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

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

Furthermore, R contains two SCCs.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Instantiation Transformation`
`       →DP Problem 2`
`         ↳Remaining`

Dependency Pair:

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

Rules:

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

Strategy:

innermost

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

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

*'(g(x'', z'''), +(f(z''), f(z''))) -> *'(g(g(x'', z'''), z''), +(f(z''), f(z'')))

The transformation is resulting in one new DP problem:

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

The following remains to be proven:
• Dependency Pair:

*'(g(x'', z'''), +(f(z''), f(z''))) -> *'(g(g(x'', z'''), z''), +(f(z''), f(z'')))

Rules:

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

Strategy:

innermost

• Dependency Pairs:

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

Rules:

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

Strategy:

innermost

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

The following remains to be proven:
• Dependency Pair:

*'(g(x'', z'''), +(f(z''), f(z''))) -> *'(g(g(x'', z'''), z''), +(f(z''), f(z'')))

Rules:

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

Strategy:

innermost

• Dependency Pairs:

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

Rules:

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

Strategy:

innermost

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