Library rt.analysis.parallel.bertogna_edf_comp
Require Import rt.util.all.
Require Import rt.analysis.parallel.bertogna_edf_theory.
From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq fintype bigop div path.
Module ResponseTimeIterationEDF.
Import ResponseTimeAnalysisEDF.
(* In this section, we define the algorithm for Bertogna and Cirinei's
response-time analysis for EDF 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.
(* 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.
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.
(* First, recall the jitter-aware interference bound for 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 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 is only valid if it is no larger
than the deadline. *)
Definition R_le_deadline (pair: task_with_response_time) :=
let (tsk, R) := pair in
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) :=
map (update_bound rt_bounds) rt_bounds.
(* 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. *)
Let max_steps (ts: seq sporadic_task) :=
\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 R_le_deadline R_values) then
Some R_values
else None.
(* 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) :=
edf_claimed_bounds ts ≠ None.
(* 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 :
∀ 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 :
∀ l k,
unzip1 (iter k edf_rta_iteration (initial_state l)) = l.
Proof.
intros l k; clear -k.
induction k; simpl.
{
unfold initial_state.
induction l; first by done.
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 :
∀ 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 :
∀ l k tsk R,
(tsk, R) ∈ (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 :
∀ ts rt_bounds tsk R,
edf_claimed_bounds ts = Some rt_bounds →
(tsk, R) ∈ rt_bounds →
R ≤ task_deadline tsk.
Proof.
intros ts rt_bounds tsk R SOME PAIR; unfold edf_claimed_bounds in SOME.
destruct (all 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 :
∀ ts rt_bounds tsk,
edf_claimed_bounds ts = Some rt_bounds →
tsk ∈ ts →
∃ R,
(tsk, R) ∈ rt_bounds.
Proof.
intros ts rt_bounds tsk SOME IN.
unfold edf_claimed_bounds in SOME.
destruct (all 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.
∃ (task_cost tsk).
by apply/mapP; ∃ 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.
∃ (edf_response_time_bound prev_state tsk R).
by apply/mapP; ∃ (tsk, R); [by done | by f_equal].
}
Qed.
End SimpleLemmas.
(* 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. *)
Section Convergence.
(* Consider any sequence of tasks with valid parameters. *)
Variable ts: seq sporadic_task.
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)).
Qed.
(* ... and transitive. *)
Lemma all_le_transitive: transitive all_le.
Proof.
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.
}
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.
}
instantiate (1 := t); intro LE.
exploit LEyz; first by apply SIZEyz.
{
rewrite size_zip SIZExy SIZEyz minnn in LTi.
by rewrite size_zip SIZEyz minnn; apply LTi.
}
by instantiate (1 := t); intro LE'; apply (leq_trans LE).
Qed.
(* 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 :
∀ step, all_le (initial_state ts) (f step).
Proof.
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.
Qed.
(* The application of the function is inductive. *)
Lemma bertogna_edf_comp_iteration_inductive (P : seq task_with_response_time → Type) :
P (initial_state ts) →
(∀ 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 :
∀ l1 l2,
all_le (initial_state ts) l1 →
all_le l1 l2 →
all_le (edf_rta_iteration l1) (edf_rta_iteration l2).
Proof.
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: ∀ l delta,
\sum_(j <- l | let '(tsk_other, _) := j in
different_task tsk_i tsk_other)
(let '(tsk_other, R_other) := j in
interference_bound_edf task_cost task_period task_deadline tsk_i delta
(tsk_other, R_other)) =
\sum_(j <- l | different_task tsk_i (fst j))
interference_bound_edf task_cost task_period task_deadline 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].
}
move: GE_COST ⇒ /allP GE_COST.
assert (LESUM: \sum_(j <- x1' | different_task tsk_i (fst j))
interference_bound_edf task_cost task_period task_deadline tsk_i delta j ≤ \sum_(j <- x2' | different_task tsk_i (fst j))
interference_bound_edf task_cost task_period task_deadline 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; ∃ (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:SUBST1,
(nth elem x2' j) as [tsk_j' R_j'] eqn:SUBST2.
rewrite SUBST1 SUBST2 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.
}
Qed.
(* It follows from the properties above that the iteration is monotonically increasing. *)
Lemma bertogna_edf_comp_iteration_monotonic: ∀ 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 :
(∃ 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 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:
∀ 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 :
∀ k,
k ≤ max_steps ts → one_lt (f k) (f k.+1).
Proof.
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: ∃ tsk: sporadic_task, True).
{
destruct ts as [|tsk0]; first by rewrite ltnn in NONEMPTY.
by ∃ tsk0.
