The Runtime Complexity (full) 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:

merge(x, nil) → x
merge(nil, y) → y
merge(++(x, y), ++(u, v)) → ++(x, merge(y, ++(u, v)))
merge(++(x, y), ++(u, v)) → ++(u, merge(++(x, y), v))

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
merge(++(x, y), ++(u, v)) →⁺ ++(x, merge(y, ++(u, v)))
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)

(3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(4) Obligation:

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

merge(x, nil) → x
merge(nil, y) → y
merge(++(x, y), ++(u, v)) → ++(x, merge(y, ++(u, v)))
merge(++(x, y), ++(u, v)) → ++(u, merge(++(x, y), v))

S is empty.
Rewrite Strategy: FULL

(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(6) Obligation:

TRS:
Rules:
merge(x, nil) → x
merge(nil, y) → y
merge(++(x, y), ++(u, v)) → ++(x, merge(y, ++(u, v)))
merge(++(x, y), ++(u, v)) → ++(u, merge(++(x, y), v))

Types:
merge :: nil:++:v → nil:++:v → nil:++:v
nil :: nil:++:v
++ :: u → nil:++:v → nil:++:v
u :: u
v :: nil:++:v
hole_nil:++:v1_0 :: nil:++:v
hole_u2_0 :: u
gen_nil:++:v3_0 :: Nat → nil:++:v

(7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
merge

(8) Obligation:

TRS:
Rules:
merge(x, nil) → x
merge(nil, y) → y
merge(++(x, y), ++(u, v)) → ++(x, merge(y, ++(u, v)))
merge(++(x, y), ++(u, v)) → ++(u, merge(++(x, y), v))

Types:
merge :: nil:++:v → nil:++:v → nil:++:v
nil :: nil:++:v
++ :: u → nil:++:v → nil:++:v
u :: u
v :: nil:++:v
hole_nil:++:v1_0 :: nil:++:v
hole_u2_0 :: u
gen_nil:++:v3_0 :: Nat → nil:++:v

Generator Equations:
gen_nil:++:v3_0(0) ⇔ nil
gen_nil:++:v3_0(+(x, 1)) ⇔ ++(u, gen_nil:++:v3_0(x))

The following defined symbols remain to be analysed:
merge

(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol merge.

(10) Obligation:

TRS:
Rules:
merge(x, nil) → x
merge(nil, y) → y
merge(++(x, y), ++(u, v)) → ++(x, merge(y, ++(u, v)))
merge(++(x, y), ++(u, v)) → ++(u, merge(++(x, y), v))

Types:
merge :: nil:++:v → nil:++:v → nil:++:v
nil :: nil:++:v
++ :: u → nil:++:v → nil:++:v
u :: u
v :: nil:++:v
hole_nil:++:v1_0 :: nil:++:v
hole_u2_0 :: u
gen_nil:++:v3_0 :: Nat → nil:++:v

Generator Equations:
gen_nil:++:v3_0(0) ⇔ nil
gen_nil:++:v3_0(+(x, 1)) ⇔ ++(u, gen_nil:++:v3_0(x))

No more defined symbols left to analyse.