Termination w.r.t. Q proof of TRS/TRCSRExAppendixB_AEL03

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:

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))
a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(cons2(X1, X2)) → cons2(X1, mark(X2))
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

Q is empty.

↳ QTRS

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

a__from(X) → cons(mark(X), from(s(X)))
a__2ndspos(0, Z) → rnil
a__2ndspos(s(N), cons(X, Z)) → a__2ndspos(s(mark(N)), cons2(X, mark(Z)))
a__2ndspos(s(N), cons2(X, cons(Y, Z))) → rcons(posrecip(mark(Y)), a__2ndsneg(mark(N), mark(Z)))
a__2ndsneg(0, Z) → rnil
a__2ndsneg(s(N), cons(X, Z)) → a__2ndsneg(s(mark(N)), cons2(X, mark(Z)))
a__2ndsneg(s(N), cons2(X, cons(Y, Z))) → rcons(negrecip(mark(Y)), a__2ndspos(mark(N), mark(Z)))
a__pi(X) → a__2ndspos(mark(X), a__from(0))
a__plus(0, Y) → mark(Y)
a__plus(s(X), Y) → s(a__plus(mark(X), mark(Y)))
a__times(0, Y) → 0
a__times(s(X), Y) → a__plus(mark(Y), a__times(mark(X), mark(Y)))
a__square(X) → a__times(mark(X), mark(X))
mark(from(X)) → a__from(mark(X))
mark(2ndspos(X1, X2)) → a__2ndspos(mark(X1), mark(X2))
mark(2ndsneg(X1, X2)) → a__2ndsneg(mark(X1), mark(X2))
mark(pi(X)) → a__pi(mark(X))
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(times(X1, X2)) → a__times(mark(X1), mark(X2))
mark(square(X)) → a__square(mark(X))
mark(0) → 0
mark(s(X)) → s(mark(X))
mark(posrecip(X)) → posrecip(mark(X))
mark(negrecip(X)) → negrecip(mark(X))
mark(nil) → nil
mark(cons(X1, X2)) → cons(mark(X1), X2)
mark(cons2(X1, X2)) → cons2(X1, mark(X2))
mark(rnil) → rnil
mark(rcons(X1, X2)) → rcons(mark(X1), mark(X2))
a__from(X) → from(X)
a__2ndspos(X1, X2) → 2ndspos(X1, X2)
a__2ndsneg(X1, X2) → 2ndsneg(X1, X2)
a__pi(X) → pi(X)
a__plus(X1, X2) → plus(X1, X2)
a__times(X1, X2) → times(X1, X2)
a__square(X) → square(X)

Q is empty.