(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