Library prosa.implementation.refinements.fast_search_space_computation

In this section, we provide definitions and lemmas to show Abstract RTA' s search space can be rewritten in an equivalent, computation-oriented way.
Let L be a constant which bounds any busy interval of task tsk.
Variable L : duration.

Consider a task set ts with valid arrivals...

... and a task tsk of ts with positive cost.
Hypothesis H_positive_cost : 0 < task_cost tsk.
Hypothesis H_tsk_in_ts : tsk \in ts.

Section Definitions.

We generically define the search space for fixed-priority tasks in the interval `[l,r)` by repeating the time steps of the task in the interval `[l*h,r*h)`.
By using the above definition, we give a concrete definition of the search space for fixed-priority tasks.
Definition search_space_arrival_curve_prefix_FP (tsk : Task) L :=
let h := get_horizon_of_task tsk in
search_space_arrival_curve_prefix_FP_h (tsk : Task) 0 (L %/h).+1.

End Definitions.

Section Facts.

We begin by showing that either each time step of an arrival curve prefix is either strictly less than the horizon, or it is the horizon.
Lemma steps_lt_horizon_last_eq_horizon :
Proof.
move: (has_valid_arrival_curve_prefix_tsk _ H_valid_task_set _ H_tsk_in_ts).
move ⇒ [arrival_curve_prefix [EQ [POSh [LARGEh [NOINF [BUR SORT]]]]]].
as [GT1 | LT1 | EQ1]; last by right.
{ left.
movestep IN.
apply leq_ltn_trans with (n:=last0 (get_time_steps_of_task tsk)) ⇒ //.
move: (time_steps_sorted ts H_valid_task_set tsk H_tsk_in_ts).
rewrite ltn_sorted_uniq_leq ⇒ /andP [TS_UNIQ TS_SORT].
apply (sorted_leq_index leq_trans leqnn TS_SORT step _ IN) ⇒ //=.
- by destruct get_time_steps_of_task; last by rewrite /last0 //=; apply mem_last.
- destruct get_time_steps_of_task; rewrite //= -index_mem in IN.
by rewrite /last0; simpl (last 0 _); rewrite index_last. }
{ exfalso. (* h is at least the last step *)
destruct get_time_steps_of_task as [|d l] eqn:TS ⇒ //.
have IN: last0 (d::l) \in d::l by rewrite /last0 //=; apply mem_last.
rewrite EQ in TS; rewrite TS in LARGEh.
apply LARGEh in IN.
by rewrite /get_horizon_of_task EQ in LT1; lia. }
Qed.

Next, we show that for each offset A in the search space for fixed-priority tasks, either (1) A+ε is zero or a multiple of the horizon, offset by one of the time steps of the arrival curve prefix, or (2) A+ε is exactly a multiple of the horizon.
Lemma structure_of_correct_search_space :
A,
A < L
( i t,
i < (L %/ get_horizon_of_task tsk).+1
A + ε = i × get_horizon_of_task tsk + t )
( i,
i < (L %/ get_horizon_of_task tsk).+1
A + ε = i × get_horizon_of_task tsk).
Proof.
move: (has_valid_arrival_curve_prefix_tsk ts H_valid_task_set tsk H_tsk_in_ts).
move⇒ [evec [EQ [POSh [LARGEh [NOINF [BUR SORT]]]]]] A LT_L NEQ.
rewrite eqn_pmul2l // /max_arrivals /MaxArrivals /ConcreteMaxArrivals
/concrete_max_arrivals EQ in NEQ.
move: (sorted_ltn_steps_imply_sorted_leq_steps_steps _ SORT NOINF) ⇒ SORT_LEQ.
unfold positive_horizon in *; set (h := horizon_of evec) in ×.
move: (extrapolated_arrival_curve_change evec POSh SORT_LEQ A NEQ) ⇒ [LT|[EQdiv LTe]].
{ right.
((A+ε) %/ h); split; first by rewrite ltnS leq_div2r // ; lia.
by symmetry; apply /eqP; rewrite -dvdn_eq; by apply ltdivn_dvdn. }
{ left.
((A + ε) %/ h), (step_at evec ((A + ε) %% h)).1; split; last split.
{ by rewrite addn1 ltnS; apply leq_div2r. }
have→ := @map_f _ _ fst (steps_of _) (last (0, 0) [seq s <- _ | s.1 (A+ε)%%h])=> //.
have EQeps: (A + ε) %% h = A %% h + ε by apply addn1_modn_commute.
rewrite EQeps in LTe.
move: (value_at_change_is_in_steps_of _ SORT_LEQ NOINF _ LTe) ⇒ [v IN].
rewrite -EQeps in IN; apply filter_last_mem; apply /hasP.
by ((A+ε) %% h, v). }
{ case (extrapolated_arrival_curve_change _ POSh SORT_LEQ _ NEQ) as [EQs|[EQs LT]].
{ apply ltdivn_dvdn in EQs; move: EQs ⇒ /dvdnP [k EQs]; rewrite EQs.
by rewrite modnMl step_at_0_is_00; [rewrite addn0 mulnK | apply SORT |]. }
{ subst h; set (h := horizon_of evec) in ×.
rewrite {1}[_ + _](divn_eq _ h).
rewrite modnD //= [h _]leqNgt addmod_le_mod //= subn0 in LT.
have EQ1: 1%%h= 1 by apply modn_small; case h as [|[|h]]; rewrite //= in EQs; lia.
rewrite EQ1 in LT; apply value_at_change_is_in_steps_of in LT ⇒ //.
case LT as [v IN].
rewrite modnD //= [h _]leqNgt addmod_le_mod //=.
apply step_at_agrees_with_steps_of in IN ⇒ //.
by rewrite subn0 !EQ1 IN. } } }
Qed.

