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

Furthermore, R contains two SCCs.

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

Dependency Pair:

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

Rules:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

There are no usable rules for innermost that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

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 3`
`             ↳Dependency Graph`
`       →DP Problem 2`
`         ↳Remaining`

Dependency Pair:

Rules:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳AFS`
`       →DP Problem 2`
`         ↳Remaining Obligation(s)`

The following remains to be proven:
Dependency Pairs:

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

Rules:

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

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:00 minutes