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Require Import Vbase TaskDefs JobDefs TaskArrivalDefs ScheduleDefs
PlatformDefs WorkloadDefs SchedulabilityDefs PriorityDefs
ResponseTimeDefs BertognaResponseTimeDefs divround helper
ssreflect ssrbool eqtype ssrnat seq fintype bigop div path tuple.
Module ResponseTimeAnalysisEDF.
Export Job SporadicTaskset Schedule Workload Schedulability ResponseTime Priority SporadicTaskArrival ResponseTimeAnalysis.
Section InterferenceBoundEDF.
Context {sporadic_task: eqType}.
Variable task_cost: sporadic_task -> nat.
Variable task_period: sporadic_task -> nat.
Variable task_deadline: sporadic_task -> nat.
(* Let tsk be the task to be analyzed. *)
Variable tsk: sporadic_task.
Let task_with_response_time := (sporadic_task * time)%type.
(* Assume a known response-time bound for each interfering task ... *)
Variable R_prev: seq task_with_response_time.
(* ... and an interval length delta. *)
Variable delta: time.
Section PerTask.
Variable tsk_R: task_with_response_time.
Let tsk_other := fst tsk_R.
Let R_other := snd tsk_R.
(* By combining Bertogna's interference bound for a work-conserving
scheduler ... *)
Let basic_interference_bound := interference_bound task_cost task_period tsk delta tsk_R.
(* ... with and EDF-specific interference bound, ... *)
Definition edf_specific_bound :=
let d_tsk := task_deadline tsk in
let e_other := task_cost tsk_other in
let p_other := task_period tsk_other in
let d_other := task_deadline tsk_other in
(div_floor d_tsk p_other) * e_other +
minn e_other ((d_tsk %% p_other) - d_other + R_other).
(* Bertogna and Cirinei define the following interference bound
under EDF scheduling. *)
minn basic_interference_bound edf_specific_bound.
End PerTask.
Section AllTasks.
Definition is_interfering_task_jlfp (tsk_other: sporadic_task) :=
tsk_other != tsk.
(* The total interference incurred by tsk is thus bounded by: *)
Definition total_interference_bound_edf :=
\sum_((tsk_other, R_other) <- R_prev | is_interfering_task_jlfp tsk_other)
interference_bound_edf (tsk_other, R_other).
End AllTasks.
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Section ResponseTimeBound.
Context {sporadic_task: eqType}.
Variable task_cost: sporadic_task -> nat.
Variable task_period: sporadic_task -> nat.
Variable task_deadline: sporadic_task -> nat.
Context {Job: eqType}.
Variable job_cost: Job -> nat.
Variable job_deadline: Job -> nat.
Variable job_task: Job -> sporadic_task.
(* Assume any job arrival sequence... *)
Context {arr_seq: arrival_sequence Job}.
(* ... in which jobs arrive sporadically and have valid parameters. *)
Hypothesis H_sporadic_tasks:
sporadic_task_model task_period arr_seq job_task.
Hypothesis H_valid_job_parameters:
forall (j: JobIn arr_seq),
valid_sporadic_job task_cost task_deadline job_cost job_deadline job_task j.
(* Consider any schedule such that...*)
Variable num_cpus: nat.
Variable rate: Job -> processor num_cpus -> nat.
Variable sched: schedule num_cpus arr_seq.
(* ...jobs do not execute before their arrival times nor 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 rate sched.
(* Also assume that jobs do not execute in parallel, processors have
unit speed, and that there exists at least one processor. *)
Hypothesis H_no_parallelism:
jobs_dont_execute_in_parallel sched.
Hypothesis H_rate_equals_one :
forall j cpu, rate j cpu = 1.
Hypothesis H_at_least_one_cpu :
num_cpus > 0.
(* Assume that we have a task set where all tasks have valid
parameters and restricted deadlines. *)
Variable ts: taskset_of sporadic_task.
Hypothesis H_valid_task_parameters:
valid_sporadic_taskset task_cost task_period task_deadline ts.
Hypothesis H_restricted_deadlines:
forall tsk, tsk \in ts -> task_deadline tsk <= task_period tsk.
Let no_deadline_is_missed_by_tsk (tsk: sporadic_task) :=
task_misses_no_deadline job_cost job_deadline job_task rate sched tsk.
