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__nats -> a__adx1(a__zeros)
a__zeros -> cons2(0, zeros)
a__incr1(cons2(X, Y)) -> cons2(s1(X), incr1(Y))
a__adx1(cons2(X, Y)) -> a__incr1(cons2(X, adx1(Y)))
a__hd1(cons2(X, Y)) -> mark1(X)
a__tl1(cons2(X, Y)) -> mark1(Y)
mark1(nats) -> a__nats
mark1(adx1(X)) -> a__adx1(mark1(X))
mark1(zeros) -> a__zeros
mark1(incr1(X)) -> a__incr1(mark1(X))
mark1(hd1(X)) -> a__hd1(mark1(X))
mark1(tl1(X)) -> a__tl1(mark1(X))
mark1(cons2(X1, X2)) -> cons2(X1, X2)
mark1(0) -> 0
mark1(s1(X)) -> s1(X)
a__nats -> nats
a__adx1(X) -> adx1(X)
a__zeros -> zeros
a__incr1(X) -> incr1(X)
a__hd1(X) -> hd1(X)
a__tl1(X) -> tl1(X)

Q is empty.


↳ QTRS
  ↳ DependencyPairsProof

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

a__nats -> a__adx1(a__zeros)
a__zeros -> cons2(0, zeros)
a__incr1(cons2(X, Y)) -> cons2(s1(X), incr1(Y))
a__adx1(cons2(X, Y)) -> a__incr1(cons2(X, adx1(Y)))
a__hd1(cons2(X, Y)) -> mark1(X)
a__tl1(cons2(X, Y)) -> mark1(Y)
mark1(nats) -> a__nats
mark1(adx1(X)) -> a__adx1(mark1(X))
mark1(zeros) -> a__zeros
mark1(incr1(X)) -> a__incr1(mark1(X))
mark1(hd1(X)) -> a__hd1(mark1(X))
mark1(tl1(X)) -> a__tl1(mark1(X))
mark1(cons2(X1, X2)) -> cons2(X1, X2)
mark1(0) -> 0
mark1(s1(X)) -> s1(X)
a__nats -> nats
a__adx1(X) -> adx1(X)
a__zeros -> zeros
a__incr1(X) -> incr1(X)
a__hd1(X) -> hd1(X)
a__tl1(X) -> tl1(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(tl1(X)) -> A__TL1(mark1(X))
MARK1(hd1(X)) -> A__HD1(mark1(X))
MARK1(tl1(X)) -> MARK1(X)
A__HD1(cons2(X, Y)) -> MARK1(X)
MARK1(adx1(X)) -> A__ADX1(mark1(X))
A__NATS -> A__ADX1(a__zeros)
MARK1(incr1(X)) -> A__INCR1(mark1(X))
MARK1(zeros) -> A__ZEROS
A__TL1(cons2(X, Y)) -> MARK1(Y)
MARK1(hd1(X)) -> MARK1(X)
MARK1(nats) -> A__NATS
A__NATS -> A__ZEROS
A__ADX1(cons2(X, Y)) -> A__INCR1(cons2(X, adx1(Y)))
MARK1(incr1(X)) -> MARK1(X)
MARK1(adx1(X)) -> MARK1(X)

The TRS R consists of the following rules:

a__nats -> a__adx1(a__zeros)
a__zeros -> cons2(0, zeros)
a__incr1(cons2(X, Y)) -> cons2(s1(X), incr1(Y))
a__adx1(cons2(X, Y)) -> a__incr1(cons2(X, adx1(Y)))
a__hd1(cons2(X, Y)) -> mark1(X)
a__tl1(cons2(X, Y)) -> mark1(Y)
mark1(nats) -> a__nats
mark1(adx1(X)) -> a__adx1(mark1(X))
mark1(zeros) -> a__zeros
mark1(incr1(X)) -> a__incr1(mark1(X))
mark1(hd1(X)) -> a__hd1(mark1(X))
mark1(tl1(X)) -> a__tl1(mark1(X))
mark1(cons2(X1, X2)) -> cons2(X1, X2)
mark1(0) -> 0
mark1(s1(X)) -> s1(X)
a__nats -> nats
a__adx1(X) -> adx1(X)
a__zeros -> zeros
a__incr1(X) -> incr1(X)
a__hd1(X) -> hd1(X)
a__tl1(X) -> tl1(X)

