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Commit 839126b3 authored by Björn Brandenburg's avatar Björn Brandenburg
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improve the EDF optimality proof by reasoning about prefixes 

Don't always "drill to the bottom" and unfold to sums; instead
explicitly make use of the fact that the EDF proof reasons about
finite identical prefixes, which allows staying at a semantically
higher level in the proof.

While at it, switch the file to using the preferred `now` tactical
when closing out proofs (rather than `by`) to avoid emacs indentation
issues.
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analysis/facts/transform/edf_opt.v
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......@@ -52,7 +52,7 @@ Section FindSwapCandidateFacts.
move: H_not_idle. rewrite scheduled_at_def => /eqP ->.
rewrite /relevant_pstate -/(has_arrived j1 t1).
move: (H_jobs_must_arrive_to_execute j1 t1) => SCHED_ARR.
by apply SCHED_ARR.
now apply SCHED_ARR.
Qed.
(** Since [t1] is relevant, we conclude that a search for a relevant
......@@ -73,7 +73,7 @@ Section FindSwapCandidateFacts.
search_arg sched (relevant_pstate t1) earlier_deadline t1 (job_deadline j1) = Some (find_swap_candidate sched t1 j1).
Proof.
move: fsc_search_successful => [t FOUND].
by rewrite /find_swap_candidate FOUND.
now rewrite /find_swap_candidate FOUND.
Qed.
(** There is a job that is scheduled at the time that
......@@ -90,7 +90,7 @@ Section FindSwapCandidateFacts.
exists j'. split => //.
rewrite /find_swap_candidate FOUND.
rewrite scheduled_at_def //.
by apply /eqP.
now apply /eqP.
Qed.
(** Since we are considering a uniprocessor model, only one job is
......@@ -104,7 +104,7 @@ Section FindSwapCandidateFacts.
Proof.
move=> j2 SCHED_j2.
move: fsc_not_idle => [j' [SCHED_j' ARR]].
by rewrite -(ideal_proc_model_is_a_uniprocessor_model _ _ _ _ SCHED_j' SCHED_j2).
now rewrite -(ideal_proc_model_is_a_uniprocessor_model _ _ _ _ SCHED_j' SCHED_j2).
Qed.
(** We observe that [find_swap_candidate] returns a value within a
......@@ -148,7 +148,7 @@ Section FindSwapCandidateFacts.
- by apply /andP; split.
- by rewrite /relevant_pstate SCHED_j.
}
by move: ED; rewrite /earlier_deadline /oapp SCHED_j SCHED_j2.
now move: ED; rewrite /earlier_deadline /oapp SCHED_j SCHED_j2.
Qed.
(** As a special case of the above lemma, we observe that the job
......@@ -222,7 +222,7 @@ Section MakeEDFAtFacts.
by rewrite scheduled_at_def; apply /eqP.
apply swapped_completed_jobs_dont_execute => //.
apply fsc_range1 => //.
by apply scheduled_job_in_sched_has_later_deadline.
now apply scheduled_job_in_sched_has_later_deadline.
Qed.
(** Importantly, [make_edf_at] does not introduce any deadline
......@@ -237,7 +237,7 @@ Section MakeEDFAtFacts.
{
apply (H_no_deadline_misses _ t).
move: SCHED.
by rewrite /sched' /make_edf_at SCHED_orig.
now rewrite /sched' /make_edf_at SCHED_orig.
}
{
have SCHED': scheduled_at sched j_orig t_edf
......@@ -253,14 +253,14 @@ Section MakeEDFAtFacts.
- move=> j1 SCHED_j1.
move: (fsc_not_idle sched H_jobs_must_arrive_to_execute j_orig t_edf SCHED' DL_orig) => [j' [SCHED_j' ARR_j']].
exists j'. split => //.
by apply scheduled_job_in_sched_has_later_deadline.
now apply scheduled_job_in_sched_has_later_deadline.
- have EX: (exists t', scheduled_at sched j t').
{
apply swap_job_scheduled with (t1 := t_edf) (t2 := find_swap_candidate sched t_edf j_orig) (t0 := t).
by move: SCHED; rewrite /sched' /make_edf_at SCHED_orig.
now move: SCHED; rewrite /sched' /make_edf_at SCHED_orig.
