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(a(a(a(x1)))) → b(b(b(b(b(b(x1))))))
b(b(b(b(x1)))) → c(c(c(c(c(c(x1))))))
c(c(c(c(x1)))) → d(d(d(d(d(d(x1))))))
b(b(x1)) → d(d(d(d(x1))))
c(c(d(d(d(d(x1)))))) → a(a(x1))
Q is empty.
↳ QTRS
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(a(a(x1)))) → b(b(b(b(b(b(x1))))))
b(b(b(b(x1)))) → c(c(c(c(c(c(x1))))))
c(c(c(c(x1)))) → d(d(d(d(d(d(x1))))))
b(b(x1)) → d(d(d(d(x1))))
c(c(d(d(d(d(x1)))))) → a(a(x1))
Q is empty.
Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:
C(c(d(d(d(d(x1)))))) → A(a(x1))
A(a(a(a(x1)))) → B(b(x1))
B(b(b(b(x1)))) → C(c(c(x1)))
B(b(b(b(x1)))) → C(c(x1))
C(c(d(d(d(d(x1)))))) → A(x1)
A(a(a(a(x1)))) → B(b(b(b(x1))))
B(b(b(b(x1)))) → C(c(c(c(c(c(x1))))))
B(b(b(b(x1)))) → C(c(c(c(x1))))
B(b(b(b(x1)))) → C(c(c(c(c(x1)))))
A(a(a(a(x1)))) → B(b(b(b(b(b(x1))))))
A(a(a(a(x1)))) → B(x1)
A(a(a(a(x1)))) → B(b(b(b(b(x1)))))
B(b(b(b(x1)))) → C(x1)
A(a(a(a(x1)))) → B(b(b(x1)))
The TRS R consists of the following rules:
a(a(a(a(x1)))) → b(b(b(b(b(b(x1))))))
b(b(b(b(x1)))) → c(c(c(c(c(c(x1))))))
c(c(c(c(x1)))) → d(d(d(d(d(d(x1))))))
b(b(x1)) → d(d(d(d(x1))))
c(c(d(d(d(d(x1)))))) → a(a(x1))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
C(c(d(d(d(d(x1)))))) → A(a(x1))
A(a(a(a(x1)))) → B(b(x1))
B(b(b(b(x1)))) → C(c(c(x1)))
B(b(b(b(x1)))) → C(c(x1))
C(c(d(d(d(d(x1)))))) → A(x1)
A(a(a(a(x1)))) → B(b(b(b(x1))))
B(b(b(b(x1)))) → C(c(c(c(c(c(x1))))))
B(b(b(b(x1)))) → C(c(c(c(x1))))
B(b(b(b(x1)))) → C(c(c(c(c(x1)))))
A(a(a(a(x1)))) → B(b(b(b(b(b(x1))))))
A(a(a(a(x1)))) → B(x1)
A(a(a(a(x1)))) → B(b(b(b(b(x1)))))
B(b(b(b(x1)))) → C(x1)
A(a(a(a(x1)))) → B(b(b(x1)))
The TRS R consists of the following rules:
a(a(a(a(x1)))) → b(b(b(b(b(b(x1))))))
b(b(b(b(x1)))) → c(c(c(c(c(c(x1))))))
c(c(c(c(x1)))) → d(d(d(d(d(d(x1))))))
b(b(x1)) → d(d(d(d(x1))))
c(c(d(d(d(d(x1)))))) → a(a(x1))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.