Termination Proof using AProVE

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.

     ↳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.

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

       →DP Problem 2

         ↳Remaining Obligation(s)

The following remains to be proven:

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
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

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

       →DP Problem 2

         ↳Remaining Obligation(s)

The following remains to be proven:

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
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