}
move: EX => [t' SCHED_t'].
by apply H_no_deadline_misses with (t := t').
now apply H_no_deadline_misses with (t := t').
}
Qed.
......@@ -343,7 +343,7 @@ Section MakeEDFAtFacts.
apply leq_trans with (n := job_deadline j_orig) => // ;
first by exact mea_guarantee_deadlines.
apply leq_trans with (n := t') => //.
by apply ltnW.
now apply ltnW.
Qed.
(** Next, we consider the more difficult case, where [t'] is
......@@ -360,23 +360,23 @@ Section MakeEDFAtFacts.
- exists (find_swap_candidate sched t_edf j_orig).
split; last by apply fsc_range => //; exact mea_guarantee_dl_orig.
subst. rewrite -(ideal_proc_model_is_a_uniprocessor_model _ _ _ _ H_sched_edf H_sched').
by rewrite scheduled_at_def FSC //=.
now rewrite scheduled_at_def FSC //=.
- case: (boolP(find_swap_candidate sched t_edf j_orig == t')) => [/eqP EQ' | /eqP NEQ'].
+ exists t_edf.
split; last by apply /andP; split => //; exact mea_guarantee_dl_orig.
rewrite -(swap_job_scheduled_t2 _ _ (find_swap_candidate sched t_edf j_orig) _).
move: H_sched'. rewrite /sched' /make_edf_at SCHED.
by rewrite EQ'.
now rewrite EQ'.
+ move: NEQ NEQ' => /eqP NEQ /eqP NEQ'. exists t'.
split; last by apply /andP; split.
rewrite -(swap_job_scheduled_other_times _ t_edf (find_swap_candidate sched t_edf j_orig)) //.
move: H_sched'.
by rewrite /sched' /make_edf_at SCHED.
now rewrite /sched' /make_edf_at SCHED.
}
move: EX => [t'' [SCHED'' RANGE]].
apply: (fsc_found_job_deadline sched _ j_orig t_edf _ _ _ _ _ t'') => // ;
first by exact mea_guarantee_dl_orig.
by rewrite scheduled_at_def FSC //=.
now rewrite scheduled_at_def FSC //=.
Qed.
End GuaranteeCaseAnalysis.
......@@ -395,7 +395,7 @@ Section MakeEDFAtFacts.
case: (boolP (t' < job_deadline j_orig)).
- by apply mea_guarantee_case_t'_before_deadline.
- rewrite -leqNgt => BOUND_t'.
by apply: (mea_guarantee_case_t'_past_deadline j_orig j_edf j' t').
now 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
......@@ -420,10 +420,10 @@ Section MakeEDFAtFacts.
- case (boolP(t_edf == t)) => [/eqP EQ'| /eqP NEQ'].
+ move=> SCHED_j.
have ARR_j: job_arrival j <= t_edf by apply fsc_found_job_arrival with (sched0 := sched) (j1 := j_orig) => //; rewrite scheduled_at_def.
by rewrite -EQ'.
now rewrite -EQ'.
+ move=> SCHED_j.
apply H_jobs_must_arrive_to_execute.
by rewrite scheduled_at_def /scheduled_in /pstate_instance.
now rewrite scheduled_at_def /scheduled_in /pstate_instance.
Qed.
(** We connect the fact that a job is scheduled in [sched'] to the
......@@ -438,7 +438,7 @@ Section MakeEDFAtFacts.
move=> j t SCHED_j.
destruct (sched t_edf) as [j_orig|] eqn:SCHED_orig; last by exists t.
eapply swap_job_scheduled.
by exact SCHED_j.
now exact SCHED_j.
Qed.
(** Conversely, if a job is scheduled in [sched], it is also
......@@ -454,7 +454,7 @@ Section MakeEDFAtFacts.
destruct (sched t_edf) as [j_orig|] eqn:SCHED_orig;
last by exists t.
eapply swap_job_scheduled_original.
by exact SCHED_j.
now exact SCHED_j.
Qed.
