YES
(VAR X XS N Y YS X1 X2)
(RULES 
from(X) -> cons(X,n__from(n__s(X)))
sel(0,cons(X,XS)) -> X
sel(s(N),cons(X,XS)) -> sel(N,activate(XS))
minus(X,0) -> 0
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)))
zWquot(XS,nil) -> nil
zWquot(nil,XS) -> nil
zWquot(cons(X,XS),cons(Y,YS)) -> cons(quot(X,Y),n__zWquot(activate(XS),activate(YS)))
from(X) -> n__from(X)
s(X) -> n__s(X)
zWquot(X1,X2) -> n__zWquot(X1,X2)
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
activate(n__zWquot(X1,X2)) -> zWquot(activate(X1),activate(X2))
activate(X) -> X
)

Proving termination of rewriting for Ex4_7_37_Bor03_FR:

-> Dependency pairs:
nF_sel(s(N),cons(X,XS)) -> nF_sel(N,activate(XS))
nF_sel(s(N),cons(X,XS)) -> nF_activate(XS)
nF_minus(s(X),s(Y)) -> nF_minus(X,Y)
nF_quot(s(X),s(Y)) -> nF_s(quot(minus(X,Y),s(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_quot(s(X),s(Y)) -> nF_s(Y)
nF_zWquot(cons(X,XS),cons(Y,YS)) -> nF_quot(X,Y)
nF_zWquot(cons(X,XS),cons(Y,YS)) -> nF_activate(XS)
nF_zWquot(cons(X,XS),cons(Y,YS)) -> nF_activate(YS)
nF_activate(n__from(X)) -> nF_from(activate(X))
nF_activate(n__from(X)) -> nF_activate(X)
nF_activate(n__s(X)) -> nF_s(activate(X))
nF_activate(n__s(X)) -> nF_activate(X)
nF_activate(n__zWquot(X1,X2)) -> nF_zWquot(activate(X1),activate(X2))
nF_activate(n__zWquot(X1,X2)) -> nF_activate(X1)
nF_activate(n__zWquot(X1,X2)) -> nF_activate(X2)

-> Proof of termination for Ex4_7_37_Bor03_FR_1_1:
-> -> Dependency pairs in cycle:
nF_sel(s(N),cons(X,XS)) -> nF_sel(N,activate(XS))

Termination proved: Cycles verify subterm criterion.

-> Proof of termination for Ex4_7_37_Bor03_FR_1_2:
-> -> Dependency pairs in cycle:
nF_activate(n__zWquot(X1,X2)) -> nF_activate(X2)
nF_activate(n__zWquot(X1,X2)) -> nF_activate(X1)
nF_activate(n__s(X)) -> nF_activate(X)
nF_activate(n__from(X)) -> nF_activate(X)
nF_zWquot(cons(X,XS),cons(Y,YS)) -> nF_activate(YS)
nF_activate(n__zWquot(X1,X2)) -> nF_zWquot(activate(X1),activate(X2))
nF_zWquot(cons(X,XS),cons(Y,YS)) -> nF_activate(XS)

UsableRules:
from(X) -> cons(X,n__from(n__s(X)))
minus(X,0) -> 0
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)))
zWquot(XS,nil) -> nil
zWquot(nil,XS) -> nil
zWquot(cons(X,XS),cons(Y,YS)) -> cons(quot(X,Y),n__zWquot(activate(XS),activate(YS)))
from(X) -> n__from(X)
s(X) -> n__s(X)
zWquot(X1,X2) -> n__zWquot(X1,X2)
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
activate(n__zWquot(X1,X2)) -> zWquot(activate(X1),activate(X2))
activate(X) -> X

Polynomial Interpretation:

[from](X) = X
[cons](X1,X2) = X2
[n__from](X) = X
[n__s](X) = X
[sel](X1,X2) = 0
[0] = 0
[s](X) = X
[activate](X) = X
[minus](X1,X2) = 0
[quot](X1,X2) = 0
[zWquot](X1,X2) = X1 + X2 + 1
[nil] = 0
[n__zWquot](X1,X2) = X1 + X2 + 1
[nF_activate](X) = X
[nF_zWquot](X1,X2) = X1 + X2 + 1

