YES
(VAR x y z)
(RULES 
minus(x,0) -> x
minus(s(x),s(y)) -> minus(x,y)
quot(0,s(y)) -> 0
quot(s(x),s(y)) -> s(quot(minus(x,y),s(y)))
plus(0,y) -> y
plus(s(x),y) -> s(plus(x,y))
plus(minus(x,s(0)),minus(y,s(s(z)))) -> plus(minus(y,s(s(z))),minus(x,s(0)))
)

Proving termination of rewriting for _3_39:

-> Dependency pairs:
nF_minus(s(x),s(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_plus(s(x),y) -> nF_plus(x,y)
nF_plus(minus(x,s(0)),minus(y,s(s(z)))) -> nF_plus(minus(y,s(s(z))),minus(x,s(0)))
nF_plus(minus(x,s(0)),minus(y,s(s(z)))) -> nF_minus(y,s(s(z)))
nF_plus(minus(x,s(0)),minus(y,s(s(z)))) -> nF_minus(x,s(0))

-> Dependency pairs narrowed: 
nF_plus(minus(x,s(0)),minus(y,s(s(z)))) -> nF_plus(minus(y,s(s(z))),minus(x,s(0)))

-> New dependency pairs: 
nF_plus(minus(x,s(0)),minus(s(x),s(s(z)))) -> nF_plus(minus(x,s(z)),minus(x,s(0)))

-> Proof of termination for _3_39_1_1:
-> -> Dependency pairs in cycle:
nF_plus(s(x),y) -> nF_plus(x,y)
nF_plus(minus(x,s(0)),minus(s(x),s(s(z)))) -> nF_plus(minus(x,s(z)),minus(x,s(0)))

UsableRules:
minus(x,0) -> x
minus(s(x),s(y)) -> minus(x,y)

Polynomial Interpretation:

[minus](X1,X2) = X1
[0] = 0
[s](X) = X + 1
[quot](X1,X2) = 0
[plus](X1,X2) = 0
[nF_plus](X1,X2) = X2

TIME: 4.633e-2

-> -> Dependency pairs in cycle:
nF_plus(s(x),y) -> nF_plus(x,y)

Termination proved: Cycles verify subterm criterion.

-> Proof of termination for _3_39_1_2:
-> -> Dependency pairs in cycle:
nF_quot(s(x),s(y)) -> nF_quot(minus(x,y),s(y))

UsableRules:
minus(x,0) -> x
minus(s(x),s(y)) -> minus(x,y)

Polynomial Interpretation:

[minus](X1,X2) = X1
[0] = 0
[s](X) = X + 1
[quot](X1,X2) = 0
[plus](X1,X2) = 0
[nF_quot](X1,X2) = X1

TIME: 4.7752e-2


-> Proof of termination for _3_39_1_3:
-> -> Dependency pairs in cycle:
nF_minus(s(x),s(y)) -> nF_minus(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.