# Library prosa.analysis.facts.job_index

In this section, we prove some properties related to job_index.
Section JobIndexLemmas.

Consider any type of tasks ...

... and any jobs associated with the tasks.
Context {Job : JobType}.
Context `{JobArrival Job}.

Consider any arrival sequence with consistent and non-duplicate arrivals, ...
... and any two jobs j1 and j2 from this arrival sequence that stem from the same task.
In the next section, we prove some basic properties about jobs with equal indices.
Section EqualJobIndex.

Assume that the jobs j1 and j2 have the same job_index in the arrival sequence.
To show that jobs j1 and j2 are equal, we'll perform case analysis on the relation between their arrival times.
Jobs with equal indices have to be equal regardless of their arrival times because of the way job_index is defined (i.e., jobs are first ordered according to their arrival times and ties are broken arbitrarily due to which no two unequal jobs have the same job_index).
In case job j2 arrives after or at the same time as j1 arrives, we show that the jobs are equal.
Lemma case_arrival_lte_implies_equal_job:
job_arrival j1 job_arrival j2 j1 = j2.
Proof.
moveARR_LT.
move: H_equal_indexIND.
try ( apply task_arrivals_up_to_prefix_cat with (arr_seq0 := arr_seq) in ARR_LT ⇒ // ) ||
apply task_arrivals_up_to_prefix_cat with (arr_seq := arr_seq) in ARR_LT ⇒ //.
move : ARR_LT ⇒ [xs ARR_CAT].
try ( apply eq_ind_in_seq with (xs0 := task_arrivals_up_to_job_arrival arr_seq j2) ⇒ // ) ||
apply eq_ind_in_seq with (xs := task_arrivals_up_to_job_arrival arr_seq j2) ⇒ //.
- by rewrite -/(job_index _ _) -IND -ARR_CAT index_cat ifT;
- rewrite -ARR_CAT mem_cat; apply /orP.
Qed.

And similarly if j1 arrives after j2, we show that the jobs are equal.
Lemma case_arrival_gt_implies_equal_job:
job_arrival j1 > job_arrival j2 j1 = j2.
Proof.
moveLT.
move: H_equal_indexIND.
apply ltnW, (task_arrivals_up_to_prefix_cat arr_seq) in LT ⇒ //.
move: LT ⇒ [xs ARR_CAT].
apply (eq_ind_in_seq _ _ (task_arrivals_up_to_job_arrival arr_seq j1)) ⇒ //.
- by rewrite -/(job_index _ _) IND -ARR_CAT index_cat ifT;
- rewrite -ARR_CAT mem_cat; apply /orP.
by left; apply arrives_in_task_arrivals_up_to ⇒ //; rt_eauto.
Qed.

And finally we show that irrespective of the relation between the arrival of job j1 and j2, j1 must be equal to j2.
Lemma equal_index_implies_equal_jobs:
j1 = j2.
Proof.
case: (ltngtP (job_arrival j1) (job_arrival j2)) ⇒ [LT|GT|EQ].
- by apply case_arrival_lte_implies_equal_job; lia.
- by apply case_arrival_gt_implies_equal_job; lia.
- by apply case_arrival_lte_implies_equal_job; lia.
Qed.

End EqualJobIndex.

We show that jobs of a task are different if and only if they have different indices.
Lemma diff_jobs_iff_diff_indices:
j1 j2
job_index arr_seq j1 job_index arr_seq j2.
Proof.
split.
+ moveNEQJ EQ.
by apply equal_index_implies_equal_jobs in EQ.
+ moveNEQ EQ.
by move: EQ NEQ ⇒ →.
Qed.

We show that job_index j can be written as a sum of size j) and index j arr_seq j) j)).
Lemma index_as_sum_size_and_index:
job_index arr_seq j1 =
+ index j1 (task_arrivals_at_job_arrival arr_seq j1).
Proof.
rewrite index_cat ifF; first by reflexivity.
apply Bool.not_true_is_false; intro T.
move : T; rewrite mem_filter ⇒ /andP [/eqP SM_TSK JB_IN_ARR].
apply mem_bigcat_nat_exists in JB_IN_ARR; move : JB_IN_ARR ⇒ [ind [JB_IN IND_INTR]].
have ARR : job_arrival j1 = ind by rt_eauto.
by lia.
Qed.

