Term Rewriting System R:
[x]
a(b(a(b(x)))) -> b(a(b(a(a(b(x))))))

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

A(b(a(b(x)))) -> A(b(a(a(b(x)))))
A(b(a(b(x)))) -> A(a(b(x)))

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

A(b(a(b(x)))) -> A(a(b(x)))
A(b(a(b(x)))) -> A(b(a(a(b(x)))))


Rule:


a(b(a(b(x)))) -> b(a(b(a(a(b(x))))))





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

A(b(a(b(x)))) -> A(b(a(a(b(x)))))
one new Dependency Pair is created:

A(b(a(b(a(b(x'')))))) -> A(b(a(b(a(b(a(a(b(x'')))))))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Narrowing Transformation


Dependency Pairs:

A(b(a(b(a(b(x'')))))) -> A(b(a(b(a(b(a(a(b(x'')))))))))
A(b(a(b(x)))) -> A(a(b(x)))


Rule:


a(b(a(b(x)))) -> b(a(b(a(a(b(x))))))





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

A(b(a(b(x)))) -> A(a(b(x)))
one new Dependency Pair is created:

A(b(a(b(a(b(x'')))))) -> A(b(a(b(a(a(b(x'')))))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 3
                 ↳Narrowing Transformation


Dependency Pairs:

A(b(a(b(a(b(x'')))))) -> A(b(a(b(a(a(b(x'')))))))
A(b(a(b(a(b(x'')))))) -> A(b(a(b(a(b(a(a(b(x'')))))))))


Rule:


a(b(a(b(x)))) -> b(a(b(a(a(b(x))))))





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

A(b(a(b(a(b(x'')))))) -> A(b(a(b(a(a(b(x'')))))))
one new Dependency Pair is created:

A(b(a(b(a(b(a(b(x')))))))) -> A(b(a(b(a(b(a(b(a(a(b(x')))))))))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 4
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

A(b(a(b(a(b(a(b(x')))))))) -> A(b(a(b(a(b(a(b(a(a(b(x')))))))))))
A(b(a(b(a(b(x'')))))) -> A(b(a(b(a(b(a(a(b(x'')))))))))


Rule:


a(b(a(b(x)))) -> b(a(b(a(a(b(x))))))




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