Require Import Vbase schedule bertogna_fp_theory util_divround util_lemmas
ssreflect ssrbool eqtype ssrnat seq fintype bigop div path
workload_bound.
Module ResponseTimeIterationFP.
Import Schedule ResponseTimeAnalysisFP WorkloadBound.
(* In this section, we define the algorithm of Bertogna and Cirinei's
response-time analysis for FP scheduling. *)
Section Analysis.
Context {sporadic_task: eqType}.
Variable task_cost: sporadic_task -> nat.
Variable task_period: sporadic_task -> nat.
Variable task_deadline: sporadic_task -> nat.
(* During the iterations of the algorithm, we pass around pairs
of tasks and computed response-time bounds. *)
Let task_with_response_time := (sporadic_task * nat)%type.
Context {Job: eqType}.
Variable job_cost: Job -> nat.
Variable job_deadline: Job -> nat.
Variable job_task: Job -> sporadic_task.
(* Consider a platform with num_cpus processors, ...*)
Variable num_cpus: nat.
(* ..., and priorities based on an FP policy. *)
Variable higher_eq_priority: fp_policy sporadic_task.
Hypothesis H_valid_policy: valid_fp_policy higher_eq_priority.
(* Next we define the fixed-point iteration for computing
Bertogna's response-time bound for any task in ts. *)
(* First, given a sequence of pairs R_prev = <..., (tsk_hp, R_hp)> of
response-time bounds for the higher-priority tasks, we define an
iteration that computes the response-time bound of the current task:
R_tsk (0) = task_cost tsk
R_tsk (step + 1) = f (R step),
where f is the response-time recurrence, step is the number of iterations,
and R_tsk (0) is the initial state. *)
Definition per_task_rta (tsk: sporadic_task)
(R_prev: seq task_with_response_time) (step: nat) :=
iter step
(fun t => task_cost tsk +
div_floor
(total_interference_bound_fp task_cost task_period tsk
R_prev t higher_eq_priority)
num_cpus)
(task_cost tsk).
(* To ensure that the iteration converges, we will apply per_task_rta
a "sufficient" number of times: task_deadline tsk - task_cost tsk.
This corresponds to the time complexity of the iteration. *)
Definition max_steps (tsk: sporadic_task) := task_deadline tsk - task_cost tsk.
(* Next we compute the response-time bounds for the entire task set.
Since high-priority tasks may not be schedulable, we allow the
computation to fail.
Thus, given the response-time bound of previous tasks, we either
(a) append the computed response-time bound (tsk, R) of the current task
to the list of pairs, or,
(b) return None if the response-time analysis failed. *)
Definition R_list_helper hp_pairs tsk :=
if hp_pairs is Some rt_bounds then
let R := per_task_rta tsk rt_bounds (max_steps tsk) in
if R <= task_deadline tsk then
Some (rcons rt_bounds (tsk, R))
else None
else None.
(* The response-time analysis for a given task set is defined
as a left-fold (reduce) based on the function above.
This either returns a list of task and response-time bounds, or None. *)
Definition R_list (ts: taskset_of sporadic_task) :=
foldl R_list_helper (Some [::]) ts.
(* The schedulability test simply checks if we got a list of
response-time bounds (i.e., if the computation did not fail). *)
Definition fp_schedulable (ts: taskset_of sporadic_task) :=
R_list ts != None.
(* In the following section, we prove several helper lemmas about the
list of response-time bounds. The results seem trivial, but must be proven
nonetheless since the list of response-time bounds is computed with
a specific algorithm and there are no lemmas in the library for that. *)
Section SimpleLemmas.
(* First, we show that R_list of the prefix is the prefix of R_list. *)
Lemma R_list_rcons_prefix :
forall ts' hp_bounds tsk1 tsk2 R,
R_list (rcons ts' tsk1) = Some (rcons hp_bounds (tsk2, R)) ->
R_list ts' = Some hp_bounds.
Proof.
intros ts hp_bounds tsk1 tsk2 R SOME.
rewrite -cats1 in SOME.
unfold R_list in *.
rewrite foldl_cat in SOME.
simpl in SOME.
unfold R_list_helper in SOME.
desf; rewrite Heq; rename H0 into EQ.
move: EQ => /eqP EQ.
rewrite eqseq_rcons in EQ.
move: EQ => /andP [/eqP EQ _].
by f_equal.
