YES
(VAR N X Y Z X1 X2)
(RULES 
t(N) -> cs(r(q(N)),nt(ns(N)))
q(0) -> 0
q(s(X)) -> s(p(q(X),d(X)))
d(0) -> 0
d(s(X)) -> s(s(d(X)))
p(0,X) -> X
p(X,0) -> X
p(s(X),s(Y)) -> s(s(p(X,Y)))
f(0,X) -> nil
f(s(X),cs(Y,Z)) -> cs(Y,nf(X,a(Z)))
t(X) -> nt(X)
s(X) -> ns(X)
f(X1,X2) -> nf(X1,X2)
a(nt(X)) -> t(a(X))
a(ns(X)) -> s(a(X))
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(X) -> X
)

Proving termination of rewriting for secret5:

-> Dependency pairs:
nF_t(N) -> nF_q(N)
nF_q(s(X)) -> nF_s(p(q(X),d(X)))
nF_q(s(X)) -> nF_p(q(X),d(X))
nF_q(s(X)) -> nF_q(X)
nF_q(s(X)) -> nF_d(X)
nF_d(s(X)) -> nF_s(s(d(X)))
nF_d(s(X)) -> nF_s(d(X))
nF_d(s(X)) -> nF_d(X)
nF_p(s(X),s(Y)) -> nF_s(s(p(X,Y)))
nF_p(s(X),s(Y)) -> nF_s(p(X,Y))
nF_p(s(X),s(Y)) -> nF_p(X,Y)
nF_f(s(X),cs(Y,Z)) -> nF_a(Z)
nF_a(nt(X)) -> nF_t(a(X))
nF_a(nt(X)) -> nF_a(X)
nF_a(ns(X)) -> nF_s(a(X))
nF_a(ns(X)) -> nF_a(X)
nF_a(nf(X1,X2)) -> nF_f(a(X1),a(X2))
nF_a(nf(X1,X2)) -> nF_a(X1)
nF_a(nf(X1,X2)) -> nF_a(X2)

-> Proof of termination for secret5_1_1:
-> -> Dependency pairs in cycle:
nF_f(s(X),cs(Y,Z)) -> nF_a(Z)
nF_a(nf(X1,X2)) -> nF_f(a(X1),a(X2))
nF_a(nf(X1,X2)) -> nF_a(X2)
nF_a(nf(X1,X2)) -> nF_a(X1)
nF_a(ns(X)) -> nF_a(X)
nF_a(nt(X)) -> nF_a(X)

UsableRules:
t(N) -> cs(r(q(N)),nt(ns(N)))
q(0) -> 0
q(s(X)) -> s(p(q(X),d(X)))
d(0) -> 0
d(s(X)) -> s(s(d(X)))
p(0,X) -> X
p(X,0) -> X
p(s(X),s(Y)) -> s(s(p(X,Y)))
f(0,X) -> nil
f(s(X),cs(Y,Z)) -> cs(Y,nf(X,a(Z)))
t(X) -> nt(X)
s(X) -> ns(X)
f(X1,X2) -> nf(X1,X2)
a(nt(X)) -> t(a(X))
a(ns(X)) -> s(a(X))
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(X) -> X

Polynomial Interpretation:

[t](X) = X + 1
[cs](X1,X2) = X2
[r](X) = 0
[q](X) = 0
[nt](X) = X + 1
[ns](X) = X
[0] = 0
[s](X) = X
[p](X1,X2) = X1 + X2
[d](X) = 0
[f](X1,X2) = X1 + X2
[nil] = 0
[nf](X1,X2) = X1 + X2
[a](X) = X
[nF_f](X1,X2) = X2
[nF_a](X) = X

TIME: 5.1975e-2

-> -> Dependency pairs in cycle:
nF_f(s(X),cs(Y,Z)) -> nF_a(Z)
nF_a(nf(X1,X2)) -> nF_f(a(X1),a(X2))
nF_a(ns(X)) -> nF_a(X)
nF_a(nf(X1,X2)) -> nF_a(X1)
nF_a(nf(X1,X2)) -> nF_a(X2)

UsableRules:
t(N) -> cs(r(q(N)),nt(ns(N)))
q(0) -> 0
q(s(X)) -> s(p(q(X),d(X)))
d(0) -> 0
d(s(X)) -> s(s(d(X)))
p(0,X) -> X
p(X,0) -> X
p(s(X),s(Y)) -> s(s(p(X,Y)))
f(0,X) -> nil
f(s(X),cs(Y,Z)) -> cs(Y,nf(X,a(Z)))
t(X) -> nt(X)
s(X) -> ns(X)
f(X1,X2) -> nf(X1,X2)
a(nt(X)) -> t(a(X))
a(ns(X)) -> s(a(X))
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(X) -> X

