Definition contiguous
             (prevLogIndex : logIndex)
             (prevLogTerm : term)
             (leaderLog entries : list entry) : Prop :=
    (prevLogIndex = 0 \/
     exists e, findAtIndex leaderLog prevLogIndex = Some e /\
          eTerm e = prevLogTerm) /\
    (forall e,
       In e leaderLog ->
       eIndex e > prevLogIndex ->
       eIndex e <= maxIndex entries ->
       In e entries) /\
    forall e e',
      eIndex e = eIndex e' ->
      eTerm e = eTerm e' ->
      In e entries ->
      In e' leaderLog ->
      e = e'.

Theorem handleTimeout_lastApplied : forall h st out st' ps, handleTimeout h st = (out, st', ps) -> lastApplied st' = lastApplied st.