Term Rewriting System R:
[x, y]
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))

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

MERGE(++(x, y), ++(u, v)) -> MERGE(y, ++(u, v))
MERGE(++(x, y), ++(u, v)) -> MERGE(++(x, y), v)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Argument Filtering and Ordering


Dependency Pair:

MERGE(++(x, y), ++(u, v)) -> MERGE(y, ++(u, v))


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))





The following dependency pair can be strictly oriented:

MERGE(++(x, y), ++(u, v)) -> MERGE(y, ++(u, v))


The following rules can be 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))


Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
nil > u
MERGE > u
{++, merge} > u
v > u

resulting in one new DP problem.
Used Argument Filtering System:
MERGE(x1, x2) -> MERGE(x1, x2)
++(x1, x2) -> ++(x1, x2)
merge(x1, x2) -> merge(x1, x2)


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 2
             ↳Dependency Graph


Dependency Pair:


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))





Using the Dependency Graph resulted in no new DP problems.

Termination of R successfully shown.
Duration:
0:00 minutes