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:

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

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

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

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:

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

The TRS R consists of the following rules:

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

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:

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

The TRS R consists of the following rules:

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

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.


+12(+2(x, *2(y, z)), *2(y, u)) -> +12(z, u)
+12(+2(x, *2(y, z)), *2(y, u)) -> *12(y, +2(z, u))
+12(x, +2(y, z)) -> +12(x, y)
The remaining pairs can at least be oriented weakly.

*12(x, +2(y, z)) -> +12(*2(x, y), *2(x, z))
+12(+2(x, *2(y, z)), *2(y, u)) -> +12(x, *2(y, +2(z, u)))
*12(x, +2(y, z)) -> *12(x, z)
*12(x, +2(y, z)) -> *12(x, y)
+12(x, +2(y, z)) -> +12(+2(x, y), z)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( +12(x1, x2) ) = x1 + x2


POL( +2(x1, x2) ) = x1 + x2 + 1


POL( *2(x1, x2) ) = x2


POL( *12(x1, x2) ) = max{0, x2 - 1}



The following usable rules [14] were oriented:

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



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

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

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

The TRS R consists of the following rules:

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

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 with 1 less node.

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

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

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

The TRS R consists of the following rules:

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

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

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

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

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

The TRS R consists of the following rules:

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

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.


*12(x, +2(y, z)) -> *12(x, z)
*12(x, +2(y, z)) -> *12(x, y)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( *12(x1, x2) ) = x2


POL( +2(x1, x2) ) = x1 + x2 + 1



The following usable rules [14] were oriented: none



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

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

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