Let is_response_time_bound (tsk: sporadic_task) :=
is_response_time_bound_of_task job_cost job_task tsk rate sched.
(* Assume a known response-time bound for any interfering task *)
Let task_with_response_time := (sporadic_task * time)%type.
(* Assume that the response-time bounds are a fixed-point of the
response-time recurrence. *)
Let I (tsk: sporadic_task) (delta: time) :=
total_interference_bound_edf task_cost task_period task_deadline tsk rt_bounds delta.
Hypothesis H_response_time_is_fixed_point :
forall tsk R,
(tsk, R) \in rt_bounds ->
R = task_cost tsk + div_floor (I tsk R) num_cpus.
(* Assume that the response-time bounds are larger than task costs. *)
Hypothesis H_response_time_bounds_ge_cost:
forall tsk_other R,
(tsk_other, R) \in rt_bounds -> R >= task_cost tsk_other.
(* Assume that no deadline is missed by any interfering task, i.e.,
response-time bound R_other <= deadline. *)
Hypothesis H_interfering_tasks_miss_no_deadlines:
forall tsk_other R,
(tsk_other, R) \in rt_bounds -> R <= task_deadline tsk_other.
(* Assume that the schedule satisfies the global scheduling
invariant, i.e., if any job of tsk is backlogged, all
the processors must be busy with jobs of equal or higher
priority. *)
Hypothesis H_global_scheduling_invariant:
job_task j = tsk ->
backlogged job_cost rate sched j t ->
count
(fun tsk_other : sporadic_task =>
is_interfering_task_jlfp tsk tsk_other &&
task_is_scheduled job_task sched tsk_other t) ts = num_cpus.
(* Next, we define Bertogna and Cirinei's response-time bound recurrence *)
Variable tsk: sporadic_task.
Variable R: time.
Hypothesis tsk_R_in_rt_bounds: (tsk, R) \in rt_bounds.
(* ..., then R bounds the response time of tsk in any schedule. *)
Theorem bertogna_cirinei_response_time_bound_edf :
is_response_time_bound tsk R.
Proof.
unfold is_response_time_bound, is_response_time_bound_of_task,
job_has_completed_by, completed,
completed_jobs_dont_execute,
valid_sporadic_job in *.
rename H_completed_jobs_dont_execute into COMP,
H_valid_job_parameters into PARAMS,
H_valid_task_parameters into TASK_PARAMS,
H_restricted_deadlines into RESTR,
H_interfering_tasks_miss_no_deadlines into NOMISS,
H_rate_equals_one into RATE,
H_global_scheduling_invariant into INVARIANT,
H_response_time_bounds_ge_cost into GE_COST,
H_response_time_is_fixed_point into FIX.
(* For simplicity, let x denote per-task interference under FP
scheduling, and let X denote the total interference. *)
set x := fun hp_tsk =>
if (hp_tsk \in ts) && is_interfering_task_jlfp tsk hp_tsk then
task_interference job_cost job_task rate sched j
(job_arrival j) (job_arrival j + R) hp_tsk
else 0.
set X := total_interference job_cost rate sched j (job_arrival j) (job_arrival j + R).
(* Let's recall the workload bound under EDF scheduling. *)
set workload_bound := fun (tup: task_with_response_time) =>
let (tsk_k, R_k) := tup in
interference_bound_edf task_cost task_period task_deadline tsk R (tsk_k, R_k) (*add EDF-term*)
else 0.
assert (INtsk: tsk \in ts).
{
admit.
}
(* Now we start the proof. Assume by contradiction that job j
is not complete at time (job_arrival j + R). *)
destruct (completed job_cost rate sched j (job_arrival j + R)) eqn:COMPLETED;
first by move: COMPLETED => /eqP COMPLETED; rewrite COMPLETED eq_refl.
apply negbT in COMPLETED; exfalso.
(* Since j has not completed, recall the time when it is not
executing is the total interference. *)
exploit (complement_of_interf_equals_service job_cost rate sched j (job_arrival j)
(job_arrival j + R));
last intro EQinterf; ins; unfold has_arrived;
first by apply leqnn.
rewrite {2}[_ + R]addnC -addnBA // subnn addn0 in EQinterf.
