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:

a2(a2(y, 0), 0) -> y
c1(c1(y)) -> y
c1(a2(c1(c1(y)), x)) -> a2(c1(c1(c1(a2(x, 0)))), y)

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

a2(a2(y, 0), 0) -> y
c1(c1(y)) -> y
c1(a2(c1(c1(y)), x)) -> a2(c1(c1(c1(a2(x, 0)))), y)

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:

C1(a2(c1(c1(y)), x)) -> C1(c1(a2(x, 0)))
C1(a2(c1(c1(y)), x)) -> A2(c1(c1(c1(a2(x, 0)))), y)
C1(a2(c1(c1(y)), x)) -> A2(x, 0)
C1(a2(c1(c1(y)), x)) -> C1(c1(c1(a2(x, 0))))
C1(a2(c1(c1(y)), x)) -> C1(a2(x, 0))

The TRS R consists of the following rules:

a2(a2(y, 0), 0) -> y
c1(c1(y)) -> y
c1(a2(c1(c1(y)), x)) -> a2(c1(c1(c1(a2(x, 0)))), y)

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:

C1(a2(c1(c1(y)), x)) -> C1(c1(a2(x, 0)))
C1(a2(c1(c1(y)), x)) -> A2(c1(c1(c1(a2(x, 0)))), y)
C1(a2(c1(c1(y)), x)) -> A2(x, 0)
C1(a2(c1(c1(y)), x)) -> C1(c1(c1(a2(x, 0))))
C1(a2(c1(c1(y)), x)) -> C1(a2(x, 0))

The TRS R consists of the following rules:

a2(a2(y, 0), 0) -> y
c1(c1(y)) -> y
c1(a2(c1(c1(y)), x)) -> a2(c1(c1(c1(a2(x, 0)))), y)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 1 SCC with 2 less nodes.

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

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

C1(a2(c1(c1(y)), x)) -> C1(c1(a2(x, 0)))
C1(a2(c1(c1(y)), x)) -> C1(c1(c1(a2(x, 0))))
C1(a2(c1(c1(y)), x)) -> C1(a2(x, 0))

The TRS R consists of the following rules:

a2(a2(y, 0), 0) -> y
c1(c1(y)) -> y
c1(a2(c1(c1(y)), x)) -> a2(c1(c1(c1(a2(x, 0)))), y)

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.


C1(a2(c1(c1(y)), x)) -> C1(c1(a2(x, 0)))
C1(a2(c1(c1(y)), x)) -> C1(a2(x, 0))
The remaining pairs can at least be oriented weakly.

C1(a2(c1(c1(y)), x)) -> C1(c1(c1(a2(x, 0))))
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( C1(x1) ) = max{0, x1 - 1}


POL( a2(x1, x2) ) = x1 + x2


POL( c1(x1) ) = x1 + 1


POL( 0 ) = 0



The following usable rules [14] were oriented:

c1(a2(c1(c1(y)), x)) -> a2(c1(c1(c1(a2(x, 0)))), y)
a2(a2(y, 0), 0) -> y
c1(c1(y)) -> y



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

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

C1(a2(c1(c1(y)), x)) -> C1(c1(c1(a2(x, 0))))

The TRS R consists of the following rules:

a2(a2(y, 0), 0) -> y
c1(c1(y)) -> y
c1(a2(c1(c1(y)), x)) -> a2(c1(c1(c1(a2(x, 0)))), y)

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