YES
(VAR x y)
(RULES 
f(c(s(x),y)) -> f(c(x,s(y)))
g(c(x,s(y))) -> g(c(s(x),y))
g(s(f(x))) -> g(f(x))
)

Proving termination of rewriting for _4_37a:

-> Dependency pairs:
nF_f(c(s(x),y)) -> nF_f(c(x,s(y)))
nF_g(c(x,s(y))) -> nF_g(c(s(x),y))
nF_g(s(f(x))) -> nF_g(f(x))
nF_g(s(f(x))) -> nF_f(x)

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

-> New dependency pairs: 
nF_g(s(f(c(s(x),y)))) -> nF_g(f(c(x,s(y))))

-> Proof of termination for _4_37a_1_1:
-> -> Dependency pairs in cycle:
nF_g(s(f(c(s(x),y)))) -> nF_g(f(c(x,s(y))))

UsableRules:
f(c(s(x),y)) -> f(c(x,s(y)))

Polynomial Interpretation:

[f](X) = 0
[c](X1,X2) = 0
[s](X) = 1
[g](X) = 0
[nF_g](X) = X

TIME: 4.5603e-2


-> Proof of termination for _4_37a_1_2:
-> -> Dependency pairs in cycle:
nF_g(c(x,s(y))) -> nF_g(c(s(x),y))

There are no usable rules.

Polynomial Interpretation:

[f](X) = 0
[c](X1,X2) = X2
[s](X) = X + 1
[g](X) = 0
[nF_g](X) = X

TIME: 4.0894e-2


-> Proof of termination for _4_37a_1_3:
-> -> Dependency pairs in cycle:
nF_f(c(s(x),y)) -> nF_f(c(x,s(y)))

There are no usable rules.

Polynomial Interpretation:

[f](X) = 0
[c](X1,X2) = X1
[s](X) = X + 1
[g](X) = 0
[nF_f](X) = X

TIME: 4.2845e-2


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.