(* In order to derive a contradiction, we first show that
the interference x_k of any task is no larger than the
workload bound W_k. *)
assert (WORKLOAD: forall tsk_k,
(tsk_k \in ts) && is_interfering_task_jlfp tsk tsk_k ->
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move => tsk_k /andP [INk INTERk] R_k HPk.
unfold x, workload_bound; rewrite INk INTERk andbT.
exploit (exists_R tsk_k); [by ins | by ins | intro INhp; des].
apply leq_trans with (n := workload job_task rate sched tsk_k
(job_arrival j) (job_arrival j + R)).
{
unfold task_interference, workload.
apply leq_sum; intros t _.
rewrite -mulnb -[\sum_(_ < _) _]mul1n.
apply leq_mul; first by apply leq_b1.
destruct (task_is_scheduled job_task sched tsk_k t) eqn:SCHED; last by ins.
unfold task_is_scheduled in SCHED.
move: SCHED =>/exists_inP SCHED.
destruct SCHED as [cpu _ HAScpu].
rewrite -> bigD1 with (j := cpu); simpl; last by ins.
apply ltn_addr; unfold service_of_task, schedules_job_of_tsk in *.
by destruct (sched cpu t);[by rewrite HAScpu mul1n RATE|by ins].
}
{
apply workload_bounded_by_W with (task_deadline0 := task_deadline) (job_cost0 := job_cost) (job_deadline0 := job_deadline); ins;
[ by rewrite RATE
| by apply TASK_PARAMS
| by apply RESTR
| by red; red; ins; apply (RESP tsk_k)
| by apply GE_COST |].
red; red; move => j' /eqP JOBtsk' _;
unfold job_misses_no_deadline.
specialize (PARAMS j'); des.
rewrite PARAMS1 JOBtsk'.
apply completion_monotonic with (t := job_arrival j' + R0); ins;
[by rewrite leq_add2l; apply NOMISS | by apply (RESP tsk_k)].
}
(* In the remaining of the proof, we show that the workload bound
W_k is less than the task interference x (contradiction!).
For that, we require a few auxiliary lemmas: *)
(* 1) We show that the total interference X >= R - e_k + 1.
Otherwise, job j would have completed on time. *)
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{
unfold completed in COMPLETED.
rewrite addn1.
move: COMPLETED; rewrite neq_ltn; move => /orP COMPLETED; des;
last first.
{
apply (leq_ltn_trans (COMP j (job_arrival j + R))) in COMPLETED.
by rewrite ltnn in COMPLETED.
}
apply leq_trans with (n := R - service rate sched j (job_arrival j + R)); last first.
{
unfold service; rewrite service_before_arrival_eq_service_during; ins.
rewrite EQinterf.
rewrite subKn; first by ins.
{
unfold total_interference.
rewrite -{1}[_ j]add0n big_addn addnC -addnBA // subnn addn0.
rewrite -{2}[R]subn0 -[R-_]mul1n -[1*_]addn0 -iter_addn.
by rewrite -big_const_nat leq_sum //; ins; apply leq_b1.
}
}
{
apply ltn_sub2l; last first.
{
apply leq_trans with (n := job_cost j); first by ins.
by rewrite -JOBtsk; specialize (PARAMS j); des; apply PARAMS0.
}
apply leq_trans with (n := job_cost j); first by ins.
apply leq_trans with (n := task_cost tsk);
first by rewrite -JOBtsk; specialize (PARAMS j); des; apply PARAMS0.
}
}
(* 2) Then, we prove that the sum of the interference of each
task is equal to the total interference multiplied by the
number of processors. This holds because interference only
occurs when all processors are busy with some task. *)
assert(ALLBUSY: \sum_(tsk_k <- ts) x tsk_k = X * num_cpus).
{
unfold x, X, total_interference, task_interference.
rewrite -big_mkcond -exchange_big big_distrl /=.
apply eq_big_nat; move => t LTt.
destruct (backlogged job_cost rate sched j t) eqn:BACK;
last by rewrite (eq_bigr (fun i => 0));
[by rewrite big_const_seq iter_addn mul0n addn0 mul0n|by ins].
rewrite big_mkcond mul1n /=.
rewrite (eq_bigr (fun i =>
task_is_scheduled job_task sched i t then 1 else 0))); last first.