Qed.
(* R_list returns the same tasks as the original task set. *)
Lemma R_list_rcons_task :
forall ts' hp_bounds tsk1 tsk2 R,
R_list (rcons ts' tsk1) = Some (rcons hp_bounds (tsk2, R)) ->
tsk1 = tsk2.
Proof.
intros ts hp_bounds tsk1 tsk2 R SOME.
rewrite -cats1 in SOME.
unfold R_list in *.
rewrite foldl_cat in SOME.
simpl in SOME.
unfold R_list_helper in SOME.
desf; rename H0 into EQ.
move: EQ => /eqP EQ.
rewrite eqseq_rcons in EQ.
move: EQ => /andP [_ /eqP EQ].
by inversion EQ.
Qed.
(* The response-time bounds computed using R_list are based on the per-task
fixed-point iteration. *)
Lemma R_list_rcons_response_time :
forall ts' hp_bounds tsk R,
R_list (rcons ts' tsk) = Some (rcons hp_bounds (tsk, R)) ->
R = per_task_rta tsk hp_bounds (max_steps tsk).
Proof.
intros ts hp_bounds tsk R SOME.
rewrite -cats1 in SOME.
unfold R_list in SOME.
rewrite foldl_cat in SOME.
simpl in SOME.
unfold R_list_helper in SOME.
desf; rename H0 into EQ; move: EQ => /eqP EQ.
rewrite eqseq_rcons in EQ; move: EQ => /andP [/eqP EQ1 /eqP EQ2].
by inversion EQ2; rewrite EQ1.
Qed.
(* If the analysis suceeds, the computed response-time bounds are no larger
than the deadline. *)
Lemma R_list_le_deadline :
forall ts' rt_bounds tsk R,
R_list ts' = Some rt_bounds ->
(tsk, R) \in rt_bounds ->
R <= task_deadline tsk.
Proof.
intros ts; induction ts as [| ts' tsk_lst] using last_ind.
{
intros rt_bounds tsk R SOME IN.
by inversion SOME; subst; rewrite in_nil in IN.
}
{
intros rt_bounds tsk_i R SOME IN.
destruct (lastP rt_bounds) as [|rt_bounds (tsk_lst', R_lst)];
first by rewrite in_nil in IN.
rewrite mem_rcons in_cons in IN; move: IN => /orP IN.
destruct IN as [LAST | FRONT].
{
move: LAST => /eqP LAST.
rewrite -cats1 in SOME.
unfold R_list in *.
rewrite foldl_cat in SOME.
simpl in SOME.
unfold R_list_helper in SOME.
desf; rename H0 into EQ.
move: EQ => /eqP EQ.
rewrite eqseq_rcons in EQ.
move: EQ => /andP [_ /eqP EQ].
inversion EQ; subst.
by apply Heq0.
}
{
apply IHts with (rt_bounds := rt_bounds); last by ins.
by apply R_list_rcons_prefix in SOME.
}
}
Qed.
(* If the analysis succeeds, the computed response-time bounds are no smaller
than the task cost. *)
Lemma R_list_ge_cost :
forall ts' rt_bounds tsk R,
R_list ts' = Some rt_bounds ->
(tsk, R) \in rt_bounds ->
R >= task_cost tsk.
Proof.
intros ts; induction ts as [| ts' tsk_lst] using last_ind.
{
intros rt_bounds tsk R SOME IN.
by inversion SOME; subst; rewrite in_nil in IN.
}
{
intros rt_bounds tsk_i R SOME IN.
destruct (lastP rt_bounds) as [|rt_bounds (tsk_lst', R_lst)];
first by rewrite in_nil in IN.
rewrite mem_rcons in_cons in IN; move: IN => /orP IN.
destruct IN as [LAST | FRONT].