Conversely, every multiple of the horizon that is strictly less than L is contained in the search space for fixed-priority tasks...
Lemma multiple_of_horizon_in_approx_ss :
A,
A < L
A \in search_space_arrival_curve_prefix_FP tsk L.
Proof.
move: (has_valid_arrival_curve_prefix_tsk ts H_valid_task_set tsk H_tsk_in_ts) ⇒ [evec [EMAX VALID]].
intros A LT DIV; rewrite /search_space_arrival_curve_prefix_FP.
destruct VALID as [POSh [LARGEh [NOINF [BUR SORT]]]].
replace A with (A + ε - ε); last by lia.
rewrite subn1; apply map_f.
set (h := get_horizon_of_task tsk) in ×.
rewrite /repeat_steps_with_offset; apply/flatten_mapP.
move: DIV ⇒ /dvdnP [k DIV]; subst A.
(k × h).
{ rewrite mulnC; apply map_f.
rewrite mem_iota; apply/andP; split; first by apply leq0n.
rewrite /specified_bursts in BUR.
by rewrite /h /get_horizon_of_task EMAX. }
have MFF := map_f (fun t0t0 + k × h); apply: MFF.
by rewrite /get_time_steps_of_task EMAX; apply BUR. }
Qed.

... and every A for which A+ε is a multiple of the horizon offset by a time step of the arrival curve prefix is also in the search space for fixed-priority tasks.
Lemma steps_in_approx_ss :
i t A,
i < (L %/ get_horizon_of_task tsk).+1
A + ε = i × get_horizon_of_task tsk + t
A \in search_space_arrival_curve_prefix_FP tsk L.
Proof.
move: (H_valid_task_set) ⇒ VALID; intros i t A LT IN EQ; rewrite /search_space_arrival_curve_prefix_FP.
replace A with (A + ε - ε); last by lia.
rewrite subn1; apply map_f.
set (h := get_horizon_of_task tsk) in ×.
rewrite /repeat_steps_with_offset; apply/flatten_mapP.
(i × h); first by rewrite mulnC; apply map_f; rewrite mem_iota; lia.
unfold time_steps_with_offset.
by apply (map_f (fun t0t0 + i × h) IN).
Qed.

