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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

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


Rules:


f(g(X)) -> g(f(f(X)))
f(h(X)) -> h(g(X))





On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

F(g(X)) -> F(f(X))
two new Dependency Pairs are created:

F(g(g(X''))) -> F(g(f(f(X''))))
F(g(h(X''))) -> F(h(g(X'')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Polynomial Ordering


Dependency Pairs:

F(g(g(X''))) -> F(g(f(f(X''))))
F(g(X)) -> F(X)


Rules:


f(g(X)) -> g(f(f(X)))
f(h(X)) -> h(g(X))





The following dependency pairs can be strictly oriented:

F(g(g(X''))) -> F(g(f(f(X''))))
F(g(X)) -> F(X)


Additionally, the following usable rules w.r.t. to the implicit AFS can be oriented:

f(g(X)) -> g(f(f(X)))
f(h(X)) -> h(g(X))


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

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Polo
             ...
               →DP Problem 3
                 ↳Dependency Graph


Dependency Pair:


Rules:


f(g(X)) -> g(f(f(X)))
f(h(X)) -> h(g(X))





Using the Dependency Graph resulted in no new DP problems.

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