{
ins; destruct ((i \in ts) && is_interfering_task_jlfp tsk i) eqn:IN;
rewrite (eq_bigr (fun i => if (i \in ts) && true then (if is_interfering_task_jlfp tsk i && task_is_scheduled job_task sched i t then 1 else 0) else 0));
last by ins; destruct (i \in ts) eqn:IN; rewrite ?andTb ?andFb.
rewrite -big_mkcond -big_seq_cond -big_mkcond sum1_count.
apply (INVARIANT tsk j); try (by done).
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(* 3) Next, we prove the auxiliary lemma from the paper. *)
assert (MINSERV: \sum_(tsk_k <- ts) x tsk_k >=
(R - task_cost tsk + 1) * num_cpus ->
\sum_(tsk_k <- ts) minn (x tsk_k) (R - task_cost tsk + 1) >=
(R - task_cost tsk + 1) * num_cpus).
{
intro SUMLESS.
set more_interf := fun tsk_k => x tsk_k >= R - task_cost tsk + 1.
rewrite [\sum_(_ <- _) minn _ _](bigID more_interf) /=.
unfold more_interf, minn.
rewrite [\sum_(_ <- _ | R - _ + _ <= _)_](eq_bigr (fun i => R - task_cost tsk + 1));
last first.
{
intros i COND; rewrite leqNgt in COND.
destruct (R - task_cost tsk + 1 > x i); ins.
}
rewrite [\sum_(_ <- _ | ~~_)_](eq_big (fun i => x i < R - task_cost tsk + 1)
(fun i => x i));
[| by red; ins; rewrite ltnNge
| by intros i COND; rewrite -ltnNge in COND; rewrite COND].
(* Case 1 |A| = 0 *)
destruct (~~ has (fun i => R - task_cost tsk + 1 <= x i) ts) eqn:HASa.
{
rewrite [\sum_(_ <- _ | _ <= _) _]big_hasC; last by apply HASa.
rewrite big_seq_cond; move: HASa => /hasPn HASa.
rewrite add0n (eq_bigl (fun i => (i \in ts) && true));
last by red; intros tsk_k; destruct (tsk_k \in ts) eqn:INk;
[by rewrite andTb ltnNge; apply HASa | by rewrite andFb].
by rewrite -big_seq_cond.
} apply negbFE in HASa.
set cardA := count (fun i => R - task_cost tsk + 1 <= x i) ts.
destruct (cardA >= num_cpus) eqn:CARD.
{
apply leq_trans with ((R - task_cost tsk + 1) * cardA);
first by rewrite leq_mul2l; apply/orP; right.
unfold cardA; rewrite -sum1_count big_distrr /=.
rewrite -[\sum_(_ <- _ | _) _]addn0.
by apply leq_add; [by apply leq_sum; ins; rewrite muln1|by ins].
} apply negbT in CARD; rewrite -ltnNge in CARD.
assert (GEsum: \sum_(i <- ts | x i < R - task_cost tsk + 1) x i >=
(R - task_cost tsk + 1) * (num_cpus - cardA)).
{
set some_interference_A := fun t =>
backlogged job_cost rate sched j t &&
((x tsk_k) >= R - task_cost tsk + 1) &&
task_is_scheduled job_task sched tsk_k t)) ts.
set total_interference_B := fun t =>
backlogged job_cost rate sched j t *
count (fun tsk_k =>
((x tsk_k) < R - task_cost tsk + 1) &&
task_is_scheduled job_task sched tsk_k t) ts.
apply leq_trans with ((\sum_(job_arrival j <= t < job_arrival j + R)
some_interference_A t) * (num_cpus - cardA)).
{
rewrite leq_mul2r; apply/orP; right.
move: HASa => /hasP HASa; destruct HASa as [tsk_a INa LEa].
apply leq_trans with (n := x tsk_a); first by apply LEa.
unfold x, task_interference, some_interference_A.
destruct ((tsk_a \in ts) && is_interfering_task_jlfp tsk tsk_a) eqn:INTERFa;
last by ins.
move: INTERFa => /andP INTERFa; des.
apply leq_sum; ins.
destruct (backlogged job_cost rate sched j i);
[rewrite 2!andTb | by ins].
destruct (task_is_scheduled job_task sched tsk_a i) eqn:SCHEDa;
[apply eq_leq; symmetry | by ins].
apply/eqP; rewrite eqb1.
apply/hasP; exists tsk_a; first by ins.
apply/andP; split; last by ins.
by apply/andP; split; ins.
