Commit fbcbb997 authored by Björn Brandenburg's avatar Björn Brandenburg

address spell-checking issues in EDF optimality proof

parent 22a76ba8
......@@ -7,9 +7,9 @@ Require Export rt.restructuring.analysis.facts.readiness.basic.
(** This file contains the main argument of the EDF optimality proof,
starting with an analysis of the individual functions that drive
the "EDF-ication" of a given reference schedule and ending with
the proofs of individual properties of the obtained EDF
schedule. *)
the piece-wise transformation of a given reference schedule in an
EDF schedule, and ending with proofs of individual properties of
the obtained EDF schedule. *)
(** Throughout this file, we assume ideal uniprocessor schedules. *)
Require Import rt.restructuring.model.processor.ideal.
......@@ -170,7 +170,7 @@ End FindSwapCandidateFacts.
(** In the next section, we analyze properties of [make_edf_at], which
we abbreviate as "mea" in the following. *)
we abbreviate as "[mea]" in the following. *)
Section MakeEDFAtFacts.
(** For any given type of jobs... *)
......@@ -279,13 +279,14 @@ Section MakeEDFAtFacts.
(** Next comes a big step in the optimality proof: we observe that
[make_edf_at] indeed ensures that [EDF_at] holds at time [t_edf] in
sched'. As this is a larger argument, we proceed by case analysis and
[sched']. As this is a larger argument, we proceed by case analysis and
first establish a couple of helper lemmas in the following section. *)
Section GuaranteeCaseAnalysis.
(** Let j_orig denote the job scheduled in sched at time t_edf, let j_edf
denote the job scheduled in sched' at time t_edf, and let j' denote any
job scheduled in sched' at some time t' after t_edf... *)
(** Let [j_orig] denote the job scheduled in [sched] at time
[t_edf], let [j_edf] denote the job scheduled in [sched'] at
time [t_edf], and let [j'] denote any job scheduled in
[sched'] at some time [t'] after [t_edf]... *)
Variable j_orig j_edf j': Job.
Variable t': instant.
......@@ -295,18 +296,18 @@ Section MakeEDFAtFacts.
Hypothesis H_sched_edf: scheduled_at sched' j_edf t_edf.
Hypothesis H_sched': scheduled_at sched' j' t'.
(** ... and that arrives before time t_edf. *)
(** ... and that arrives before time [t_edf]. *)
Hypothesis H_arrival_j' : job_arrival j' <= t_edf.
(** We begin by observing three simple facts that will be used repeatedly in
the case analysis. *)
(** First, the deadline of j_orig is later than t_edf. *)
(** First, the deadline of [j_orig] is later than [t_edf]. *)
Fact mea_guarantee_dl_orig: t_edf < job_deadline j_orig.
Proof. by apply (scheduled_job_in_sched_has_later_deadline j_orig t_edf H_sched_orig). Qed.
(** Second, by the definition of sched', j_edf is scheduled in sched at the time
returned by [find_swap_candidate]. *)
(** Second, by the definition of [sched'], [j_edf] is scheduled in
[sched] at the time returned by [find_swap_candidate]. *)
Fact mea_guarantee_fsc_is_j_edf: sched (find_swap_candidate sched t_edf j_orig) = Some j_edf.
Proof.
move: (H_sched_orig). rewrite scheduled_at_def => /eqP SCHED.
......@@ -317,7 +318,8 @@ Section MakeEDFAtFacts.
- by rewrite ifT // => /eqP.
Qed.
(** Third, the deadline of j_edf is no later than the deadline of j_orig. *)
(** Third, the deadline of [j_edf] is no later than the deadline
of [j_orig]. *)
Fact mea_guarantee_deadlines: job_deadline j_edf <= job_deadline j_orig.
Proof.
apply: (fsc_no_later_deadline sched _ _ t_edf) => //.
......@@ -327,11 +329,12 @@ Section MakeEDFAtFacts.
(** With the setup in place, we are now ready to begin the case analysis. *)
(** First, we consider the simpler case where t' is no earlier than the
deadline of j_orig. This case is simpler because t' being no earlier
than j_orig's deadline implies that j' has deadline no earlier than
j_orig (since no scheduled job in sched misses a deadline), which in
turn has a deadline no earlier than j_edf. *)
(** First, we consider the simpler case where [t'] is no earlier
than the deadline of [j_orig]. This case is simpler because
[t'] being no earlier than [j_orig]'s deadline implies that
[j'] has deadline no earlier than [j_orig] (since no scheduled
job in [sched] misses a deadline), which in turn has a
deadline no earlier than [j_edf]. *)
Lemma mea_guarantee_case_t'_past_deadline:
job_deadline j_orig <= t' ->
job_deadline j_edf <= job_deadline j'.
