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

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

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

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

The TRS R consists of the following rules:

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

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:

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

The TRS R consists of the following rules:

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

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

POL(C(x1)) = x_1   
POL(c(x1)) = 1/2 + (2)x_1   
POL(a(x1)) = 3/2 + (4)x_1   
POL(A(x1)) = 1/2 + (2)x_1   
POL(b(x1)) = 1/2 + (2)x_1   
The value of delta used in the strict ordering is 1/2.
The following usable rules [17] were oriented:

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ QDPOrderProof
        ↳ QDP

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

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

The TRS R consists of the following rules:

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

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