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

Furthermore, R contains one SCC. 


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


Dependency Pair:

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


Rule:


f(a, f(x, a)) -> f(a, f(f(a, a), f(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
                 ↳Argument Filtering and Ordering


Dependency Pair:

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


Rule:

none


Strategy:

innermost




The following dependency pair can be strictly oriented:

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


There are no usable rules w.r.t. the AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Precedence:
trivial

resulting in one new DP problem.
Used Argument Filtering System:
F(x1, x2) -> F(x1, x2)
f(x1, x2) -> f(x1, x2)


   R
     ↳OC
       →TRS2
         ↳DPs
           →DP Problem 1
             ↳UsableRules
             ...
               →DP Problem 3
                 ↳Dependency Graph


Dependency Pair:


Rule:

none


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

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