......@@ -343,8 +346,8 @@ Section MakeEDFAtFacts.
by apply ltnW.
Qed.
(** Next, we consider the more difficult case, where t' is before the
deadline of j_orig. *)
(** Next, we consider the more difficult case, where [t'] is
before the deadline of [j_orig]. *)
Lemma mea_guarantee_case_t'_before_deadline:
t' < job_deadline j_orig ->
job_deadline j_edf <= job_deadline j'.
......@@ -378,8 +381,9 @@ Section MakeEDFAtFacts.
End GuaranteeCaseAnalysis.
(** Finally, putting the preceding case analysis together, we obtain the
result that [make_edf_at] establishes [EDF_at] at time [t_edf]. *)
(** Finally, putting the preceding cases together, we obtain the
result that [make_edf_at] establishes [EDF_at] at time
[t_edf]. *)
Lemma make_edf_at_guarantee:
EDF_at sched' t_edf.
Proof.
......@@ -394,8 +398,8 @@ Section MakeEDFAtFacts.
by apply: (mea_guarantee_case_t'_past_deadline j_orig j_edf j' t').
Qed.
(** We observe that [make_edf_at] maintains the property that jobs must arrive
to execute. *)
(** We observe that [make_edf_at] maintains the property that jobs
must arrive to execute. *)
Lemma mea_jobs_must_arrive:
jobs_must_arrive_to_execute sched'.
Proof.
......@@ -535,7 +539,7 @@ Section EDFPrefixFacts.
(** Consider any point in time, denoted [horizon], and... *)
Variable horizon: instant.
(** ...let [sched'] denote the schedule obtained by EDF-ifying
(** ...let [sched'] denote the schedule obtained by transforming
[sched] up to the horizon. *)
Let sched' := edf_transform_prefix sched horizon.
......@@ -707,8 +711,8 @@ Section EDFPrefixInclusion.
End EDFPrefixInclusion.
(** In the following section, we finally establish properties of the overall
EDF-ication operation [edf_transform]. *)
(** In the following section, we finally establish properties of the
overall EDF transformation[edf_transform]. *)
Section EDFTransformFacts.
(** For any given type of jobs... *)
......
......@@ -21,7 +21,7 @@ Section ReplaceAtFacts.
(** ...and a replacement state [state]. *)
Variable nstate: PState.
(** In the following, let sched' denote the schedule with replacement at time
(** In the following, let [sched'] denote the schedule with replacement at time
t'. *)
Let sched' := replace_at sched t' nstate.
......@@ -36,8 +36,8 @@ Section ReplaceAtFacts.
by move/eqP in TT; rewrite TT in DIFF; contradiction.
Qed.
(** As a result, the service in intervals that do not intersect with t' is
invariant, too. *)
(** As a result, the service in intervals that do not intersect with
[t'] is invariant, too. *)
Lemma service_at_other_times_invariant:
forall t1 t2,
t2 <= t' \/ t' < t1 ->
......@@ -54,10 +54,10 @@ Section ReplaceAtFacts.
apply /andP; by split.
Qed.
(** Next, we consider the amount of service received in intervals that do span
across the replacement point. We can relate the service received in the
original and the new schedules by adding the service in the respective
"missing" state... *)
(** Next, we consider the amount of service received in intervals
that do span across the replacement point. We can relate the
service received in the original and the new schedules by adding
the service in the respective "missing" state... *)
Lemma service_delta:
forall t1 t2,
t1 <= t' < t2 ->
......@@ -82,10 +82,11 @@ Section ReplaceAtFacts.
service_during sched j t1 t2 + service_at sched' j t' - service_at sched j t'.
Proof. move => t1 t2 ORDER j. by rewrite service_delta// addnK. Qed.
(** As another simple invariant (useful for case analysis), we observe that if
a job is scheduled neither in the replaced nor in the new state, then at
any time it receives exactly the same amount of service in the new
schedule with replacements as in the original schedule. *)
(** As another simple invariant (useful for case analysis), we
observe that if a job is scheduled neither in the replaced nor
in the new state, then at any time it receives exactly the same
amount of service in the new schedule with replacements as in
the original schedule. *)
Lemma service_at_of_others_invariant (j: Job):
~~ scheduled_in j (sched' t') ->
~~ scheduled_in j (sched t') ->
......@@ -99,9 +100,10 @@ Section ReplaceAtFacts.
