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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

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


Rule:


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


Strategy:

innermost




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

A(f, a(f, a(g, a(g, x)))) -> A(f, a(f, a(f, x)))
two new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem: 



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




The following remains to be proven:
Dependency Pairs:

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


Rule:


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


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:01 minutes