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

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Forward Instantiation Transformation


Dependency Pairs:

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


Rule:


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


Strategy:

innermost




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

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

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

The transformation is resulting in one new DP problem: 



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




The following remains to be proven:
Dependency Pairs:

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


Rule:


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


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:01 minutes