{
move: LAST => /eqP LAST.
rewrite -cats1 in SOME.
unfold R_list in *.
rewrite foldl_cat in SOME.
simpl in SOME.
unfold R_list_helper in SOME.
desf; rename H0 into EQ.
move: EQ => /eqP EQ.
rewrite eqseq_rcons in EQ.
move: EQ => /andP [_ /eqP EQ].
inversion EQ; subst.
by destruct (max_steps tsk_lst');
[by apply leqnn | by apply leq_addr].
}
{
apply IHts with (rt_bounds := rt_bounds); last by ins.
by apply R_list_rcons_prefix in SOME.
}
}
Qed.
(* R_list contains a response-time bound for every tasks in the original task set. *)
Lemma R_list_non_empty :
forall ts' rt_bounds tsk,
R_list ts' = Some rt_bounds ->
(tsk \in ts' <->
exists R,
(tsk, R) \in rt_bounds).
Proof.
intros ts; induction ts as [| ts' tsk_lst] using last_ind.
{
intros rt_bounds tsk SOME.
inversion SOME; rewrite in_nil; split; first by ins.
by intro EX; des; rewrite in_nil in EX.
}
{
intros rt_bounds tsk_i SOME.
destruct (lastP rt_bounds) as [|rt_bounds (tsk_lst', R_lst)].
{
split; last first; intro EX; des; first by rewrite in_nil in EX.
unfold R_list in *.
rewrite -cats1 foldl_cat in SOME.
simpl in SOME.
unfold R_list_helper in *; desf; rename H0 into EQ.
destruct l; first by ins.
by rewrite rcons_cons in EQ; inversion EQ.
}
split.
{
intros IN; rewrite mem_rcons in_cons in IN; move: IN => /orP IN.
destruct IN as [LAST | FRONT].
{
move: LAST => /eqP LAST; subst tsk_i.
generalize SOME; apply R_list_rcons_task in SOME; subst tsk_lst'; intro SOME.
exists R_lst.
by rewrite mem_rcons in_cons; apply/orP; left.
}
{
apply R_list_rcons_prefix in SOME.
exploit (IHts rt_bounds tsk_i); [by ins | intro EX].
apply EX in FRONT; des.
by exists R; rewrite mem_rcons in_cons; apply/orP; right.
}
}
{
intro IN; des.
rewrite mem_rcons in_cons in IN; move: IN => /orP IN.
destruct IN as [LAST | FRONT].
{
move: LAST => /eqP LAST.
inversion LAST; subst tsk_i R; clear LAST.
apply R_list_rcons_task in SOME; subst.
by rewrite mem_rcons in_cons; apply/orP; left.
}
{
rewrite mem_rcons in_cons; apply/orP; right.
exploit (IHts rt_bounds tsk_i);
[by apply R_list_rcons_prefix in SOME | intro EX].
by apply EX; exists R.
}
}
}
Qed.
(* Short lemma about unfolding the iteration one step. *)
Lemma per_task_rta_fold :
forall tsk rt_bounds,
task_cost tsk +
div_floor (total_interference_bound_fp task_cost task_period tsk rt_bounds
(per_task_rta tsk rt_bounds (max_steps tsk)) higher_eq_priority) num_cpus
= per_task_rta tsk rt_bounds (max_steps tsk).+1.
Proof.
by done.
Qed.
End SimpleLemmas.
(* In this section, we prove that if the task set is sorted by priority,
the tasks in R_list are interfering tasks. *)
Section HighPriorityTasks.
(* Consider a list of previous tasks and a task tsk to be analyzed. *)
Variable ts_hp: taskset_of sporadic_task.
Variable tsk: sporadic_task.
(* Assume that the task set doesn't contain duplicates and is sorted by priority, ... *)
Hypothesis H_task_set_is_a_set: uniq (rcons ts_hp tsk).
Hypothesis H_task_set_is_sorted: sorted higher_eq_priority (rcons ts_hp tsk).
(* ...the priority order is transitive, ...*)
Hypothesis H_priority_transitive: transitive higher_eq_priority.
(* ... and that the response-time analysis succeeds. *)
Variable hp_bounds: seq task_with_response_time.
Variable R: time.
Hypothesis H_analysis_succeeds: R_list (rcons ts_hp tsk) = Some (rcons hp_bounds (tsk, R)).