}
apply leq_trans with (\sum_(job_arrival j <= t < job_arrival j + R)
total_interference_B t).
{
rewrite big_distrl /=.
apply leq_sum; intros t _.
unfold some_interference_A, total_interference_B.
destruct (backlogged job_cost rate sched j t) eqn:BACK;
[rewrite andTb mul1n | by ins].
destruct (has (fun tsk_k : sporadic_task =>
(R - task_cost tsk + 1 <= x tsk_k) &&
task_is_scheduled job_task sched tsk_k t) ts) eqn:HAS;
last by ins.
rewrite mul1n; move: HAS => /hasP HAS.
destruct HAS as [tsk_k INk H].
move: H => /andP [/andP [INTERFk LEk] SCHEDk].
exploit INVARIANT;
[by apply INtsk | by apply JOBtsk | by apply BACK | intro COUNT].
unfold cardA.
set interfering_tasks_at_t :=
task_is_scheduled job_task sched tsk_k t].
rewrite -(count_filter (fun i => true)) in COUNT.
fold interfering_tasks_at_t in COUNT.
rewrite count_predT in COUNT.
apply leq_trans with (n := num_cpus -
(x i >= R - task_cost tsk + 1) &&
task_is_scheduled job_task sched i t) ts).
{
apply leq_sub2l.
rewrite -2!sum1_count big_mkcond /=.
rewrite [\sum_(_ <- _ | _ <= _)_]big_mkcond /=.
apply leq_sum; intros i _.
[rewrite andTb | by rewrite 2!andFb].
destruct (task_is_scheduled job_task sched i t);
[by rewrite andbT | by rewrite andbF].
}
rewrite leq_subLR.
rewrite -count_predUI.
apply leq_trans with (n :=
count (predU (fun i : sporadic_task =>
(R - task_cost tsk + 1 <= x i) &&
task_is_scheduled job_task sched i t)
(fun tsk_k0 : sporadic_task =>
(x tsk_k0 < R - task_cost tsk + 1) &&
task_is_scheduled job_task sched tsk_k0 t))
ts); last by apply leq_addr.
apply leq_trans with (n := size interfering_tasks_at_t);
first by rewrite COUNT.
unfold interfering_tasks_at_t.
rewrite -count_predT count_filter.
rewrite leq_eqVlt; apply/orP; left; apply/eqP.
apply eq_count; red; simpl.
intros i.
(task_is_scheduled job_task sched i t);
rewrite 3?andTb ?andFb ?andbF ?andbT /=; try ins.
by rewrite leqNgt orNb.
}
{
unfold x at 2, task_interference.
rewrite [\sum_(i <- ts | _) _](eq_bigr
(fun i => \sum_(job_arrival j <= t < job_arrival j + R)
backlogged job_cost rate sched j t &&
task_is_scheduled job_task sched i t));
last first.
{
ins; destruct ((i \in ts) && is_interfering_task_jlfp tsk i) eqn:INTERi;
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first by move: INTERi => /andP [_ INTERi]; apply eq_bigr; ins; rewrite INTERi andTb.
by rewrite (eq_bigr (fun i => 0));
[by rewrite big_const_nat iter_addn mul0n addn0 | by ins].
}
{
rewrite exchange_big /=; apply leq_sum; intros t _.
unfold total_interference_B.
destruct (backlogged job_cost rate sched j t); last by ins.
rewrite mul1n -sum1_count.
rewrite big_seq_cond big_mkcond [\sum_(i <- ts | _ < _) _]big_mkcond.
apply leq_sum; ins; destruct (x i<R - task_cost tsk + 1);
[by rewrite 2!andbT andbA | by rewrite 2!andbF].
}
}
}
rewrite big_const_seq iter_addn addn0; fold cardA.
apply leq_trans with (n := (R-task_cost tsk+1)*cardA +
(R-task_cost tsk+1)*(num_cpus-cardA));
last by rewrite leq_add2l.
by rewrite -mulnDr subnKC //; apply ltnW.
}
(* 4) Now, we prove that the Bertogna's interference bound
is not enough to cover the sum of the "minimum" term over
all tasks (artifact of the proof by contradiction). *)
assert (SUM: \sum_(k <- rt_bounds | is_interfering_task_jlfp tsk (fst k))
(minn
(interference_bound_edf task_cost task_period task_deadline
tsk R k )
(minn (x (fst k)) ((snd k) - task_cost tsk + 1)))
>
workload_bound (tsk, R)).