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:

MARK1(tl1(X)) -> A__TL1(mark1(X))
MARK1(hd1(X)) -> A__HD1(mark1(X))
MARK1(tl1(X)) -> MARK1(X)
A__HD1(cons2(X, Y)) -> MARK1(X)
MARK1(adx1(X)) -> A__ADX1(mark1(X))
A__NATS -> A__ADX1(a__zeros)
MARK1(incr1(X)) -> A__INCR1(mark1(X))
MARK1(zeros) -> A__ZEROS
A__TL1(cons2(X, Y)) -> MARK1(Y)
MARK1(hd1(X)) -> MARK1(X)
MARK1(nats) -> A__NATS
A__NATS -> A__ZEROS
A__ADX1(cons2(X, Y)) -> A__INCR1(cons2(X, adx1(Y)))
MARK1(incr1(X)) -> MARK1(X)
MARK1(adx1(X)) -> MARK1(X)

The TRS R consists of the following rules:

a__nats -> a__adx1(a__zeros)
a__zeros -> cons2(0, zeros)
a__incr1(cons2(X, Y)) -> cons2(s1(X), incr1(Y))
a__adx1(cons2(X, Y)) -> a__incr1(cons2(X, adx1(Y)))
a__hd1(cons2(X, Y)) -> mark1(X)
a__tl1(cons2(X, Y)) -> mark1(Y)
mark1(nats) -> a__nats
mark1(adx1(X)) -> a__adx1(mark1(X))
mark1(zeros) -> a__zeros
mark1(incr1(X)) -> a__incr1(mark1(X))
mark1(hd1(X)) -> a__hd1(mark1(X))
mark1(tl1(X)) -> a__tl1(mark1(X))
mark1(cons2(X1, X2)) -> cons2(X1, X2)
mark1(0) -> 0
mark1(s1(X)) -> s1(X)
a__nats -> nats
a__adx1(X) -> adx1(X)
a__zeros -> zeros
a__incr1(X) -> incr1(X)
a__hd1(X) -> hd1(X)
a__tl1(X) -> tl1(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 7 less nodes.

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

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

MARK1(tl1(X)) -> A__TL1(mark1(X))
MARK1(hd1(X)) -> A__HD1(mark1(X))
MARK1(tl1(X)) -> MARK1(X)
A__HD1(cons2(X, Y)) -> MARK1(X)
MARK1(incr1(X)) -> MARK1(X)
A__TL1(cons2(X, Y)) -> MARK1(Y)
MARK1(hd1(X)) -> MARK1(X)
MARK1(adx1(X)) -> MARK1(X)

The TRS R consists of the following rules:

a__nats -> a__adx1(a__zeros)
a__zeros -> cons2(0, zeros)
a__incr1(cons2(X, Y)) -> cons2(s1(X), incr1(Y))
a__adx1(cons2(X, Y)) -> a__incr1(cons2(X, adx1(Y)))
a__hd1(cons2(X, Y)) -> mark1(X)
a__tl1(cons2(X, Y)) -> mark1(Y)
mark1(nats) -> a__nats
mark1(adx1(X)) -> a__adx1(mark1(X))
mark1(zeros) -> a__zeros
mark1(incr1(X)) -> a__incr1(mark1(X))
mark1(hd1(X)) -> a__hd1(mark1(X))
mark1(tl1(X)) -> a__tl1(mark1(X))
mark1(cons2(X1, X2)) -> cons2(X1, X2)
mark1(0) -> 0
mark1(s1(X)) -> s1(X)
a__nats -> nats
a__adx1(X) -> adx1(X)
a__zeros -> zeros
a__incr1(X) -> incr1(X)
a__hd1(X) -> hd1(X)
a__tl1(X) -> tl1(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(tl1(X)) -> MARK1(X)
A__TL1(cons2(X, Y)) -> MARK1(Y)
The remaining pairs can at least be oriented weakly.

