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Commit f7a31570 authored by Björn Brandenburg's avatar Björn Brandenburg
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sporadic tasks analysis: remove unnecessary offset parameter 

Arguments about sporadic tasks do not reason about offsets as future
arrival times are unknown.
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analysis/facts/sporadic.v
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......@@ -3,68 +3,61 @@ Require Export prosa.analysis.facts.job_index.
(** * The Sporadic Model *)
(** In this section we prove a few basic properties involving
(** In this section we prove a few basic properties involving
job indices in context of the sporadic model. *)
Section SporadicModelIndexLemmas.
(** Consider sporadic tasks with an offset ... *)
Context {Task : TaskType}.
Context `{TaskOffset Task}.
Context `{TaskMaxInterArrival Task}.
Context `{SporadicModel Task}.
(** Consider sporadic tasks ... *)
Context {Task : TaskType} `{SporadicModel Task}.
(** ... and any type of jobs associated with these tasks. *)
Context {Job : JobType}.
Context `{JobTask Job Task}.
Context `{JobArrival Job}.
Context {Job : JobType} `{JobTask Job Task} `{JobArrival Job}.
(** Consider any unique arrival sequence with consistent arrivals, ... *)
Variable arr_seq : arrival_sequence Job.
Hypothesis H_consistent_arrivals: consistent_arrival_times arr_seq.
Hypothesis H_uniq_arrseq: arrival_sequence_uniq arr_seq.
(** ... and any sporadic task [tsk] that is to be analyzed. *)
Variable tsk : Task.
Hypothesis H_sporadic_model: respects_sporadic_task_model arr_seq tsk.
Hypothesis H_valid_inter_min_arrival: valid_task_min_inter_arrival_time tsk.
(** Consider any two jobs from the arrival sequence that stem
from task [tsk]. *)
(** Consider any two jobs from the arrival sequence that stem
from task [tsk]. *)
Variable j1 : Job.
Variable j2 : Job.
Hypothesis H_j1_from_arrseq: arrives_in arr_seq j1.
Hypothesis H_j2_from_arrseq: arrives_in arr_seq j2.
Hypothesis H_j1_task: job_task j1 = tsk.
Hypothesis H_j2_task: job_task j2 = tsk.
(** We first show that for any two jobs [j1] and [j2], [j2] arrives after [j1]
provided [job_index] of [j2] strictly exceeds the [job_index] of [j1]. *)
(** We first show that for any two jobs [j1] and [j2], [j2] arrives after [j1]
provided [job_index] of [j2] strictly exceeds the [job_index] of [j1]. *)
Lemma lower_index_implies_earlier_arrival:
job_index arr_seq j1 < job_index arr_seq j2 ->
job_arrival j1 < job_arrival j2.
Proof.
intro IND.
move: (H_sporadic_model j1 j2) => SPORADIC; feed_n 6 SPORADIC => //.
- rewrite -> diff_jobs_iff_diff_indices => //; eauto.
+ now lia.
+ now rewrite H_j1_task.
move=> LT_IND.
move: (H_sporadic_model j1 j2) => SPORADIC; feed_n 6 SPORADIC => //.
- rewrite -> diff_jobs_iff_diff_indices => //; eauto; first by lia.
by subst.
- apply (index_lte_implies_arrival_lte arr_seq); try eauto.
now rewrite H_j1_task.
by subst.
- have POS_IA : task_min_inter_arrival_time tsk > 0 by auto.
now lia.
by lia.
Qed.
End SporadicModelIndexLemmas.
(** ** Different jobs have different arrival times. *)
(** ** Different jobs have different arrival times. *)
Section DifferentJobsImplyDifferentArrivalTimes.
(** Consider sporadic tasks with an offset ... *)
(** Consider sporadic tasks ... *)
Context {Task : TaskType}.
Context `{TaskOffset Task}.
Context `{TaskMaxInterArrival Task}.
Context `{SporadicModel Task}.
(** ... and any type of jobs associated with these tasks. *)
Context {Job : JobType}.
Context `{JobTask Job Task}.
