YES
(VAR y x)
(RULES 
le(0,y) -> true
le(s(x),0) -> false
le(s(x),s(y)) -> le(x,y)
minus(0,y) -> 0
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
if_minus(true,s(x),y) -> 0
if_minus(false,s(x),y) -> s(minus(x,y))
quot(0,s(y)) -> 0
quot(s(x),s(y)) -> s(quot(minus(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 _3_8b:

-> Dependency pairs:
nF_le(s(x),s(y)) -> nF_le(x,y)
nF_minus(s(x),y) -> nF_if_minus(le(s(x),y),s(x),y)
nF_minus(s(x),y) -> nF_le(s(x),y)
nF_if_minus(false,s(x),y) -> nF_minus(x,y)
nF_quot(s(x),s(y)) -> nF_quot(minus(x,y),s(y))
nF_quot(s(x),s(y)) -> nF_minus(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 _3_8b_1_1:
-> -> Dependency pairs in cycle:
nF_log(s(s(x))) -> nF_log(s(quot(x,s(s(0)))))

UsableRules:
le(0,y) -> true
le(s(x),0) -> false
le(s(x),s(y)) -> le(x,y)
minus(0,y) -> 0
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
if_minus(true,s(x),y) -> 0
if_minus(false,s(x),y) -> s(minus(x,y))
quot(0,s(y)) -> 0
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))

Polynomial Interpretation:

[le](X1,X2) = X1 + X2
[0] = 1
[true] = 0
[s](X) = X + 1
[false] = 0
[minus](X1,X2) = X1
[if_minus](X1,X2,X3) = X2
[quot](X1,X2) = X1
[log](X) = 0
[nF_log](X) = X

TIME: 5.2857e-2


-> Proof of termination for _3_8b_1_2:
-> -> Dependency pairs in cycle:
nF_quot(s(x),s(y)) -> nF_quot(minus(x,y),s(y))

UsableRules:
le(0,y) -> true
le(s(x),0) -> false
le(s(x),s(y)) -> le(x,y)
minus(0,y) -> 0
minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
if_minus(true,s(x),y) -> 0
if_minus(false,s(x),y) -> s(minus(x,y))

Polynomial Interpretation:

[le](X1,X2) = X1 + X2
[0] = 1
[true] = 0
[s](X) = X + 1
[false] = 0
[minus](X1,X2) = X1
[if_minus](X1,X2,X3) = X2
[quot](X1,X2) = 0
[log](X) = 0
[nF_quot](X1,X2) = X1

TIME: 4.9742e-2


-> Proof of termination for _3_8b_1_3:
-> -> Dependency pairs in cycle:
nF_minus(s(x),y) -> nF_if_minus(le(s(x),y),s(x),y)
nF_if_minus(false,s(x),y) -> nF_minus(x,y)

Termination proved: Cycles verify subterm criterion.

-> Proof of termination for _3_8b_1_4:
-> -> Dependency pairs in cycle:
nF_le(s(x),s(y)) -> nF_le(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.