MARK1(tl1(X)) -> A__TL1(mark1(X))
MARK1(hd1(X)) -> A__HD1(mark1(X))
A__HD1(cons2(X, Y)) -> MARK1(X)
MARK1(incr1(X)) -> MARK1(X)
MARK1(hd1(X)) -> MARK1(X)
MARK1(adx1(X)) -> MARK1(X)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( mark1(x1) ) = x1 + 1


POL( zeros ) = max{0, -1}


POL( a__zeros ) = 1


POL( hd1(x1) ) = x1


POL( MARK1(x1) ) = x1 + 1


POL( 0 ) = max{0, -1}


POL( a__nats ) = 1


POL( A__HD1(x1) ) = x1


POL( adx1(x1) ) = x1


POL( cons2(x1, x2) ) = x1 + x2 + 1


POL( a__hd1(x1) ) = x1


POL( a__adx1(x1) ) = x1


POL( incr1(x1) ) = x1


POL( A__TL1(x1) ) = x1 + 1


POL( nats ) = max{0, -1}


POL( tl1(x1) ) = x1 + 1


POL( a__incr1(x1) ) = x1


POL( s1(x1) ) = x1


POL( a__tl1(x1) ) = x1 + 1



The following usable rules [14] were oriented:

a__adx1(cons2(X, Y)) -> a__incr1(cons2(X, adx1(Y)))
mark1(nats) -> a__nats
a__tl1(X) -> tl1(X)
mark1(0) -> 0
a__nats -> a__adx1(a__zeros)
a__hd1(X) -> hd1(X)
mark1(incr1(X)) -> a__incr1(mark1(X))
a__incr1(cons2(X, Y)) -> cons2(s1(X), incr1(Y))
mark1(s1(X)) -> s1(X)
mark1(hd1(X)) -> a__hd1(mark1(X))
a__hd1(cons2(X, Y)) -> mark1(X)
mark1(tl1(X)) -> a__tl1(mark1(X))
a__tl1(cons2(X, Y)) -> mark1(Y)
a__incr1(X) -> incr1(X)
a__zeros -> cons2(0, zeros)
mark1(zeros) -> a__zeros
a__nats -> nats
a__zeros -> zeros
mark1(adx1(X)) -> a__adx1(mark1(X))
mark1(cons2(X1, X2)) -> cons2(X1, X2)
a__adx1(X) -> adx1(X)



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

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

MARK1(hd1(X)) -> A__HD1(mark1(X))
MARK1(tl1(X)) -> A__TL1(mark1(X))
A__HD1(cons2(X, Y)) -> MARK1(X)
MARK1(incr1(X)) -> MARK1(X)
MARK1(adx1(X)) -> MARK1(X)
MARK1(hd1(X)) -> MARK1(X)

The TRS R consists of the following rules:

a__nats -> a__adx1(a__zeros)
a__zeros -> cons2(0, zeros)
a__incr1(cons2(X, Y)) -> cons2(s1(X), incr1(Y))
a__adx1(cons2(X, Y)) -> a__incr1(cons2(X, adx1(Y)))
a__hd1(cons2(X, Y)) -> mark1(X)
a__tl1(cons2(X, Y)) -> mark1(Y)
mark1(nats) -> a__nats
mark1(adx1(X)) -> a__adx1(mark1(X))
mark1(zeros) -> a__zeros
mark1(incr1(X)) -> a__incr1(mark1(X))
mark1(hd1(X)) -> a__hd1(mark1(X))
mark1(tl1(X)) -> a__tl1(mark1(X))
mark1(cons2(X1, X2)) -> cons2(X1, X2)
mark1(0) -> 0
mark1(s1(X)) -> s1(X)
a__nats -> nats
a__adx1(X) -> adx1(X)
a__zeros -> zeros
a__incr1(X) -> incr1(X)
a__hd1(X) -> hd1(X)
a__tl1(X) -> tl1(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
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ QDP
                  ↳ QDPOrderProof

