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
         ↳Argument Filtering and Ordering


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))





The following dependency pairs can be strictly oriented:

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


The following rules can be oriented:

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


Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
a > G > F > b
a > G > g > b
a > s > b
f > b

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


   R
     ↳DPs
       →DP Problem 1
         ↳AFS
           →DP Problem 2
             ↳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