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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(x) -> F(f(x))
F(x) -> F(x)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Rewriting Transformation


Dependency Pairs:

F(x) -> F(x)
F(x) -> F(f(x))


Rule:


f(x) -> f(f(x))


Strategy:

innermost




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

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

F(x) -> F(f(f(x)))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Rw
           →DP Problem 2
             ↳Rewriting Transformation


Dependency Pairs:

F(x) -> F(f(f(x)))
F(x) -> F(x)


Rule:


f(x) -> f(f(x))


Strategy:

innermost




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

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

F(x) -> F(f(f(f(x))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Rw
           →DP Problem 2
             ↳Rw
             ...
               →DP Problem 3
                 ↳Rewriting Transformation


Dependency Pairs:

F(x) -> F(f(f(f(x))))
F(x) -> F(x)


Rule:


f(x) -> f(f(x))


Strategy:

innermost




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

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

F(x) -> F(f(f(f(f(x)))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Rw
           →DP Problem 2
             ↳Rw
             ...
               →DP Problem 4
                 ↳Rewriting Transformation


Dependency Pairs:

F(x) -> F(f(f(f(f(x)))))
F(x) -> F(x)


Rule:


f(x) -> f(f(x))


Strategy:

innermost




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

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

F(x) -> F(f(f(f(f(f(x))))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Rw
           →DP Problem 2
             ↳Rw
             ...
               →DP Problem 5
                 ↳Rewriting Transformation


Dependency Pairs:

F(x) -> F(f(f(f(f(f(x))))))
F(x) -> F(x)


Rule:


f(x) -> f(f(x))


Strategy:

innermost




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

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

F(x) -> F(f(f(f(f(f(f(x)))))))

The transformation is resulting in one new DP problem: 



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




The following remains to be proven:
Dependency Pairs:

F(x) -> F(f(f(f(f(f(f(x)))))))
F(x) -> F(x)


Rule:


f(x) -> f(f(x))


Strategy:

innermost



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