(** Next, we observe that if all jobs in [sched] come from a given
......@@ -474,7 +474,7 @@ Section MakeEDFAtFacts.
rewrite /sched' /make_edf_at.
destruct (sched t_edf) as [j_orig|] eqn:SCHED_orig;
last by done.
by apply swapped_jobs_come_from_arrival_sequence.
now apply swapped_jobs_come_from_arrival_sequence.
Qed.
End ArrivalSequence.
......@@ -503,7 +503,7 @@ Section MakeEDFAtFacts.
{
apply ltn_leq_trans with (n := t_edf) => //.
apply fsc_range1 => //.
by apply scheduled_job_in_sched_has_later_deadline.
now apply scheduled_job_in_sched_has_later_deadline.
}
move: SCHED_j.
have ->: scheduled_at (swapped sched t_edf (find_swap_candidate sched t_edf j_orig)) j t = scheduled_at sched j t
......@@ -597,7 +597,7 @@ Section EDFPrefixFacts.
last by move=> t SCHED; exists t.
move=> sched'' t'' EX t''' SCHED_mea.
move: (mea_job_scheduled _ _ _ _ SCHED_mea) => [t'''' SCHED''''].
by apply: (EX t'''' SCHED'''').
now apply: (EX t'''' SCHED'''').
Qed.
(** Conversely, if a job is scheduled in the original schedule, it is
......@@ -612,7 +612,7 @@ Section EDFPrefixFacts.
apply prefix_map_property_invariance; last by exists t.
move=> schedX tx [t' SCHEDX_j].
eapply mea_job_scheduled'.
by exact SCHEDX_j.
now exact SCHEDX_j.
Qed.
(** Next, we note that [edf_transform_prefix] maintains the
......@@ -633,7 +633,7 @@ Section EDFPrefixFacts.
rewrite /sched' /edf_transform_prefix.
apply prefix_map_property_invariance; last by done.
move => schedX t ARR.
by apply mea_jobs_come_from_arrival_sequence.
now apply mea_jobs_come_from_arrival_sequence.
Qed.
End ArrivalSequence.
......@@ -661,7 +661,7 @@ Section EDFPrefixFacts.
+ by apply mea_jobs_must_arrive => //.
+ by apply mea_no_deadline_misses => //.
- move=> schedX t_ref [COMP [ARR DL]].
by apply mea_EDF_widen.
now apply mea_EDF_widen.
Qed.
End EDFPrefixFacts.
......@@ -688,11 +688,9 @@ Section EDFPrefixInclusion.
Lemma edf_prefix_inclusion:
forall h1 h2,
h1 <= h2 ->
forall t,
t < h1 ->
(edf_transform_prefix sched h1) t = (edf_transform_prefix sched h2) t.
identical_prefix (edf_transform_prefix sched h1) (edf_transform_prefix sched h2) h1.
Proof.
move=> h1 h2 LE_h1_h2 t LT_t_h1.
move=> h1 h2 LE_h1_h2. rewrite /identical_prefix => t LT_t_h1.
induction h2; first by move: (ltn_leq_trans LT_t_h1 LE_h1_h2).
move: LE_h1_h2. rewrite leq_eqVlt => /orP [/eqP ->|LT]; first by done.
move: LT. rewrite ltnS => LE_h1_h2.
......@@ -732,44 +730,46 @@ Section EDFTransformFacts.
transforming the given reference schedule. *)
Let sched_edf := edf_transform sched.
(** We begin with the first key property: the resulting schedule is actually
an EDF schedule. *)
(** We begin with a simple lemma relating [sched_edf] to its definition that
allows us to easily look at any finite prefix of the EDF-transformed
scheduled. *)
Lemma edf_finite_prefix:
forall h,
identical_prefix sched_edf (edf_transform_prefix sched h) h.
Proof.
move=> h. rewrite /sched_edf/edf_transform /identical_prefix => t LT_h.
now apply edf_prefix_inclusion.
Qed.
(** From this, we move on to the defining property of the transformation: the
resulting schedule is actually an EDF schedule. *)
Theorem edf_transform_ensures_edf:
EDF_schedule sched_edf.
