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

Innermost 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`
`         ↳Forward Instantiation Transformation`

Dependency Pair:

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

Rules:

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

Strategy:

innermost

On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

*'(x, *(y, z)) -> *'(x, y)
one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳Polynomial Ordering`

Dependency Pair:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

Additionally, the following usable rules for innermost w.r.t. to the implicit 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.

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

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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