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:

b(b(a(a(b(x1))))) → b(a(a(b(b(x1)))))
b(a(a(a(b(b(x1)))))) → b(b(a(a(a(b(x1))))))
b(a(b(a(a(b(x1)))))) → b(a(b(b(a(a(a(b(x1))))))))
b(a(a(a(b(a(b(x1))))))) → b(a(a(b(a(b(x1))))))

Q is empty.


QTRS
  ↳ DependencyPairsProof
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

b(b(a(a(b(x1))))) → b(a(a(b(b(x1)))))
b(a(a(a(b(b(x1)))))) → b(b(a(a(a(b(x1))))))
b(a(b(a(a(b(x1)))))) → b(a(b(b(a(a(a(b(x1))))))))
b(a(a(a(b(a(b(x1))))))) → b(a(a(b(a(b(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:

B(a(b(a(a(b(x1)))))) → B(a(b(b(a(a(a(b(x1))))))))
B(a(b(a(a(b(x1)))))) → B(a(a(a(b(x1)))))
B(b(a(a(b(x1))))) → B(b(x1))
B(a(a(a(b(a(b(x1))))))) → B(a(a(b(a(b(x1))))))
B(b(a(a(b(x1))))) → B(a(a(b(b(x1)))))
B(a(b(a(a(b(x1)))))) → B(b(a(a(a(b(x1))))))
B(a(a(a(b(b(x1)))))) → B(a(a(a(b(x1)))))
B(a(a(a(b(b(x1)))))) → B(b(a(a(a(b(x1))))))

The TRS R consists of the following rules:

b(b(a(a(b(x1))))) → b(a(a(b(b(x1)))))
b(a(a(a(b(b(x1)))))) → b(b(a(a(a(b(x1))))))
b(a(b(a(a(b(x1)))))) → b(a(b(b(a(a(a(b(x1))))))))
b(a(a(a(b(a(b(x1))))))) → b(a(a(b(a(b(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:

B(a(b(a(a(b(x1)))))) → B(a(b(b(a(a(a(b(x1))))))))
B(a(b(a(a(b(x1)))))) → B(a(a(a(b(x1)))))
B(b(a(a(b(x1))))) → B(b(x1))
B(a(a(a(b(a(b(x1))))))) → B(a(a(b(a(b(x1))))))
B(b(a(a(b(x1))))) → B(a(a(b(b(x1)))))
B(a(b(a(a(b(x1)))))) → B(b(a(a(a(b(x1))))))
B(a(a(a(b(b(x1)))))) → B(a(a(a(b(x1)))))
B(a(a(a(b(b(x1)))))) → B(b(a(a(a(b(x1))))))

The TRS R consists of the following rules:

b(b(a(a(b(x1))))) → b(a(a(b(b(x1)))))
b(a(a(a(b(b(x1)))))) → b(b(a(a(a(b(x1))))))
b(a(b(a(a(b(x1)))))) → b(a(b(b(a(a(a(b(x1))))))))
b(a(a(a(b(a(b(x1))))))) → b(a(a(b(a(b(x1))))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 3 SCCs with 5 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
QDP
            ↳ SemLabProof
          ↳ QDP
          ↳ QDP
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

B(b(a(a(b(x1))))) → B(b(x1))

The TRS R consists of the following rules:

b(b(a(a(b(x1))))) → b(a(a(b(b(x1)))))
b(a(a(a(b(b(x1)))))) → b(b(a(a(a(b(x1))))))
b(a(b(a(a(b(x1)))))) → b(a(b(b(a(a(a(b(x1))))))))
b(a(a(a(b(a(b(x1))))))) → b(a(a(b(a(b(x1))))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We found the following model for the rules of the TRS R. Interpretation over the domain with elements from 0 to 1.B: 0
a: 1
b: 0
By semantic labelling [33] we obtain the following labelled TRS:Q DP problem:
The TRS P consists of the following rules:

B.0(b.1(a.1(a.0(b.1(x1))))) → B.0(b.1(x1))
B.0(b.1(a.1(a.0(b.0(x1))))) → B.0(b.0(x1))

