The Runtime Complexity (full) 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: FULL

(1) CpxTrsToCpxRelTrsProof (BOTH BOUNDS(ID, ID) transformation)

Transformed TRS to relative TRS where S is empty.

(2) Obligation:

Runtime Complexity Relative 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

S is empty.
Rewrite Strategy: FULL

(3) SlicingProof (LOWER BOUND(ID) transformation)

Sliced the following arguments:
./0
</0
</1

(4) Obligation:

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

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

S is empty.
Rewrite Strategy: FULL

(5) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(2ⁿ):
The rewrite sequence
merge(.(y), .(v)) →⁺ if(<, .(merge(y, .(v))), .(merge(.(y), v)))
gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0].
The pumping substitution is [y / .(y)].
The result substitution is [ ].

The rewrite sequence
merge(.(y), .(v)) →⁺ if(<, .(merge(y, .(v))), .(merge(.(y), v)))
gives rise to a decreasing loop by considering the right hand sides subterm at position [2,0].
The pumping substitution is [v / .(v)].
The result substitution is [ ].

(6) BOUNDS(2^n, INF)