YES
(VAR X X1 X2)
(RULES 
active(f(0)) -> mark(cons(0,f(s(0))))
active(f(s(0))) -> mark(f(p(s(0))))
active(p(s(X))) -> mark(X)
active(f(X)) -> f(active(X))
active(cons(X1,X2)) -> cons(active(X1),X2)
active(s(X)) -> s(active(X))
active(p(X)) -> p(active(X))
f(mark(X)) -> mark(f(X))
cons(mark(X1),X2) -> mark(cons(X1,X2))
s(mark(X)) -> mark(s(X))
p(mark(X)) -> mark(p(X))
proper(f(X)) -> f(proper(X))
proper(0) -> ok(0)
proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
proper(s(X)) -> s(proper(X))
proper(p(X)) -> p(proper(X))
f(ok(X)) -> ok(f(X))
cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
s(ok(X)) -> ok(s(X))
p(ok(X)) -> ok(p(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))
)

Proving termination of rewriting for ExProp7_Luc06_C:

-> Dependency pairs:
nF_active(f(0)) -> nF_cons(0,f(s(0)))
nF_active(f(0)) -> nF_f(s(0))
nF_active(f(0)) -> nF_s(0)
nF_active(f(s(0))) -> nF_f(p(s(0)))
nF_active(f(s(0))) -> nF_p(s(0))
nF_active(f(s(0))) -> nF_s(0)
nF_active(f(X)) -> nF_f(active(X))
nF_active(f(X)) -> nF_active(X)
nF_active(cons(X1,X2)) -> nF_cons(active(X1),X2)
nF_active(cons(X1,X2)) -> nF_active(X1)
nF_active(s(X)) -> nF_s(active(X))
nF_active(s(X)) -> nF_active(X)
nF_active(p(X)) -> nF_p(active(X))
nF_active(p(X)) -> nF_active(X)
nF_f(mark(X)) -> nF_f(X)
nF_cons(mark(X1),X2) -> nF_cons(X1,X2)
nF_s(mark(X)) -> nF_s(X)
nF_p(mark(X)) -> nF_p(X)
nF_proper(f(X)) -> nF_f(proper(X))
nF_proper(f(X)) -> nF_proper(X)
nF_proper(cons(X1,X2)) -> nF_cons(proper(X1),proper(X2))
nF_proper(cons(X1,X2)) -> nF_proper(X1)
nF_proper(cons(X1,X2)) -> nF_proper(X2)
nF_proper(s(X)) -> nF_s(proper(X))
nF_proper(s(X)) -> nF_proper(X)
nF_proper(p(X)) -> nF_p(proper(X))
nF_proper(p(X)) -> nF_proper(X)
nF_f(ok(X)) -> nF_f(X)
nF_cons(ok(X1),ok(X2)) -> nF_cons(X1,X2)
nF_s(ok(X)) -> nF_s(X)
nF_p(ok(X)) -> nF_p(X)
nF_top(mark(X)) -> nF_top(proper(X))
nF_top(mark(X)) -> nF_proper(X)
nF_top(ok(X)) -> nF_top(active(X))
nF_top(ok(X)) -> nF_active(X)

-> Proof of termination for ExProp7_Luc06_C_1_1:
-> -> Dependency pairs in cycle:
nF_top(mark(X)) -> nF_top(proper(X))
nF_top(ok(X)) -> nF_top(active(X))

UsableRules:
active(f(0)) -> mark(cons(0,f(s(0))))
active(f(s(0))) -> mark(f(p(s(0))))
active(p(s(X))) -> mark(X)
active(f(X)) -> f(active(X))
active(cons(X1,X2)) -> cons(active(X1),X2)
active(s(X)) -> s(active(X))
active(p(X)) -> p(active(X))
f(mark(X)) -> mark(f(X))
cons(mark(X1),X2) -> mark(cons(X1,X2))
s(mark(X)) -> mark(s(X))
p(mark(X)) -> mark(p(X))
proper(f(X)) -> f(proper(X))
proper(0) -> ok(0)
proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
proper(s(X)) -> s(proper(X))
proper(p(X)) -> p(proper(X))
f(ok(X)) -> ok(f(X))
cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
s(ok(X)) -> ok(s(X))
p(ok(X)) -> ok(p(X))

