The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

++(nil, y) → y
++(x, nil) → x
++(.(x, y), z) → .(x, ++(y, z))
++(++(x, y), z) → ++(x, ++(y, z))

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
++(.(x, y), z) →⁺ .(x, ++(y, z))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
The pumping substitution is [y / .(x, y)].
The result substitution is [ ].

(2) BOUNDS(n^1, INF)