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:

f2(s1(X), X) -> f2(X, a1(X))
f2(X, c1(X)) -> f2(s1(X), X)
f2(X, X) -> c1(X)

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

f2(s1(X), X) -> f2(X, a1(X))
f2(X, c1(X)) -> f2(s1(X), X)
f2(X, X) -> c1(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:

F2(X, c1(X)) -> F2(s1(X), X)
F2(s1(X), X) -> F2(X, a1(X))

The TRS R consists of the following rules:

f2(s1(X), X) -> f2(X, a1(X))
f2(X, c1(X)) -> f2(s1(X), X)
f2(X, X) -> c1(X)

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:

F2(X, c1(X)) -> F2(s1(X), X)
F2(s1(X), X) -> F2(X, a1(X))

The TRS R consists of the following rules:

f2(s1(X), X) -> f2(X, a1(X))
f2(X, c1(X)) -> f2(s1(X), X)
f2(X, X) -> c1(X)

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

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

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

F2(s1(X), X) -> F2(X, a1(X))

The TRS R consists of the following rules:

f2(s1(X), X) -> f2(X, a1(X))
f2(X, c1(X)) -> f2(s1(X), X)
f2(X, X) -> c1(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.


F2(s1(X), X) -> F2(X, a1(X))
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( s1(x1) ) = x1 + 1


POL( F2(x1, x2) ) = x1 + x2 + 1


POL( a1(x1) ) = max{0, -1}



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ PisEmptyProof
          ↳ QDP

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

f2(s1(X), X) -> f2(X, a1(X))
f2(X, c1(X)) -> f2(s1(X), X)
f2(X, X) -> c1(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
            ↳ QDPOrderProof

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

F2(X, c1(X)) -> F2(s1(X), X)

The TRS R consists of the following rules:

f2(s1(X), X) -> f2(X, a1(X))
f2(X, c1(X)) -> f2(s1(X), X)
f2(X, X) -> c1(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.


F2(X, c1(X)) -> F2(s1(X), X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( c1(x1) ) = x1 + 1


POL( s1(x1) ) = max{0, -1}


POL( F2(x1, x2) ) = x1 + x2 + 1



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ PisEmptyProof

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

f2(s1(X), X) -> f2(X, a1(X))
f2(X, c1(X)) -> f2(s1(X), X)
f2(X, X) -> c1(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.