}
des; clear DUMMY.
move: DIFF ⇒ /eqP DIFF; apply DIFF.
apply eq_from_nth with (x0 := (tsk, 0));
first by simpl; rewrite size_map.
{
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.
}
Qed.
(* 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 :
∀ k,
k ≤ max_steps ts →
\sum_((tsk, R) <- f k) (R - task_cost tsk) + 1 > k.
Proof.
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: ∃ tsk: sporadic_task, True).
{
destruct ts as [|tsk0]; first by rewrite ltnn in NONEMPTY.
by ∃ tsk0.
} destruct DUMMY as [elem _].
induction step; first 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 subh1; 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 subh1; first by rewrite -addnBA // subnn addn0.
apply (edf_claimed_bounds_ge_cost ts step.+1).
by rewrite iterS; apply/mapP; ∃ (s, t).
}
rewrite -2!big_seq_cond.
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].
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.
}
}
Qed.
End DerivingContradiction.
(* 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 :
∀ 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)).
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 ([∃ 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.
∃ 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 subh1; 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 subh1; 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 R_le_deadline in LE.
by specialize (LE i IN); destruct i.
}
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 :
∀ 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 :
∀ tsk R rt_bounds,
edf_claimed_bounds ts = Some rt_bounds →
(tsk, R) ∈ 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.
End Convergence.
Section MainProof.
(* Consider a task set ts where... *)
Variable ts: taskset_of sporadic_task.
(* ...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.
(* 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 a work-conserving scheduler with EDF policy. *)
Hypothesis H_work_conserving: work_conserving job_cost sched.
Hypothesis H_edf_policy: enforces_JLDP_policy job_cost sched (EDF job_deadline).
Definition no_deadline_missed_by_task (tsk: sporadic_task) :=
task_misses_no_deadline job_cost job_deadline job_task sched tsk.
Definition 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 edf_claimed_bounds is safe. The proof follows by direct application of
the main Theorem from bertogna_edf_theory.v. *)
Theorem edf_analysis_yields_response_time_bounds :
∀ tsk R,
(tsk, R) ∈ edf_claimed_bounds ts →
response_time_bounded_by tsk R.
Proof.
intros tsk R IN j JOBj.
destruct (edf_claimed_bounds ts) as [rt_bounds |] eqn:SOME; last by done.
unfold edf_rta_iteration in ×.
have BOUND := bertogna_cirinei_response_time_bound_edf.
unfold is_response_time_bound_of_task in ×.
apply BOUND with (task_cost := task_cost) (task_period := task_period)
(task_deadline := task_deadline) (job_deadline := job_deadline)
(job_task := job_task) (ts := ts) (tsk := tsk) (rt_bounds := rt_bounds); try (by ins).
by unfold edf_claimed_bounds in SOME; desf; rewrite edf_claimed_bounds_unzip1_iteration.
by ins; apply edf_claimed_bounds_finds_fixed_point_for_each_bound with (ts := ts).
by ins; rewrite (edf_claimed_bounds_le_deadline ts rt_bounds).
Qed.
(* Therefore, if the schedulability test suceeds, ...*)
Hypothesis H_test_succeeds: edf_schedulable ts.
(*... no task misses its deadline. *)
Theorem taskset_schedulable_by_edf_rta :
∀ tsk, tsk ∈ ts → no_deadline_missed_by_task tsk.
Proof.
have RLIST := (edf_analysis_yields_response_time_bounds).
have DL := (edf_claimed_bounds_le_deadline ts).
have HAS := (edf_claimed_bounds_has_R_for_every_task ts).
unfold no_deadline_missed_by_task, task_misses_no_deadline,
job_misses_no_deadline, completed,
edf_schedulable,
valid_sporadic_job in ×.
rename H_valid_job_parameters into JOBPARAMS,
H_valid_task_parameters into TASKPARAMS,
H_constrained_deadlines into RESTR,
H_completed_jobs_dont_execute into COMP,
H_jobs_must_arrive_to_execute into MUSTARRIVE,
H_all_jobs_from_taskset into ALLJOBS,
H_test_succeeds into TEST.
move ⇒ tsk INtsk j JOBtsk.
destruct (edf_claimed_bounds ts) as [rt_bounds |] eqn:SOME; last by ins.
exploit (HAS rt_bounds tsk); [by ins | by ins | clear HAS; intro HAS; des].
have COMPLETED := RLIST tsk R HAS j JOBtsk.
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 conclude that no job misses
its deadline. *)
Theorem jobs_schedulable_by_edf_rta :
∀ (j: JobIn arr_seq), no_deadline_missed_by_job j.
Proof.
intros j.
have SCHED := taskset_schedulable_by_edf_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 ResponseTimeIterationEDF.
