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(l(x1)) → l(a(x1))
a(c(x1)) → c(a(x1))
c(a(r(x1))) → r(a(x1))
l(r(a(a(x1)))) → a(a(l(c(c(c(r(x1)))))))
Q is empty.
↳ QTRS
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
a(l(x1)) → l(a(x1))
a(c(x1)) → c(a(x1))
c(a(r(x1))) → r(a(x1))
l(r(a(a(x1)))) → a(a(l(c(c(c(r(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:
L(r(a(a(x1)))) → A(l(c(c(c(r(x1))))))
A(l(x1)) → L(a(x1))
L(r(a(a(x1)))) → C(r(x1))
C(a(r(x1))) → A(x1)
L(r(a(a(x1)))) → A(a(l(c(c(c(r(x1)))))))
A(c(x1)) → A(x1)
L(r(a(a(x1)))) → L(c(c(c(r(x1)))))
L(r(a(a(x1)))) → C(c(r(x1)))
A(l(x1)) → A(x1)
A(c(x1)) → C(a(x1))
L(r(a(a(x1)))) → C(c(c(r(x1))))
The TRS R consists of the following rules:
a(l(x1)) → l(a(x1))
a(c(x1)) → c(a(x1))
c(a(r(x1))) → r(a(x1))
l(r(a(a(x1)))) → a(a(l(c(c(c(r(x1)))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
L(r(a(a(x1)))) → A(l(c(c(c(r(x1))))))
A(l(x1)) → L(a(x1))
L(r(a(a(x1)))) → C(r(x1))
C(a(r(x1))) → A(x1)
L(r(a(a(x1)))) → A(a(l(c(c(c(r(x1)))))))
A(c(x1)) → A(x1)
L(r(a(a(x1)))) → L(c(c(c(r(x1)))))
L(r(a(a(x1)))) → C(c(r(x1)))
A(l(x1)) → A(x1)
A(c(x1)) → C(a(x1))
L(r(a(a(x1)))) → C(c(c(r(x1))))
The TRS R consists of the following rules:
a(l(x1)) → l(a(x1))
a(c(x1)) → c(a(x1))
c(a(r(x1))) → r(a(x1))
l(r(a(a(x1)))) → a(a(l(c(c(c(r(x1)))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 4 less nodes.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
L(r(a(a(x1)))) → A(l(c(c(c(r(x1))))))
A(l(x1)) → L(a(x1))
C(a(r(x1))) → A(x1)
A(c(x1)) → A(x1)
L(r(a(a(x1)))) → A(a(l(c(c(c(r(x1)))))))
A(l(x1)) → A(x1)
A(c(x1)) → C(a(x1))
The TRS R consists of the following rules:
a(l(x1)) → l(a(x1))
a(c(x1)) → c(a(x1))
c(a(r(x1))) → r(a(x1))
l(r(a(a(x1)))) → a(a(l(c(c(c(r(x1)))))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.