Given jobs j1 and j2 in arrivals_between_P arr_seq P t1 t2, the fact that j2 arrives strictly before j1 implies that j2 also belongs in the sequence arrivals_between_P arr_seq P t1 j1).
Lemma arrival_lt_implies_job_in_arrivals_between_P:
(P : Job bool) (t1 t2 : instant),
(j1 \in arrivals_between_P arr_seq P t1 t2)
(j2 \in arrivals_between_P arr_seq P t1 t2)
job_arrival j2 < job_arrival j1
j2 \in arrivals_between_P arr_seq P t1 (job_arrival j1).
Proof.
intros × J1_IN J2_IN ARR_LT.
rewrite mem_filter in J2_IN; move : J2_IN ⇒ /andP [PJ2 J2ARR] ⇒ //.
rewrite mem_filter; apply /andP; split ⇒ //.
apply mem_bigcat_nat_exists in J2ARR; move : J2ARR ⇒ [i [J2_IN INEQ]].
apply mem_bigcat_nat with (j := i) ⇒ //.
apply job_arrival_at in J2_IN; rt_eauto.
by lia.
Qed.

We show that jobs in the sequence arrivals_between_P are ordered by their arrival times, i.e., if index of a job j2 is greater than or equal to index of any other job j1 in the sequence, then job_arrival j2 must be greater than or equal to job_arrival j1.
Lemma index_lte_implies_arrival_lte_P:
(P : Job bool) (t1 t2 : instant),
(j1 \in arrivals_between_P arr_seq P t1 t2)
(j2 \in arrivals_between_P arr_seq P t1 t2)
index j1 (arrivals_between_P arr_seq P t1 t2) index j2 (arrivals_between_P arr_seq P t1 t2)
job_arrival j1 job_arrival j2.
Proof.
intros × IN1 IN2 LE.
move_neq_up LT; move_neq_down LE.
rewritearrivals_P_cat with (t := job_arrival j1); last by apply job_arrival_between_P in IN1; rt_eauto.
rewrite !index_cat ifT; last by eapply arrival_lt_implies_job_in_arrivals_between_P; eauto.
rewrite ifF.
- eapply leq_trans; [ | by erewrite leq_addr].
rewrite index_mem.
by eapply arrival_lt_implies_job_in_arrivals_between_P; eauto.
- apply Bool.not_true_is_false; intro T.
by apply job_arrival_between_P in T; try lia; rt_eauto.
Qed.

We observe that index of job j1 is same in the sequences task_arrivals_up_to_job_arrival j1 and task_arrivals_up_to_job_arrival j2 provided j2 arrives after j1.
job_arrival j1 job_arrival j2
Proof.
rewrite leq_eqVlt ⇒ /orP [/eqP LT|LT]; first by rewrite /task_arrivals_up_to_job_arrival LT H_same_task.
have CONSISTENT : consistent_arrival_times arr_seq by rt_eauto.
specialize (arrival_lt_implies_strict_prefix arr_seq CONSISTENT (job_task j1) j1 j2) ⇒ SUB.
feed_n 5 SUB ⇒ //; move: SUB ⇒ [xs2 [NEMPT2 CAT2]].
rewrite -CAT2 index_cat ifT //=.
Qed.

We show that the job_index of a job j1 is strictly less than the size of task_arrivals_up_to_job_arrival arr_seq j1.
Finally, we show that a lower job index implies an earlier arrival time.
Lemma index_lte_implies_arrival_lte:
job_index arr_seq j2 job_index arr_seq j1
job_arrival j2 job_arrival j1.
Proof.
intros IND_LT.
rewrite /job_index in IND_LT.
move_neq_up ARR_GT; move_neq_down IND_LT.
rewritetask_arrivals_cat with (t_m := job_arrival j1); try by lia.
rewrite ifF.
- by eapply leq_trans;
- apply Bool.not_true_is_false; intro T.
by apply job_arrival_between_P in T; try lia; rt_eauto.
Qed.