The TRS R consists of the following rules:

b.1(a.0(b.1(a.1(a.0(b.0(x1)))))) → b.1(a.0(b.0(b.1(a.1(a.1(a.0(b.0(x1))))))))
b.1(a.0(b.1(a.1(a.0(b.1(x1)))))) → b.1(a.0(b.0(b.1(a.1(a.1(a.0(b.1(x1))))))))
b.1(a.1(a.1(a.0(b.1(a.0(b.0(x1))))))) → b.1(a.1(a.0(b.1(a.0(b.0(x1))))))
b.0(b.1(a.1(a.0(b.1(x1))))) → b.1(a.1(a.0(b.0(b.1(x1)))))
b.1(a.1(a.1(a.0(b.1(a.0(b.1(x1))))))) → b.1(a.1(a.0(b.1(a.0(b.1(x1))))))
b.0(b.1(a.1(a.0(b.0(x1))))) → b.1(a.1(a.0(b.0(b.0(x1)))))
b.1(a.1(a.1(a.0(b.0(b.1(x1)))))) → b.0(b.1(a.1(a.1(a.0(b.1(x1))))))
b.1(a.1(a.1(a.0(b.0(b.0(x1)))))) → b.0(b.1(a.1(a.1(a.0(b.0(x1))))))

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ SemLabProof
QDP
                ↳ RuleRemovalProof
          ↳ QDP
          ↳ QDP
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

B.0(b.1(a.1(a.0(b.1(x1))))) → B.0(b.1(x1))
B.0(b.1(a.1(a.0(b.0(x1))))) → B.0(b.0(x1))

The TRS R consists of the following rules:

b.1(a.0(b.1(a.1(a.0(b.0(x1)))))) → b.1(a.0(b.0(b.1(a.1(a.1(a.0(b.0(x1))))))))
b.1(a.0(b.1(a.1(a.0(b.1(x1)))))) → b.1(a.0(b.0(b.1(a.1(a.1(a.0(b.1(x1))))))))
b.1(a.1(a.1(a.0(b.1(a.0(b.0(x1))))))) → b.1(a.1(a.0(b.1(a.0(b.0(x1))))))
b.0(b.1(a.1(a.0(b.1(x1))))) → b.1(a.1(a.0(b.0(b.1(x1)))))
b.1(a.1(a.1(a.0(b.1(a.0(b.1(x1))))))) → b.1(a.1(a.0(b.1(a.0(b.1(x1))))))
b.0(b.1(a.1(a.0(b.0(x1))))) → b.1(a.1(a.0(b.0(b.0(x1)))))
b.1(a.1(a.1(a.0(b.0(b.1(x1)))))) → b.0(b.1(a.1(a.1(a.0(b.1(x1))))))
b.1(a.1(a.1(a.0(b.0(b.0(x1)))))) → b.0(b.1(a.1(a.1(a.0(b.0(x1))))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

B.0(b.1(a.1(a.0(b.1(x1))))) → B.0(b.1(x1))
B.0(b.1(a.1(a.0(b.0(x1))))) → B.0(b.0(x1))


Used ordering: POLO with Polynomial interpretation [25]:

POL(B.0(x1)) = x1   
POL(a.0(x1)) = 1 + x1   
POL(a.1(x1)) = x1   
POL(b.0(x1)) = x1   
POL(b.1(x1)) = x1   



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ SemLabProof
              ↳ QDP
                ↳ RuleRemovalProof
QDP
                    ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

b.1(a.0(b.1(a.1(a.0(b.0(x1)))))) → b.1(a.0(b.0(b.1(a.1(a.1(a.0(b.0(x1))))))))
b.1(a.0(b.1(a.1(a.0(b.1(x1)))))) → b.1(a.0(b.0(b.1(a.1(a.1(a.0(b.1(x1))))))))
b.1(a.1(a.1(a.0(b.1(a.0(b.0(x1))))))) → b.1(a.1(a.0(b.1(a.0(b.0(x1))))))
b.0(b.1(a.1(a.0(b.1(x1))))) → b.1(a.1(a.0(b.0(b.1(x1)))))
b.1(a.1(a.1(a.0(b.1(a.0(b.1(x1))))))) → b.1(a.1(a.0(b.1(a.0(b.1(x1))))))
b.0(b.1(a.1(a.0(b.0(x1))))) → b.1(a.1(a.0(b.0(b.0(x1)))))
b.1(a.1(a.1(a.0(b.0(b.1(x1)))))) → b.0(b.1(a.1(a.1(a.0(b.1(x1))))))
b.1(a.1(a.1(a.0(b.0(b.0(x1)))))) → b.0(b.1(a.1(a.1(a.0(b.0(x1))))))