(* Then, the tasks in the prefix of R_list are exactly interfering tasks
under FP scheduling.*)
Lemma R_list_unzip1 :
[seq tsk_hp <- rcons ts_hp tsk | is_interfering_task_fp higher_eq_priority tsk tsk_hp] =
unzip1 hp_bounds.
Proof.
rename H_priority_transitive into TRANS,
H_task_set_is_a_set into UNIQ,
H_analysis_succeeds into SOME,
H_task_set_is_sorted into SORT.
revert tsk hp_bounds R UNIQ SORT SOME.
induction ts_hp as [| ts_hp' tsk_lst] using last_ind.
{
intros tsk hp_bounds R UNIQ SORT SOME; simpl in *.
unfold is_interfering_task_fp.
rewrite eq_refl andbF.
destruct hp_bounds; first by ins.
unfold R_list in SOME; inversion SOME; desf.
by destruct l.
}
{
intros tsk hp_bounds R UNIQ SORTED SOME.
destruct (lastP hp_bounds) as [| hp_bounds (tsk_lst', R_lst)].
{
apply R_list_rcons_prefix in SOME.
unfold R_list in SOME.
rewrite -cats1 foldl_cat in SOME.
unfold R_list_helper in SOME.
inversion SOME; desf.
by destruct l.
}
generalize SOME; apply R_list_rcons_prefix, R_list_rcons_task in SOME;
subst tsk_lst'; intro SOME.
specialize (IHt tsk_lst hp_bounds R_lst).
rewrite filter_rcons in IHt.
unfold is_interfering_task_fp in IHt; rewrite eq_refl andbF in IHt.
assert (NOTHP: is_interfering_task_fp higher_eq_priority tsk tsk = false).
{
by unfold is_interfering_task_fp; rewrite eq_refl andbF.
} rewrite filter_rcons NOTHP; clear NOTHP.
assert (HP: is_interfering_task_fp higher_eq_priority tsk tsk_lst).
{
unfold is_interfering_task_fp; apply/andP; split.
{
apply order_sorted_rcons with (x := tsk_lst) in SORTED; try (by ins).
by rewrite mem_rcons in_cons; apply/orP; left.
}
{
rewrite 2!rcons_uniq mem_rcons in_cons negb_or in UNIQ.
move : UNIQ => /andP [/andP [UNIQ _] _].
by rewrite eq_sym in UNIQ.
}
} rewrite filter_rcons HP; clear HP.
unfold unzip1; rewrite map_rcons /=; f_equal.
assert (SHIFT: [seq tsk_hp <- ts_hp' | is_interfering_task_fp
higher_eq_priority tsk tsk_hp] =
[seq tsk_hp <- ts_hp' | is_interfering_task_fp
higher_eq_priority tsk_lst tsk_hp]).
{
apply eq_in_filter; red.
unfold is_interfering_task_fp; intros x INx.
rewrite 2!rcons_uniq mem_rcons in_cons negb_or in UNIQ.
move: UNIQ => /andP [/andP [NEQ NOTIN] /andP [NOTIN' UNIQ]].
destruct (x == tsk) eqn:EQtsk;
first by move: EQtsk => /eqP EQtsk; subst; rewrite INx in NOTIN.
destruct (x == tsk_lst) eqn:EQlst;
first by move: EQlst => /eqP EQlst; subst; rewrite INx in NOTIN'.
rewrite 2!andbT.
generalize SORTED; intro SORTED'.
apply order_sorted_rcons with (x0 := x) in SORTED; try (by ins);
last by rewrite mem_rcons in_cons; apply/orP; right.
rewrite SORTED.
apply sorted_rcons_prefix in SORTED'.
by apply order_sorted_rcons with (x0 := x) in SORTED'.
} rewrite SHIFT; clear SHIFT.
apply IHt.
by rewrite rcons_uniq in UNIQ; move: UNIQ => /andP [_ UNIQ].
by apply sorted_rcons_prefix in SORTED.
by apply R_list_rcons_prefix in SOME.
}
Qed.
End HighPriorityTasks.
(* In this section, we show that the fixed-point iteration converges. *)
Section Convergence.
(* Consider any valid set of higher-priority tasks. *)
Variable ts_hp: taskset_of sporadic_task.
