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

The TRS is an overlay system and all critical pairs are trivial, thus termination of innermost rewriting is equivalent to termination of rewriting.

Proving termination of innermost rewriting for _4_37_1:

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

-> Proof of termination for _4_37_1:
-> -> 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
[nF_f](X) = X

TIME: 4.3654e-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

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

The TRS is an overlay system and all critical pairs are trivial, thus termination of innermost rewriting is equivalent to termination of rewriting.

Proving termination of innermost rewriting for _4_37_2:

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

-> Proof of termination for _4_37_2:
-> -> Dependency pairs in cycle:
nF_g(c(x,s(y))) -> nF_g(c(s(x),y))

There are no usable rules.

Polynomial Interpretation:

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

TIME: 4.172e-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.