Library rt.restructuring.analysis.transform.facts.replace_at

Section ReplaceAtFacts.

  (* For any given type of jobs... *)
  Context {Job : JobType}.

  (* ... and any given type of processor states, ... *)
  Context {PState: eqType}.
  Context `{ProcessorState Job PState}.

  (* ...consider any given reference schedule. *)
  Variable sched: schedule PState.

  (* Suppose we are given a specific time [t'] ... *)
  Variable t': instant.

  (* ...and a replacement state [state]. *)
  Variable nstate: PState.

  (* In the following, let sched' denote the schedule with replacement at time
     t'. *)
  Let sched' := replace_at sched t' nstate.

  (* We begin with the trivial observation that the schedule doesn't change at
     other times. *)
  Lemma rest_of_schedule_invariant:
    ∀ t, t ≠ t' → sched' t = sched t.

  (* As a result, the service in intervals that do not intersect with t' is
     invariant, too. *)
  Lemma service_at_other_times_invariant:
    ∀ t1 t2,
      t2 ≤ t' ∨ t' < t1 →
      ∀ j,
        service_during sched j t1 t2
        =
        service_during sched' j t1 t2.

  (* 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:
    ∀ t1 t2,
      t1 ≤ t' < t2 →
      ∀ j,
        service_during sched j t1 t2 + service_at sched' j t'
        =
        service_during sched' j t1 t2 + service_at sched j t'.

  (* ...which we can also rewrite as follows. *)
  Corollary service_in_replaced:
    ∀ t1 t2,
      t1 ≤ t' < t2 →
      ∀ j,
        service_during sched' j t1 t2
        =
        service_during sched j t1 t2 + service_at sched' j t' - service_at sched j t'.

  (* 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') →
    ∀ t,
      service_at sched j t = service_at sched' j t.

  (* 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') →
    ∀ t1 t2,
      service_during sched j t1 t2 = service_during sched' j t1 t2.

End ReplaceAtFacts.