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

MARK1(hd1(X)) -> A__HD1(mark1(X))
A__HD1(cons2(X, Y)) -> MARK1(X)
MARK1(incr1(X)) -> MARK1(X)
MARK1(hd1(X)) -> MARK1(X)
MARK1(adx1(X)) -> MARK1(X)

The TRS R consists of the following rules:

a__nats -> a__adx1(a__zeros)
a__zeros -> cons2(0, zeros)
a__incr1(cons2(X, Y)) -> cons2(s1(X), incr1(Y))
a__adx1(cons2(X, Y)) -> a__incr1(cons2(X, adx1(Y)))
a__hd1(cons2(X, Y)) -> mark1(X)
a__tl1(cons2(X, Y)) -> mark1(Y)
mark1(nats) -> a__nats
mark1(adx1(X)) -> a__adx1(mark1(X))
mark1(zeros) -> a__zeros
mark1(incr1(X)) -> a__incr1(mark1(X))
mark1(hd1(X)) -> a__hd1(mark1(X))
mark1(tl1(X)) -> a__tl1(mark1(X))
mark1(cons2(X1, X2)) -> cons2(X1, X2)
mark1(0) -> 0
mark1(s1(X)) -> s1(X)
a__nats -> nats
a__adx1(X) -> adx1(X)
a__zeros -> zeros
a__incr1(X) -> incr1(X)
a__hd1(X) -> hd1(X)
a__tl1(X) -> tl1(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(adx1(X)) -> MARK1(X)
The remaining pairs can at least be oriented weakly.

MARK1(hd1(X)) -> A__HD1(mark1(X))
A__HD1(cons2(X, Y)) -> MARK1(X)
MARK1(incr1(X)) -> MARK1(X)
MARK1(hd1(X)) -> MARK1(X)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( mark1(x1) ) = x1


POL( zeros ) = max{0, -1}


POL( a__zeros ) = max{0, -1}


POL( hd1(x1) ) = x1


POL( MARK1(x1) ) = x1


POL( 0 ) = max{0, -1}


POL( a__nats ) = 1


POL( A__HD1(x1) ) = x1


POL( adx1(x1) ) = x1 + 1


POL( cons2(x1, x2) ) = x1 + x2


POL( a__hd1(x1) ) = x1


POL( a__adx1(x1) ) = x1 + 1


POL( incr1(x1) ) = x1


POL( nats ) = 1


POL( tl1(x1) ) = x1


POL( a__incr1(x1) ) = x1


POL( s1(x1) ) = max{0, -1}


POL( a__tl1(x1) ) = x1



The following usable rules [14] were oriented:

mark1(nats) -> a__nats
a__adx1(cons2(X, Y)) -> a__incr1(cons2(X, adx1(Y)))
a__tl1(X) -> tl1(X)
mark1(0) -> 0
a__nats -> a__adx1(a__zeros)
mark1(incr1(X)) -> a__incr1(mark1(X))
a__hd1(X) -> hd1(X)
a__incr1(cons2(X, Y)) -> cons2(s1(X), incr1(Y))
mark1(s1(X)) -> s1(X)
mark1(hd1(X)) -> a__hd1(mark1(X))
a__hd1(cons2(X, Y)) -> mark1(X)
mark1(tl1(X)) -> a__tl1(mark1(X))
a__tl1(cons2(X, Y)) -> mark1(Y)
a__incr1(X) -> incr1(X)
a__zeros -> cons2(0, zeros)
mark1(zeros) -> a__zeros
a__nats -> nats
a__zeros -> zeros
mark1(adx1(X)) -> a__adx1(mark1(X))
mark1(cons2(X1, X2)) -> cons2(X1, X2)
a__adx1(X) -> adx1(X)