Require Import rt.analysis.parallel.bertogna_edf_theory.
From mathcomp Require Import ssreflect ssrbool eqtype ssrnat seq fintype bigop div path.
Module ResponseTimeIterationEDF.
Import ResponseTimeAnalysisEDF.
(* In this section, we define the algorithm for Bertogna and Cirinei's
response-time analysis for EDF 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.
(* 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.
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.
(* First, recall the jitter-aware interference bound for 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 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 is only valid if it is no larger
than the deadline. *)
Definition R_le_deadline (pair: task_with_response_time) :=
let (tsk, R) := pair in
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) :=
map (update_bound rt_bounds) rt_bounds.
(* 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. *)
Let max_steps (ts: seq sporadic_task) :=
\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 R_le_deadline R_values) then
Some R_values
else None.
(* 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) :=
edf_claimed_bounds ts ≠ None.
(* 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 :
∀ 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 :
∀ l k,
unzip1 (iter k edf_rta_iteration (initial_state l)) = l.
Proof.
intros l k; clear -k.
induction k; simpl.
{
unfold initial_state.
induction l; first by done.
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 :
∀ 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 :
∀ l k tsk R,
(tsk, R) ∈ (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 :
∀ ts rt_bounds tsk R,
edf_claimed_bounds ts = Some rt_bounds →
(tsk, R) ∈ rt_bounds →
R ≤ task_deadline tsk.
Proof.
intros ts rt_bounds tsk R SOME PAIR; unfold edf_claimed_bounds in SOME.
destruct (all 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 :
∀ ts rt_bounds tsk,
edf_claimed_bounds ts = Some rt_bounds →
tsk ∈ ts →
∃ R,
(tsk, R) ∈ rt_bounds.
Proof.
intros ts rt_bounds tsk SOME IN.
unfold edf_claimed_bounds in SOME.
destruct (all 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.
∃ (task_cost tsk).
by apply/mapP; ∃ 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.
∃ (edf_response_time_bound prev_state tsk R).
by apply/mapP; ∃ (tsk, R); [by done | by f_equal].
}
Qed.
End SimpleLemmas.
(* 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. *)
Section Convergence.
(* Consider any sequence of tasks with valid parameters. *)
Variable ts: seq sporadic_task.
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)).
Qed.
(* ... and transitive. *)
Lemma all_le_transitive: transitive all_le.
Proof.
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.
}
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.
}
instantiate (1 := t); intro LE.
exploit LEyz; first by apply SIZEyz.
{
rewrite size_zip SIZExy SIZEyz minnn in LTi.
by rewrite size_zip SIZEyz minnn; apply LTi.
}
by instantiate (1 := t); intro LE'; apply (leq_trans LE).
Qed.
(* 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 :
∀ step, all_le (initial_state ts) (f step).
Proof.
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.
Qed.
(* The application of the function is inductive. *)
Lemma bertogna_edf_comp_iteration_inductive (P : seq task_with_response_time → Type) :
P (initial_state ts) →
(∀ 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 :
∀ l1 l2,
all_le (initial_state ts) l1 →
all_le l1 l2 →
all_le (edf_rta_iteration l1) (edf_rta_iteration l2).
Proof.
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: ∀ l delta,
\sum_(j <- l | let '(tsk_other, _) := j in
different_task tsk_i tsk_other)
(let '(tsk_other, R_other) := j in
interference_bound_edf task_cost task_period task_deadline tsk_i delta
(tsk_other, R_other)) =
\sum_(j <- l | different_task tsk_i (fst j))
interference_bound_edf task_cost task_period task_deadline 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].
}
move: GE_COST ⇒ /allP GE_COST.
assert (LESUM: \sum_(j <- x1' | different_task tsk_i (fst j))
interference_bound_edf task_cost task_period task_deadline tsk_i delta j ≤ \sum_(j <- x2' | different_task tsk_i (fst j))
interference_bound_edf task_cost task_period task_deadline 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; ∃ (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:SUBST1,
(nth elem x2' j) as [tsk_j' R_j'] eqn:SUBST2.
rewrite SUBST1 SUBST2 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.
}
Qed.
(* It follows from the properties above that the iteration is monotonically increasing. *)
Lemma bertogna_edf_comp_iteration_monotonic: ∀ 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 :
(∃ 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 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:
∀ 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 :
∀ k,
k ≤ max_steps ts → one_lt (f k) (f k.+1).
Proof.
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: ∃ tsk: sporadic_task, True).
{
destruct ts as [|tsk0]; first by rewrite ltnn in NONEMPTY.
by ∃ tsk0.
