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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Forward Instantiation Transformation


Dependency Pairs:

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


Rule:


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





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

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



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳Narrowing Transformation


Dependency Pair:

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


Rule:


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





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

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

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 3
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pair:

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


Rule:


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




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