SmartThings by itself can do this, but it doesn’t play nice with other zwave controllers, so I’m afraid you may not get the results that you want. ( in particular, it may not always be notified of the first light coming on) If you want to remove your Lighting devices from control of your ADT pulse system and put them on SmartThings instead, that would work for those devices.
So you can set it up as a parallel system with its own devices. Setting it up as a secondary to your ADT system has a lot of other challenges.
SmartThings strongly discourages adding the Hub to another Z-Wave network. We cannot offer support for disconnected Z-Wave devices or the inability to add devices through the Hub as a result of including the Hub into another Z-Wave network.
There are some community members who have successfully added a SmartThings hub as a secondary, but they all seem to have the newest generation of the ADT controller. See the following thread for more discussion: