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

Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.


   R
DPs
       →DP Problem 1
Narrowing Transformation


Dependency Pairs:

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


Rule:


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





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

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



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




The following remains to be proven:
Dependency Pair:

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


Rule:


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




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