Termination w.r.t. Q proof of TRS/TRCSREx5_7_Luc97

Termination w.r.t. Q of the following Term Rewriting System could not be shown:

Q restricted rewrite system:
The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(n__dbl(activate(X)), n__dbls(activate(Y)))
sel(0, cons(X, Y)) → activate(X)
sel(s(X), cons(Y, Z)) → sel(activate(X), activate(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(n__sel(activate(X), activate(Z)), n__indx(activate(Y), activate(Z)))
from(X) → cons(activate(X), n__from(n__s(activate(X))))
dbl1(0) → 01
dbl1(s(X)) → s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) → activate(X)
sel1(s(X), cons(Y, Z)) → sel1(activate(X), activate(Z))
quote(0) → 01
quote(s(X)) → s1(quote(activate(X)))
quote(dbl(X)) → dbl1(X)
quote(sel(X, Y)) → sel1(X, Y)
s(X) → n__s(X)
dbl(X) → n__dbl(X)
dbls(X) → n__dbls(X)
sel(X1, X2) → n__sel(X1, X2)
indx(X1, X2) → n__indx(X1, X2)
from(X) → n__from(X)
activate(n__s(X)) → s(X)
activate(n__dbl(X)) → dbl(X)
activate(n__dbls(X)) → dbls(X)
activate(n__sel(X1, X2)) → sel(X1, X2)
activate(n__indx(X1, X2)) → indx(X1, X2)
activate(n__from(X)) → from(X)
activate(X) → X

Q is empty.

↳ QTRS

Q restricted rewrite system:
The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(n__dbl(activate(X)), n__dbls(activate(Y)))
sel(0, cons(X, Y)) → activate(X)
sel(s(X), cons(Y, Z)) → sel(activate(X), activate(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(n__sel(activate(X), activate(Z)), n__indx(activate(Y), activate(Z)))
from(X) → cons(activate(X), n__from(n__s(activate(X))))
dbl1(0) → 01
dbl1(s(X)) → s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) → activate(X)
sel1(s(X), cons(Y, Z)) → sel1(activate(X), activate(Z))
quote(0) → 01
quote(s(X)) → s1(quote(activate(X)))
quote(dbl(X)) → dbl1(X)
quote(sel(X, Y)) → sel1(X, Y)
s(X) → n__s(X)
dbl(X) → n__dbl(X)
dbls(X) → n__dbls(X)
sel(X1, X2) → n__sel(X1, X2)
indx(X1, X2) → n__indx(X1, X2)
from(X) → n__from(X)
activate(n__s(X)) → s(X)
activate(n__dbl(X)) → dbl(X)
activate(n__dbls(X)) → dbls(X)
activate(n__sel(X1, X2)) → sel(X1, X2)
activate(n__indx(X1, X2)) → indx(X1, X2)
activate(n__from(X)) → from(X)
activate(X) → X

Q is empty.