Hypothesis valid_task_parameters:
valid_sporadic_taskset task_cost task_period task_deadline ts_hp.
(* Assume that the response-time analysis succeeds. *)
Variable rt_bounds: seq task_with_response_time.
Hypothesis H_test_succeeds: R_list ts_hp = Some rt_bounds.
(* Consider any task tsk to be analyzed. *)
Variable tsk: sporadic_task.
(* To simplify, let f denote the fixed-point iteration. *)
Let f := per_task_rta tsk rt_bounds.
(* Assume that f (max_steps tsk) is no larger than the deadline. *)
Hypothesis H_no_larger_than_deadline: f (max_steps tsk) <= task_deadline tsk.
(* First, we show that f is monotonically increasing. *)
Lemma bertogna_fp_comp_f_monotonic :
forall x1 x2, x1 <= x2 -> f x1 <= f x2.
Proof.
unfold valid_sporadic_taskset, is_valid_sporadic_task in *.
rename H_test_succeeds into SOME,
valid_task_parameters into VALID.
intros x1 x2 LEx; unfold f, per_task_rta.
apply fun_mon_iter_mon; [by ins | by ins; apply leq_addr |].
clear LEx x1 x2; intros x1 x2 LEx.
unfold div_floor, total_interference_bound_fp.
rewrite big_seq_cond [\sum_(i <- _ | let '(tsk_other, _) := i in
_ && (tsk_other != tsk))_]big_seq_cond.
rewrite leq_add2l leq_div2r // leq_sum //.
intros i; destruct (i \in rt_bounds) eqn:HP; last by rewrite andFb.
destruct i as [i R]; intros _.
have GE_COST := (R_list_ge_cost ts_hp rt_bounds i R).
have INts := (R_list_non_empty ts_hp rt_bounds i SOME).
destruct INts as [_ EX]; exploit EX; [by exists R | intro IN].
unfold interference_bound; simpl.
rewrite leq_min; apply/andP; split.
{
apply leq_trans with (n := W task_cost task_period i R x1); first by apply geq_minl.
specialize (VALID i IN); des.
by apply W_monotonic; try (by ins);
[by apply GE_COST | by apply leqnn].
}
{
apply leq_trans with (n := x1 - task_cost tsk + 1); first by apply geq_minr.
by rewrite leq_add2r leq_sub2r //.
}
Qed.
(* If the iteration converged at an earlier step, then it remains stable. *)
Lemma bertogna_fp_comp_f_converges_early :
(exists k, k < max_steps tsk /\ f k = f k.+1) ->
f (max_steps tsk) = f (max_steps tsk).+1.
Proof.
by intros EX; des; apply iter_fix with (k := k); [by done | by apply ltnW].
Qed.
(* Else, we derive a contradiction. *)
Section DerivingContradiction.
(* Assume instead that the iteration continued to diverge. *)
Hypothesis H_keeps_diverging:
forall k,
k < max_steps tsk -> f k != f k.+1.
(* By monotonicity, it follows that the value always increases. *)
Lemma bertogna_fp_comp_f_increases :
forall k,
k < max_steps tsk ->
f k < f k.+1.
Proof.
intros k LT.
rewrite ltn_neqAle; apply/andP; split.
by apply H_keeps_diverging.
by apply bertogna_fp_comp_f_monotonic, leqnSn.
Qed.
(* In the end, the response-time bound must exceed the deadline. Contradiction! *)
Lemma bertogna_fp_comp_rt_exceeds_deadline :
f (max_steps tsk) > task_deadline tsk.
Proof.
unfold max_steps.
induction (task_deadline tsk).
{
simpl.
admit.
}
admit.
Qed.
End DerivingContradiction.
(* Using the lemmas above, we prove the convergence of the iteration after max_steps. *)
Lemma per_task_rta_converges:
f (max_steps tsk) = f (max_steps tsk).+1.
Proof.
unfold valid_sporadic_taskset, is_valid_sporadic_task in *.
rename H_no_larger_than_deadline into LE.
(* Either f converges by the deadline or not. *)
destruct ([exists k in 'I_(max_steps tsk), f k == f k.+1]) eqn:EX.
