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.

Using Dependency Pairs [1,13] we result in the following initial DP problem:
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
      ↳ QDPOrderProof

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.
We use the reduction pair processor [13].


The following pairs can be strictly oriented and are deleted.


A1(a1(x)) -> B1(x)
B1(b1(a1(x))) -> B1(x)
B1(b1(a1(x))) -> B1(b1(x))
The remaining pairs can at least by weakly be oriented.

B1(b1(a1(x))) -> A1(b1(b1(x)))
A1(a1(x)) -> B1(b1(x))
Used ordering: Combined order from the following AFS and order.
A1(x1)  =  A1(x1)
a1(x1)  =  a1(x1)
B1(x1)  =  B1(x1)
b1(x1)  =  b1(x1)

Lexicographic Path Order [19].
Precedence:
[A1, B1] > [a1, b1]


The following usable rules [14] were oriented:

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ QDPOrderProof
QDP
          ↳ QDPOrderProof

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

B1(b1(a1(x))) -> A1(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.
We use the reduction pair processor [13].


The following pairs can be strictly oriented and are deleted.


B1(b1(a1(x))) -> A1(b1(b1(x)))
A1(a1(x)) -> B1(b1(x))
The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
B1(x1)  =  B1(x1)
b1(x1)  =  x1
a1(x1)  =  a1(x1)
A1(x1)  =  A1(x1)

Lexicographic Path Order [19].
Precedence:
[B1, a1, A1]


The following usable rules [14] were oriented:

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ QDPOrderProof
        ↳ QDP
          ↳ QDPOrderProof
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.