Term Rewriting System R:
[x, y, z]
*(i(x), x) -> 1
*(1, y) -> y
*(x, 0) -> 0
*(*(x, y), z) -> *(x, *(y, z))

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) -> *'(y, z)

Furthermore, R contains one SCC.

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

Dependency Pairs:

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

Rules:

*(i(x), x) -> 1
*(1, y) -> y
*(x, 0) -> 0
*(*(x, y), z) -> *(x, *(y, z))

The following dependency pair can be strictly oriented:

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

The following usable rules using the Ce-refinement can be oriented:

*(i(x), x) -> 1
*(1, y) -> y
*(x, 0) -> 0
*(*(x, y), z) -> *(x, *(y, z))

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

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

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

Dependency Pair:

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

Rules:

*(i(x), x) -> 1
*(1, y) -> y
*(x, 0) -> 0
*(*(x, y), z) -> *(x, *(y, z))

The following dependency pair can be strictly oriented:

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

The following usable rules using the Ce-refinement can be oriented:

*(i(x), x) -> 1
*(1, y) -> y
*(x, 0) -> 0
*(*(x, y), z) -> *(x, *(y, z))

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

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

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

Dependency Pair:

Rules:

*(i(x), x) -> 1
*(1, y) -> y
*(x, 0) -> 0
*(*(x, y), z) -> *(x, *(y, z))

Using the Dependency Graph resulted in no new DP problems.

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