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.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
QDP
            ↳ QDPToSRSProof
          ↳ QDP
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

B(a(a(a(b(b(x1)))))) → B(a(a(a(b(x1)))))

The TRS R consists of the following rules:

b(b(a(a(b(x1))))) → b(a(a(b(b(x1)))))
b(a(a(a(b(b(x1)))))) → b(b(a(a(a(b(x1))))))
b(a(b(a(a(b(x1)))))) → b(a(b(b(a(a(a(b(x1))))))))
b(a(a(a(b(a(b(x1))))))) → b(a(a(b(a(b(x1))))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The finiteness of this DP problem is implied by strong termination of a SRS due to [12].


↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ QDPToSRSProof
QTRS
                ↳ QTRS Reverse
          ↳ QDP
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

b(b(a(a(b(x1))))) → b(a(a(b(b(x1)))))
b(a(a(a(b(b(x1)))))) → b(b(a(a(a(b(x1))))))
b(a(b(a(a(b(x1)))))) → b(a(b(b(a(a(a(b(x1))))))))
b(a(a(a(b(a(b(x1))))))) → b(a(a(b(a(b(x1))))))
B(a(a(a(b(b(x1)))))) → B(a(a(a(b(x1)))))

Q is empty.

We have reversed the following QTRS:
The set of rules R is

b(b(a(a(b(x1))))) → b(a(a(b(b(x1)))))
b(a(a(a(b(b(x1)))))) → b(b(a(a(a(b(x1))))))
b(a(b(a(a(b(x1)))))) → b(a(b(b(a(a(a(b(x1))))))))
b(a(a(a(b(a(b(x1))))))) → b(a(a(b(a(b(x1))))))
B(a(a(a(b(b(x1)))))) → B(a(a(a(b(x1)))))

The set Q is empty.
We have obtained the following QTRS:

b(a(a(b(b(x))))) → b(b(a(a(b(x)))))
b(b(a(a(a(b(x)))))) → b(a(a(a(b(b(x))))))
b(a(a(b(a(b(x)))))) → b(a(a(a(b(b(a(b(x))))))))
b(a(b(a(a(a(b(x))))))) → b(a(b(a(a(b(x))))))
b(b(a(a(a(B(x)))))) → b(a(a(a(B(x)))))

The set Q is empty.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ QDPToSRSProof
              ↳ QTRS
                ↳ QTRS Reverse
QTRS
                    ↳ QTRS Reverse
                    ↳ DependencyPairsProof
                    ↳ QTRS Reverse
          ↳ QDP
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

b(a(a(b(b(x))))) → b(b(a(a(b(x)))))
b(b(a(a(a(b(x)))))) → b(a(a(a(b(b(x))))))
b(a(a(b(a(b(x)))))) → b(a(a(a(b(b(a(b(x))))))))
b(a(b(a(a(a(b(x))))))) → b(a(b(a(a(b(x))))))
b(b(a(a(a(B(x)))))) → b(a(a(a(B(x)))))

Q is empty.

We have reversed the following QTRS:
The set of rules R is

b(a(a(b(b(x))))) → b(b(a(a(b(x)))))
b(b(a(a(a(b(x)))))) → b(a(a(a(b(b(x))))))
b(a(a(b(a(b(x)))))) → b(a(a(a(b(b(a(b(x))))))))
b(a(b(a(a(a(b(x))))))) → b(a(b(a(a(b(x))))))
b(b(a(a(a(B(x)))))) → b(a(a(a(B(x)))))

The set Q is empty.
We have obtained the following QTRS:

b(b(a(a(b(x))))) → b(a(a(b(b(x)))))
b(a(a(a(b(b(x)))))) → b(b(a(a(a(b(x))))))
b(a(b(a(a(b(x)))))) → b(a(b(b(a(a(a(b(x))))))))
b(a(a(a(b(a(b(x))))))) → b(a(a(b(a(b(x))))))
B(a(a(a(b(b(x)))))) → B(a(a(a(b(x)))))

