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
Narrowing 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 Narrowing 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
Nar
             ...
               →DP Problem 3
Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

F(x'') -> 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