(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) QDPSizeChangeProof (EQUIVALENT transformation)
We used the following order and afs together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem.
Order:Homeomorphic Embedding Order
AFS:
u = u
v = v
++(x1, x2) = ++(x2)
From the DPs we obtained the following set of size-change graphs:
- MERGE(++(x, y), ++(u, v)) → MERGE(y, ++(u, v)) (allowed arguments on rhs = {1, 2})
The graph contains the following edges 1 > 1, 2 >= 2
We oriented the following set of usable rules [AAECC05,FROCOS05].
none
(6) TRUE