The set Q is empty.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ QDPToSRSProof
              ↳ QTRS
                ↳ QTRS Reverse
                  ↳ QTRS
                    ↳ QTRS Reverse
QTRS
                    ↳ DependencyPairsProof
                    ↳ QTRS Reverse
          ↳ QDP
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

b(b(a(a(b(x))))) → b(a(a(b(b(x)))))
b(a(a(a(b(b(x)))))) → b(b(a(a(a(b(x))))))
b(a(b(a(a(b(x)))))) → b(a(b(b(a(a(a(b(x))))))))
b(a(a(a(b(a(b(x))))))) → b(a(a(b(a(b(x))))))
B(a(a(a(b(b(x)))))) → B(a(a(a(b(x)))))

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:

B1(a(a(b(a(b(x)))))) → B1(b(a(b(x))))
B1(a(a(b(a(b(x)))))) → B1(a(a(a(b(b(a(b(x))))))))
B1(a(b(a(a(a(b(x))))))) → B1(a(b(a(a(b(x))))))
B1(b(a(a(a(b(x)))))) → B1(b(x))
B1(b(a(a(a(b(x)))))) → B1(a(a(a(b(b(x))))))
B1(a(b(a(a(a(b(x))))))) → B1(a(a(b(x))))
B1(a(a(b(b(x))))) → B1(a(a(b(x))))
B1(a(a(b(b(x))))) → B1(b(a(a(b(x)))))

The TRS R consists of the following rules:

b(a(a(b(b(x))))) → b(b(a(a(b(x)))))
b(b(a(a(a(b(x)))))) → b(a(a(a(b(b(x))))))
b(a(a(b(a(b(x)))))) → b(a(a(a(b(b(a(b(x))))))))
b(a(b(a(a(a(b(x))))))) → b(a(b(a(a(b(x))))))
b(b(a(a(a(B(x)))))) → b(a(a(a(B(x)))))

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ QDPToSRSProof
              ↳ QTRS
                ↳ QTRS Reverse
                  ↳ QTRS
                    ↳ QTRS Reverse
                    ↳ DependencyPairsProof
QDP
                        ↳ DependencyGraphProof
                    ↳ QTRS Reverse
          ↳ QDP
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

B1(a(a(b(a(b(x)))))) → B1(b(a(b(x))))
B1(a(a(b(a(b(x)))))) → B1(a(a(a(b(b(a(b(x))))))))
B1(a(b(a(a(a(b(x))))))) → B1(a(b(a(a(b(x))))))
B1(b(a(a(a(b(x)))))) → B1(b(x))
B1(b(a(a(a(b(x)))))) → B1(a(a(a(b(b(x))))))
B1(a(b(a(a(a(b(x))))))) → B1(a(a(b(x))))
B1(a(a(b(b(x))))) → B1(a(a(b(x))))
B1(a(a(b(b(x))))) → B1(b(a(a(b(x)))))

The TRS R consists of the following rules:

b(a(a(b(b(x))))) → b(b(a(a(b(x)))))
b(b(a(a(a(b(x)))))) → b(a(a(a(b(b(x))))))
b(a(a(b(a(b(x)))))) → b(a(a(a(b(b(a(b(x))))))))
b(a(b(a(a(a(b(x))))))) → b(a(b(a(a(b(x))))))
b(b(a(a(a(B(x)))))) → b(a(a(a(B(x)))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 3 SCCs with 5 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ QDPToSRSProof
              ↳ QTRS
                ↳ QTRS Reverse
                  ↳ QTRS
                    ↳ QTRS Reverse
                    ↳ DependencyPairsProof
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ AND
QDP
                              ↳ SemLabProof
                            ↳ QDP
                            ↳ QDP
                    ↳ QTRS Reverse
          ↳ QDP
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

B1(b(a(a(a(b(x)))))) → B1(b(x))

The TRS R consists of the following rules:

b(a(a(b(b(x))))) → b(b(a(a(b(x)))))
b(b(a(a(a(b(x)))))) → b(a(a(a(b(b(x))))))
b(a(a(b(a(b(x)))))) → b(a(a(a(b(b(a(b(x))))))))
b(a(b(a(a(a(b(x))))))) → b(a(b(a(a(b(x))))))
b(b(a(a(a(B(x)))))) → b(a(a(a(B(x)))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We found the following model for the rules of the TRS R. Interpretation over the domain with elements from 0 to 1.B: 0
a: 1
B1: 0
b: 0
By semantic labelling [33] we obtain the following labelled TRS:Q DP problem:
The TRS P consists of the following rules:

