The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(2^n, INF).

(0) Obligation:

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

merge(nil, y) → y
merge(x, nil) → x
merge(.(x, y), .(u, v)) → if(<(x, u), .(x, merge(y, .(u, v))), .(u, merge(.(x, y), v)))
++(nil, y) → y
++(.(x, y), z) → .(x, ++(y, z))
if(true, x, y) → x
if(false, x, y) → x

Rewrite Strategy: INNERMOST

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(2ⁿ):
The rewrite sequence
merge(.(x, y), .(u, v)) →⁺ if(<(x, u), .(x, merge(y, .(u, v))), .(u, merge(.(x, y), v)))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1,1].
The pumping substitution is [y / .(x, y)].
The result substitution is [ ].

The rewrite sequence
merge(.(x, y), .(u, v)) →⁺ if(<(x, u), .(x, merge(y, .(u, v))), .(u, merge(.(x, y), v)))
gives rise to a decreasing loop by considering the right hand sides subterm at position [2,1].
The pumping substitution is [v / .(u, v)].
The result substitution is [ ].

(2) BOUNDS(2^n, INF)