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 pairs can be strictly oriented:

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

The following rules can be oriented:

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

Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
{*', *, 1}

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`
`             ↳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