{
move: EX => /exists_inP EX; destruct EX as [k _ ITERk].
apply bertogna_fp_comp_f_converges_early.
by exists k; split; [by apply ltn_ord | by apply/eqP].
}
(* If not, then we reach a contradiction *)
apply negbT in EX; rewrite negb_exists_in in EX.
move: EX => /forall_inP EX.
rewrite leqNgt in LE; move: LE => /negP LE.
exfalso; apply LE.
by apply bertogna_fp_comp_rt_exceeds_deadline.
Qed.
End Convergence.
Section MainProof.
(* Consider a task set ts. *)
Variable ts: taskset_of sporadic_task.
(* Assume that higher_eq_priority is a total order.
TODO: it doesn't have to be total over the entire domain, but
only within the task set.
But to weaken the hypothesis, we need to re-prove some lemmas
from ssreflect. *)
Hypothesis H_reflexive: reflexive higher_eq_priority.
Hypothesis H_transitive: transitive higher_eq_priority.
Hypothesis H_unique_priorities: antisymmetric higher_eq_priority.
Hypothesis H_total: total higher_eq_priority.
(* Assume the task set has no duplicates, ... *)
Hypothesis H_ts_is_a_set: uniq ts.
(* ...all tasks have valid parameters, ... *)
Hypothesis H_valid_task_parameters:
valid_sporadic_taskset task_cost task_period task_deadline ts.
(* ...restricted deadlines, ...*)
Hypothesis H_restricted_deadlines:
forall tsk, tsk \in ts -> task_deadline tsk <= task_period tsk.
(* ...and tasks are ordered by increasing priorities. *)
Hypothesis H_sorted_ts: sorted higher_eq_priority ts.
(* 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: JobIn arr_seq), job_task j \in ts.
(* ...they have valid parameters,...*)
Hypothesis H_valid_job_parameters:
forall (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 and unit
unit execution rate, where...*)
Variable rate: Job -> processor num_cpus -> nat.
Variable sched: schedule num_cpus arr_seq.
Hypothesis H_at_least_one_cpu :
num_cpus > 0.
Hypothesis H_rate_equals_one :
forall j cpu, rate j cpu = 1.
(* ...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 rate sched.
(* ...and do not execute in parallel. *)
Hypothesis H_no_parallelism:
jobs_dont_execute_in_parallel sched.
(* Assume the platform satisfies the global scheduling invariant. *)
Hypothesis H_global_scheduling_invariant:
FP_scheduling_invariant_holds job_cost job_task num_cpus rate sched ts higher_eq_priority.
Definition no_deadline_missed_by_task (tsk: sporadic_task) :=
task_misses_no_deadline job_cost job_deadline job_task rate sched tsk.
Definition no_deadline_missed_by_job :=
job_misses_no_deadline job_cost job_deadline rate sched.
(* In the following lemma, we prove that any response-time bound contained
in R_list is safe. The proof follows by induction on the task set:
Induction hypothesis: all higher-priority tasks have safe response-time bounds.
Inductive step: We prove that the response-time bound of the current task is safe.
Note that the inductive step is a direct application of the main Theorem from
bertogna_fp_theory.v.
The proof is only long because of the dozens of hypothesis that we need to supply,
so there's no clean way of breaking this down into small lemmas. *)
Lemma R_list_has_response_time_bounds :
forall rt_bounds tsk R,
R_list ts = Some rt_bounds ->
(tsk, R) \in rt_bounds ->
forall j : JobIn arr_seq,
job_task j = tsk ->
completed job_cost rate sched j (job_arrival j + R).
Proof.
rename H_valid_job_parameters into JOBPARAMS, H_valid_task_parameters into TASKPARAMS,
H_restricted_deadlines into RESTR, H_completed_jobs_dont_execute into COMP,
H_jobs_must_arrive_to_execute into MUSTARRIVE, H_sorted_ts into SORT,
H_global_scheduling_invariant into INVARIANT, H_transitive into TRANS,
H_unique_priorities into UNIQ, H_total into TOTAL,
H_all_jobs_from_taskset into ALLJOBS, H_ts_is_a_set into SET.
clear ALLJOBS.
unfold fp_schedulable, R_list in *.
induction ts as [| ts' tsk_i IH] using last_ind.
