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

Proving termination of rewriting for _3_49:

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

-> Proof of termination for _3_49_1:
-> -> Dependency pairs in cycle:
nF_f(c(s(x),y)) -> nF_f(c(x,s(y)))
nF_g(c(s(x),s(y))) -> nF_f(c(x,y))
nF_g(c(x,s(y))) -> nF_g(c(s(x),y))
nF_f(c(s(x),s(y))) -> nF_g(c(x,y))

There are no usable rules.

Polynomial Interpretation:

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

TIME: 5.7147e-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: 4.8772e-2

-> -> 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: 4.9841e-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.