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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

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


Rules:


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





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

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

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

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

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


Rules:


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





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

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

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

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


Rules:


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





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

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

A(a(f, x'''), g) -> A(f, a(g, a(f, 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 4
                 ↳Narrowing Transformation


Dependency Pairs:

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


Rules:


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





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

A(g, g) -> A(f, a(g, a(f, a(g, a(f, f)))))
no new Dependency Pairs are created.
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:

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


Rules:


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





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

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

A(a(f, x'''), g) -> A(g, a(f, 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 6
                 ↳Narrowing Transformation


Dependency Pairs:

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


Rules:


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





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

A(g, g) -> A(g, a(f, a(g, a(f, f))))
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 7
                 ↳Forward Instantiation Transformation


Dependency Pairs:

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


Rules:


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





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

A(a(f, x'''), g) -> A(g, a(f, a(g, a(f, x'''))))
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

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


Rules:


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





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

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

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

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:

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


Rules:


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





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

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

A(a(f, a(f, x'')), g) -> A(f, a(g, a(f, a(g, a(f, 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 10
                 ↳Narrowing Transformation


Dependency Pairs:

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


Rules:


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





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

A(a(f, g), g) -> A(f, a(g, a(f, a(g, a(f, a(g, a(f, f)))))))
no new Dependency Pairs are created.
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:

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


Rules:


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





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

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

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

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:

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


Rules:


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





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

A(a(f, a(f, g)), g) -> A(f, a(g, a(f, a(g, a(f, a(g, a(f, a(g, a(f, f)))))))))
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 13
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

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


Rules:


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




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