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

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.

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

Dependency Pairs:

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

Rules:

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

The following dependency pair can be strictly oriented:

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

The following usable rules w.r.t. to the AFS can be oriented:

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

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

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

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`           →DP Problem 2`
`             ↳Argument Filtering and Ordering`

Dependency Pair:

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

Rules:

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

The following dependency pair can be strictly oriented:

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

There are no usable rules w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(*(x1, x2)) =  1 + x1 + x2

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

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`           →DP Problem 2`
`             ↳AFS`
`             ...`
`               →DP Problem 3`
`                 ↳Dependency Graph`

Dependency Pair:

Rules:

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

Using the Dependency Graph resulted in no new DP problems.

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