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