YES
(VAR X Z Y X1 X2)
(RULES 
from(X) -> cons(X,n__from(n__s(X)))
first(0,Z) -> nil
first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
sel(0,cons(X,Z)) -> X
sel(s(X),cons(Y,Z)) -> sel(X,activate(Z))
from(X) -> n__from(X)
s(X) -> n__s(X)
first(X1,X2) -> n__first(X1,X2)
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(X) -> X
)

Proving termination of rewriting for Ex1_Luc02b_FR:

-> Dependency pairs:
nF_first(s(X),cons(Y,Z)) -> nF_activate(Z)
nF_sel(s(X),cons(Y,Z)) -> nF_sel(X,activate(Z))
nF_sel(s(X),cons(Y,Z)) -> nF_activate(Z)
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__first(X1,X2)) -> nF_first(activate(X1),activate(X2))
nF_activate(n__first(X1,X2)) -> nF_activate(X1)
nF_activate(n__first(X1,X2)) -> nF_activate(X2)

-> Proof of termination for Ex1_Luc02b_FR_1_1:
-> -> Dependency pairs in cycle:
nF_sel(s(X),cons(Y,Z)) -> nF_sel(X,activate(Z))

Termination proved: Cycles verify subterm criterion.

-> Proof of termination for Ex1_Luc02b_FR_1_2:
-> -> Dependency pairs in cycle:
nF_first(s(X),cons(Y,Z)) -> nF_activate(Z)
nF_activate(n__first(X1,X2)) -> nF_first(activate(X1),activate(X2))
nF_activate(n__first(X1,X2)) -> nF_activate(X2)
nF_activate(n__first(X1,X2)) -> nF_activate(X1)
nF_activate(n__s(X)) -> nF_activate(X)
nF_activate(n__from(X)) -> nF_activate(X)

UsableRules:
from(X) -> cons(X,n__from(n__s(X)))
first(0,Z) -> nil
first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
from(X) -> n__from(X)
s(X) -> n__s(X)
first(X1,X2) -> n__first(X1,X2)
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(X) -> X

Polynomial Interpretation:

[from](X) = X + 1
[cons](X1,X2) = X2
[n__from](X) = X + 1
[n__s](X) = X
[first](X1,X2) = X1 + X2
[0] = 1
[nil] = 0
[s](X) = X
[n__first](X1,X2) = X1 + X2
[activate](X) = X
[sel](X1,X2) = 0
[nF_first](X1,X2) = X2
[nF_activate](X) = X

TIME: 6.734e-2

-> -> Dependency pairs in cycle:
nF_first(s(X),cons(Y,Z)) -> nF_activate(Z)
nF_activate(n__first(X1,X2)) -> nF_first(activate(X1),activate(X2))
nF_activate(n__s(X)) -> nF_activate(X)
nF_activate(n__first(X1,X2)) -> nF_activate(X1)
nF_activate(n__first(X1,X2)) -> nF_activate(X2)

UsableRules:
from(X) -> cons(X,n__from(n__s(X)))
first(0,Z) -> nil
first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
from(X) -> n__from(X)
s(X) -> n__s(X)
first(X1,X2) -> n__first(X1,X2)
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(X) -> X

Polynomial Interpretation:

[from](X) = 0
[cons](X1,X2) = X2
[n__from](X) = 0
[n__s](X) = X
[first](X1,X2) = X1 + X2 + 1
[0] = 1
[nil] = 0
[s](X) = X
[n__first](X1,X2) = X1 + X2 + 1
[activate](X) = X
[sel](X1,X2) = 0
[nF_first](X1,X2) = X2
[nF_activate](X) = X

TIME: 5.7406e-2

-> -> Dependency pairs in cycle:
nF_first(s(X),cons(Y,Z)) -> nF_activate(Z)
nF_activate(n__first(X1,X2)) -> nF_first(activate(X1),activate(X2))
nF_activate(n__first(X1,X2)) -> nF_activate(X1)
nF_activate(n__s(X)) -> nF_activate(X)

UsableRules:
from(X) -> cons(X,n__from(n__s(X)))
first(0,Z) -> nil
first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
from(X) -> n__from(X)
s(X) -> n__s(X)
first(X1,X2) -> n__first(X1,X2)
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(X) -> X

Polynomial Interpretation:

[from](X) = 0
[cons](X1,X2) = X2
[n__from](X) = 0
[n__s](X) = X + 1
[first](X1,X2) = X1 + X2
[0] = 1
[nil] = 0
[s](X) = X + 1
[n__first](X1,X2) = X1 + X2
[activate](X) = X
[sel](X1,X2) = 0
[nF_first](X1,X2) = X2
[nF_activate](X) = X

TIME: 5.71e-2

-> -> Dependency pairs in cycle:
nF_first(s(X),cons(Y,Z)) -> nF_activate(Z)
nF_activate(n__first(X1,X2)) -> nF_first(activate(X1),activate(X2))
nF_activate(n__first(X1,X2)) -> nF_activate(X1)

UsableRules:
from(X) -> cons(X,n__from(n__s(X)))
first(0,Z) -> nil
first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
from(X) -> n__from(X)
s(X) -> n__s(X)
first(X1,X2) -> n__first(X1,X2)
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(X) -> X

Polynomial Interpretation:

[from](X) = 0
[cons](X1,X2) = X2
[n__from](X) = 0
[n__s](X) = X
[first](X1,X2) = X1 + X2 + 1
[0] = 1
[nil] = 0
[s](X) = X
[n__first](X1,X2) = X1 + X2 + 1
[activate](X) = X
[sel](X1,X2) = 0
[nF_first](X1,X2) = X2
[nF_activate](X) = X

TIME: 5.8584e-2

-> -> Dependency pairs in cycle:
nF_first(s(X),cons(Y,Z)) -> nF_activate(Z)
nF_activate(n__first(X1,X2)) -> nF_first(activate(X1),activate(X2))

UsableRules:
from(X) -> cons(X,n__from(n__s(X)))
first(0,Z) -> nil
first(s(X),cons(Y,Z)) -> cons(Y,n__first(X,activate(Z)))
from(X) -> n__from(X)
s(X) -> n__s(X)
first(X1,X2) -> n__first(X1,X2)
activate(n__from(X)) -> from(activate(X))
activate(n__s(X)) -> s(activate(X))
activate(n__first(X1,X2)) -> first(activate(X1),activate(X2))
activate(X) -> X

Polynomial Interpretation:

[from](X) = 0
[cons](X1,X2) = X2
[n__from](X) = 0
[n__s](X) = X + 1
[first](X1,X2) = X1 + X2
[0] = 0
[nil] = 0
[s](X) = X + 1
[n__first](X1,X2) = X1 + X2
[activate](X) = X
[sel](X1,X2) = 0
[nF_first](X1,X2) = X1 + X2
[nF_activate](X) = X + 1

TIME: 6.4614e-2


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.