YES
(VAR x)
(RULES 
f(f(x)) -> f(x)
f(s(x)) -> f(x)
g(s(0)) -> g(f(s(0)))
)

Proving termination of rewriting for _4_20a:

-> Dependency pairs:
nF_f(f(x)) -> nF_f(x)
nF_f(s(x)) -> nF_f(x)
nF_g(s(0)) -> nF_g(f(s(0)))
nF_g(s(0)) -> nF_f(s(0))

-> Dependency pairs narrowed: 
nF_g(s(0)) -> nF_g(f(s(0)))

-> New dependency pairs: 
nF_g(s(0)) -> nF_g(f(0))

-> Proof of termination for _4_20a_1_1:
-> -> Dependency pairs in cycle:
nF_g(s(0)) -> nF_g(f(0))

UsableRules:
f(f(x)) -> f(x)
f(s(x)) -> f(x)

Polynomial Interpretation:

[f](X) = 0
[s](X) = X
[g](X) = 0
[0] = 1
[nF_g](X) = X

TIME: 4.5264e-2


-> Proof of termination for _4_20a_1_2:
-> -> Dependency pairs in cycle:
nF_f(f(x)) -> nF_f(x)
nF_f(s(x)) -> nF_f(x)

Termination proved: Cycles verify subterm criterion.

SETTINGS:
Base ordering: Polynomial ordering
Proof mode: SCCs in DG + base ordering
Upper bound for coeffs: 1
Rationals below 1 for all non-replacing args: No
Polynomial interpretation: Linear
Coeffs in polynomials: No rationals
Delta: automatic



Termination was proved succesfully.