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))

Innermost 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`
`         ↳Argument Filtering and Ordering`

Dependency Pairs:

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

Strategy:

innermost

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 for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(*'(x1, x2)) =  x1 + x2 POL(oplus(x1, x2)) =  1 + x1 + x2 POL(+(x1, x2)) =  x1 + x2

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

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

Dependency Pairs:

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

Strategy:

innermost

The following dependency pairs can be strictly oriented:

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

There are no usable rules for innermost w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(*'(x1, x2)) =  x1 + x2 POL(+(x1, x2)) =  1 + x1 + x2

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

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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