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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

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


Rule:


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


Strategy:

innermost




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

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

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


Rule:


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


Strategy:

innermost




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

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

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

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


Rule:


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


Strategy:

innermost




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

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

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

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


Rule:


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


Strategy:

innermost




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

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

F(a, f(f(f(x', a), a), a)) -> F(f(a, a), f(a, f(f(a, a), f(a, f(f(a, a), 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
                 ↳Forward Instantiation Transformation


Dependency Pairs:

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


Rule:


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


Strategy:

innermost




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

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

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 6
                 ↳Forward Instantiation Transformation


Dependency Pairs:

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


Rule:


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


Strategy:

innermost




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

F(a, f(f(f(x', a), a), a)) -> F(f(a, a), f(a, f(f(a, a), f(a, f(f(a, a), f(a, 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 7
                 ↳Narrowing Transformation


Dependency Pairs:

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


Rule:


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


Strategy:

innermost




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

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

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

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:

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


Rule:


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


Strategy:

innermost




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

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

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


Rule:


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


Strategy:

innermost




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

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

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 10
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

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


Rule:


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


Strategy:

innermost



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