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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(g(a)) -> F(s(g(b)))
F(g(a)) -> G(b)
G(x) -> F(g(x))
G(x) -> G(x)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Forward Instantiation Transformation


Dependency Pairs:

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


Rules:


f(g(a)) -> f(s(g(b)))
f(f(x)) -> b
g(x) -> f(g(x))





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

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

G(a) -> F(g(a))

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

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


Rules:


f(g(a)) -> f(s(g(b)))
f(f(x)) -> b
g(x) -> f(g(x))





The following dependency pair can be strictly oriented:

G(a) -> F(g(a))


There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(g(x1))=  0  
  POL(G(x1))=  x1  
  POL(b)=  0  
  POL(s(x1))=  0  
  POL(a)=  1  
  POL(f(x1))=  0  
  POL(F(x1))=  0  

resulting in one new DP problem. 



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


Dependency Pairs:

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


Rules:


f(g(a)) -> f(s(g(b)))
f(f(x)) -> b
g(x) -> f(g(x))





Using the Dependency Graph the DP problem was split into 1 DP problems.


   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳Polo
             ...
               →DP Problem 4
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pair:

G(x) -> G(x)


Rules:


f(g(a)) -> f(s(g(b)))
f(f(x)) -> b
g(x) -> f(g(x))




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