Proof.
rewrite /EDF_schedule /sched_edf /edf_transform => t.
rewrite /EDF_schedule => t.
rewrite /EDF_at //= => j SCHED_j t' j' LE_t_t' SCHED_j' ARR_j'.
move: SCHED_j.
rewrite scheduled_at_def.
have ->: edf_transform_prefix sched t.+1 t = edf_transform_prefix sched t'.+1 t
by apply edf_prefix_inclusion.
rewrite -scheduled_at_def.
move=> SCHED_j.
move: (edf_prefix_guarantee sched H_jobs_must_arrive_to_execute H_completed_jobs_dont_execute H_no_deadline_misses t'.+1 t) => EDF.
feed EDF; first by done.
by apply (EDF j SCHED_j t' j' LE_t_t').
set S := (edf_transform_prefix sched t'.+1).
have EDF: EDF_at S t by apply edf_prefix_guarantee.
apply EDF with (t' := t') => //.
rewrite -(identical_prefix_scheduled_at sched_edf _ t'.+1) //.
now apply edf_finite_prefix.
Qed.
(** Next, we observe that completed jobs still don't execute in the resulting
EDF schedule. This observation is needed to establish that the resulting
EDF schedule is valid. *)
EDF schedule. This observation is needed to establish that the resulting
EDF schedule is valid. *)
Lemma edf_transform_completed_jobs_dont_execute:
completed_jobs_dont_execute sched_edf.
Proof.
move=> j t.
rewrite /sched_edf /edf_transform /service /service_during.
have ->: \sum_(0 <= t0 < t) service_at (fun t1 : instant => edf_transform_prefix sched t1.+1 t1) j t0 =
\sum_(0 <= t0 < t) service_at (edf_transform_prefix sched t.+1) j t0.
{
apply eq_big_nat => t' /andP [_ LT_t].
rewrite /service_at.
rewrite (edf_prefix_inclusion _ _ _ _ t'.+1 t.+1) => //.
by apply ltn_trans with (n := t).
}
set S := (edf_transform_prefix sched t.+1).
rewrite -/(service_during S j 0 t) -/(service S j t) {}/S.
move: (edf_prefix_well_formedness sched H_jobs_must_arrive_to_execute H_completed_jobs_dont_execute H_no_deadline_misses t.+1) => [COMP _].
by apply COMP.
rewrite (identical_prefix_scheduled_at _ S t.+1) //;
last by apply edf_finite_prefix.
rewrite (identical_prefix_service _ S t) //;
last by apply (identical_prefix_inclusion _ _ t.+1) => //; apply edf_finite_prefix.
move: (edf_prefix_well_formedness sched H_jobs_must_arrive_to_execute
H_completed_jobs_dont_execute H_no_deadline_misses t.+1) => [COMP _].
now apply COMP.
Qed.
(** Similarly, we observe that no job is scheduled prior to its arrival. *)
......@@ -779,7 +779,7 @@ Section EDFTransformFacts.
move=> j t.
rewrite /sched_edf /edf_transform.
move: (edf_prefix_well_formedness sched H_jobs_must_arrive_to_execute H_completed_jobs_dont_execute H_no_deadline_misses t.+1) => [_ [ARR _]].
by apply ARR.
now apply ARR.
Qed.
(** We next establish the second key property: in the transformed EDF
......@@ -788,22 +788,17 @@ Section EDFTransformFacts.
all_deadlines_met sched_edf.
Proof.
move=> j t.
rewrite /sched_edf /edf_transform /job_meets_deadline /completed_by /service /service_during => SCHED.
have LT_t_dl: t < job_deadline j
by apply edf_prefix_scheduled_job_has_later_deadline with (sched0 := sched) (horizon := t.+1).
rewrite /sched_edf /job_meets_deadline.
set t_dl := (job_deadline j).
have ->: \sum_(0 <= t0 < t_dl) service_at (fun t1 : instant => edf_transform_prefix sched t1.+1 t1) j t0 =
\sum_(0 <= t0 < t_dl) service_at (edf_transform_prefix sched t_dl) j t0.
