Term Rewriting System R:
[x, y, z]
+(-(x, y), z) -> -(+(x, z), y)
-(+(x, y), y) -> x

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

+'(-(x, y), z) -> -'(+(x, z), y)
+'(-(x, y), z) -> +'(x, z)

Furthermore, R contains one SCC.

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

Dependency Pair:

+'(-(x, y), z) -> +'(x, z)

Rules:

+(-(x, y), z) -> -(+(x, z), y)
-(+(x, y), y) -> x

The following dependency pair can be strictly oriented:

+'(-(x, y), z) -> +'(x, z)

The following rules can be oriented:

+(-(x, y), z) -> -(+(x, z), y)
-(+(x, y), y) -> x

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(-(x1, x2)) =  1 + x1 + x2 POL(+(x1, x2)) =  x1 + x2 POL(+'(x1, x2)) =  x1 + x2

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

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

Dependency Pair:

Rules:

+(-(x, y), z) -> -(+(x, z), y)
-(+(x, y), y) -> x

Using the Dependency Graph resulted in no new DP problems.

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