We show that if job j1 arrives earlier than job j2 then job_index arr_seq j1 is strictly less than job_index arr_seq j2.
Lemma earlier_arrival_implies_lower_index:
job_arrival j1 < job_arrival j2
job_index arr_seq j1 < job_index arr_seq j2.
Proof.
intros ARR_LT.
move_neq_up IND_LT; move_neq_down ARR_LT.
by apply index_lte_implies_arrival_lte.
Qed.

We prove a weaker version of the lemma index_job_lt_size_task_arrivals_up_to_job, given that the job_index of j is positive.
Since task_arrivals_up_to_job_arrival arr_seq j has at least the job j in it, its size has to be positive.
size (task_arrivals_up_to_job_arrival arr_seq j1) > 0.
Proof.
rewrite lt0n; apply /eqP; intro Z.
apply size0nil in Z.
have J_IN : (j1 \in task_arrivals_up_to_job_arrival arr_seq j1)
by apply arrives_in_task_arrivals_up_to ⇒ //; rt_eauto.
by rewrite Z in J_IN.
Qed.

End JobIndexLemmas.

In this section, we prove a few basic properties of function prev_job.
Section PreviousJob.

Consider any type of tasks ...

... and any jobs associated with the tasks.
Context {Job : JobType}.
Context `{JobArrival Job}.

Consider any unique arrival sequence with consistent arrivals ...
... and an arbitrary job with a positive job_index.
Variable j : Job.
Hypothesis H_arrives_in_arr_seq: arrives_in arr_seq j.
Hypothesis H_positive_job_index: job_index arr_seq j > 0.

We observe that the fact that job j is in the arrival sequence implies that job prev_job j is in the arrival sequence.
Lemma prev_job_arr:
arrives_in arr_seq (prev_job arr_seq j).
Proof.
destruct (default_or_in (job_index arr_seq j - 1) j (task_arrivals_up_to_job_arrival arr_seq j)) as [EQ|IN].
+ by rewrite /prev_job EQ.
+ apply in_arrivals_implies_arrived with (t1 := 0) (t2 := (job_arrival j).+1).
by move: IN; rewrite mem_filter ⇒ /andP [_ IN].
Qed.

We show that the index of prev_job j in task arrivals up to j is one less than job_index arr_seq j.
Lemma prev_job_index:
index (prev_job arr_seq j) (task_arrivals_up_to_job_arrival arr_seq j) = job_index arr_seq j - 1.
Proof.
apply index_uniq; last by apply uniq_task_arrivals; rt_eauto.
apply leq_ltn_trans with (n := job_index arr_seq j); try by lia.
Qed.

Observe that job j and prev_job j stem from the same task.
Proof.
specialize (job_index_minus_one_lt_size_task_arrivals_up_to arr_seq H_valid_arrival_sequence j H_arrives_in_arr_seq) ⇒ SIZEL.
rewrite -prev_job_index index_mem mem_filter in SIZEL.
by move : SIZEL ⇒ /andP [/eqP TSK PREV_IN].
Qed.

prev_job arr_seq j \in task_arrivals_up_to_job_arrival arr_seq j.
Proof.
rewrite /prev_job -index_mem index_uniq;
Qed.

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_lte:
job_arrival (prev_job arr_seq j) job_arrival j.
Proof.
rewrite mem_filter in PREV_JOB_IN; move : PREV_JOB_IN ⇒ /andP [TSK PREV_IN].
apply mem_bigcat_nat_exists in PREV_IN; move : PREV_IN ⇒ [i [PREV_IN INEQ]].
move : H_valid_arrival_sequence ⇒ [CONSISTENT UNIQ].
apply CONSISTENT in PREV_IN; rewrite -PREV_IN in INEQ.
by lia.
Qed.

