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:

sum2(cons2(s1(n), x), cons2(m, y)) -> sum2(cons2(n, x), cons2(s1(m), y))
sum2(cons2(0, x), y) -> sum2(x, y)
sum2(nil, y) -> y
weight1(cons2(n, cons2(m, x))) -> weight1(sum2(cons2(n, cons2(m, x)), cons2(0, x)))
weight1(cons2(n, nil)) -> n

Q is empty.


QTRS
  ↳ DependencyPairsProof

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

sum2(cons2(s1(n), x), cons2(m, y)) -> sum2(cons2(n, x), cons2(s1(m), y))
sum2(cons2(0, x), y) -> sum2(x, y)
sum2(nil, y) -> y
weight1(cons2(n, cons2(m, x))) -> weight1(sum2(cons2(n, cons2(m, x)), cons2(0, x)))
weight1(cons2(n, nil)) -> n

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:

WEIGHT1(cons2(n, cons2(m, x))) -> SUM2(cons2(n, cons2(m, x)), cons2(0, x))
WEIGHT1(cons2(n, cons2(m, x))) -> WEIGHT1(sum2(cons2(n, cons2(m, x)), cons2(0, x)))
SUM2(cons2(0, x), y) -> SUM2(x, y)
SUM2(cons2(s1(n), x), cons2(m, y)) -> SUM2(cons2(n, x), cons2(s1(m), y))

The TRS R consists of the following rules:

sum2(cons2(s1(n), x), cons2(m, y)) -> sum2(cons2(n, x), cons2(s1(m), y))
sum2(cons2(0, x), y) -> sum2(x, y)
sum2(nil, y) -> y
weight1(cons2(n, cons2(m, x))) -> weight1(sum2(cons2(n, cons2(m, x)), cons2(0, x)))
weight1(cons2(n, nil)) -> n

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:

WEIGHT1(cons2(n, cons2(m, x))) -> SUM2(cons2(n, cons2(m, x)), cons2(0, x))
WEIGHT1(cons2(n, cons2(m, x))) -> WEIGHT1(sum2(cons2(n, cons2(m, x)), cons2(0, x)))
SUM2(cons2(0, x), y) -> SUM2(x, y)
SUM2(cons2(s1(n), x), cons2(m, y)) -> SUM2(cons2(n, x), cons2(s1(m), y))

The TRS R consists of the following rules:

sum2(cons2(s1(n), x), cons2(m, y)) -> sum2(cons2(n, x), cons2(s1(m), y))
sum2(cons2(0, x), y) -> sum2(x, y)
sum2(nil, y) -> y
weight1(cons2(n, cons2(m, x))) -> weight1(sum2(cons2(n, cons2(m, x)), cons2(0, x)))
weight1(cons2(n, nil)) -> n

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
      ↳ DependencyGraphProof
        ↳ AND
QDP
            ↳ QDPOrderProof
          ↳ QDP

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

SUM2(cons2(s1(n), x), cons2(m, y)) -> SUM2(cons2(n, x), cons2(s1(m), y))
SUM2(cons2(0, x), y) -> SUM2(x, y)

The TRS R consists of the following rules:

sum2(cons2(s1(n), x), cons2(m, y)) -> sum2(cons2(n, x), cons2(s1(m), y))
sum2(cons2(0, x), y) -> sum2(x, y)
sum2(nil, y) -> y
weight1(cons2(n, cons2(m, x))) -> weight1(sum2(cons2(n, cons2(m, x)), cons2(0, x)))
weight1(cons2(n, nil)) -> n

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.


SUM2(cons2(0, x), y) -> SUM2(x, y)
The remaining pairs can at least be oriented weakly.

SUM2(cons2(s1(n), x), cons2(m, y)) -> SUM2(cons2(n, x), cons2(s1(m), y))
Used ordering: Polynomial interpretation [21]:

POL(0) = 3   
POL(SUM2(x1, x2)) = 3·x1   
POL(cons2(x1, x2)) = 3·x1 + 3·x1·x2 + 3·x2   
POL(s1(x1)) = x1   

The following usable rules [14] were oriented: none



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

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

SUM2(cons2(s1(n), x), cons2(m, y)) -> SUM2(cons2(n, x), cons2(s1(m), y))

The TRS R consists of the following rules:

sum2(cons2(s1(n), x), cons2(m, y)) -> sum2(cons2(n, x), cons2(s1(m), y))
sum2(cons2(0, x), y) -> sum2(x, y)
sum2(nil, y) -> y
weight1(cons2(n, cons2(m, x))) -> weight1(sum2(cons2(n, cons2(m, x)), cons2(0, x)))
weight1(cons2(n, nil)) -> n

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.


SUM2(cons2(s1(n), x), cons2(m, y)) -> SUM2(cons2(n, x), cons2(s1(m), y))
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(SUM2(x1, x2)) = 3·x1 + x2   
POL(cons2(x1, x2)) = 3·x1   
POL(s1(x1)) = 1 + x1   

The following usable rules [14] were oriented: none



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

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

sum2(cons2(s1(n), x), cons2(m, y)) -> sum2(cons2(n, x), cons2(s1(m), y))
sum2(cons2(0, x), y) -> sum2(x, y)
sum2(nil, y) -> y
weight1(cons2(n, cons2(m, x))) -> weight1(sum2(cons2(n, cons2(m, x)), cons2(0, x)))
weight1(cons2(n, nil)) -> n

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
            ↳ QDPOrderProof

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

WEIGHT1(cons2(n, cons2(m, x))) -> WEIGHT1(sum2(cons2(n, cons2(m, x)), cons2(0, x)))

The TRS R consists of the following rules:

sum2(cons2(s1(n), x), cons2(m, y)) -> sum2(cons2(n, x), cons2(s1(m), y))
sum2(cons2(0, x), y) -> sum2(x, y)
sum2(nil, y) -> y
weight1(cons2(n, cons2(m, x))) -> weight1(sum2(cons2(n, cons2(m, x)), cons2(0, x)))
weight1(cons2(n, nil)) -> n

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.


WEIGHT1(cons2(n, cons2(m, x))) -> WEIGHT1(sum2(cons2(n, cons2(m, x)), cons2(0, x)))
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(0) = 0   
POL(WEIGHT1(x1)) = x1 + x12   
POL(cons2(x1, x2)) = 2 + x1 + 3·x1·x2 + 3·x2   
POL(nil) = 0   
POL(s1(x1)) = 0   
POL(sum2(x1, x2)) = 3 + 2·x2   

The following usable rules [14] were oriented:

sum2(cons2(0, x), y) -> sum2(x, y)
sum2(cons2(s1(n), x), cons2(m, y)) -> sum2(cons2(n, x), cons2(s1(m), y))
sum2(nil, y) -> y



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

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

sum2(cons2(s1(n), x), cons2(m, y)) -> sum2(cons2(n, x), cons2(s1(m), y))
sum2(cons2(0, x), y) -> sum2(x, y)
sum2(nil, y) -> y
weight1(cons2(n, cons2(m, x))) -> weight1(sum2(cons2(n, cons2(m, x)), cons2(0, x)))
weight1(cons2(n, nil)) -> n

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.