Definition lifted_prevLog_leader_sublog (net : network) : Prop :=
    forall leader p t leaderId prevLogIndex prevLogTerm entries leaderCommit,
      type (snd (nwState net leader)) = Leader ->
      In p (nwPackets net) ->
      snd (pBody p) = AppendEntries t leaderId prevLogIndex prevLogTerm entries leaderCommit ->
      currentTerm (snd (nwState net leader)) = prevLogTerm ->
      0 < prevLogIndex ->
      0 < prevLogTerm ->
      exists ple, eIndex ple = prevLogIndex /\
             eTerm ple = prevLogTerm /\
             In ple (log (snd (nwState net leader))).

Lemma handleTimeout_preserves_committed : forall h (net net' : ghost_log_network) out d' l e t, handleTimeout h (snd (nwState net h)) = (out, d', l) -> (forall h', nwState net' h' = update name_eq_dec (nwState net) h (update_elections_data_timeout h (nwState net h), d') h') -> lifted_committed net e t -> lifted_committed net' e t.