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(x1) → x1
a(a(x1)) → a(b(c(c(x1))))
c(x1) → b(x1)
c(b(x1)) → a(x1)
Q is empty.
↳ QTRS
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
a(x1) → x1
a(a(x1)) → a(b(c(c(x1))))
c(x1) → b(x1)
c(b(x1)) → 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(b(x1)) → A(x1)
A(a(x1)) → C(x1)
A(a(x1)) → C(c(x1))
A(a(x1)) → A(b(c(c(x1))))
The TRS R consists of the following rules:
a(x1) → x1
a(a(x1)) → a(b(c(c(x1))))
c(x1) → b(x1)
c(b(x1)) → a(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:
C(b(x1)) → A(x1)
A(a(x1)) → C(x1)
A(a(x1)) → C(c(x1))
A(a(x1)) → A(b(c(c(x1))))
The TRS R consists of the following rules:
a(x1) → x1
a(a(x1)) → a(b(c(c(x1))))
c(x1) → b(x1)
c(b(x1)) → a(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 1 less node.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
C(b(x1)) → A(x1)
A(a(x1)) → C(x1)
A(a(x1)) → C(c(x1))
The TRS R consists of the following rules:
a(x1) → x1
a(a(x1)) → a(b(c(c(x1))))
c(x1) → b(x1)
c(b(x1)) → a(x1)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.