ts).
fold s in ITER, IN.
move: IN => /(nthP (tsk,0)) IN; destruct IN as [i LT EQ].
generalize EQ; rewrite ITER iterS in EQ; intro EQ'.
fold s in EQ.
unfold edf_rta_iteration in EQ.
have MAP := @nth_map _ (tsk,0) _ _ (update_bound s).
by rewrite MAP // EQ' in EQ; rewrite EQ.
Qed.
End Convergence.
Section MainProof.
(* ...all tasks have valid parameters... *)
Hypothesis H_valid_task_parameters:
valid_sporadic_taskset task_cost task_period task_deadline ts.
Hypothesis H_constrained_deadlines:
forall tsk, tsk \in ts -> task_deadline tsk <= task_period tsk.
(* 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:
forall j, arrives_in arr_seq j -> job_task j \in ts.
(* ...they have valid parameters,...*)
Hypothesis H_valid_job_parameters:
forall j,
arrives_in arr_seq j ->
valid_sporadic_job_with_jitter task_cost task_deadline task_jitter job_cost
job_deadline job_task job_jitter j.
(* ... and satisfy the sporadic task model.*)
Hypothesis H_sporadic_tasks:
sporadic_task_model task_period job_arrival job_task arr_seq.
(* Then, consider any schedule of this arrival sequence such that... *)
Variable sched: schedule Job num_cpus.
Hypothesis H_at_least_one_cpu: num_cpus > 0.
Hypothesis H_jobs_come_from_arrival_sequence:
jobs_come_from_arrival_sequence sched arr_seq.
(* ...jobs only execute after jitter and no longer than their execution costs. *)
Hypothesis H_jobs_execute_after_jitter:
jobs_execute_after_jitter job_arrival job_jitter sched.
completed_jobs_dont_execute job_cost sched.
(* Also assume that jobs are sequential. *)
Hypothesis H_sequential_jobs: sequential_jobs sched.
(* Assume that we have a work-conserving EDF scheduler. *)
Hypothesis H_work_conserving: work_conserving job_arrival job_cost job_jitter arr_seq sched.
Hypothesis H_edf_policy: respects_JLFP_policy job_arrival job_cost job_jitter arr_seq sched
(EDF job_arrival job_deadline).
Let no_deadline_missed_by_task (tsk: sporadic_task) :=
task_misses_no_deadline job_arrival job_cost job_deadline job_task arr_seq sched tsk.
Let no_deadline_missed_by_job :=
job_misses_no_deadline job_arrival job_cost job_deadline sched.