Termination w.r.t. Q of the following Term Rewriting System could be proven:

Q restricted rewrite system:
The TRS R consists of the following rules:

a1(a1(x)) -> b1(b1(x))
b1(b1(a1(x))) -> a1(b1(b1(x)))

Q is empty.


QTRS
  ↳ DependencyPairsProof

Q restricted rewrite system:
The TRS R consists of the following rules:

a1(a1(x)) -> b1(b1(x))
b1(b1(a1(x))) -> a1(b1(b1(x)))

Q is empty.

Q DP problem:
The TRS P consists of the following rules:

A1(a1(x)) -> B1(x)
B1(b1(a1(x))) -> A1(b1(b1(x)))
B1(b1(a1(x))) -> B1(x)
B1(b1(a1(x))) -> B1(b1(x))
A1(a1(x)) -> B1(b1(x))

The TRS R consists of the following rules:

a1(a1(x)) -> b1(b1(x))
b1(b1(a1(x))) -> a1(b1(b1(x)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

↳ QTRS
  ↳ DependencyPairsProof
QDP
      ↳ QDPAfsSolverProof

Q DP problem:
The TRS P consists of the following rules:

A1(a1(x)) -> B1(x)
B1(b1(a1(x))) -> A1(b1(b1(x)))
B1(b1(a1(x))) -> B1(x)
B1(b1(a1(x))) -> B1(b1(x))
A1(a1(x)) -> B1(b1(x))

The TRS R consists of the following rules:

a1(a1(x)) -> b1(b1(x))
b1(b1(a1(x))) -> a1(b1(b1(x)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using an argument filtering and a montonic ordering, at least one Dependency Pair of this SCC can be strictly oriented.

A1(a1(x)) -> B1(x)
B1(b1(a1(x))) -> A1(b1(b1(x)))
B1(b1(a1(x))) -> B1(x)
B1(b1(a1(x))) -> B1(b1(x))
A1(a1(x)) -> B1(b1(x))
Used argument filtering: A1(x1)  =  x1
a1(x1)  =  a1(x1)
B1(x1)  =  x1
b1(x1)  =  x1
Used ordering: Quasi Precedence: trivial


↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ QDPAfsSolverProof
QDP
          ↳ PisEmptyProof

Q DP problem:
P is empty.
The TRS R consists of the following rules:

a1(a1(x)) -> b1(b1(x))
b1(b1(a1(x))) -> a1(b1(b1(x)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.