}
des; clear DUMMY.
move: DIFF ⇒ /eqP DIFF; apply DIFF.
apply eq_from_nth with (x0 := (tsk, 0));
first by simpl; rewrite size_map.
{
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.
}
Qed.
(* 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 :
∀ k,
k ≤ max_steps ts →
\sum_((tsk, R) <- f k) (R - task_cost tsk) + 1 > k.
Proof.
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: ∃ tsk: sporadic_task, True).
{
destruct ts as [|tsk0]; first by rewrite ltnn in NONEMPTY.
by ∃ tsk0.
} destruct DUMMY as [elem _].
induction step; first 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 subh1; 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 subh1; first by rewrite -addnBA // subnn addn0.
apply (edf_claimed_bounds_ge_cost ts step.+1).
by rewrite iterS; apply/mapP; ∃ (s, t).
}
rewrite -2!big_seq_cond.
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].
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.
}
}
Qed.
End DerivingContradiction.
(* 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 :
∀ 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)).
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 ([∃ 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.
∃ 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 subh1; 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 subh1; 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 R_le_deadline in LE.
by specialize (LE i IN); destruct i.
}
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 :
∀ 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 :
∀ tsk R rt_bounds,
edf_claimed_bounds ts = Some rt_bounds →
(tsk, R) ∈ 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.
End Convergence.
Section MainProof.
(* Consider a task set ts where... *)
Variable ts: taskset_of sporadic_task.
(* ...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.
(* 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 a work-conserving scheduler with EDF policy. *)
Hypothesis H_work_conserving: work_conserving job_cost sched.
Hypothesis H_edf_policy: enforces_JLDP_policy job_cost sched (EDF job_deadline).
Definition no_deadline_missed_by_task (tsk: sporadic_task) :=
task_misses_no_deadline job_cost job_deadline job_task sched tsk.
Definition 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 edf_claimed_bounds is safe. The proof follows by direct application of
the main Theorem from bertogna_edf_theory.v. *)
Theorem edf_analysis_yields_response_time_bounds :
∀ tsk R,
(tsk, R) ∈ edf_claimed_bounds ts →
response_time_bounded_by tsk R.
Proof.
intros tsk R IN j JOBj.
destruct (edf_claimed_bounds ts) as [rt_bounds |] eqn:SOME; last by done.
unfold edf_rta_iteration in ×.
have BOUND := bertogna_cirinei_response_time_bound_edf.
unfold is_response_time_bound_of_task in ×.
apply BOUND with (task_cost := task_cost) (task_period := task_period)
(task_deadline := task_deadline) (job_deadline := job_deadline)
(job_task := job_task) (ts := ts) (tsk := tsk) (rt_bounds := rt_bounds); try (by ins).
by unfold edf_claimed_bounds in SOME; desf; rewrite edf_claimed_bounds_unzip1_iteration.
by ins; apply edf_claimed_bounds_finds_fixed_point_for_each_bound with (ts := ts).
by ins; rewrite (edf_claimed_bounds_le_deadline ts rt_bounds).
Qed.
(* Therefore, if the schedulability test suceeds, ...*)
Hypothesis H_test_succeeds: edf_schedulable ts.
(*... no task misses its deadline. *)
Theorem taskset_schedulable_by_edf_rta :
∀ tsk, tsk ∈ ts → no_deadline_missed_by_task tsk.
Proof.
have RLIST := (edf_analysis_yields_response_time_bounds).
have DL := (edf_claimed_bounds_le_deadline ts).
have HAS := (edf_claimed_bounds_has_R_for_every_task ts).
unfold no_deadline_missed_by_task, task_misses_no_deadline,
job_misses_no_deadline, completed,
edf_schedulable,
valid_sporadic_job in ×.
rename H_valid_job_parameters into JOBPARAMS,
H_valid_task_parameters into TASKPARAMS,
H_constrained_deadlines into RESTR,
H_completed_jobs_dont_execute into COMP,
H_jobs_must_arrive_to_execute into MUSTARRIVE,
H_all_jobs_from_taskset into ALLJOBS,
H_test_succeeds into TEST.
move ⇒ tsk INtsk j JOBtsk.
destruct (edf_claimed_bounds ts) as [rt_bounds |] eqn:SOME; last by ins.
exploit (HAS rt_bounds tsk); [by ins | by ins | clear HAS; intro HAS; des].
have COMPLETED := RLIST tsk R HAS j JOBtsk.
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 conclude that no job misses
its deadline. *)
Theorem jobs_schedulable_by_edf_rta :
∀ (j: JobIn arr_seq), no_deadline_missed_by_job j.
Proof.
intros j.
have SCHED := taskset_schedulable_by_edf_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 ResponseTimeIterationEDF.