YES
(VAR X Y)
(RULES 
min(X,0) -> X
min(s(X),s(Y)) -> min(X,Y)
quot(0,s(Y)) -> 0
quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))
log(s(0)) -> 0
log(s(s(X))) -> s(log(s(quot(X,s(s(0))))))
)

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 logarquot:

-> Dependency pairs:
nF_min(s(X),s(Y)) -> nF_min(X,Y)
nF_quot(s(X),s(Y)) -> nF_quot(min(X,Y),s(Y))
nF_quot(s(X),s(Y)) -> nF_min(X,Y)
nF_log(s(s(X))) -> nF_log(s(quot(X,s(s(0)))))
nF_log(s(s(X))) -> nF_quot(X,s(s(0)))

-> Proof of termination for logarquot_1_1:
-> -> Dependency pairs in cycle:
nF_log(s(s(X))) -> nF_log(s(quot(X,s(s(0)))))

UsableRules:
min(X,0) -> X
min(s(X),s(Y)) -> min(X,Y)
quot(0,s(Y)) -> 0
quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y)))

Polynomial Interpretation:

[min](X1,X2) = X1
[0] = 1
[s](X) = X + 1
[quot](X1,X2) = X1
[log](X) = 0
[nF_log](X) = X

TIME: 0.106655


-> Proof of termination for logarquot_1_2:
-> -> Dependency pairs in cycle:
nF_quot(s(X),s(Y)) -> nF_quot(min(X,Y),s(Y))

UsableRules:
min(X,0) -> X
min(s(X),s(Y)) -> min(X,Y)

Polynomial Interpretation:

[min](X1,X2) = X1
[0] = 0
[s](X) = X + 1
[quot](X1,X2) = 0
[log](X) = 0
[nF_quot](X1,X2) = X1

TIME: 4.0802e-2


-> Proof of termination for logarquot_1_3:
-> -> Dependency pairs in cycle:
nF_min(s(X),s(Y)) -> nF_min(X,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.