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

Innermost Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.


   R
DPs
       →DP Problem 1
Narrowing Transformation


Dependency Pairs:

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


Rule:


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


Strategy:

innermost




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

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

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


Rule:


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


Strategy:

innermost




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

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

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


Dependency Pairs:

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


Rule:


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


Strategy:

innermost




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

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

A(f, a(f, a(g, a(g, a(g, a(g, a(f, a(g, a(g, x'))))))))) -> A(f, a(g, a(g, a(g, a(g, a(g, a(g, a(f, a(f, a(f, a(g, a(f, a(f, 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
Forward Instantiation Transformation


Dependency Pairs:

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


Rule:


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


Strategy:

innermost




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

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

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


Dependency Pairs:

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


Rule:


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


Strategy:

innermost




The following dependency pairs can be strictly oriented:

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


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

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


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(g)=  0  
  POL(a(x1, x2))=  x1  
  POL(f)=  1  
  POL(A(x1, x2))=  x2  

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 Pairs:

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


Rule:


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


Strategy:

innermost



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