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

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

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

A(a(c(c(x1)))) → A(a(x1))
A(a(c(c(x1)))) → C(a(a(x1)))
A(a(b(b(b(b(a(a(x1)))))))) → C(a(a(b(b(x1)))))
A(a(c(c(x1)))) → C(c(c(c(a(a(x1))))))
A(a(c(c(x1)))) → C(c(c(a(a(x1)))))
C(c(c(c(c(c(x1)))))) → C(c(b(b(x1))))
A(a(b(b(b(b(a(a(x1)))))))) → C(c(a(a(b(b(x1))))))
C(c(c(c(c(c(x1)))))) → C(b(b(x1)))
A(a(b(b(b(b(a(a(x1)))))))) → A(a(c(c(a(a(b(b(x1))))))))
A(a(b(b(b(b(a(a(x1)))))))) → A(c(c(a(a(b(b(x1)))))))
A(a(c(c(x1)))) → C(c(a(a(x1))))
A(a(c(c(x1)))) → A(x1)
A(a(b(b(b(b(a(a(x1)))))))) → A(a(b(b(x1))))
A(a(b(b(b(b(a(a(x1)))))))) → A(b(b(x1)))

The TRS R consists of the following rules:

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

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

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof

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

A(a(c(c(x1)))) → A(a(x1))
A(a(c(c(x1)))) → C(a(a(x1)))
A(a(b(b(b(b(a(a(x1)))))))) → C(a(a(b(b(x1)))))
A(a(c(c(x1)))) → C(c(c(c(a(a(x1))))))
A(a(c(c(x1)))) → C(c(c(a(a(x1)))))
C(c(c(c(c(c(x1)))))) → C(c(b(b(x1))))
A(a(b(b(b(b(a(a(x1)))))))) → C(c(a(a(b(b(x1))))))
C(c(c(c(c(c(x1)))))) → C(b(b(x1)))
A(a(b(b(b(b(a(a(x1)))))))) → A(a(c(c(a(a(b(b(x1))))))))
A(a(b(b(b(b(a(a(x1)))))))) → A(c(c(a(a(b(b(x1)))))))
A(a(c(c(x1)))) → C(c(a(a(x1))))
A(a(c(c(x1)))) → A(x1)
A(a(b(b(b(b(a(a(x1)))))))) → A(a(b(b(x1))))
A(a(b(b(b(b(a(a(x1)))))))) → A(b(b(x1)))

The TRS R consists of the following rules:

a(a(b(b(b(b(a(a(x1)))))))) → a(a(c(c(a(a(b(b(x1))))))))
a(a(c(c(x1)))) → c(c(c(c(a(a(x1))))))
c(c(c(c(c(c(x1)))))) → b(b(c(c(b(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 1 SCC with 9 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof

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

A(a(c(c(x1)))) → A(a(x1))
A(a(b(b(b(b(a(a(x1)))))))) → A(a(c(c(a(a(b(b(x1))))))))
A(a(b(b(b(b(a(a(x1)))))))) → A(c(c(a(a(b(b(x1)))))))
A(a(c(c(x1)))) → A(x1)
A(a(b(b(b(b(a(a(x1)))))))) → A(a(b(b(x1))))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].


The following pairs can be oriented strictly and are deleted.


A(a(b(b(b(b(a(a(x1)))))))) → A(c(c(a(a(b(b(x1)))))))
A(a(c(c(x1)))) → A(x1)
A(a(b(b(b(b(a(a(x1)))))))) → A(a(b(b(x1))))
The remaining pairs can at least be oriented weakly.

A(a(c(c(x1)))) → A(a(x1))
A(a(b(b(b(b(a(a(x1)))))))) → A(a(c(c(a(a(b(b(x1))))))))
Used ordering: Polynomial interpretation [25,35]:

POL(c(x1)) = x_1   
POL(a(x1)) = 1/4 + x_1   
POL(A(x1)) = (1/4)x_1   
POL(b(x1)) = x_1   
The value of delta used in the strict ordering is 1/16.
The following usable rules [17] were oriented:

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP

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

A(a(c(c(x1)))) → A(a(x1))
A(a(b(b(b(b(a(a(x1)))))))) → A(a(c(c(a(a(b(b(x1))))))))

The TRS R consists of the following rules:

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

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