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

0 QTRS
↳1 QTRSRRRProof (⇔)
↳2 QTRS
↳3 QTRSRRRProof (⇔)
↳4 QTRS
↳5 QTRSRRRProof (⇔)
↳6 QTRS
  ↳7 RisEmptyProof (⇔)
    ↳8 TRUE
  ↳9 RisEmptyProof (⇔)
    ↳10 TRUE
  ↳11 RisEmptyProof (⇔)
    ↳12 TRUE

(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) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(++(x1, x2)) = x1 + x2   
POL(merge(x1, x2)) = 1 + x1 + x2   
POL(nil) = 0   
POL(u) = 0   
POL(v) = 0   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

merge(x, nil) → x
merge(nil, y) → y


(2) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

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

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(++(x1, x2)) = 1 + x1 + x2   
POL(merge(x1, x2)) = 1 + 2·x1 + x2   
POL(u) = 0   
POL(v) = 0   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

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


(4) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

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

Q is empty.

(5) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(++(x1, x2)) = 2·x1 + x2   
POL(merge(x1, x2)) = 2·x1 + 2·x2   
POL(u) = 2   
POL(v) = 0   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

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


(6) Obligation:

Q restricted rewrite system:
R is empty.
Q is empty.

(7) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(8) TRUE

(9) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(10) TRUE

(11) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(12) TRUE