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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains two SCCs. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation
       →DP Problem 2
         ↳Nar


Dependency Pair:

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


Rules:


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





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

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

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 3
             ↳Narrowing Transformation
       →DP Problem 2
         ↳Nar


Dependency Pair:

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


Rules:


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





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

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

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 3
             ↳Nar
             ...
               →DP Problem 4
                 ↳Forward Instantiation Transformation
       →DP Problem 2
         ↳Nar


Dependency Pair:

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


Rules:


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





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

F(g(f(g(f(g(f(x'))))))) -> F(f(f(f(x'))))
no new Dependency Pairs are created.
The transformation is resulting in no new DP problems. 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Narrowing Transformation


Dependency Pair:

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


Rules:


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





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

H(f(f(x))) -> H(f(g(f(x))))
two new Dependency Pairs are created:

H(f(f(x''))) -> H(f(f(x'')))
H(f(f(g(f(x''))))) -> H(f(g(f(f(x'')))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Nar
           →DP Problem 5
             ↳Narrowing Transformation


Dependency Pairs:

H(f(f(g(f(x''))))) -> H(f(g(f(f(x'')))))
H(f(f(x''))) -> H(f(f(x'')))


Rules:


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





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

H(f(f(g(f(x''))))) -> H(f(g(f(f(x'')))))
two new Dependency Pairs are created:

H(f(f(g(f(x'''))))) -> H(f(f(f(x'''))))
H(f(f(g(f(g(f(x'))))))) -> H(f(g(f(f(f(x'))))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Nar
           →DP Problem 5
             ↳Nar
             ...
               →DP Problem 6
                 ↳Polynomial Ordering


Dependency Pairs:

H(f(f(g(f(g(f(x'))))))) -> H(f(g(f(f(f(x'))))))
H(f(f(g(f(x'''))))) -> H(f(f(f(x'''))))
H(f(f(x''))) -> H(f(f(x'')))


Rules:


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





The following dependency pairs can be strictly oriented:

H(f(f(g(f(g(f(x'))))))) -> H(f(g(f(f(f(x'))))))
H(f(f(g(f(x'''))))) -> H(f(f(f(x'''))))


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

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


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

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
       →DP Problem 2
         ↳Nar
           →DP Problem 5
             ↳Nar
             ...
               →DP Problem 7
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pair:

H(f(f(x''))) -> H(f(f(x'')))


Rules:


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




Termination of R could not be shown.
Duration:
0:00 minutes