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(x, *2(y, z)) -> *2(otimes2(x, y), z)
*2(1, y) -> y
*2(+2(x, y), z) -> oplus2(*2(x, z), *2(y, z))
*2(x, oplus2(y, z)) -> oplus2(*2(x, y), *2(x, z))

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

*2(x, *2(y, z)) -> *2(otimes2(x, y), z)
*2(1, y) -> y
*2(+2(x, y), z) -> oplus2(*2(x, z), *2(y, z))
*2(x, oplus2(y, z)) -> oplus2(*2(x, y), *2(x, z))

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, y), z) -> *12(x, z)
*12(x, oplus2(y, z)) -> *12(x, y)
*12(x, oplus2(y, z)) -> *12(x, z)
*12(x, *2(y, z)) -> *12(otimes2(x, y), z)
*12(+2(x, y), z) -> *12(y, z)

The TRS R consists of the following rules:

*2(x, *2(y, z)) -> *2(otimes2(x, y), z)
*2(1, y) -> y
*2(+2(x, y), z) -> oplus2(*2(x, z), *2(y, z))
*2(x, oplus2(y, z)) -> oplus2(*2(x, y), *2(x, z))

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, y), z) -> *12(x, z)
*12(x, oplus2(y, z)) -> *12(x, y)
*12(x, oplus2(y, z)) -> *12(x, z)
*12(x, *2(y, z)) -> *12(otimes2(x, y), z)
*12(+2(x, y), z) -> *12(y, z)

The TRS R consists of the following rules:

*2(x, *2(y, z)) -> *2(otimes2(x, y), z)
*2(1, y) -> y
*2(+2(x, y), z) -> oplus2(*2(x, z), *2(y, z))
*2(x, oplus2(y, z)) -> oplus2(*2(x, y), *2(x, z))

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, oplus2(y, z)) -> *12(x, y)
*12(x, oplus2(y, z)) -> *12(x, z)
The remaining pairs can at least be oriented weakly.

*12(+2(x, y), z) -> *12(x, z)
*12(x, *2(y, z)) -> *12(otimes2(x, y), z)
*12(+2(x, y), z) -> *12(y, z)
Used ordering: Polynomial interpretation [21]:

POL(*2(x1, x2)) = 2·x2   
POL(*12(x1, x2)) = 2·x2   
POL(+2(x1, x2)) = 0   
POL(oplus2(x1, x2)) = 2 + 2·x1 + 2·x2   
POL(otimes2(x1, x2)) = 0   

The following usable rules [14] were oriented: none



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

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

*12(+2(x, y), z) -> *12(x, z)
*12(x, *2(y, z)) -> *12(otimes2(x, y), z)
*12(+2(x, y), z) -> *12(y, z)

The TRS R consists of the following rules:

*2(x, *2(y, z)) -> *2(otimes2(x, y), z)
*2(1, y) -> y
*2(+2(x, y), z) -> oplus2(*2(x, z), *2(y, z))
*2(x, oplus2(y, z)) -> oplus2(*2(x, y), *2(x, z))

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
      ↳ QDPOrderProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
                ↳ QDPOrderProof
              ↳ QDP

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

*12(x, *2(y, z)) -> *12(otimes2(x, y), z)

The TRS R consists of the following rules:

*2(x, *2(y, z)) -> *2(otimes2(x, y), z)
*2(1, y) -> y
*2(+2(x, y), z) -> oplus2(*2(x, z), *2(y, z))
*2(x, oplus2(y, z)) -> oplus2(*2(x, y), *2(x, z))

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(otimes2(x, y), z)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(*2(x1, x2)) = 2 + 2·x2   
POL(*12(x1, x2)) = 2·x2   
POL(otimes2(x1, x2)) = 0   

The following usable rules [14] were oriented: none



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

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

*2(x, *2(y, z)) -> *2(otimes2(x, y), z)
*2(1, y) -> y
*2(+2(x, y), z) -> oplus2(*2(x, z), *2(y, z))
*2(x, oplus2(y, z)) -> oplus2(*2(x, y), *2(x, z))

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
      ↳ QDPOrderProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ AND
              ↳ QDP
              ↳ QDP
                ↳ QDPOrderProof

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

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

The TRS R consists of the following rules:

*2(x, *2(y, z)) -> *2(otimes2(x, y), z)
*2(1, y) -> y
*2(+2(x, y), z) -> oplus2(*2(x, z), *2(y, z))
*2(x, oplus2(y, z)) -> oplus2(*2(x, y), *2(x, z))

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

POL(*12(x1, x2)) = 2·x1   
POL(+2(x1, x2)) = 1 + 2·x1 + 2·x2   

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(otimes2(x, y), z)
*2(1, y) -> y
*2(+2(x, y), z) -> oplus2(*2(x, z), *2(y, z))
*2(x, oplus2(y, z)) -> oplus2(*2(x, y), *2(x, z))

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.