Termination w.r.t. Q of the given QTRS could be proven:

(0) Obligation:

Q restricted rewrite system:
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))

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MERGE(++(x, y), ++(u, v)) → MERGE(y, ++(u, v))
MERGE(++(x, y), ++(u, v)) → MERGE(++(x, y), v)

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))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(4) Obligation:

Q DP problem:
The TRS P consists of the following rules:

MERGE(++(x, y), ++(u, v)) → MERGE(y, ++(u, v))

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))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(5) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MERGE(++(x, y), ++(u, v)) → MERGE(y, ++(u, v))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MERGE(x1, x2)  =  MERGE(x1)
++(x1, x2)  =  ++(x2)
u  =  u
v  =  v
merge(x1, x2)  =  merge(x1, x2)
nil  =  nil

Recursive path order with status [RPO].
Quasi-Precedence:

merge₂ > [++₁, u] > [MERGE₁, v]
nil > [MERGE₁, v]

Status:

MERGE₁: multiset
++₁: [1]
u: multiset
v: multiset
merge₂: [1,2]
nil: multiset

The following usable rules [FROCOS05] were oriented:

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))

(6) Obligation:

Q DP problem:
P is empty.
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))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(7) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(8) TRUE