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

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) -> *'(y, z)

Furthermore, R contains one SCC.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Forward Instantiation Transformation`

Dependency Pair:

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

Rules:

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

Strategy:

innermost

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

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

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

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

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

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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