Term Rewriting System R:
[x]
f(g(x), s(0)) -> f(g(x), g(x))
g(s(x)) -> s(g(x))
g(0) -> 0

Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

F(g(x), s(0)) -> F(g(x), g(x))
G(s(x)) -> G(x)

Furthermore, R contains two SCCs.


   R
DPs
       →DP Problem 1
Argument Filtering and Ordering
       →DP Problem 2
Remaining


Dependency Pair:

G(s(x)) -> G(x)


Rules:


f(g(x), s(0)) -> f(g(x), g(x))
g(s(x)) -> s(g(x))
g(0) -> 0





The following dependency pair can be strictly oriented:

G(s(x)) -> G(x)


There are no usable rules w.r.t. to the AFS that need to be oriented.
Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

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


   R
DPs
       →DP Problem 1
AFS
           →DP Problem 3
Dependency Graph
       →DP Problem 2
Remaining


Dependency Pair:


Rules:


f(g(x), s(0)) -> f(g(x), g(x))
g(s(x)) -> s(g(x))
g(0) -> 0





Using the Dependency Graph resulted in no new DP problems.


   R
DPs
       →DP Problem 1
AFS
       →DP Problem 2
Remaining Obligation(s)




The following remains to be proven:
Dependency Pair:

F(g(x), s(0)) -> F(g(x), g(x))


Rules:


f(g(x), s(0)) -> f(g(x), g(x))
g(s(x)) -> s(g(x))
g(0) -> 0




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