B1.0(b.1(a.1(a.1(a.0(b.1(x)))))) → B1.0(b.1(x))
B1.0(b.1(a.1(a.1(a.0(b.0(x)))))) → B1.0(b.0(x))

The TRS R consists of the following rules:

b.0(b.1(a.1(a.1(a.0(b.0(x)))))) → b.1(a.1(a.1(a.0(b.0(b.0(x))))))
b.1(a.1(a.0(b.0(b.0(x))))) → b.0(b.1(a.1(a.0(b.0(x)))))
b.1(a.0(b.1(a.1(a.1(a.0(b.0(x))))))) → b.1(a.0(b.1(a.1(a.0(b.0(x))))))
b.1(a.1(a.0(b.0(b.1(x))))) → b.0(b.1(a.1(a.0(b.1(x)))))
b.1(a.1(a.0(b.1(a.0(b.0(x)))))) → b.1(a.1(a.1(a.0(b.0(b.1(a.0(b.0(x))))))))
b.0(b.1(a.1(a.1(a.0(B.1(x)))))) → b.1(a.1(a.1(a.0(B.1(x)))))
b.1(a.1(a.0(b.1(a.0(b.1(x)))))) → b.1(a.1(a.1(a.0(b.0(b.1(a.0(b.1(x))))))))
b.1(a.0(b.1(a.1(a.1(a.0(b.1(x))))))) → b.1(a.0(b.1(a.1(a.0(b.1(x))))))
b.0(b.1(a.1(a.1(a.0(B.0(x)))))) → b.1(a.1(a.1(a.0(B.0(x)))))
b.0(b.1(a.1(a.1(a.0(b.1(x)))))) → b.1(a.1(a.1(a.0(b.0(b.1(x))))))

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ QDPToSRSProof
              ↳ QTRS
                ↳ QTRS Reverse
                  ↳ QTRS
                    ↳ QTRS Reverse
                    ↳ DependencyPairsProof
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ AND
                            ↳ QDP
                              ↳ SemLabProof
QDP
                                  ↳ RuleRemovalProof
                            ↳ QDP
                            ↳ QDP
                    ↳ QTRS Reverse
          ↳ QDP
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

B1.0(b.1(a.1(a.1(a.0(b.1(x)))))) → B1.0(b.1(x))
B1.0(b.1(a.1(a.1(a.0(b.0(x)))))) → B1.0(b.0(x))

The TRS R consists of the following rules:

b.0(b.1(a.1(a.1(a.0(b.0(x)))))) → b.1(a.1(a.1(a.0(b.0(b.0(x))))))
b.1(a.1(a.0(b.0(b.0(x))))) → b.0(b.1(a.1(a.0(b.0(x)))))
b.1(a.0(b.1(a.1(a.1(a.0(b.0(x))))))) → b.1(a.0(b.1(a.1(a.0(b.0(x))))))
b.1(a.1(a.0(b.0(b.1(x))))) → b.0(b.1(a.1(a.0(b.1(x)))))
b.1(a.1(a.0(b.1(a.0(b.0(x)))))) → b.1(a.1(a.1(a.0(b.0(b.1(a.0(b.0(x))))))))
b.0(b.1(a.1(a.1(a.0(B.1(x)))))) → b.1(a.1(a.1(a.0(B.1(x)))))
b.1(a.1(a.0(b.1(a.0(b.1(x)))))) → b.1(a.1(a.1(a.0(b.0(b.1(a.0(b.1(x))))))))
b.1(a.0(b.1(a.1(a.1(a.0(b.1(x))))))) → b.1(a.0(b.1(a.1(a.0(b.1(x))))))
b.0(b.1(a.1(a.1(a.0(B.0(x)))))) → b.1(a.1(a.1(a.0(B.0(x)))))
b.0(b.1(a.1(a.1(a.0(b.1(x)))))) → b.1(a.1(a.1(a.0(b.0(b.1(x))))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

B1.0(b.1(a.1(a.1(a.0(b.1(x)))))) → B1.0(b.1(x))
B1.0(b.1(a.1(a.1(a.0(b.0(x)))))) → B1.0(b.0(x))


Used ordering: POLO with Polynomial interpretation [25]:

