To enable tooling like autocompletion or documentation for a given lua execution environment (like lua 5.1, gideros, MOAI ...), you need to add the support of this one to LDT like this.
You could find some execution environment support on our wiki here.
Currently, nobody has contributed a Corona Execution Environment. We hope for a community contribution. You could also ask to Corona developpers to provide this Execution Environment for its users.
If you want to test the potencial of execution envionment in LDT you could try the Lua 5.1 one.