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
Polynomial 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))


There are no usable rules using the Ce-refinement that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(s(x1))=  1 + x1  
  POL(F(x1, x2, x3))=  x1 + x2  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
Polo
           →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