Next, we show that if the horizon of the arrival curve prefix divides A+ε, then A is not contained in the search space for fixed-priority tasks.
Lemma constant_max_arrivals :
A,
get_horizon_of_task tsk %| (A + ε)
max_arrivals tsk A = max_arrivals tsk (A + ε).
Proof.
move: (has_valid_arrival_curve_prefix_tsk ts H_valid_task_set tsk H_tsk_in_ts).
move⇒ [evec [EMAX [POSh [LARGEh [NOINF [BUR SORT]]]]]] A LTH DIV.
rewrite /max_arrivals /ConcreteMaxArrivals /concrete_max_arrivals EMAX.
rewrite /get_horizon_of_task EMAX in DIV; move: (DIV) ⇒ /eqP MOD0.
rewrite /extrapolated_arrival_curve.
set (h := horizon_of evec) in *; set (vec := value_at evec) in ×.
destruct (ltngtP 1 h) as [GT1|LT1|EQ1].
{ move: NOINF ⇒ /eqP NOINF.
have → : vec (A %% h) = vec h.
{ rewrite /vec /value_at.
have → : ((step_at evec (A %% h)) = (step_at evec h)) ⇒ //.
rewrite (pred_Sn A) -addn1 modn_pred; [|repeat destruct h as [|h]=> //|lia].
rewrite /step_at DIV.
have → : [seq step <- steps_of evec | step.1 h] = steps_of evec.
{ apply /all_filterP /allP ⇒ [step IN]; apply ltnW; subst h.
move: LTH; rewrite /get_horizon_of_task EMAX ⇒ → //.
by apply (map_f fst). }
have → : [seq step <- steps_of evec | step.1 h.-1] = steps_of evec ⇒ //.
apply /all_filterP /allP ⇒ [step IN]; subst h.
specialize (VALID step.1).
feed VALID; first by rewrite /get_time_steps_of_task EMAX; apply (map_f fst).
by destruct (horizon_of evec).
}
rewrite -mulSnr {1}(pred_Sn A) divn_pred -(addn1 A) DIV subn1 prednK //=.
move: DIV ⇒ /dvdnP [k EQk]; rewrite EQk.
by destruct k;[lia | rewrite mulnK]. }
{ replace h with 0; rewrite /positive_horizon in POSh; lia. }
{ exfalso.
specialize (LTH _ BUR).
move: LTH; rewrite ltnNge ⇒ /negP CONTR; apply: CONTR.
by unfold h in *; rewrite /get_horizon_of_task EMAX -EQ1. }
Qed.

Finally, we show that steps in the request-bound function correspond to points in the search space for fixed-priority tasks.
A,
A < L
A \in search_space_arrival_curve_prefix_FP tsk L.
Proof.
intros A LT_L IN.
first by apply multiple_of_horizon_in_approx_ss.
move: (structure_of_correct_search_space _ LT_L IN).
move⇒ [[i [t [LT [INt EQ]]]] | [i [LT EQ]]]; first by eapply steps_in_approx_ss; eauto.
destruct (steps_lt_horizon_last_eq_horizon) as [LTh | EQh].
{ exfalso. (* Can't be in search space if h > last step *)
move: (H_valid_task_set) ⇒ VALID; specialize (VALID _ H_tsk_in_ts).
move: (has_valid_arrival_curve_prefix_tsk ts H_valid_task_set tsk H_tsk_in_ts).
move ⇒ [evec [EMAXeq [POSh [LARGEh [NOINF [BUR SORT]]]]]].
/MaxArrivals /ConcreteMaxArrivals /concrete_max_arrivals
EMAXeq eqn_mul2l negb_or in IN.
move: IN ⇒ /andP [_ NEQ].
move: (sorted_ltn_steps_imply_sorted_leq_steps_steps _ SORT NOINF) ⇒ SORT_LEQ.
move: (constant_max_arrivals A LTh) ⇒ EQmax.
feed EQmax; first by rewrite EQ; apply dvdn_mull, dvdnn.
rewrite /max_arrivals /MaxArrivals /ConcreteMaxArrivals
/concrete_max_arrivals EMAXeq in EQmax.
by rewrite EQmax in NEQ; move: NEQ ⇒ /eqP. }
{ replace A with (A + ε - ε); last by lia.
rewrite subn1; apply map_f; rewrite EQ.
set (h := get_horizon_of_task tsk) in ×.
apply /flatten_mapP; ((i-1)*h); first by rewrite mulnC map_f // mem_iota; lia.
replace (i × h) with (h + (i - 1) × h); last first.
{ destruct (posnP i) as [Z|POS]; first by subst i; lia.
by rewrite mulnBl mul1n; apply subnKC, leq_pmull. }
have MFF := map_f (fun t0t0 + (i - 1) × h); apply: MFF.
rewrite -EQh.
move: (has_valid_arrival_curve_prefix_tsk ts H_valid_task_set tsk H_tsk_in_ts).
move ⇒ [evec [EMAXeq [POSh [LARGEh [NOINF [BUR SORT]]]]]].
rewrite /get_time_steps_of_task EMAXeq; unfold specified_bursts in ×.
destruct (time_steps_of evec) ⇒ //=.
by rewrite /last0 //=; apply mem_last. }
Qed.

End Facts.

End FastSearchSpaceComputation.