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(b(x1)) → b(b(c(a(x1))))
b(x1) → x1
c(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(b(x1)) → b(b(c(a(x1))))
b(x1) → x1
c(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:
A(b(x1)) → B(b(c(a(x1))))
A(b(x1)) → B(c(a(x1)))
C(c(b(x1))) → A(x1)
A(b(x1)) → C(a(x1))
A(b(x1)) → A(x1)
The TRS R consists of the following rules:
a(x1) → x1
a(b(x1)) → b(b(c(a(x1))))
b(x1) → x1
c(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:
A(b(x1)) → B(b(c(a(x1))))
A(b(x1)) → B(c(a(x1)))
C(c(b(x1))) → A(x1)
A(b(x1)) → C(a(x1))
A(b(x1)) → A(x1)
The TRS R consists of the following rules:
a(x1) → x1
a(b(x1)) → b(b(c(a(x1))))
b(x1) → x1
c(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 2 less nodes.
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
C(c(b(x1))) → A(x1)
A(b(x1)) → C(a(x1))
A(b(x1)) → A(x1)
The TRS R consists of the following rules:
a(x1) → x1
a(b(x1)) → b(b(c(a(x1))))
b(x1) → x1
c(c(b(x1))) → a(x1)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule A(b(x1)) → C(a(x1)) at position [0] we obtained the following new rules:
A(b(b(x0))) → C(b(b(c(a(x0)))))
A(b(x0)) → C(x0)
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Narrowing
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
A(b(b(x0))) → C(b(b(c(a(x0)))))
C(c(b(x1))) → A(x1)
A(b(x0)) → C(x0)
A(b(x1)) → A(x1)
The TRS R consists of the following rules:
a(x1) → x1
a(b(x1)) → b(b(c(a(x1))))
b(x1) → x1
c(c(b(x1))) → a(x1)
Q is empty.
We have to consider all minimal (P,Q,R)-chains.