{
apply eq_big_nat => t' /andP [_ LT_t].
rewrite /service_at.
by rewrite (edf_prefix_inclusion _ _ _ _ t'.+1 t_dl).
}
move: SCHED. rewrite scheduled_at_def (edf_prefix_inclusion _ _ _ _ t.+1 t_dl) => // SCHED.
move: (edf_prefix_well_formedness sched H_jobs_must_arrive_to_execute H_completed_jobs_dont_execute H_no_deadline_misses t_dl) => [_ [_ DL]]. move: DL.
rewrite /all_deadlines_met /job_meets_deadline /completed_by /service /service_during => DL.
apply: (DL j t)=>//=.
by rewrite scheduled_at_def.
rewrite (identical_prefix_completed_by _ (edf_transform_prefix sched t_dl.+1) t_dl) //;
last by apply (identical_prefix_inclusion _ _ t_dl.+1) => //; apply edf_finite_prefix.
move=> SCHED_AT; move: (SCHED_AT).
rewrite (identical_prefix_scheduled_at _ (edf_transform_prefix sched t_dl.+1) t_dl).
- move: (edf_prefix_well_formedness sched H_jobs_must_arrive_to_execute
H_completed_jobs_dont_execute H_no_deadline_misses t_dl.+1) => [_ [_ DL]].
now apply (DL j t).
- by apply (identical_prefix_inclusion _ _ t_dl.+1) => //; apply edf_finite_prefix.
- by apply edf_prefix_scheduled_job_has_later_deadline with (sched0 := sched) (horizon := t.+1).
Qed.
(** We observe that no new jobs are introduced: any job scheduled in the EDF
......@@ -828,7 +823,7 @@ Section EDFTransformFacts.
exists t'.
rewrite /sched_edf /edf_transform scheduled_at_def.
rewrite (edf_prefix_inclusion _ _ _ _ t'.+1 (job_deadline j)) -?scheduled_at_def=> //.
by apply edf_prefix_scheduled_job_has_later_deadline with (sched0 := sched) (horizon := job_deadline j).
now apply edf_prefix_scheduled_job_has_later_deadline with (sched0 := sched) (horizon := job_deadline j).
Qed.
(** Next, we note that [edf_transform] maintains the property that all jobs
......@@ -849,7 +844,7 @@ Section EDFTransformFacts.
rewrite /sched_edf /edf_transform.
move=> j t.
rewrite scheduled_at_def - scheduled_at_def.
by apply (edf_prefix_jobs_come_from_arrival_sequence sched t.+1 arr_seq H_from_arr_seq).
now apply (edf_prefix_jobs_come_from_arrival_sequence sched t.+1 arr_seq H_from_arr_seq).
Qed.
End ArrivalSequence.
......@@ -902,7 +897,7 @@ Section Optimality.
move: H_sched_valid => [COME READY].
have ARR := jobs_must_arrive_to_be_ready sched READY.
have COMP := completed_jobs_are_not_ready sched READY.
by apply edf_transform_deadlines_met.
now apply edf_transform_deadlines_met.
Qed.
End AllDeadlinesMet.
......@@ -928,14 +923,14 @@ Section Optimality.
destruct (job_cost j == 0) eqn:COST.
- move: COST => /eqP COST.
rewrite /job_meets_deadline /completed_by COST.
by apply leq0n.
now apply leq0n.
- move: (neq0_lt0n COST) => NONZERO.
move: (H_no_deadline_misses_of_arrivals j ARR_j). rewrite {1}/job_meets_deadline => COMP_j.
move: (completed_implies_scheduled_before sched j NONZERO ARR (job_deadline j) COMP_j) => [t' [_ SCHED']].
move: (all_deadlines_met_in_valid_schedule arr_seq sched COME H_no_deadline_misses_of_arrivals) => NO_MISSES.
move: (edf_transform_job_scheduled' sched ARR COMP NO_MISSES j t' SCHED') => [t'' SCHED''].
move: (edf_schedule_meets_all_deadlines NO_MISSES) => DL_MET.
by apply: (DL_MET j t'' SCHED'').
now apply: (DL_MET j t'' SCHED'').
Qed.
End AllDeadlinesOfArrivalsMet.
......
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