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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Argument Filtering and Ordering


Dependency Pair:

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


Rule:


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





The following dependency pair can be strictly oriented:

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


The following usable rule using the Ce-refinement can be oriented:

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


Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi 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
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 2
             ↳Dependency Graph


Dependency Pair:


Rule:


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





Using the Dependency Graph resulted in no new DP problems.

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