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

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.

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

Dependency Pairs:

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

Rule:

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

The following dependency pair can be strictly oriented:

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

The following rule can be oriented:

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

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

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

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

Dependency Pair:

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

Rule:

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

The following dependency pair can be strictly oriented:

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

The following rule can be oriented:

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

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(*'(x1, x2)) =  x1 + x2 POL(*(x1, x2)) =  1 + x1 + x2 POL(minus) =  0

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

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

Dependency Pair:

Rule:

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

Using the Dependency Graph resulted in no new DP problems.

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