......@@ -74,7 +67,7 @@ Section DifferentJobsImplyDifferentArrivalTimes.
Variable arr_seq : arrival_sequence Job.
Hypothesis H_consistent_arrivals: consistent_arrival_times arr_seq.
Hypothesis H_uniq_arrseq: arrival_sequence_uniq arr_seq.
(** ... and any sporadic task [tsk] that is to be analyzed. *)
Variable tsk : Task.
Hypothesis H_sporadic_model: respects_sporadic_task_model arr_seq tsk.
......@@ -98,7 +91,7 @@ Section DifferentJobsImplyDifferentArrivalTimes.
rewrite -> diff_jobs_iff_diff_indices in UNEQ => //; eauto; last by rewrite H_j1_task.
move /neqP: UNEQ; rewrite neq_ltn => /orP [LT|LT].
all: try ( now apply lower_index_implies_earlier_arrival with (tsk0 := tsk) in LT => //; lia ) ||
now apply lower_index_implies_earlier_arrival with (tsk := tsk) in LT => //; lia.
now apply lower_index_implies_earlier_arrival with (tsk := tsk) in LT => //; lia.
Qed.
(** We prove a stronger version of the above lemma by showing
......@@ -115,23 +108,18 @@ Section DifferentJobsImplyDifferentArrivalTimes.
move /neqP: NEQ; rewrite neq_ltn => /orP [LT|LT].
all: try ( now apply lower_index_implies_earlier_arrival with (tsk0 := tsk) in LT => //; lia ) || now apply lower_index_implies_earlier_arrival with (tsk := tsk) in LT => //; lia.
Qed.
End DifferentJobsImplyDifferentArrivalTimes.
(** In this section we prove a few properties regarding task arrivals
in context of the sporadic task model. *)
Section SporadicArrivals.
(** Consider sporadic tasks with an offset ... *)
Context {Task : TaskType}.
Context `{TaskOffset Task}.
Context `{SporadicModel Task}.
Context `{TaskMaxInterArrival Task}.
(** Consider sporadic tasks ... *)
Context {Task : TaskType} `{SporadicModel Task} `{TaskMaxInterArrival Task}.
(** ... and any type of jobs associated with these tasks. *)
Context {Job : JobType}.
Context `{JobTask Job Task}.
Context `{JobArrival Job}.
Context {Job : JobType} `{JobTask Job Task} `{JobArrival Job}.
(** Consider any unique arrival sequence with consistent arrivals, ... *)
Variable arr_seq : arrival_sequence Job.
......@@ -140,15 +128,14 @@ Section SporadicArrivals.
(** ... and any sporadic task [tsk] to be analyzed. *)
Variable tsk : Task.
(** Assume all tasks have valid minimum inter-arrival times, valid offsets, and respect the
(** Assume all tasks have valid minimum inter-arrival times, valid offsets, and respect the
sporadic task model. *)
Hypothesis H_sporadic_model: respects_sporadic_task_model arr_seq tsk.
Hypothesis H_valid_inter_min_arrival: valid_task_min_inter_arrival_time tsk.
Hypothesis H_valid_offset: valid_offset arr_seq tsk.
(** Consider any two jobs from the arrival sequence that stem
from task [tsk]. *)
(** Consider any two jobs from the arrival sequence that stem
from task [tsk]. *)
Variable j1 j2 : Job.
Hypothesis H_j1_from_arrival_sequence: arrives_in arr_seq j1.
Hypothesis H_j2_from_arrival_sequence: arrives_in arr_seq j2.
......@@ -180,7 +167,7 @@ Section SporadicArrivals.
(** We show that no jobs of the task [tsk] other than [j1] arrive at
the same time as [j1], and thus the task arrivals at [job arrival j1]
consists only of job [j1]. *)
consists only of job [j1]. *)
Lemma only_j_in_task_arrivals_at_j:
task_arrivals_at_job_arrival arr_seq j1 = [::j1].
Proof.
......@@ -190,7 +177,7 @@ Section SporadicArrivals.
by intros; now destruct (size seq) as [ | [ | ]]; try auto.
move: SIZE_CASE => [Z|[ONE|GTONE]].
