YES
(VAR X)
(RULES 
f(a,a) -> f(a,b)
f(a,b) -> f(s(a),c)
f(s(X),c) -> f(X,c)
f(c,c) -> f(a,a)
)

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 test4:

-> Dependency pairs:
nF_f(a,a) -> nF_f(a,b)
nF_f(a,b) -> nF_f(s(a),c)
nF_f(s(X),c) -> nF_f(X,c)
nF_f(c,c) -> nF_f(a,a)

-> Proof of termination for test4_1:
-> -> Dependency pairs in cycle:
nF_f(a,a) -> nF_f(a,b)
nF_f(c,c) -> nF_f(a,a)
nF_f(s(X),c) -> nF_f(X,c)
nF_f(a,b) -> nF_f(s(a),c)

There are no usable rules.

Polynomial Interpretation:

[f](X1,X2) = 0
[a] = 0
[b] = 0
[s](X) = X
[c] = 1
[nF_f](X1,X2) = X1

TIME: 5.7147e-2

-> -> Dependency pairs in cycle:
nF_f(s(X),c) -> nF_f(X,c)

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.