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

Termination of R to be shown. 



   R
     ↳Overlay and local confluence Check



The TRS is overlay and locally confluent (all critical pairs are trivially joinable).Hence, we can switch to innermost. 


   R
     ↳OC
       →TRS2
         ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


   R
     ↳OC
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳Usable Rules (Innermost)


Dependency Pair:

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


Rule:


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


Strategy:

innermost




As we are in the innermost case, we can delete all 1 non-usable-rules.


   R
     ↳OC
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳UsableRules
             ...
               →DP Problem 2
                 ↳Instantiation Transformation


Dependency Pair:

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


Rule:

none


Strategy:

innermost




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

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

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

The transformation is resulting in no new DP problems.


Termination of R successfully shown.
Duration:
0:00 minutes