We show that for any job j the job index of prev_job j is one less than the job index of j.
Lemma prev_job_index_j:
job_index arr_seq j > 0
job_index arr_seq (prev_job arr_seq j) = job_index arr_seq j - 1.
Proof.
intros NZ_IND.
rewrite -prev_job_index.
- by apply prev_job_arr.
- by apply prev_job_arr_lte.
Qed.

We also show that for any job j there are no task arrivals between prev_job j and j.
Lemma no_jobs_between_consecutive_jobs:
job_index arr_seq j > 0
(job_arrival (prev_job arr_seq j)).+1 (job_arrival j) = [::].
Proof.
movePOS_IND.
move : (prev_job_arr_lte); rewrite leq_eqVlt; move ⇒ /orP [/eqP EQ | LT].
- rewrite EQ.
apply/eqP/negPn/negP; rewrite -has_filter ⇒ /hasP [j' IN /eqP TASK].
apply mem_bigcat_nat_exists in IN; move : IN ⇒ [i [J'_IN ARR_INEQ]].
by lia.
- apply/eqP/negPn/negP; rewrite -has_filter ⇒ /hasP [j3 IN /eqP TASK].
apply mem_bigcat_nat_exists in IN; move : IN ⇒ [i [J3_IN ARR_INEQ]].
have J3_ARR : (arrives_in arr_seq j3) by apply in_arrseq_implies_arrives with (t := i).
apply job_arrival_at in J3_IN; rt_eauto; rewrite -J3_IN in ARR_INEQ.
move : ARR_INEQ ⇒ /andP [PJ_L_J3 J3_L_J].
apply (earlier_arrival_implies_lower_index arr_seq H_valid_arrival_sequence _ _) in PJ_L_J3 ⇒ //; try by
+ apply (earlier_arrival_implies_lower_index arr_seq H_valid_arrival_sequence _ _) in J3_L_J ⇒ //.
by rewrite prev_job_index_j in PJ_L_J3; try by lia.
+ by apply prev_job_arr.
Qed.

We show that there always exists a job of lesser job_index than a job with a positive job_index that arrives in the arrival sequence.
Lemma exists_jobs_before_j:
k,
k < job_index arr_seq j
j',
j j'
arrives_in arr_seq j'
job_index arr_seq j' = k.
Proof.
clear H_positive_job_index.
intros k K_LT.
(nth j (task_arrivals_up_to_job_arrival arr_seq j) k).
set (jk := nth j (task_arrivals_up_to_job_arrival arr_seq j) k).
have K_LT_SIZE : (k < size (task_arrivals_up_to_job_arrival arr_seq j)) by
apply leq_trans with (n := job_index arr_seq j) ⇒ //; first by apply index_size.
have JK_IN : (jk \in task_arrivals_up_to_job_arrival arr_seq j) by rewrite /jk; apply mem_nth ⇒ //.
rewrite mem_filter in JK_IN; move : JK_IN ⇒ /andP [/eqP TSK JK_IN].
apply mem_bigcat_nat_exists in JK_IN; move : JK_IN ⇒ [i [JK_IN I_INEQ]].
have JK_ARR : (arrives_in arr_seq jk) by apply in_arrseq_implies_arrives with (t := i) ⇒ //.
have INDJK : (index jk (task_arrivals_up_to_job_arrival arr_seq j) = k).
{ apply index_uniq ⇒ //.
by apply uniq_task_arrivals ⇒ //; rt_eauto. }
repeat split ⇒ //.
{ rewritediff_jobs_iff_diff_indices ⇒ //; eauto.
rewrite /job_index; rewrite [in X in _ X] (job_index_same_in_task_arrivals _ _ jk j) ⇒ //.
- rewrite index_uniq -/(job_index arr_seq j)=> //; last by apply uniq_task_arrivals; rt_eauto.
by lia.
- by apply job_arrival_at in JK_IN; rt_eauto; rewrite -JK_IN in I_INEQ. }
{ rewrite /job_index; rewrite [in X in X = _] (job_index_same_in_task_arrivals _ _ jk j) ⇒ //.
by apply job_arrival_at in JK_IN; rt_eauto; rewrite -JK_IN in I_INEQ. }
Qed.

End PreviousJob.