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))))

Termination of R to be shown. 



   R
     ↳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 one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

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





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

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

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

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

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





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

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

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





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

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

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 4
                 ↳Narrowing Transformation


Dependency Pairs:

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





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

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

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 5
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

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




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