- rewrite rest_of_schedule_invariant//.
Qed.
(** Based on the previous observation, we can trivially lift the invariant
that jobs not involved in the replacement receive equal service to the
cumulative service received in any interval. *)
(** Based on the previous observation, we can trivially lift the
invariant that jobs not involved in the replacement receive
equal service to the cumulative service received in any
interval. *)
Corollary service_during_of_others_invariant (j: Job):
~~ scheduled_in j (sched' t') ->
~~ scheduled_in j (sched t') ->
......
......@@ -298,7 +298,7 @@ Section SwappedScheduleProperties.
[t1] and [t2] have been swapped. *)
Let sched' := swapped sched t1 t2.
(** First, we observe that if jobs never accomulate more service than
(** First, we observe that if jobs never accumulate more service than
required, then that's still the case after the swap. *)
Lemma swapped_service_bound:
(forall j t, service sched j t <= job_cost j) ->
......
Require Export rt.restructuring.analysis.transform.prefix.
Require Export rt.restructuring.analysis.transform.swap.
(** In this file we define the "EDF-ification" of a schedule, the
operation at the core of the EDF optimality proof. *)
(** In this file we define the EDF transformation of a schedule, which turns a
(finite prefix of a) schedule into an EDF schedule. This operation is at
the core of the EDF optimality proof. *)
Section EDFTransformation.
(** Consider any given type of jobs... *)
......@@ -12,37 +13,34 @@ Section EDFTransformation.
Let PState := ideal.processor_state Job.
Let SchedType := schedule (PState).
(** We say that a state s1 "has an earlier or equal deadline" than a
state s2 if the job scheduled in state s1 has has an earlier or
equal deadline than the job scheduled in state s2. This function
is never used on idle states, so the default values are
irrelevant. *)
(** We say that a state [s1] "has an earlier or equal deadline" than a state
[s2] if the job scheduled in state [s1] has has an earlier or equal
deadline than the job scheduled in state [s2]. This function is never
used on idle states, so the default values are irrelevant. *)
Definition earlier_deadline (s1 s2: PState) :=
(oapp job_deadline 0 s1) <= (oapp job_deadline 0 s2).
(** We say that a state is relevant (for the purpose of the
transformation) if it is not idle and if the job scheduled in it
has arrived prior to some given reference time. *)
(** We say that a state is relevant (for the purpose of the transformation)
if it is not idle and if the job scheduled in it has arrived prior to
some given reference time. *)
Definition relevant_pstate (reference_time: instant) (s: PState) :=
match s with
| None => false
| Some j' => job_arrival j' <= reference_time
end.
(** Next, we define a central element of the "EDF-ification"
procedure: given a job scheduled at a time t1, find a later time
t2 before the job's deadlineat which a job with an
earlier-or-equal deadline is scheduled. In other words, find a
job that causes the [EDF_at] property to be violated at time
t1. *)
(** Next, we define a central element of the EDF transformation procedure:
given a job scheduled at a time [t1], find a later time [t2] before the
job's deadline at which a job with an earlier-or-equal deadline is
scheduled. In other words, find a job that causes the [EDF_at] property
to be violated at time [t1]. *)
Definition find_swap_candidate (sched: SchedType) (t1: instant) (j: Job) :=
if search_arg sched (relevant_pstate t1) earlier_deadline t1 (job_deadline j) is Some t
then t
else 0.
(** The point-wise EDF-ification procedure: given a schedule and a
time t1, ensure that the schedule satisfies [EDF_at] at time
t1. *)
(** The point-wise EDF transformation procedure: given a schedule and a time
[t1], ensure that the schedule satisfies [EDF_at] at time [t1]. *)
Definition make_edf_at (sched: SchedType) (t1: instant): SchedType :=
match sched t1 with
| None => sched (* leave idle instants alone *)
......@@ -57,10 +55,10 @@ Section EDFTransformation.
Definition edf_transform_prefix (sched: SchedType) (horizon: instant): SchedType :=
prefix_map sched make_edf_at horizon.
(** Finally, a full EDF schedule (i.e., one that satisfies [EDF_at]
at any time) is obtained by first computing an EDF prefix up to
and including the requested time t, and by then looking at the
last point of the prefix. *)
(** Finally, a full EDF schedule (i.e., one that satisfies [EDF_at] at any
time) is obtained by first computing an EDF prefix up to and including
the requested time [t], and by then looking at the last point of the
prefix. *)
Definition edf_transform (sched: SchedType) (t: instant): ideal.processor_state Job :=
let
edf_prefix := edf_transform_prefix sched t.+1
......
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