YES
(VAR x)
(RULES 
g(h(g(x))) -> g(x)
g(g(x)) -> g(h(g(x)))
h(h(x)) -> h(f(h(x),x))
)

Proving termination of rewriting for motivation:

-> Dependency pairs:
nF_g(h(g(x))) -> nF_g(x)
nF_g(g(x)) -> nF_g(h(g(x)))
nF_g(g(x)) -> nF_h(g(x))
nF_g(g(x)) -> nF_g(x)
nF_h(h(x)) -> nF_h(f(h(x),x))
nF_h(h(x)) -> nF_h(x)

-> Proof of termination for motivation_1_1:
-> -> Dependency pairs in cycle:
nF_g(h(g(x))) -> nF_g(x)
nF_g(g(x)) -> nF_g(x)
nF_g(g(x)) -> nF_g(h(g(x)))

UsableRules:
g(h(g(x))) -> g(x)
g(g(x)) -> g(h(g(x)))
h(h(x)) -> h(f(h(x),x))

Polynomial Interpretation:

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

TIME: 5.6827e-2

-> -> Dependency pairs in cycle:
nF_g(h(g(x))) -> nF_g(x)
nF_g(g(x)) -> nF_g(h(g(x)))

UsableRules:
g(h(g(x))) -> g(x)
g(g(x)) -> g(h(g(x)))
h(h(x)) -> h(f(h(x),x))

Polynomial Interpretation:

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

TIME: 5.2818e-2

-> -> Dependency pairs in cycle:
nF_g(g(x)) -> nF_g(h(g(x)))

UsableRules:
g(h(g(x))) -> g(x)
g(g(x)) -> g(h(g(x)))
h(h(x)) -> h(f(h(x),x))

Polynomial Interpretation:

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

TIME: 5.8681e-2


-> Proof of termination for motivation_1_2:
-> -> Dependency pairs in cycle:
nF_h(h(x)) -> nF_h(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



Termination was proved succesfully.