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

Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

*'(x, *(y, z)) -> *'(otimes(x, y), z)
*'(+(x, y), z) -> *'(x, z)
*'(+(x, y), z) -> *'(y, z)
*'(x, oplus(y, z)) -> *'(x, y)
*'(x, oplus(y, z)) -> *'(x, z)

Furthermore, R contains one SCC.

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

Dependency Pairs:

*'(x, oplus(y, z)) -> *'(x, z)
*'(+(x, y), z) -> *'(y, z)
*'(+(x, y), z) -> *'(x, z)
*'(x, oplus(y, z)) -> *'(x, y)
*'(x, *(y, z)) -> *'(otimes(x, y), z)

Rules:

*(x, *(y, z)) -> *(otimes(x, y), z)
*(1, y) -> y
*(+(x, y), z) -> oplus(*(x, z), *(y, z))
*(x, oplus(y, z)) -> oplus(*(x, y), *(x, z))

The following dependency pairs can be strictly oriented:

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

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

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

resulting in one new DP problem.

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

Dependency Pairs:

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

Rules:

*(x, *(y, z)) -> *(otimes(x, y), z)
*(1, y) -> y
*(+(x, y), z) -> oplus(*(x, z), *(y, z))
*(x, oplus(y, z)) -> oplus(*(x, y), *(x, z))

The following dependency pairs can be strictly oriented:

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

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

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

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`           →DP Problem 2`
`             ↳Polo`
`             ...`
`               →DP Problem 3`
`                 ↳Polynomial Ordering`

Dependency Pair:

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

Rules:

*(x, *(y, z)) -> *(otimes(x, y), z)
*(1, y) -> y
*(+(x, y), z) -> oplus(*(x, z), *(y, z))
*(x, oplus(y, z)) -> oplus(*(x, y), *(x, z))

The following dependency pair can be strictly oriented:

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

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

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

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Polo`
`           →DP Problem 2`
`             ↳Polo`
`             ...`
`               →DP Problem 4`
`                 ↳Dependency Graph`

Dependency Pair:

Rules:

*(x, *(y, z)) -> *(otimes(x, y), z)
*(1, y) -> y
*(+(x, y), z) -> oplus(*(x, z), *(y, z))
*(x, oplus(y, z)) -> oplus(*(x, y), *(x, z))

Using the Dependency Graph resulted in no new DP problems.

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