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 oriented strictly and are deleted.


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 remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(A1(x1)) = 2 + 3·x1   
POL(B1(x1)) = 1 + 3·x1   
POL(a1(x1)) = 3 + 3·x1   
POL(b1(x1)) = 1 + 3·x1   

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
          ↳ 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.