NO Termination w.r.t. Q proof of /home/cern_httpd/provide/research/cycsrs/tpdb/TPDB-d9b80194f163/SRS_Standard/Zantema_04/z121-shift.srs

Termination w.r.t. Q of the given QTRS could be disproven:

0 QTRS
↳1 NonTerminationProof (⇒, 1081 ms)
↳2 NO

(0) Obligation:

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

B(x) → W(M(M(V(x))))
M(x) → x
M(V(f(x))) → V(Xf(x))
M(V(b(x))) → V(Xb(x))
M(V(a(x))) → V(Xa(x))
M(V(c(x))) → V(Xc(x))
M(V(d(x))) → V(Xd(x))
Xf(f(x)) → f(Xf(x))
Xf(b(x)) → b(Xf(x))
Xf(a(x)) → a(Xf(x))
Xf(c(x)) → c(Xf(x))
Xf(d(x)) → d(Xf(x))
Xb(f(x)) → f(Xb(x))
Xb(b(x)) → b(Xb(x))
Xb(a(x)) → a(Xb(x))
Xb(c(x)) → c(Xb(x))
Xb(d(x)) → d(Xb(x))
Xa(f(x)) → f(Xa(x))
Xa(b(x)) → b(Xa(x))
Xa(a(x)) → a(Xa(x))
Xa(c(x)) → c(Xa(x))
Xa(d(x)) → d(Xa(x))
Xc(f(x)) → f(Xc(x))
Xc(b(x)) → b(Xc(x))
Xc(a(x)) → a(Xc(x))
Xc(c(x)) → c(Xc(x))
Xc(d(x)) → d(Xc(x))
Xd(f(x)) → f(Xd(x))
Xd(b(x)) → b(Xd(x))
Xd(a(x)) → a(Xd(x))
Xd(c(x)) → c(Xd(x))
Xd(d(x)) → d(Xd(x))
Xf(E(x)) → f(E(x))
Xb(E(x)) → b(E(x))
Xa(E(x)) → a(E(x))
Xc(E(x)) → c(E(x))
Xd(E(x)) → d(E(x))
W(V(x)) → R(L(x))
L(f(x)) → Yf(L(x))
L(b(x)) → Yb(L(x))
L(a(x)) → Ya(L(x))
L(c(x)) → Yc(L(x))
L(d(x)) → Yd(L(x))
L(f(f(x))) → D(b(b(b(x))))
L(a(f(x))) → D(f(a(a(x))))
L(b(b(x))) → D(c(c(a(c(x)))))
L(d(b(x))) → D(d(a(b(x))))
L(c(c(x))) → D(d(d(d(x))))
L(b(d(x))) → D(d(b(x)))
L(c(d(d(x)))) → D(f(x))
Yf(D(x)) → D(f(x))
Yb(D(x)) → D(b(x))
Ya(D(x)) → D(a(x))
Yc(D(x)) → D(c(x))
Yd(D(x)) → D(d(x))
R(D(x)) → B(x)

Q is empty.

(1) NonTerminationProof (COMPLETE transformation)

We used the non-termination processor [OPPELT08] to show that the SRS problem is infinite.

Found the self-embedding DerivationStructure:
W V b d E → W V b d E

W V b d E → W V b d E
by OverlapClosure OC 3
W V b d E → W M V b d E
by OverlapClosure OC 2
W V b d → W M V b Xd
by OverlapClosure OC 2
W V b d → W M V Xd b
by OverlapClosure OC 3
W V b d → B d b
by OverlapClosure OC 3
W V b d → R D d b
by OverlapClosure OC 2
W V → R L
by original rule (OC 1)
L b d → D d b
by original rule (OC 1)
R D → B
by original rule (OC 1)
B d → W M V Xd
by OverlapClosure OC 2
B → W M M V
by original rule (OC 1)
M V d → V Xd
by original rule (OC 1)
Xd b → b Xd
by original rule (OC 1)
Xd E → d E
by original rule (OC 1)
M →
by original rule (OC 1)

(2) NO