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

Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.


   R
DPs
       →DP Problem 1
Argument Filtering and Ordering


Dependency Pair:

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


Rules:


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





The following dependency pair can be strictly oriented:

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


The following rules can be oriented:

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


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

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


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


Dependency Pair:


Rules:


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





Using the Dependency Graph resulted in no new DP problems.

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