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

Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.


   R
DPs
       →DP Problem 1
Narrowing Transformation


Dependency Pairs:

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


Rules:


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





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

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

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

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


Rules:


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





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

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

F(b, f(c, f(c, x''))) -> F(c, f(c, f(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:

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


Rules:


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





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

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

F(c, f(a, f(a, x''))) -> F(a, f(a, f(c, 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:

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


Rules:


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





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

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

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

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


Rules:


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





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

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

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

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
Nar
             ...
               →DP Problem 6
Narrowing Transformation


Dependency Pairs:

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


Rules:


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





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

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

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

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
Nar
             ...
               →DP Problem 7
Polynomial Ordering


Dependency Pairs:

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


Rules:


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





The following dependency pairs can be strictly oriented:

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


Additionally, the following usable rules w.r.t. to the implicit AFS can be oriented:

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


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(c)=  0  
  POL(b)=  0  
  POL(a)=  0  
  POL(f(x1, x2))=  1 + x2  
  POL(F(x1, x2))=  x2  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
Nar
             ...
               →DP Problem 8
Narrowing Transformation


Dependency Pairs:

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


Rules:


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





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

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

F(a, f(b, f(b, f(b, f(b, x''))))) -> F(b, f(b, 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 9
Narrowing Transformation


Dependency Pairs:

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


Rules:


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





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

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

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

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
Nar
             ...
               →DP Problem 10
Narrowing Transformation


Dependency Pairs:

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


Rules:


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





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

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

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

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
Nar
             ...
               →DP Problem 11
Narrowing Transformation


Dependency Pairs:

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


Rules:


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





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

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

F(a, f(b, f(b, f(b, f(b, f(b, x')))))) -> F(b, f(b, f(b, 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 12
Narrowing Transformation


Dependency Pairs:

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


Rules:


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





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

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

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

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
Nar
             ...
               →DP Problem 13
Narrowing Transformation


Dependency Pairs:

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


Rules:


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





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

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

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

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
Nar
             ...
               →DP Problem 14
Narrowing Transformation


Dependency Pairs:

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


Rules:


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





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

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

F(a, f(b, f(b, f(b, f(b, f(b, f(b, x''))))))) -> F(b, f(b, f(b, f(b, 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 15
Narrowing Transformation


Dependency Pairs:

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


Rules:


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





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

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

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

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
Nar
             ...
               →DP Problem 16
Narrowing Transformation


Dependency Pairs:

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


Rules:


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





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

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

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

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Nar
           →DP Problem 2
Nar
             ...
               →DP Problem 17
Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

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


Rules:


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




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