Polynomial Interpretation:

[t](X) = 0
[cs](X1,X2) = X2
[r](X) = 0
[q](X) = 0
[nt](X) = 0
[ns](X) = X
[0] = 0
[s](X) = X
[p](X1,X2) = X1 + X2
[d](X) = 0
[f](X1,X2) = X1 + X2 + 1
[nil] = 0
[nf](X1,X2) = X1 + X2 + 1
[a](X) = X
[nF_f](X1,X2) = X2
[nF_a](X) = X

TIME: 5.1907e-2

-> -> Dependency pairs in cycle:
nF_f(s(X),cs(Y,Z)) -> nF_a(Z)
nF_a(nf(X1,X2)) -> nF_f(a(X1),a(X2))
nF_a(nf(X1,X2)) -> nF_a(X1)
nF_a(ns(X)) -> nF_a(X)

UsableRules:
t(N) -> cs(r(q(N)),nt(ns(N)))
q(0) -> 0
q(s(X)) -> s(p(q(X),d(X)))
d(0) -> 0
d(s(X)) -> s(s(d(X)))
p(0,X) -> X
p(X,0) -> X
p(s(X),s(Y)) -> s(s(p(X,Y)))
f(0,X) -> nil
f(s(X),cs(Y,Z)) -> cs(Y,nf(X,a(Z)))
t(X) -> nt(X)
s(X) -> ns(X)
f(X1,X2) -> nf(X1,X2)
a(nt(X)) -> t(a(X))
a(ns(X)) -> s(a(X))
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(X) -> X

Polynomial Interpretation:

[t](X) = 0
[cs](X1,X2) = X2
[r](X) = 0
[q](X) = 0
[nt](X) = 0
[ns](X) = X
[0] = 0
[s](X) = X
[p](X1,X2) = X1 + X2
[d](X) = 0
[f](X1,X2) = X1 + X2 + 1
[nil] = 0
[nf](X1,X2) = X1 + X2 + 1
[a](X) = X
[nF_f](X1,X2) = X2
[nF_a](X) = X

TIME: 5.3641e-2

-> -> Dependency pairs in cycle:
nF_f(s(X),cs(Y,Z)) -> nF_a(Z)
nF_a(nf(X1,X2)) -> nF_f(a(X1),a(X2))
nF_a(ns(X)) -> nF_a(X)

UsableRules:
t(N) -> cs(r(q(N)),nt(ns(N)))
q(0) -> 0
q(s(X)) -> s(p(q(X),d(X)))
d(0) -> 0
d(s(X)) -> s(s(d(X)))
p(0,X) -> X
p(X,0) -> X
p(s(X),s(Y)) -> s(s(p(X,Y)))
f(0,X) -> nil
f(s(X),cs(Y,Z)) -> cs(Y,nf(X,a(Z)))
t(X) -> nt(X)
s(X) -> ns(X)
f(X1,X2) -> nf(X1,X2)
a(nt(X)) -> t(a(X))
a(ns(X)) -> s(a(X))
a(nf(X1,X2)) -> f(a(X1),a(X2))
a(X) -> X

Polynomial Interpretation:

[t](X) = 0
[cs](X1,X2) = X2
[r](X) = 0
[q](X) = 0
[nt](X) = 0
[ns](X) = X
[0] = 0
[s](X) = X
[p](X1,X2) = X1 + X2
[d](X) = 0
[f](X1,X2) = X2 + 1
[nil] = 0
[nf](X1,X2) = X2 + 1
[a](X) = X
[nF_f](X1,X2) = X2
[nF_a](X) = X

TIME: 5.015e-2

-> -> Dependency pairs in cycle:
nF_a(ns(X)) -> nF_a(X)

Termination proved: Cycles verify subterm criterion.

-> Proof of termination for secret5_1_2:
-> -> Dependency pairs in cycle:
nF_q(s(X)) -> nF_q(X)

Termination proved: Cycles verify subterm criterion.

-> Proof of termination for secret5_1_3:
-> -> Dependency pairs in cycle:
nF_p(s(X),s(Y)) -> nF_p(X,Y)

Termination proved: Cycles verify subterm criterion.

-> Proof of termination for secret5_1_4:
-> -> Dependency pairs in cycle:
nF_d(s(X)) -> nF_d(X)

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.