TIME: 7.4129e-2

-> -> Dependency pairs in cycle:
nF_activate(n__zWquot(X1,X2)) -> nF_activate(X2)
nF_zWquot(cons(X,XS),cons(Y,YS)) -> nF_activate(YS)
nF_activate(n__zWquot(X1,X2)) -> nF_zWquot(activate(X1),activate(X2))
nF_activate(n__from(X)) -> nF_activate(X)
nF_activate(n__s(X)) -> nF_activate(X)
nF_activate(n__zWquot(X1,X2)) -> nF_activate(X1)

UsableRules:
from(X) -> cons(X,n__from(n__s(X)))
minus(X,0) -> 0
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)))
zWquot(XS,nil) -> nil
zWquot(nil,XS) -> nil
zWquot(cons(X,XS),cons(Y,YS)) -> cons(quot(X,Y),n__zWquot(activate(XS),activate(YS)))
from(X) -> n__from(X)
s(X) -> n__s(X)
zWquot(X1,X2) -> n__zWquot(X1,X2)
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
activate(n__zWquot(X1,X2)) -> zWquot(activate(X1),activate(X2))
activate(X) -> X

Polynomial Interpretation:

[from](X) = X
[cons](X1,X2) = X2
[n__from](X) = X
[n__s](X) = X
[sel](X1,X2) = 0
[0] = 1
[s](X) = X
[activate](X) = X
[minus](X1,X2) = X2
[quot](X1,X2) = 1
[zWquot](X1,X2) = X1 + X2 + 1
[nil] = 0
[n__zWquot](X1,X2) = X1 + X2 + 1
[nF_activate](X) = X
[nF_zWquot](X1,X2) = X2

TIME: 5.7713e-2

-> -> Dependency pairs in cycle:
nF_activate(n__zWquot(X1,X2)) -> nF_activate(X2)
nF_activate(n__s(X)) -> nF_activate(X)
nF_activate(n__from(X)) -> nF_activate(X)
nF_zWquot(cons(X,XS),cons(Y,YS)) -> nF_activate(YS)
nF_activate(n__zWquot(X1,X2)) -> nF_zWquot(activate(X1),activate(X2))

UsableRules:
from(X) -> cons(X,n__from(n__s(X)))
minus(X,0) -> 0
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)))
zWquot(XS,nil) -> nil
zWquot(nil,XS) -> nil
zWquot(cons(X,XS),cons(Y,YS)) -> cons(quot(X,Y),n__zWquot(activate(XS),activate(YS)))
from(X) -> n__from(X)
s(X) -> n__s(X)
zWquot(X1,X2) -> n__zWquot(X1,X2)
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
activate(n__zWquot(X1,X2)) -> zWquot(activate(X1),activate(X2))
activate(X) -> X

Polynomial Interpretation:

[from](X) = X
[cons](X1,X2) = X2
[n__from](X) = X
[n__s](X) = X
[sel](X1,X2) = 0
[0] = 1
[s](X) = X
[activate](X) = X
[minus](X1,X2) = X2
[quot](X1,X2) = 1
[zWquot](X1,X2) = X2 + 1
[nil] = 0
[n__zWquot](X1,X2) = X2 + 1
[nF_activate](X) = X
[nF_zWquot](X1,X2) = X2

TIME: 5.8856e-2

-> -> Dependency pairs in cycle:
nF_activate(n__zWquot(X1,X2)) -> nF_activate(X2)
nF_activate(n__from(X)) -> nF_activate(X)
nF_activate(n__s(X)) -> nF_activate(X)

Termination proved: Cycles verify subterm criterion.

-> Proof of termination for Ex4_7_37_Bor03_FR_1_3:
-> -> Dependency pairs in cycle:
nF_quot(s(X),s(Y)) -> nF_quot(minus(X,Y),s(Y))

UsableRules:
minus(X,0) -> 0
minus(s(X),s(Y)) -> minus(X,Y)
s(X) -> n__s(X)

Polynomial Interpretation:

[from](X) = 0
[cons](X1,X2) = 0
[n__from](X) = 0
[n__s](X) = 0
[sel](X1,X2) = 0
[0] = 0
[s](X) = 1
[activate](X) = 0
[minus](X1,X2) = 0
[quot](X1,X2) = 0
[zWquot](X1,X2) = 0
[nil] = 0
[n__zWquot](X1,X2) = 0
[nF_quot](X1,X2) = X1

TIME: 4.5595e-2


-> Proof of termination for Ex4_7_37_Bor03_FR_1_4:
-> -> 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.