Term Rewriting System R:
[X, Y]
f(g(X), Y) -> f(X, f(g(X), Y))

Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

F(g(X), Y) -> F(X, f(g(X), Y))
F(g(X), Y) -> F(g(X), Y)

Furthermore, R contains one SCC.


   R
DPs
       →DP Problem 1
Polynomial Ordering


Dependency Pairs:

F(g(X), Y) -> F(g(X), Y)
F(g(X), Y) -> F(X, f(g(X), Y))


Rule:


f(g(X), Y) -> f(X, f(g(X), Y))





The following dependency pair can be strictly oriented:

F(g(X), Y) -> F(X, f(g(X), Y))


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

f(g(X), Y) -> f(X, f(g(X), Y))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(g(x1))=  1 + x1  
  POL(f(x1, x2))=  0  
  POL(F(x1, x2))=  x1  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
Polo
           →DP Problem 2
Remaining Obligation(s)




The following remains to be proven:
Dependency Pair:

F(g(X), Y) -> F(g(X), Y)


Rule:


f(g(X), Y) -> f(X, f(g(X), Y))




Termination of R could not be shown.
Duration:
0:00 minutes