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:

plus2(plus2(X, Y), Z) -> plus2(X, plus2(Y, Z))
times2(X, s1(Y)) -> plus2(X, times2(Y, X))

Q is empty.


QTRS
  ↳ DependencyPairsProof

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

plus2(plus2(X, Y), Z) -> plus2(X, plus2(Y, Z))
times2(X, s1(Y)) -> plus2(X, times2(Y, X))

Q is empty.

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

PLUS2(plus2(X, Y), Z) -> PLUS2(X, plus2(Y, Z))
PLUS2(plus2(X, Y), Z) -> PLUS2(Y, Z)
TIMES2(X, s1(Y)) -> TIMES2(Y, X)
TIMES2(X, s1(Y)) -> PLUS2(X, times2(Y, X))

The TRS R consists of the following rules:

plus2(plus2(X, Y), Z) -> plus2(X, plus2(Y, Z))
times2(X, s1(Y)) -> plus2(X, times2(Y, 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:

PLUS2(plus2(X, Y), Z) -> PLUS2(X, plus2(Y, Z))
PLUS2(plus2(X, Y), Z) -> PLUS2(Y, Z)
TIMES2(X, s1(Y)) -> TIMES2(Y, X)
TIMES2(X, s1(Y)) -> PLUS2(X, times2(Y, X))

The TRS R consists of the following rules:

plus2(plus2(X, Y), Z) -> plus2(X, plus2(Y, Z))
times2(X, s1(Y)) -> plus2(X, times2(Y, X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph contains 2 SCCs with 1 less node.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
QDP
            ↳ QDPAfsSolverProof
          ↳ QDP

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

PLUS2(plus2(X, Y), Z) -> PLUS2(X, plus2(Y, Z))
PLUS2(plus2(X, Y), Z) -> PLUS2(Y, Z)

The TRS R consists of the following rules:

plus2(plus2(X, Y), Z) -> plus2(X, plus2(Y, Z))
times2(X, s1(Y)) -> plus2(X, times2(Y, X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using an argument filtering and a montonic ordering, at least one Dependency Pair of this SCC can be strictly oriented.

PLUS2(plus2(X, Y), Z) -> PLUS2(X, plus2(Y, Z))
PLUS2(plus2(X, Y), Z) -> PLUS2(Y, Z)
Used argument filtering: PLUS2(x1, x2)  =  x1
plus2(x1, x2)  =  plus2(x1, x2)
Used ordering: Quasi Precedence: trivial


↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPAfsSolverProof
QDP
                ↳ PisEmptyProof
          ↳ QDP

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

plus2(plus2(X, Y), Z) -> plus2(X, plus2(Y, Z))
times2(X, s1(Y)) -> plus2(X, times2(Y, 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

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

TIMES2(X, s1(Y)) -> TIMES2(Y, X)

The TRS R consists of the following rules:

plus2(plus2(X, Y), Z) -> plus2(X, plus2(Y, Z))
times2(X, s1(Y)) -> plus2(X, times2(Y, X))

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