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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(a, f(b, f(a, x))) -> F(a, f(b, f(b, f(a, x))))
F(a, f(b, f(a, x))) -> F(b, f(b, f(a, x)))

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pair:

F(a, f(b, f(a, x))) -> F(a, f(b, f(b, f(a, x))))


Rules:


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





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

F(a, f(b, f(a, x))) -> F(a, f(b, f(b, f(a, x))))
one new Dependency Pair is created:

F(a, f(b, f(a, f(b, f(a, x''))))) -> F(a, f(b, f(b, f(a, f(b, f(b, f(a, x'')))))))

The transformation is resulting in one new DP problem: 



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


Dependency Pair:

F(a, f(b, f(a, f(b, f(a, x''))))) -> F(a, f(b, f(b, f(a, f(b, f(b, f(a, x'')))))))


Rules:


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





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

F(a, f(b, f(a, f(b, f(a, x''))))) -> F(a, f(b, f(b, f(a, f(b, f(b, f(a, x'')))))))
one new Dependency Pair is created:

F(a, f(b, f(a, f(b, f(a, f(b, f(a, x'))))))) -> F(a, f(b, f(b, f(a, f(b, f(b, f(a, f(b, f(b, f(a, 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 Pair:

F(a, f(b, f(a, f(b, f(a, f(b, f(a, x'))))))) -> F(a, f(b, f(b, f(a, f(b, f(b, f(a, f(b, f(b, f(a, x'))))))))))


Rules:


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





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

F(a, f(b, f(a, f(b, f(a, f(b, f(a, x'))))))) -> F(a, f(b, f(b, f(a, f(b, f(b, f(a, f(b, f(b, f(a, x'))))))))))
one new Dependency Pair is created:

F(a, f(b, f(a, f(b, f(a, f(b, f(a, f(b, f(a, x''))))))))) -> F(a, f(b, f(b, f(a, f(b, f(b, f(a, f(b, f(b, f(a, f(b, f(b, f(a, 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 Pair:

F(a, f(b, f(a, f(b, f(a, f(b, f(a, f(b, f(a, x''))))))))) -> F(a, f(b, f(b, f(a, f(b, f(b, f(a, f(b, f(b, f(a, f(b, f(b, f(a, x'')))))))))))))


Rules:


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





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

F(a, f(b, f(a, f(b, f(a, f(b, f(a, f(b, f(a, x''))))))))) -> F(a, f(b, f(b, f(a, f(b, f(b, f(a, f(b, f(b, f(a, f(b, f(b, f(a, x'')))))))))))))
one new Dependency Pair is created:

F(a, f(b, f(a, f(b, f(a, f(b, f(a, f(b, f(a, f(b, f(a, x'))))))))))) -> F(a, f(b, f(b, f(a, f(b, f(b, f(a, f(b, f(b, f(a, f(b, f(b, f(a, f(b, f(b, f(a, x'))))))))))))))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 5
                 ↳Narrowing Transformation


Dependency Pair:

F(a, f(b, f(a, f(b, f(a, f(b, f(a, f(b, f(a, f(b, f(a, x'))))))))))) -> F(a, f(b, f(b, f(a, f(b, f(b, f(a, f(b, f(b, f(a, f(b, f(b, f(a, f(b, f(b, f(a, x'))))))))))))))))


Rules:


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





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

F(a, f(b, f(a, f(b, f(a, f(b, f(a, f(b, f(a, f(b, f(a, x'))))))))))) -> F(a, f(b, f(b, f(a, f(b, f(b, f(a, f(b, f(b, f(a, f(b, f(b, f(a, f(b, f(b, f(a, x'))))))))))))))))
one new Dependency Pair is created:

F(a, f(b, f(a, f(b, f(a, f(b, f(a, f(b, f(a, f(b, f(a, f(b, f(a, x''))))))))))))) -> F(a, f(b, f(b, f(a, f(b, f(b, f(a, f(b, f(b, f(a, f(b, f(b, f(a, f(b, f(b, f(a, f(b, f(b, f(a, x'')))))))))))))))))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 6
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pair:

F(a, f(b, f(a, f(b, f(a, f(b, f(a, f(b, f(a, f(b, f(a, f(b, f(a, x''))))))))))))) -> F(a, f(b, f(b, f(a, f(b, f(b, f(a, f(b, f(b, f(a, f(b, f(b, f(a, f(b, f(b, f(a, f(b, f(b, f(a, x'')))))))))))))))))))


Rules:


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




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