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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(s(x), y, y) -> F(y, x, s(x))

Furthermore, R contains one SCC. 


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


Dependency Pair:

F(s(x), y, y) -> F(y, x, s(x))


Rules:


g(x, y) -> x
g(x, y) -> y
f(s(x), y, y) -> f(y, x, s(x))





The following dependency pair can be strictly oriented:

F(s(x), y, y) -> F(y, x, s(x))


The following rules can be oriented:

g(x, y) -> x
g(x, y) -> y
f(s(x), y, y) -> f(y, x, s(x))


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

resulting in one new DP problem.
Used Argument Filtering System:
F(x1, x2, x3) -> F(x1, x2)
s(x1) -> s(x1)
g(x1, x2) -> g(x1, x2)
f(x1, x2, x3) -> f


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


Dependency Pair:


Rules:


g(x, y) -> x
g(x, y) -> y
f(s(x), y, y) -> f(y, x, s(x))





Using the Dependency Graph resulted in no new DP problems.

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