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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(f(x)) -> G(f(x))
G(g(x)) -> F(x)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

G(g(x)) -> F(x)
F(f(x)) -> G(f(x))


Rules:


f(f(x)) -> g(f(x))
g(g(x)) -> f(x)





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

F(f(x)) -> G(f(x))
one new Dependency Pair is created:

F(f(f(x''))) -> G(g(f(x'')))

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

F(f(f(x''))) -> G(g(f(x'')))
G(g(x)) -> F(x)


Rules:


f(f(x)) -> g(f(x))
g(g(x)) -> f(x)





The following dependency pair can be strictly oriented:

G(g(x)) -> F(x)


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

g(g(x)) -> f(x)
f(f(x)) -> g(f(x))


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

resulting in one new DP problem. 



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


Dependency Pair:

F(f(f(x''))) -> G(g(f(x'')))


Rules:


f(f(x)) -> g(f(x))
g(g(x)) -> f(x)





Using the Dependency Graph resulted in no new DP problems.

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