POL(B.0(x1)) = x1   
POL(B.1(x1)) = x1   
POL(B1.0(x1)) = x1   
POL(a.0(x1)) = 1 + x1   
POL(a.1(x1)) = x1   
POL(b.0(x1)) = x1   
POL(b.1(x1)) = x1   



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ QDPToSRSProof
              ↳ QTRS
                ↳ QTRS Reverse
                  ↳ QTRS
                    ↳ QTRS Reverse
                    ↳ DependencyPairsProof
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ AND
                            ↳ QDP
                              ↳ SemLabProof
                                ↳ QDP
                                  ↳ RuleRemovalProof
QDP
                                      ↳ PisEmptyProof
                            ↳ QDP
                            ↳ QDP
                    ↳ QTRS Reverse
          ↳ QDP
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

b.0(b.1(a.1(a.1(a.0(b.0(x)))))) → b.1(a.1(a.1(a.0(b.0(b.0(x))))))
b.1(a.1(a.0(b.0(b.0(x))))) → b.0(b.1(a.1(a.0(b.0(x)))))
b.1(a.0(b.1(a.1(a.1(a.0(b.0(x))))))) → b.1(a.0(b.1(a.1(a.0(b.0(x))))))
b.1(a.1(a.0(b.0(b.1(x))))) → b.0(b.1(a.1(a.0(b.1(x)))))
b.1(a.1(a.0(b.1(a.0(b.0(x)))))) → b.1(a.1(a.1(a.0(b.0(b.1(a.0(b.0(x))))))))
b.0(b.1(a.1(a.1(a.0(B.1(x)))))) → b.1(a.1(a.1(a.0(B.1(x)))))
b.1(a.1(a.0(b.1(a.0(b.1(x)))))) → b.1(a.1(a.1(a.0(b.0(b.1(a.0(b.1(x))))))))
b.1(a.0(b.1(a.1(a.1(a.0(b.1(x))))))) → b.1(a.0(b.1(a.1(a.0(b.1(x))))))
b.0(b.1(a.1(a.1(a.0(B.0(x)))))) → b.1(a.1(a.1(a.0(B.0(x)))))
b.0(b.1(a.1(a.1(a.0(b.1(x)))))) → b.1(a.1(a.1(a.0(b.0(b.1(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.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ QDPToSRSProof
              ↳ QTRS
                ↳ QTRS Reverse
                  ↳ QTRS
                    ↳ QTRS Reverse
                    ↳ DependencyPairsProof
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ AND
                            ↳ QDP
QDP
                            ↳ QDP
                    ↳ QTRS Reverse
          ↳ QDP
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

B1(a(a(b(b(x))))) → B1(a(a(b(x))))

The TRS R consists of the following rules:

b(a(a(b(b(x))))) → b(b(a(a(b(x)))))
b(b(a(a(a(b(x)))))) → b(a(a(a(b(b(x))))))
b(a(a(b(a(b(x)))))) → b(a(a(a(b(b(a(b(x))))))))
b(a(b(a(a(a(b(x))))))) → b(a(b(a(a(b(x))))))
b(b(a(a(a(B(x)))))) → b(a(a(a(B(x)))))

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ QDPToSRSProof
              ↳ QTRS
                ↳ QTRS Reverse
                  ↳ QTRS
                    ↳ QTRS Reverse
                    ↳ DependencyPairsProof
                      ↳ QDP
                        ↳ DependencyGraphProof
                          ↳ AND
                            ↳ QDP
                            ↳ QDP
QDP
                    ↳ QTRS Reverse
          ↳ QDP
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

B1(a(b(a(a(a(b(x))))))) → B1(a(b(a(a(b(x))))))

The TRS R consists of the following rules:

b(a(a(b(b(x))))) → b(b(a(a(b(x)))))
b(b(a(a(a(b(x)))))) → b(a(a(a(b(b(x))))))
b(a(a(b(a(b(x)))))) → b(a(a(a(b(b(a(b(x))))))))
b(a(b(a(a(a(b(x))))))) → b(a(b(a(a(b(x))))))
b(b(a(a(a(B(x)))))) → b(a(a(a(B(x)))))

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

b(a(a(b(b(x))))) → b(b(a(a(b(x)))))
b(b(a(a(a(b(x)))))) → b(a(a(a(b(b(x))))))
b(a(a(b(a(b(x)))))) → b(a(a(a(b(b(a(b(x))))))))
b(a(b(a(a(a(b(x))))))) → b(a(b(a(a(b(x))))))
b(b(a(a(a(B(x)))))) → b(a(a(a(B(x)))))