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

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

MARK1(hd1(X)) -> A__HD1(mark1(X))
A__HD1(cons2(X, Y)) -> MARK1(X)
MARK1(incr1(X)) -> MARK1(X)
MARK1(hd1(X)) -> MARK1(X)

The TRS R consists of the following rules:

a__nats -> a__adx1(a__zeros)
a__zeros -> cons2(0, zeros)
a__incr1(cons2(X, Y)) -> cons2(s1(X), incr1(Y))
a__adx1(cons2(X, Y)) -> a__incr1(cons2(X, adx1(Y)))
a__hd1(cons2(X, Y)) -> mark1(X)
a__tl1(cons2(X, Y)) -> mark1(Y)
mark1(nats) -> a__nats
mark1(adx1(X)) -> a__adx1(mark1(X))
mark1(zeros) -> a__zeros
mark1(incr1(X)) -> a__incr1(mark1(X))
mark1(hd1(X)) -> a__hd1(mark1(X))
mark1(tl1(X)) -> a__tl1(mark1(X))
mark1(cons2(X1, X2)) -> cons2(X1, X2)
mark1(0) -> 0
mark1(s1(X)) -> s1(X)
a__nats -> nats
a__adx1(X) -> adx1(X)
a__zeros -> zeros
a__incr1(X) -> incr1(X)
a__hd1(X) -> hd1(X)
a__tl1(X) -> tl1(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.


A__HD1(cons2(X, Y)) -> MARK1(X)
MARK1(hd1(X)) -> MARK1(X)
The remaining pairs can at least be oriented weakly.

MARK1(hd1(X)) -> A__HD1(mark1(X))
MARK1(incr1(X)) -> MARK1(X)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( mark1(x1) ) = x1


POL( zeros ) = max{0, -1}


POL( a__zeros ) = 0


POL( hd1(x1) ) = x1 + 1


POL( MARK1(x1) ) = x1


POL( 0 ) = max{0, -1}


POL( a__nats ) = 1


POL( A__HD1(x1) ) = x1 + 1


POL( adx1(x1) ) = x1


POL( cons2(x1, x2) ) = x1 + x2


POL( a__hd1(x1) ) = x1 + 1


POL( a__adx1(x1) ) = x1


POL( incr1(x1) ) = x1


POL( nats ) = 1


POL( tl1(x1) ) = x1


POL( a__incr1(x1) ) = x1


POL( s1(x1) ) = x1


POL( a__tl1(x1) ) = x1



The following usable rules [14] were oriented:

mark1(nats) -> a__nats
a__adx1(cons2(X, Y)) -> a__incr1(cons2(X, adx1(Y)))
a__tl1(X) -> tl1(X)
mark1(0) -> 0
a__nats -> a__adx1(a__zeros)
mark1(incr1(X)) -> a__incr1(mark1(X))
a__hd1(X) -> hd1(X)
a__incr1(cons2(X, Y)) -> cons2(s1(X), incr1(Y))
mark1(s1(X)) -> s1(X)
mark1(hd1(X)) -> a__hd1(mark1(X))
a__hd1(cons2(X, Y)) -> mark1(X)
mark1(tl1(X)) -> a__tl1(mark1(X))
a__tl1(cons2(X, Y)) -> mark1(Y)
a__incr1(X) -> incr1(X)
a__zeros -> cons2(0, zeros)
mark1(zeros) -> a__zeros
a__nats -> nats
a__zeros -> zeros
mark1(adx1(X)) -> a__adx1(mark1(X))
mark1(cons2(X1, X2)) -> cons2(X1, X2)
a__adx1(X) -> adx1(X)



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

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

MARK1(hd1(X)) -> A__HD1(mark1(X))
MARK1(incr1(X)) -> MARK1(X)

