Term Rewriting System R:
[a, b, c, d]
f(f(f(f(j, a), b), c), d) -> f(f(a, b), f(f(a, d), c))

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(f(f(f(j, a), b), c), d) -> F(f(a, b), f(f(a, d), c))
F(f(f(f(j, a), b), c), d) -> F(a, b)
F(f(f(f(j, a), b), c), d) -> F(f(a, d), c)
F(f(f(f(j, a), b), c), d) -> F(a, d)

Furthermore, R contains one SCC. 


   R
     ↳OC
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

F(f(f(f(j, a), b), c), d) -> F(a, d)
F(f(f(f(j, a), b), c), d) -> F(f(a, d), c)
F(f(f(f(j, a), b), c), d) -> F(a, b)
F(f(f(f(j, a), b), c), d) -> F(f(a, b), f(f(a, d), c))


Rule:


f(f(f(f(j, a), b), c), d) -> f(f(a, b), f(f(a, d), c))


Strategy:

innermost



The Proof could not be continued due to a Timeout.
Termination of R could not be shown.
Duration:
1:04 minutes