There are older papers that describes Lean's meta theory, https://leanprover.github.io/publications/, the first paper, and the one on elaboration. Unfortunately they both describe the version present in Lean 2, which still had the HoTT support we removed in Lean 3.
The new tactic paper, set to appear at ICFP '17, also briefly describes Lean's meta-theory. https://leanprover.github.io/papers/tactic.pdf
At a high level our meta-theory is very similar to Coq's with a few important departures, a big difference the remove of pattern matching and fixpoints from the kernel, we instead have primitive recursors/elminators/recursion principles.
The lead author of Lean (Leo De Moura) has built some of the worlds most advanced automated reasoning tools, and his goal has always to unify type theory and automation. Our goal is to have the benefits of dependent type theory coupled with the high level of automation present in SMT and from tools like Isabelle.