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

Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

F(x, y) -> G(x, y)
G(h(x), y) -> F(x, y)
G(h(x), y) -> G(x, y)

Furthermore, R contains one SCC.


   R
DPs
       →DP Problem 1
Polynomial Ordering


Dependency Pairs:

G(h(x), y) -> G(x, y)
G(h(x), y) -> F(x, y)
F(x, y) -> G(x, y)


Rules:


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





The following dependency pairs can be strictly oriented:

G(h(x), y) -> G(x, y)
G(h(x), y) -> F(x, y)


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

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

resulting in one new DP problem.



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


Dependency Pair:

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


Rules:


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





Using the Dependency Graph resulted in no new DP problems.

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