We are left with following problem, upon which TcT provides the
certificate YES(?,O(n^1)).

Strict Trs:
  { ++(x, nil()) -> x
  , ++(++(x, y), z) -> ++(x, ++(y, z))
  , ++(nil(), y) -> y
  , ++(.(x, y), z) -> .(x, ++(y, z)) }
Obligation:
  innermost runtime complexity
Answer:
  YES(?,O(n^1))

The problem is match-bounded by 1. The enriched problem is
compatible with the following automaton.
{ ++_0(2, 2) -> 1
, ++_1(2, 2) -> 3
, nil_0() -> 1
, nil_0() -> 2
, nil_0() -> 3
, ._0(2, 2) -> 1
, ._0(2, 2) -> 2
, ._0(2, 2) -> 3
, ._1(2, 3) -> 1
, ._1(2, 3) -> 3 }

Hurray, we answered YES(?,O(n^1))