The set Q is empty.
We have obtained the following QTRS:

b(b(a(a(b(x))))) → b(a(a(b(b(x)))))
b(a(a(a(b(b(x)))))) → b(b(a(a(a(b(x))))))
b(a(b(a(a(b(x)))))) → b(a(b(b(a(a(a(b(x))))))))
b(a(a(a(b(a(b(x))))))) → b(a(a(b(a(b(x))))))
B(a(a(a(b(b(x)))))) → B(a(a(a(b(x)))))

The set Q is empty.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ QDPToSRSProof
              ↳ QTRS
                ↳ QTRS Reverse
                  ↳ QTRS
                    ↳ QTRS Reverse
                    ↳ DependencyPairsProof
                    ↳ QTRS Reverse
QTRS
          ↳ QDP
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

b(b(a(a(b(x))))) → b(a(a(b(b(x)))))
b(a(a(a(b(b(x)))))) → b(b(a(a(a(b(x))))))
b(a(b(a(a(b(x)))))) → b(a(b(b(a(a(a(b(x))))))))
b(a(a(a(b(a(b(x))))))) → b(a(a(b(a(b(x))))))
B(a(a(a(b(b(x)))))) → B(a(a(a(b(x)))))

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
QDP
  ↳ QTRS Reverse
  ↳ QTRS Reverse

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

B(a(b(a(a(b(x1)))))) → B(a(b(b(a(a(a(b(x1))))))))

The TRS R consists of the following rules:

b(b(a(a(b(x1))))) → b(a(a(b(b(x1)))))
b(a(a(a(b(b(x1)))))) → b(b(a(a(a(b(x1))))))
b(a(b(a(a(b(x1)))))) → b(a(b(b(a(a(a(b(x1))))))))
b(a(a(a(b(a(b(x1))))))) → b(a(a(b(a(b(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

b(b(a(a(b(x1))))) → b(a(a(b(b(x1)))))
b(a(a(a(b(b(x1)))))) → b(b(a(a(a(b(x1))))))
b(a(b(a(a(b(x1)))))) → b(a(b(b(a(a(a(b(x1))))))))
b(a(a(a(b(a(b(x1))))))) → b(a(a(b(a(b(x1))))))

The set Q is empty.
We have obtained the following QTRS:

b(a(a(b(b(x))))) → b(b(a(a(b(x)))))
b(b(a(a(a(b(x)))))) → b(a(a(a(b(b(x))))))
b(a(a(b(a(b(x)))))) → b(a(a(a(b(b(a(b(x))))))))
b(a(b(a(a(a(b(x))))))) → b(a(b(a(a(b(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:

b(a(a(b(b(x))))) → b(b(a(a(b(x)))))
b(b(a(a(a(b(x)))))) → b(a(a(a(b(b(x))))))
b(a(a(b(a(b(x)))))) → b(a(a(a(b(b(a(b(x))))))))
b(a(b(a(a(a(b(x))))))) → b(a(b(a(a(b(x))))))

Q is empty.

We have reversed the following QTRS:
The set of rules R is

b(b(a(a(b(x1))))) → b(a(a(b(b(x1)))))
b(a(a(a(b(b(x1)))))) → b(b(a(a(a(b(x1))))))
b(a(b(a(a(b(x1)))))) → b(a(b(b(a(a(a(b(x1))))))))
b(a(a(a(b(a(b(x1))))))) → b(a(a(b(a(b(x1))))))

The set Q is empty.
We have obtained the following QTRS:

b(a(a(b(b(x))))) → b(b(a(a(b(x)))))
b(b(a(a(a(b(x)))))) → b(a(a(a(b(b(x))))))
b(a(a(b(a(b(x)))))) → b(a(a(a(b(b(a(b(x))))))))
b(a(b(a(a(a(b(x))))))) → b(a(b(a(a(b(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:

b(a(a(b(b(x))))) → b(b(a(a(b(x)))))
b(b(a(a(a(b(x)))))) → b(a(a(a(b(b(x))))))
b(a(a(b(a(b(x)))))) → b(a(a(a(b(b(a(b(x))))))))
b(a(b(a(a(a(b(x))))))) → b(a(b(a(a(b(x))))))

Q is empty.