The TRS R consists of the following rules:

a__nats -> a__adx1(a__zeros)
a__zeros -> cons2(0, zeros)
a__incr1(cons2(X, Y)) -> cons2(s1(X), incr1(Y))
a__adx1(cons2(X, Y)) -> a__incr1(cons2(X, adx1(Y)))
a__hd1(cons2(X, Y)) -> mark1(X)
a__tl1(cons2(X, Y)) -> mark1(Y)
mark1(nats) -> a__nats
mark1(adx1(X)) -> a__adx1(mark1(X))
mark1(zeros) -> a__zeros
mark1(incr1(X)) -> a__incr1(mark1(X))
mark1(hd1(X)) -> a__hd1(mark1(X))
mark1(tl1(X)) -> a__tl1(mark1(X))
mark1(cons2(X1, X2)) -> cons2(X1, X2)
mark1(0) -> 0
mark1(s1(X)) -> s1(X)
a__nats -> nats
a__adx1(X) -> adx1(X)
a__zeros -> zeros
a__incr1(X) -> incr1(X)
a__hd1(X) -> hd1(X)
a__tl1(X) -> tl1(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
      ↳ DependencyGraphProof
        ↳ QDP
          ↳ QDPOrderProof
            ↳ QDP
              ↳ DependencyGraphProof
                ↳ QDP
                  ↳ QDPOrderProof
                    ↳ QDP
                      ↳ QDPOrderProof
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ QDPOrderProof

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

MARK1(incr1(X)) -> MARK1(X)

The TRS R consists of the following rules:

a__nats -> a__adx1(a__zeros)
a__zeros -> cons2(0, zeros)
a__incr1(cons2(X, Y)) -> cons2(s1(X), incr1(Y))
a__adx1(cons2(X, Y)) -> a__incr1(cons2(X, adx1(Y)))
a__hd1(cons2(X, Y)) -> mark1(X)
a__tl1(cons2(X, Y)) -> mark1(Y)
mark1(nats) -> a__nats
mark1(adx1(X)) -> a__adx1(mark1(X))
mark1(zeros) -> a__zeros
mark1(incr1(X)) -> a__incr1(mark1(X))
mark1(hd1(X)) -> a__hd1(mark1(X))
mark1(tl1(X)) -> a__tl1(mark1(X))
mark1(cons2(X1, X2)) -> cons2(X1, X2)
mark1(0) -> 0
mark1(s1(X)) -> s1(X)
a__nats -> nats
a__adx1(X) -> adx1(X)
a__zeros -> zeros
a__incr1(X) -> incr1(X)
a__hd1(X) -> hd1(X)
a__tl1(X) -> tl1(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(incr1(X)) -> MARK1(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( incr1(x1) ) = x1 + 1


POL( MARK1(x1) ) = x1



The following usable rules [14] were oriented: none



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

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

a__nats -> a__adx1(a__zeros)
a__zeros -> cons2(0, zeros)
a__incr1(cons2(X, Y)) -> cons2(s1(X), incr1(Y))
a__adx1(cons2(X, Y)) -> a__incr1(cons2(X, adx1(Y)))
a__hd1(cons2(X, Y)) -> mark1(X)
a__tl1(cons2(X, Y)) -> mark1(Y)
mark1(nats) -> a__nats
mark1(adx1(X)) -> a__adx1(mark1(X))
mark1(zeros) -> a__zeros
mark1(incr1(X)) -> a__incr1(mark1(X))
mark1(hd1(X)) -> a__hd1(mark1(X))
mark1(tl1(X)) -> a__tl1(mark1(X))
mark1(cons2(X1, X2)) -> cons2(X1, X2)
mark1(0) -> 0
mark1(s1(X)) -> s1(X)
a__nats -> nats
a__adx1(X) -> adx1(X)
a__zeros -> zeros
a__incr1(X) -> incr1(X)
a__hd1(X) -> hd1(X)
a__tl1(X) -> tl1(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.