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(b(d(b(d(a(x1))))))) → a(a(c(a(a(b(d(x1)))))))
a(a(c(x1))) → c(c(a(a(x1))))
c(c(c(x1))) → b(d(c(b(d(x1)))))
Q is empty.
↳ QTRS
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
a(a(b(d(b(d(a(x1))))))) → a(a(c(a(a(b(d(x1)))))))
a(a(c(x1))) → c(c(a(a(x1))))
c(c(c(x1))) → b(d(c(b(d(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:
A(a(b(d(b(d(a(x1))))))) → C(a(a(b(d(x1)))))
A(a(c(x1))) → C(a(a(x1)))
A(a(b(d(b(d(a(x1))))))) → A(b(d(x1)))
C(c(c(x1))) → C(b(d(x1)))
A(a(b(d(b(d(a(x1))))))) → A(a(c(a(a(b(d(x1)))))))
A(a(b(d(b(d(a(x1))))))) → A(c(a(a(b(d(x1))))))
A(a(c(x1))) → A(x1)
A(a(b(d(b(d(a(x1))))))) → A(a(b(d(x1))))
A(a(c(x1))) → A(a(x1))
A(a(c(x1))) → C(c(a(a(x1))))
The TRS R consists of the following rules:
a(a(b(d(b(d(a(x1))))))) → a(a(c(a(a(b(d(x1)))))))
a(a(c(x1))) → c(c(a(a(x1))))
c(c(c(x1))) → b(d(c(b(d(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:
A(a(b(d(b(d(a(x1))))))) → C(a(a(b(d(x1)))))
A(a(c(x1))) → C(a(a(x1)))
A(a(b(d(b(d(a(x1))))))) → A(b(d(x1)))
C(c(c(x1))) → C(b(d(x1)))
A(a(b(d(b(d(a(x1))))))) → A(a(c(a(a(b(d(x1)))))))
A(a(b(d(b(d(a(x1))))))) → A(c(a(a(b(d(x1))))))
A(a(c(x1))) → A(x1)
A(a(b(d(b(d(a(x1))))))) → A(a(b(d(x1))))
A(a(c(x1))) → A(a(x1))
A(a(c(x1))) → C(c(a(a(x1))))
The TRS R consists of the following rules:
a(a(b(d(b(d(a(x1))))))) → a(a(c(a(a(b(d(x1)))))))
a(a(c(x1))) → c(c(a(a(x1))))
c(c(c(x1))) → b(d(c(b(d(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 5 less nodes.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
A(a(b(d(b(d(a(x1))))))) → A(a(c(a(a(b(d(x1)))))))
A(a(b(d(b(d(a(x1))))))) → A(c(a(a(b(d(x1))))))
A(a(c(x1))) → A(x1)
A(a(b(d(b(d(a(x1))))))) → A(a(b(d(x1))))
A(a(c(x1))) → A(a(x1))
The TRS R consists of the following rules:
a(a(b(d(b(d(a(x1))))))) → a(a(c(a(a(b(d(x1)))))))
a(a(c(x1))) → c(c(a(a(x1))))
c(c(c(x1))) → b(d(c(b(d(x1)))))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.