Polynomial Interpretation:

[active](X) = X
[f](X) = 2.X.X + 2.X + 1
[0] = 2
[mark](X) = 2.X.X + 2.X + 1
[cons](X1,X2) = X1
[s](X) = 2.X.X + 2.X + 1
[p](X) = X.X + 2.X
[proper](X) = X.X + X
[ok](X) = X + 1
[top](X) = 0
[nF_top](X) = X

TIME: 5.822207

-> -> Dependency pairs in cycle:
nF_top(mark(X)) -> nF_top(proper(X))

UsableRules:
f(mark(X)) -> mark(f(X))
cons(mark(X1),X2) -> mark(cons(X1,X2))
s(mark(X)) -> mark(s(X))
p(mark(X)) -> mark(p(X))
proper(f(X)) -> f(proper(X))
proper(0) -> ok(0)
proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
proper(s(X)) -> s(proper(X))
proper(p(X)) -> p(proper(X))
f(ok(X)) -> ok(f(X))
cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
s(ok(X)) -> ok(s(X))
p(ok(X)) -> ok(p(X))

Polynomial Interpretation:

[active](X) = 0
[f](X) = X.X
[0] = 0
[mark](X) = 1
[cons](X1,X2) = X1
[s](X) = X.X
[p](X) = X
[proper](X) = 0
[ok](X) = 0
[top](X) = 0
[nF_top](X) = X

TIME: 2.865e-3


-> Proof of termination for ExProp7_Luc06_C_1_2:
-> -> Dependency pairs in cycle:
nF_proper(f(X)) -> nF_proper(X)
nF_proper(p(X)) -> nF_proper(X)
nF_proper(s(X)) -> nF_proper(X)
nF_proper(cons(X1,X2)) -> nF_proper(X2)
nF_proper(cons(X1,X2)) -> nF_proper(X1)

Termination proved: Cycles verify subterm criterion.

-> Proof of termination for ExProp7_Luc06_C_1_3:
-> -> Dependency pairs in cycle:
nF_active(f(X)) -> nF_active(X)
nF_active(p(X)) -> nF_active(X)
nF_active(s(X)) -> nF_active(X)
nF_active(cons(X1,X2)) -> nF_active(X1)

Termination proved: Cycles verify subterm criterion.

-> Proof of termination for ExProp7_Luc06_C_1_4:
-> -> Dependency pairs in cycle:
nF_cons(ok(X1),ok(X2)) -> nF_cons(X1,X2)
nF_cons(mark(X1),X2) -> nF_cons(X1,X2)

Termination proved: Cycles verify subterm criterion.

-> Proof of termination for ExProp7_Luc06_C_1_5:
-> -> Dependency pairs in cycle:
nF_s(ok(X)) -> nF_s(X)
nF_s(mark(X)) -> nF_s(X)

Termination proved: Cycles verify subterm criterion.

-> Proof of termination for ExProp7_Luc06_C_1_6:
-> -> Dependency pairs in cycle:
nF_p(ok(X)) -> nF_p(X)
nF_p(mark(X)) -> nF_p(X)

Termination proved: Cycles verify subterm criterion.

-> Proof of termination for ExProp7_Luc06_C_1_7:
-> -> Dependency pairs in cycle:
nF_f(ok(X)) -> nF_f(X)
nF_f(mark(X)) -> nF_f(X)

Termination proved: Cycles verify subterm criterion.

SETTINGS:
Base ordering: Polynomial ordering
Proof mode: SCCs in DG + base ordering
Upper bound for coeffs: 2
Rationals below 1 for all non-replacing args: No
Polynomial interpretation: Simple mixed
Coeffs in polynomials: No rationals
Delta: automatic



Termination was proved succesfully.