YES
(VAR X)
(RULES 
f(s(X)) -> f(X)
)

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 test830_1:

-> Dependency pairs:
nF_f(s(X)) -> nF_f(X)

-> Proof of termination for test830_1:
-> -> Dependency pairs in cycle:
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

(VAR X Y)
(RULES 
g(cons(s(X),Y)) -> s(X)
g(cons(0,Y)) -> g(Y)
h(cons(X,Y)) -> h(g(cons(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 test830_2:

-> Dependency pairs:
nF_g(cons(0,Y)) -> nF_g(Y)
nF_h(cons(X,Y)) -> nF_h(g(cons(X,Y)))
nF_h(cons(X,Y)) -> nF_g(cons(X,Y))

-> Proof of termination for test830_2_1:
-> -> Dependency pairs in cycle:
nF_h(cons(X,Y)) -> nF_h(g(cons(X,Y)))

UsableRules:
g(cons(s(X),Y)) -> s(X)
g(cons(0,Y)) -> g(Y)

Polynomial Interpretation:

[g](X) = 0
[cons](X1,X2) = 1
[s](X) = 0
[0] = 0
[h](X) = 0
[nF_h](X) = X

TIME: 5.644e-2


-> Proof of termination for test830_2_2:
-> -> Dependency pairs in cycle:
nF_g(cons(0,Y)) -> nF_g(Y)

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.