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:

+2(*2(x, y), *2(x, z)) -> *2(x, +2(y, z))
+2(+2(x, y), z) -> +2(x, +2(y, z))
+2(*2(x, y), +2(*2(x, z), u)) -> +2(*2(x, +2(y, z)), u)

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

+2(*2(x, y), *2(x, z)) -> *2(x, +2(y, z))
+2(+2(x, y), z) -> +2(x, +2(y, z))
+2(*2(x, y), +2(*2(x, z), u)) -> +2(*2(x, +2(y, z)), u)

Q is empty.

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

+12(+2(x, y), z) -> +12(y, z)
+12(*2(x, y), +2(*2(x, z), u)) -> +12(y, z)
+12(*2(x, y), *2(x, z)) -> +12(y, z)
+12(*2(x, y), +2(*2(x, z), u)) -> +12(*2(x, +2(y, z)), u)
+12(+2(x, y), z) -> +12(x, +2(y, z))

The TRS R consists of the following rules:

+2(*2(x, y), *2(x, z)) -> *2(x, +2(y, z))
+2(+2(x, y), z) -> +2(x, +2(y, z))
+2(*2(x, y), +2(*2(x, z), u)) -> +2(*2(x, +2(y, z)), u)

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:

+12(+2(x, y), z) -> +12(y, z)
+12(*2(x, y), +2(*2(x, z), u)) -> +12(y, z)
+12(*2(x, y), *2(x, z)) -> +12(y, z)
+12(*2(x, y), +2(*2(x, z), u)) -> +12(*2(x, +2(y, z)), u)
+12(+2(x, y), z) -> +12(x, +2(y, z))

The TRS R consists of the following rules:

+2(*2(x, y), *2(x, z)) -> *2(x, +2(y, z))
+2(+2(x, y), z) -> +2(x, +2(y, z))
+2(*2(x, y), +2(*2(x, z), u)) -> +2(*2(x, +2(y, z)), u)

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

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

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

+12(+2(x, y), z) -> +12(y, z)
+12(*2(x, y), +2(*2(x, z), u)) -> +12(y, z)
+12(*2(x, y), *2(x, z)) -> +12(y, z)
+12(+2(x, y), z) -> +12(x, +2(y, z))

The TRS R consists of the following rules:

+2(*2(x, y), *2(x, z)) -> *2(x, +2(y, z))
+2(+2(x, y), z) -> +2(x, +2(y, z))
+2(*2(x, y), +2(*2(x, z), u)) -> +2(*2(x, +2(y, z)), u)

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.

+12(+2(x, y), z) -> +12(y, z)
+12(+2(x, y), z) -> +12(x, +2(y, z))
Used argument filtering: +12(x1, x2)  =  x1
+2(x1, x2)  =  +2(x1, x2)
*2(x1, x2)  =  x2
Used ordering: Quasi Precedence: trivial 


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

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

+12(*2(x, y), *2(x, z)) -> +12(y, z)
+12(*2(x, y), +2(*2(x, z), u)) -> +12(y, z)

The TRS R consists of the following rules:

+2(*2(x, y), *2(x, z)) -> *2(x, +2(y, z))
+2(+2(x, y), z) -> +2(x, +2(y, z))
+2(*2(x, y), +2(*2(x, z), u)) -> +2(*2(x, +2(y, z)), u)

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.

+12(*2(x, y), +2(*2(x, z), u)) -> +12(y, z)
Used argument filtering: +12(x1, x2)  =  x2
*2(x1, x2)  =  x2
+2(x1, x2)  =  +1(x1)
u  =  u
Used ordering: Quasi Precedence: trivial 


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

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

+12(*2(x, y), *2(x, z)) -> +12(y, z)

The TRS R consists of the following rules:

+2(*2(x, y), *2(x, z)) -> *2(x, +2(y, z))
+2(+2(x, y), z) -> +2(x, +2(y, z))
+2(*2(x, y), +2(*2(x, z), u)) -> +2(*2(x, +2(y, z)), u)

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.

+12(*2(x, y), *2(x, z)) -> +12(y, z)
Used argument filtering: +12(x1, x2)  =  x2
*2(x1, x2)  =  *1(x2)
Used ordering: Quasi Precedence: trivial 


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

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

+2(*2(x, y), *2(x, z)) -> *2(x, +2(y, z))
+2(+2(x, y), z) -> +2(x, +2(y, z))
+2(*2(x, y), +2(*2(x, z), u)) -> +2(*2(x, +2(y, z)), u)

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.