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), z) -> +2(x, +2(y, z))
+2(f1(x), f1(y)) -> f1(+2(x, y))
+2(f1(x), +2(f1(y), z)) -> +2(f1(+2(x, y)), z)

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

+2(+2(x, y), z) -> +2(x, +2(y, z))
+2(f1(x), f1(y)) -> f1(+2(x, y))
+2(f1(x), +2(f1(y), z)) -> +2(f1(+2(x, y)), 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(f1(x), f1(y)) -> +12(x, y)
+12(f1(x), +2(f1(y), z)) -> +12(x, y)
+12(+2(x, y), z) -> +12(y, z)
+12(f1(x), +2(f1(y), z)) -> +12(f1(+2(x, y)), z)
+12(+2(x, y), z) -> +12(x, +2(y, z))

The TRS R consists of the following rules:

+2(+2(x, y), z) -> +2(x, +2(y, z))
+2(f1(x), f1(y)) -> f1(+2(x, y))
+2(f1(x), +2(f1(y), z)) -> +2(f1(+2(x, y)), 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(f1(x), f1(y)) -> +12(x, y)
+12(f1(x), +2(f1(y), z)) -> +12(x, y)
+12(+2(x, y), z) -> +12(y, z)
+12(f1(x), +2(f1(y), z)) -> +12(f1(+2(x, y)), z)
+12(+2(x, y), z) -> +12(x, +2(y, z))

The TRS R consists of the following rules:

+2(+2(x, y), z) -> +2(x, +2(y, z))
+2(f1(x), f1(y)) -> f1(+2(x, y))
+2(f1(x), +2(f1(y), z)) -> +2(f1(+2(x, y)), 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(f1(x), f1(y)) -> +12(x, y)
+12(f1(x), +2(f1(y), z)) -> +12(x, y)
+12(+2(x, y), z) -> +12(y, z)
The remaining pairs can at least be oriented weakly.

+12(f1(x), +2(f1(y), z)) -> +12(f1(+2(x, y)), z)
+12(+2(x, y), z) -> +12(x, +2(y, z))
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( +12(x1, x2) ) = max{0, 2x1 + 2x2 - 1}


POL( +2(x1, x2) ) = x1 + x2 + 3


POL( f1(x1) ) = 2x1 + 3



The following usable rules [14] were oriented:

+2(f1(x), f1(y)) -> f1(+2(x, y))
+2(f1(x), +2(f1(y), z)) -> +2(f1(+2(x, y)), z)
+2(+2(x, y), z) -> +2(x, +2(y, z))



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

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

+12(f1(x), +2(f1(y), z)) -> +12(f1(+2(x, y)), z)
+12(+2(x, y), z) -> +12(x, +2(y, z))

The TRS R consists of the following rules:

+2(+2(x, y), z) -> +2(x, +2(y, z))
+2(f1(x), f1(y)) -> f1(+2(x, y))
+2(f1(x), +2(f1(y), z)) -> +2(f1(+2(x, y)), 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(f1(x), +2(f1(y), z)) -> +12(f1(+2(x, y)), z)

The TRS R consists of the following rules:

+2(+2(x, y), z) -> +2(x, +2(y, z))
+2(f1(x), f1(y)) -> f1(+2(x, y))
+2(f1(x), +2(f1(y), z)) -> +2(f1(+2(x, y)), 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(f1(x), +2(f1(y), z)) -> +12(f1(+2(x, y)), z)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( +12(x1, x2) ) = max{0, 2x2 - 1}


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


POL( f1(x1) ) = max{0, 2x1 - 3}



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(+2(x, y), z) -> +2(x, +2(y, z))
+2(f1(x), f1(y)) -> f1(+2(x, y))
+2(f1(x), +2(f1(y), z)) -> +2(f1(+2(x, y)), 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, +2(y, z))

The TRS R consists of the following rules:

+2(+2(x, y), z) -> +2(x, +2(y, z))
+2(f1(x), f1(y)) -> f1(+2(x, y))
+2(f1(x), +2(f1(y), z)) -> +2(f1(+2(x, y)), 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, +2(y, z))
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

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


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


POL( f1(x1) ) = 3x1



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(+2(x, y), z) -> +2(x, +2(y, z))
+2(f1(x), f1(y)) -> f1(+2(x, y))
+2(f1(x), +2(f1(y), z)) -> +2(f1(+2(x, y)), 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.