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:

a__2nd1(cons2(X, cons2(Y, Z))) -> mark1(Y)
a__from1(X) -> cons2(mark1(X), from1(s1(X)))
mark1(2nd1(X)) -> a__2nd1(mark1(X))
mark1(from1(X)) -> a__from1(mark1(X))
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
mark1(s1(X)) -> s1(mark1(X))
a__2nd1(X) -> 2nd1(X)
a__from1(X) -> from1(X)

Q is empty.


QTRS
  ↳ DependencyPairsProof

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

a__2nd1(cons2(X, cons2(Y, Z))) -> mark1(Y)
a__from1(X) -> cons2(mark1(X), from1(s1(X)))
mark1(2nd1(X)) -> a__2nd1(mark1(X))
mark1(from1(X)) -> a__from1(mark1(X))
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
mark1(s1(X)) -> s1(mark1(X))
a__2nd1(X) -> 2nd1(X)
a__from1(X) -> from1(X)

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:

MARK1(s1(X)) -> MARK1(X)
MARK1(2nd1(X)) -> MARK1(X)
MARK1(from1(X)) -> MARK1(X)
A__2ND1(cons2(X, cons2(Y, Z))) -> MARK1(Y)
MARK1(from1(X)) -> A__FROM1(mark1(X))
MARK1(cons2(X1, X2)) -> MARK1(X1)
MARK1(2nd1(X)) -> A__2ND1(mark1(X))
A__FROM1(X) -> MARK1(X)

The TRS R consists of the following rules:

a__2nd1(cons2(X, cons2(Y, Z))) -> mark1(Y)
a__from1(X) -> cons2(mark1(X), from1(s1(X)))
mark1(2nd1(X)) -> a__2nd1(mark1(X))
mark1(from1(X)) -> a__from1(mark1(X))
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
mark1(s1(X)) -> s1(mark1(X))
a__2nd1(X) -> 2nd1(X)
a__from1(X) -> from1(X)

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:

MARK1(s1(X)) -> MARK1(X)
MARK1(2nd1(X)) -> MARK1(X)
MARK1(from1(X)) -> MARK1(X)
A__2ND1(cons2(X, cons2(Y, Z))) -> MARK1(Y)
MARK1(from1(X)) -> A__FROM1(mark1(X))
MARK1(cons2(X1, X2)) -> MARK1(X1)
MARK1(2nd1(X)) -> A__2ND1(mark1(X))
A__FROM1(X) -> MARK1(X)

The TRS R consists of the following rules:

a__2nd1(cons2(X, cons2(Y, Z))) -> mark1(Y)
a__from1(X) -> cons2(mark1(X), from1(s1(X)))
mark1(2nd1(X)) -> a__2nd1(mark1(X))
mark1(from1(X)) -> a__from1(mark1(X))
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
mark1(s1(X)) -> s1(mark1(X))
a__2nd1(X) -> 2nd1(X)
a__from1(X) -> from1(X)

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.


MARK1(from1(X)) -> MARK1(X)
MARK1(from1(X)) -> A__FROM1(mark1(X))
The remaining pairs can at least be oriented weakly.

MARK1(s1(X)) -> MARK1(X)
MARK1(2nd1(X)) -> MARK1(X)
A__2ND1(cons2(X, cons2(Y, Z))) -> MARK1(Y)
MARK1(cons2(X1, X2)) -> MARK1(X1)
MARK1(2nd1(X)) -> A__2ND1(mark1(X))
A__FROM1(X) -> MARK1(X)
Used ordering: Polynomial interpretation [21]:

POL(2nd1(x1)) = x1 + x12   
POL(A__2ND1(x1)) = x12   
POL(A__FROM1(x1)) = x12   
POL(MARK1(x1)) = x12   
POL(a__2nd1(x1)) = x1 + x12   
POL(a__from1(x1)) = 1 + 3·x1 + 3·x12   
POL(cons2(x1, x2)) = x1 + x2   
POL(from1(x1)) = 1 + 2·x1 + 2·x12   
POL(mark1(x1)) = x1 + x12   
POL(s1(x1)) = x1   

The following usable rules [14] were oriented:

a__from1(X) -> cons2(mark1(X), from1(s1(X)))
a__2nd1(X) -> 2nd1(X)
a__from1(X) -> from1(X)
mark1(from1(X)) -> a__from1(mark1(X))
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
mark1(2nd1(X)) -> a__2nd1(mark1(X))
a__2nd1(cons2(X, cons2(Y, Z))) -> mark1(Y)
mark1(s1(X)) -> s1(mark1(X))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ QDPOrderProof
QDP
          ↳ DependencyGraphProof

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

MARK1(2nd1(X)) -> MARK1(X)
MARK1(s1(X)) -> MARK1(X)
A__2ND1(cons2(X, cons2(Y, Z))) -> MARK1(Y)
A__FROM1(X) -> MARK1(X)
MARK1(2nd1(X)) -> A__2ND1(mark1(X))
MARK1(cons2(X1, X2)) -> MARK1(X1)

The TRS R consists of the following rules:

a__2nd1(cons2(X, cons2(Y, Z))) -> mark1(Y)
a__from1(X) -> cons2(mark1(X), from1(s1(X)))
mark1(2nd1(X)) -> a__2nd1(mark1(X))
mark1(from1(X)) -> a__from1(mark1(X))
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
mark1(s1(X)) -> s1(mark1(X))
a__2nd1(X) -> 2nd1(X)
a__from1(X) -> from1(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 1 SCC with 1 less node.

