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

Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.


   R
DPs
       →DP Problem 1
Argument Filtering and Ordering


Dependency Pair:

F(g(x)) -> F(x)


Rule:


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





The following dependency pair can be strictly oriented:

F(g(x)) -> F(x)


The following rule can be oriented:

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


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(g(x1))=  1 + x1  
  POL(F(x1))=  1 + x1  
  POL(f(x1))=  x1  

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


   R
DPs
       →DP Problem 1
AFS
           →DP Problem 2
Dependency Graph


Dependency Pair:


Rule:


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





Using the Dependency Graph resulted in no new DP problems.

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