{
apply leq_trans with (n := \sum_(tsk_k <- ts) minn (x tsk_k) (R - task_cost tsk + 1));
last first.
{ apply leq_trans with (n := \sum_(i<- rt_bounds | is_interfering_task_jlfp tsk (fst i))
(minn (x (fst i)) ((snd(i) - task_cost tsk + 1))));
last first.
apply leq_sum.
intros i _.
destruct i as [i R_i].
rewrite leq_min; apply/andP; split; last by done.
{
apply leq_trans with (n := x i);
first by apply geq_minl.
admit.
}
}
rewrite (eq_bigr (fun i => minn (x (fst i)) (R - task_cost tsk + 1))).
admit.
admit.
}
admit.
}
admit.
(* last by ins; destruct i.
apply leq_trans with (n := \sum_(tsk_k <- ts | is_interfering_task_jlfp tsk tsk_k) minn (x tsk_k) (R - task_cost tsk + 1)).
{
rewrite [\sum_(_ <- _ | is_interfering_task_jlfp tsk _)_]big_mkcond eq_leq //.
by destruct (is_interfering_task_jlfp tsk i); rewrite ?andbT ?andbF ?min0n.
}
have MAP := big_map (fun x => fst x) (fun i => true) (fun i => minn (x i) (R - task_cost tsk + 1)).
}
apply ltn_div_trunc with (d := num_cpus);
first by apply H_at_least_one_cpu.
rewrite -(ltn_add2l (task_cost tsk)) -REC.
rewrite -addn1 -leq_subLR.
rewrite -[R + 1 - _]subh1; last by rewrite REC; apply leq_addr.
rewrite leq_divRL; last by apply H_at_least_one_cpu.
apply MINSERV.
apply leq_trans with (n := X * num_cpus); last by rewrite ALLBUSY.
by rewrite leq_mul2r; apply/orP; right; apply INTERF.
(* 5) This implies that there exists a tuple (tsk_k, R_k) such that
min (x_k, R - e_i + 1) > min (W_k, R - e_i + 1). *)
(minn (x tsk_k) (R - task_cost tsk + 1) >
minn (workload_bound (tsk_k, snd tup)) (R - task_cost tsk + 1)))
{
apply/negP; unfold not; intro NOTHAS.
move: NOTHAS => /negP /hasPn ALL.
rewrite -[_ < _]negbK in SUM.
move: SUM => /negP SUM; apply SUM; rewrite -leqNgt.
rewrite [\sum_(i <- _ | let '(tsk_other, _) := i in _)_]big_mkcond.
rewrite big_seq_cond [\sum_(i <- _ | true) _]big_seq_cond.
apply leq_sum; move => tsk_k /andP [HPk _]; destruct tsk_k as [tsk_k R_k].
specialize (ALL (tsk_k, R_k) HPk).
unfold interference_bound, workload_bound, x in *.
fold (interferes_with_tsk); destruct (interferes_with_tsk tsk_k) eqn:INTERFk;
[rewrite andbT in ALL; rewrite andbT | by rewrite andbF min0n].
destruct (tsk_k \in ts) eqn:INk; last by rewrite min0n.
by rewrite andTb -leqNgt in ALL.
}
(* For this particular task, we show that x_k > W_k.
This contradicts the previous claim. *)
move: EX => /hasP EX; destruct EX as [tup_k HPk LTmin].
destruct tup_k as [tsk_k R_k]; simpl in LTmin.
move: LTmin => /andP [INTERFk LTmin]; move: (INTERFk) => /andP [INk INTERFk'].
rewrite INTERFk' in LTmin; unfold minn at 1 in LTmin.
destruct (W task_cost task_period tsk_k R_k R < R - task_cost tsk + 1); rewrite leq_min in LTmin;
last by move: LTmin => /andP [_ BUG]; rewrite ltnn in BUG.
move: LTmin => /andP [BUG _]; des.
specialize (WORKLOAD tsk_k INTERFk R_k HPk).
apply leq_ltn_trans with (p := x tsk_k) in WORKLOAD; first by rewrite ltnn in WORKLOAD.
by unfold workload_bound; rewrite INTERFk'; apply BUG.