{
intros rt_bounds tsk R SOME IN.
by inversion SOME; subst; rewrite in_nil in IN.
}
{
intros rt_bounds tsk R SOME IN j JOBj.
destruct (lastP rt_bounds) as [| hp_bounds (tsk_lst, R_lst)];
first by rewrite in_nil in IN.
rewrite mem_rcons in_cons in IN; move: IN => /orP IN.
destruct IN as [LAST | BEGINNING]; last first.
{
apply IH with (rt_bounds := hp_bounds) (tsk := tsk); try (by ins).
by rewrite rcons_uniq in SET; move: SET => /andP [_ SET].
by ins; red; ins; apply TASKPARAMS; rewrite mem_rcons in_cons; apply/orP; right.
by ins; apply RESTR; rewrite mem_rcons in_cons; apply/orP; right.
by apply sorted_rcons_prefix in SORT.
{
intros tsk0 j0 t IN0 JOB0 BACK0.
exploit (INVARIANT tsk0 j0 t); try (by ins);
[by rewrite mem_rcons in_cons; apply/orP; right | intro INV].
rewrite -cats1 count_cat /= in INV.
unfold is_interfering_task_fp in INV.
generalize SOME; apply R_list_rcons_task in SOME; subst tsk_i; intro SOME.
assert (HP: higher_eq_priority tsk_lst tsk0 = false).
{
apply order_sorted_rcons with (x := tsk0) in SORT; [|by ins | by ins].
apply negbTE; apply/negP; unfold not; intro BUG.
exploit UNIQ; [by apply/andP; split; [by apply SORT | by ins] | intro EQ].
by rewrite rcons_uniq -EQ IN0 in SET.
}
by rewrite HP 2!andFb 2!addn0 in INV.
}
by apply R_list_rcons_prefix in SOME.
}
{
move: LAST => /eqP LAST.
inversion LAST as [[EQ1 EQ2]].
rewrite -> EQ1 in *; rewrite -> EQ2 in *; clear EQ1 EQ2 LAST.
generalize SOME; apply R_list_rcons_task in SOME; subst tsk_i; intro SOME.
generalize SOME; apply R_list_rcons_prefix in SOME; intro SOME'.
have BOUND := bertogna_cirinei_response_time_bound_fp.
unfold is_response_time_bound_of_task in BOUND.
apply BOUND with (task_cost := task_cost) (task_period := task_period) (task_deadline := task_deadline) (job_deadline := job_deadline) (job_task := job_task) (tsk := tsk_lst)
(ts := rcons ts' tsk_lst) (hp_bounds := hp_bounds)
(higher_eq_priority := higher_eq_priority); clear BOUND; try (by ins).
by rewrite mem_rcons in_cons eq_refl orTb.
by apply R_list_unzip1 with (R := R_lst).
{
intros hp_tsk R0 HP j0 JOB0.
apply IH with (rt_bounds := hp_bounds) (tsk := hp_tsk); try (by ins).
by rewrite rcons_uniq in SET; move: SET => /andP [_ SET].
by red; ins; apply TASKPARAMS; rewrite mem_rcons in_cons; apply/orP; right.
by ins; apply RESTR; rewrite mem_rcons in_cons; apply/orP; right.
by apply sorted_rcons_prefix in SORT.
{
intros tsk0 j1 t IN0 JOB1 BACK0.
exploit (INVARIANT tsk0 j1 t); try (by ins);
[by rewrite mem_rcons in_cons; apply/orP; right | intro INV].
rewrite -cats1 count_cat /= addn0 in INV.
unfold is_interfering_task_fp in INV.
assert (NOINTERF: higher_eq_priority tsk_lst tsk0 = false).
{
apply order_sorted_rcons with (x := tsk0) in SORT; [|by ins | by ins].
apply negbTE; apply/negP; unfold not; intro BUG.
exploit UNIQ; [by apply/andP; split; [by apply BUG | by ins] | intro EQ].
by rewrite rcons_uniq EQ IN0 in SET.
