YES
(VAR x)
(RULES 
half(0) -> 0
half(s(s(x))) -> s(half(x))
log(s(0)) -> 0
log(s(s(x))) -> s(log(s(half(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 _3_7:

-> Dependency pairs:
nF_half(s(s(x))) -> nF_half(x)
nF_log(s(s(x))) -> nF_log(s(half(x)))
nF_log(s(s(x))) -> nF_half(x)

-> Proof of termination for _3_7_1_1:
-> -> Dependency pairs in cycle:
nF_log(s(s(x))) -> nF_log(s(half(x)))

UsableRules:
half(0) -> 0
half(s(s(x))) -> s(half(x))

Polynomial Interpretation:

[half](X) = X
[0] = 1
[s](X) = X + 1
[log](X) = 0
[nF_log](X) = X

TIME: 4.1592e-2


-> Proof of termination for _3_7_1_2:
-> -> Dependency pairs in cycle:
nF_half(s(s(x))) -> nF_half(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.