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

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

MARK1(s1(X)) -> MARK1(X)
MARK1(2nd1(X)) -> MARK1(X)
A__2ND1(cons2(X, cons2(Y, Z))) -> MARK1(Y)
MARK1(cons2(X1, X2)) -> MARK1(X1)
MARK1(2nd1(X)) -> A__2ND1(mark1(X))

The TRS R consists of the following rules:

a__2nd1(cons2(X, cons2(Y, Z))) -> mark1(Y)
a__from1(X) -> cons2(mark1(X), from1(s1(X)))
mark1(2nd1(X)) -> a__2nd1(mark1(X))
mark1(from1(X)) -> a__from1(mark1(X))
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
mark1(s1(X)) -> s1(mark1(X))
a__2nd1(X) -> 2nd1(X)
a__from1(X) -> from1(X)

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.


MARK1(2nd1(X)) -> MARK1(X)
MARK1(2nd1(X)) -> A__2ND1(mark1(X))
The remaining pairs can at least be oriented weakly.

MARK1(s1(X)) -> MARK1(X)
A__2ND1(cons2(X, cons2(Y, Z))) -> MARK1(Y)
MARK1(cons2(X1, X2)) -> MARK1(X1)
Used ordering: Polynomial interpretation [21]:

POL(2nd1(x1)) = 1 + x1 + x12   
POL(A__2ND1(x1)) = x12   
POL(MARK1(x1)) = x12   
POL(a__2nd1(x1)) = 1 + x1 + x12   
POL(a__from1(x1)) = 1 + 3·x1 + 3·x12   
POL(cons2(x1, x2)) = x1 + x2   
POL(from1(x1)) = 1 + 2·x1 + 2·x12   
POL(mark1(x1)) = x1 + x12   
POL(s1(x1)) = x1   

The following usable rules [14] were oriented:

a__from1(X) -> cons2(mark1(X), from1(s1(X)))
a__2nd1(X) -> 2nd1(X)
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
mark1(2nd1(X)) -> a__2nd1(mark1(X))
a__2nd1(cons2(X, cons2(Y, Z))) -> mark1(Y)
a__from1(X) -> from1(X)
mark1(s1(X)) -> s1(mark1(X))
mark1(from1(X)) -> a__from1(mark1(X))



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

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

MARK1(s1(X)) -> MARK1(X)
A__2ND1(cons2(X, cons2(Y, Z))) -> MARK1(Y)
MARK1(cons2(X1, X2)) -> MARK1(X1)

The TRS R consists of the following rules:

a__2nd1(cons2(X, cons2(Y, Z))) -> mark1(Y)
a__from1(X) -> cons2(mark1(X), from1(s1(X)))
mark1(2nd1(X)) -> a__2nd1(mark1(X))
mark1(from1(X)) -> a__from1(mark1(X))
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
mark1(s1(X)) -> s1(mark1(X))
a__2nd1(X) -> 2nd1(X)
a__from1(X) -> from1(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 1 SCC with 1 less node.

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

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

MARK1(s1(X)) -> MARK1(X)
MARK1(cons2(X1, X2)) -> MARK1(X1)

The TRS R consists of the following rules:

a__2nd1(cons2(X, cons2(Y, Z))) -> mark1(Y)
a__from1(X) -> cons2(mark1(X), from1(s1(X)))
mark1(2nd1(X)) -> a__2nd1(mark1(X))
mark1(from1(X)) -> a__from1(mark1(X))
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
mark1(s1(X)) -> s1(mark1(X))
a__2nd1(X) -> 2nd1(X)
a__from1(X) -> from1(X)

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.


MARK1(cons2(X1, X2)) -> MARK1(X1)
The remaining pairs can at least be oriented weakly.

MARK1(s1(X)) -> MARK1(X)
Used ordering: Polynomial interpretation [21]:

POL(MARK1(x1)) = x12   
POL(cons2(x1, x2)) = 1 + x1   
POL(s1(x1)) = x1   

The following usable rules [14] were oriented: none



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

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

MARK1(s1(X)) -> MARK1(X)

The TRS R consists of the following rules:

a__2nd1(cons2(X, cons2(Y, Z))) -> mark1(Y)
a__from1(X) -> cons2(mark1(X), from1(s1(X)))
mark1(2nd1(X)) -> a__2nd1(mark1(X))
mark1(from1(X)) -> a__from1(mark1(X))
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
mark1(s1(X)) -> s1(mark1(X))
a__2nd1(X) -> 2nd1(X)
a__from1(X) -> from1(X)

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.


MARK1(s1(X)) -> MARK1(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(MARK1(x1)) = x12   
POL(s1(x1)) = 1 + x12   

The following usable rules [14] were oriented: none



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

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

a__2nd1(cons2(X, cons2(Y, Z))) -> mark1(Y)
a__from1(X) -> cons2(mark1(X), from1(s1(X)))
mark1(2nd1(X)) -> a__2nd1(mark1(X))
mark1(from1(X)) -> a__from1(mark1(X))
mark1(cons2(X1, X2)) -> cons2(mark1(X1), X2)
mark1(s1(X)) -> s1(mark1(X))
a__2nd1(X) -> 2nd1(X)
a__from1(X) -> from1(X)

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.