}
by rewrite NOINTERF 2!andFb addn0 in INV.
}
}
by ins; apply R_list_ge_cost with (ts' := ts') (rt_bounds := hp_bounds).
by ins; apply R_list_le_deadline with (ts' := ts') (rt_bounds := hp_bounds).
{
rewrite [R_lst](R_list_rcons_response_time ts' hp_bounds tsk_lst); last by ins.
rewrite per_task_rta_fold; apply per_task_rta_converges.
apply R_list_le_deadline with (ts' := rcons ts' tsk_lst)
(rt_bounds := rcons hp_bounds (tsk_lst, R_lst));
first by apply SOME'.
rewrite mem_rcons in_cons; apply/orP; left; apply/eqP.
f_equal; symmetry.
by apply R_list_rcons_response_time with (ts' := ts').
}
}
}
Qed.
(* Finally, if the schedulability test suceeds, ...*)
Hypothesis H_test_succeeds: fp_schedulable ts.
(*..., no task misses its deadline,... *)
Theorem taskset_schedulable_by_fp_rta :
forall tsk, tsk \in ts -> no_deadline_missed_by_task tsk.
Proof.
unfold no_deadline_missed_by_task, task_misses_no_deadline,
job_misses_no_deadline, completed,
fp_schedulable,
valid_sporadic_job in *.
rename H_valid_job_parameters into JOBPARAMS,
H_valid_task_parameters into TASKPARAMS,
H_restricted_deadlines into RESTR,
H_completed_jobs_dont_execute into COMP,
H_jobs_must_arrive_to_execute into MUSTARRIVE,
H_global_scheduling_invariant into INVARIANT,
H_sorted_ts into SORT,
H_transitive into TRANS,
H_unique_priorities into UNIQ,
H_total into TOTAL,
H_all_jobs_from_taskset into ALLJOBS,
H_test_succeeds into TEST.
move => tsk INtsk j JOBtsk.
have RLIST := (R_list_has_response_time_bounds).
have NONEMPTY := (R_list_non_empty ts).
have DL := (R_list_le_deadline ts).
destruct (R_list ts) as [rt_bounds |]; last by ins.
exploit (NONEMPTY rt_bounds tsk); [by ins | intros [EX _]; specialize (EX INtsk); des].
exploit (RLIST rt_bounds tsk R); [by ins | by ins | by apply JOBtsk | intro COMPLETED].
exploit (DL rt_bounds tsk R); [by ins | by ins | clear DL; intro DL].
rewrite eqn_leq; apply/andP; split; first by apply cumulative_service_le_job_cost.
apply leq_trans with (n := service rate 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.
(* ..., and the schedulability test yields safe response-time
bounds for each task. *)
Theorem fp_schedulability_test_yields_response_time_bounds :
forall tsk,
tsk \in ts ->
exists R,
R <= task_deadline tsk /\
forall (j: JobIn arr_seq),
job_task j = tsk ->
completed job_cost rate sched j (job_arrival j + R).
Proof.
intros tsk IN.
unfold fp_schedulable in *.
have TASKS := R_list_non_empty ts.
have BOUNDS := (R_list_has_response_time_bounds).
have DL := (R_list_le_deadline ts).
destruct (R_list ts) as [rt_bounds |]; last by ins.
exploit (TASKS rt_bounds tsk); [by ins | clear TASKS; intro EX].
destruct EX as [EX _]; specialize (EX IN); des.
exists R; split.
by apply DL with (rt_bounds0 := rt_bounds).
by ins; apply (BOUNDS rt_bounds tsk).
Qed.
(* For completeness, since all jobs of the arrival sequence
are spawned by the task set, we also conclude that no job in
the schedule misses its deadline. *)
Theorem jobs_schedulable_by_fp_rta :
forall (j: JobIn arr_seq), no_deadline_missed_by_job j.
Proof.
intros j.
have SCHED := taskset_schedulable_by_fp_rta.
unfold no_deadline_missed_by_task, task_misses_no_deadline in *.
apply SCHED with (tsk := job_task j); last by done.
by apply H_all_jobs_from_taskset.
Qed.
End MainProof.
End Analysis.
End ResponseTimeIterationFP.