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# Library prosa.behavior.arrival_sequence

Require Export prosa.behavior.job.
Require Export prosa.util.notation.
This module contains basic definitions and properties of job arrival sequences.

# Notion of an Arrival Sequence

We begin by defining a job arrival sequence.
Section ArrivalSequence.
Given any job type with decidable equality, ...
Variable Job : JobType.
..., an arrival sequence is a mapping from any time to a (finite) sequence of jobs.
Definition arrival_sequence := instant → seq Job.
End ArrivalSequence.

# Arrival of a Job

Next, we define properties of jobs in a given arrival sequence.
Section JobProperties.
Consider any job arrival sequence.
Context {Job : JobType}.
Variable arr_seq : arrival_sequence Job.
We say that job j arrives at a given time t iff it belongs to the corresponding sequence.
Definition arrives_at (j : Job) (t : instant) := j \in arr_seq t.
Similarly, we define whether job j arrives at some (unknown) time t, i.e., whether it belongs to the arrival sequence.
Definition arrives_in (j : Job) := ∃ t, j \in arr_seq t.
End JobProperties.

# Validity of an Arrival Sequence

Next, we define valid arrival sequences.
Section ValidArrivalSequence.
Assume that job arrival times are known.
Context {Job : JobType}.
Context `{JobArrival Job}.
Consider any job arrival sequence.
Variable arr_seq : arrival_sequence Job.
We say that arrival times are consistent if any job that arrives in the sequence has the corresponding arrival time.
Definition consistent_arrival_times :=
∀ j t,
arrives_at arr_seq j t → job_arrival j = t.
We say that the arrival sequence is a set iff it doesn't contain duplicate jobs at any given time.
Definition arrival_sequence_uniq := ∀ t, uniq (arr_seq t).
We say that the arrival sequence is valid iff it is a set and arrival times are consistent
Definition valid_arrival_sequence :=
consistent_arrival_times ∧ arrival_sequence_uniq.
End ValidArrivalSequence.

# Arrival Time Predicates

Next, we define properties of job arrival times.
Section ArrivalTimeProperties.
Assume that job arrival times are known.
Context {Job : JobType}.
Context `{JobArrival Job}.
Let j be any job.
Variable j : Job.
We say that job j has arrived at time t iff it arrives at some time t_0 with t_0 <= t.
Definition has_arrived (t : instant) := job_arrival j ≤ t.
Next, we say that job j arrived before t iff it arrives at some time t_0 with t_0 < t.
Definition arrived_before (t : instant) := job_arrival j < t.
Finally, we say that job j arrives between t1 and t2 iff it arrives at some time t with t1 <= t < t2.
Definition arrived_between (t1 t2 : instant) := t1 ≤ job_arrival j < t2.
End ArrivalTimeProperties.

# Finite Arrival Sequence Prefixes

In this section, we define arrival sequence prefixes, which are useful to define (computable) properties over sets of jobs in the schedule.
Section ArrivalSequencePrefix.
Assume that job arrival times are known.
Context {Job : JobType}.
Context `{JobArrival Job}.
Consider any job arrival sequence.
Variable arr_seq : arrival_sequence Job.
By concatenation, we construct the list of jobs that arrived in the interval `[t1, t2)`.
Definition arrivals_between (t1 t2 : instant) :=
\cat_(t1 ≤ t < t2) arr_seq t.
Based on that, we define the list of jobs that arrived up to time t, ...
Definition arrivals_up_to (t : instant) := arrivals_between 0 t.+1.
... the list of jobs that arrived strictly before time t, ...
Definition arrivals_before (t : instant) := arrivals_between 0 t.
... and the list of jobs that arrive in the interval `[t1, t2)` and satisfy a certain predicate P.
Definition arrivals_between_P (P : Job → bool) (t1 t2 : instant) :=
[seq j <- arrivals_between t1 t2 | P j].
End ArrivalSequencePrefix.