- apply size0nil in Z.
now rewrite Z in J_IN_FILTER.
now rewrite Z in J_IN_FILTER.
- repeat (destruct seq; try by done).
rewrite mem_seq1 in J_IN_FILTER; move : J_IN_FILTER => /eqP J1_S.
now rewrite J1_S.
......@@ -202,35 +189,35 @@ Section SporadicArrivals.
(** We show that no jobs of the task [tsk] other than [j1] arrive at
the same time as [j1], and thus the task arrivals at [job arrival j1]
consists only of job [j1]. *)
consists only of job [j1]. *)
Lemma only_j_at_job_arrival_j:
forall t,
job_arrival j1 = t ->
task_arrivals_at arr_seq tsk t = [::j1].
task_arrivals_at arr_seq tsk t = [::j1].
Proof.
intros t ARR.
rewrite -ARR.
specialize (only_j_in_task_arrivals_at_j) => J_AT.
now rewrite /task_arrivals_at_job_arrival H_j1_task in J_AT.
Qed.
(** We show that a job [j1] is the first job that arrives
in task arrivals at [job_arrival j1] by showing that the
(** We show that a job [j1] is the first job that arrives
in task arrivals at [job_arrival j1] by showing that the
index of job [j1] in [task_arrivals_at_job_arrival arr_seq j1] is 0. *)
Lemma index_j_in_task_arrivals_at:
index j1 (task_arrivals_at_job_arrival arr_seq j1) = 0.
Proof.
now rewrite only_j_in_task_arrivals_at_j //= eq_refl.
Qed.
(** We observe that for any job [j] the arrival time of [prev_job j] is
(** We observe that for any job [j] the arrival time of [prev_job j] is
strictly less than the arrival time of [j] in context of periodic tasks. *)
Lemma prev_job_arr_lt:
job_index arr_seq j1 > 0 ->
job_arrival (prev_job arr_seq j1) < job_arrival j1.
Proof.
intros IND.
have PREV_ARR_LTE : job_arrival (prev_job arr_seq j1) <= job_arrival j1 by apply prev_job_arr_lte => //.
have PREV_ARR_LTE : job_arrival (prev_job arr_seq j1) <= job_arrival j1 by apply prev_job_arr_lte => //.
rewrite ltn_neqAle; apply /andP.
split => //; apply /eqP.
try ( apply uneq_job_uneq_arr with (arr_seq0 := arr_seq) (tsk0 := job_task j1) => //; try by rewrite H_j1_task ) ||
......@@ -242,8 +229,8 @@ Section SporadicArrivals.
now lia.
Qed.
(** We show that task arrivals at [job_arrival j1] is the
same as task arrivals that arrive between [job_arrival j1]
(** We show that task arrivals at [job_arrival j1] is the
same as task arrivals that arrive between [job_arrival j1]
and [job_arrival j1 + 1]. *)
Lemma task_arrivals_at_as_task_arrivals_between:
task_arrivals_at_job_arrival arr_seq j1 = task_arrivals_between arr_seq tsk (job_arrival j1) (job_arrival j1).+1.
......@@ -251,9 +238,9 @@ Section SporadicArrivals.
rewrite /task_arrivals_at_job_arrival /task_arrivals_at /task_arrivals_between /arrivals_between.
now rewrite big_nat1 H_j1_task.
Qed.
(** We show that the task arrivals up to the previous job [j1] concatenated with
the sequence [::j1] (the sequence containing only the job [j1]) is same as
(** We show that the task arrivals up to the previous job [j1] concatenated with
the sequence [::j1] (the sequence containing only the job [j1]) is same as
task arrivals up to [job_arrival j1]. *)
Lemma prev_job_cat:
job_index arr_seq j1 > 0 ->
......@@ -268,5 +255,5 @@ Section SporadicArrivals.
rewrite no_jobs_between_consecutive_jobs => //.
now rewrite cat0s H_j1_task.
Qed.
End SporadicArrivals.
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