Termination Proof using AProVE

Term Rewriting System R:
[x]
a(f, a(g, a(f, x))) -> a(f, a(g, a(g, a(f, x))))
a(g, a(f, a(g, x))) -> a(g, a(f, a(f, a(g, x))))

Innermost Termination of R to be shown.

     ↳Dependency Pair Analysis

R contains the following Dependency Pairs:

A(f, a(g, a(f, x))) -> A(f, a(g, a(g, a(f, x))))
A(f, a(g, a(f, x))) -> A(g, a(g, a(f, x)))
A(g, a(f, a(g, x))) -> A(g, a(f, a(f, a(g, x))))
A(g, a(f, a(g, x))) -> A(f, a(f, a(g, x)))

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:
A(g, a(f, a(g, x))) -> A(g, a(f, a(f, a(g, x))))

Rules:

a(f, a(g, a(f, x))) -> a(f, a(g, a(g, a(f, x))))
a(g, a(f, a(g, x))) -> a(g, a(f, a(f, a(g, x))))

Strategy:
innermost
Dependency Pair:
A(f, a(g, a(f, x))) -> A(f, a(g, a(g, a(f, x))))

Rules:

a(f, a(g, a(f, x))) -> a(f, a(g, a(g, a(f, x))))
a(g, a(f, a(g, x))) -> a(g, a(f, a(f, a(g, x))))

Strategy:
innermost

     ↳DPs

       →DP Problem 1

         ↳Remaining Obligation(s)

       →DP Problem 2

         ↳Remaining Obligation(s)

The following remains to be proven:

Dependency Pair:
A(g, a(f, a(g, x))) -> A(g, a(f, a(f, a(g, x))))

Rules:

a(f, a(g, a(f, x))) -> a(f, a(g, a(g, a(f, x))))
a(g, a(f, a(g, x))) -> a(g, a(f, a(f, a(g, x))))

Strategy:
innermost
Dependency Pair:
A(f, a(g, a(f, x))) -> A(f, a(g, a(g, a(f, x))))

Rules:

a(f, a(g, a(f, x))) -> a(f, a(g, a(g, a(f, x))))
a(g, a(f, a(g, x))) -> a(g, a(f, a(f, a(g, x))))

Strategy:
innermost

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