Termination w.r.t. Q proof of /home/nowonder/forschung/aprove/satrung/aprove/lpar16/TRCSRSLASHEx4_7_37_Bor03

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:

from₁(X) -> cons₂(X, n__from₁(n__s₁(X)))
sel₂(0, cons₂(X, XS)) -> X
sel₂(s₁(N), cons₂(X, XS)) -> sel₂(N, activate₁(XS))
minus₂(X, 0) -> 0
minus₂(s₁(X), s₁(Y)) -> minus₂(X, Y)
quot₂(0, s₁(Y)) -> 0
quot₂(s₁(X), s₁(Y)) -> s₁(quot₂(minus₂(X, Y), s₁(Y)))
zWquot₂(XS, nil) -> nil
zWquot₂(nil, XS) -> nil
zWquot₂(cons₂(X, XS), cons₂(Y, YS)) -> cons₂(quot₂(X, Y), n__zWquot₂(activate₁(XS), activate₁(YS)))
from₁(X) -> n__from₁(X)
s₁(X) -> n__s₁(X)
zWquot₂(X1, X2) -> n__zWquot₂(X1, X2)
activate₁(n__from₁(X)) -> from₁(activate₁(X))
activate₁(n__s₁(X)) -> s₁(activate₁(X))
activate₁(n__zWquot₂(X1, X2)) -> zWquot₂(activate₁(X1), activate₁(X2))
activate₁(X) -> X

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

from₁(X) -> cons₂(X, n__from₁(n__s₁(X)))
sel₂(0, cons₂(X, XS)) -> X
sel₂(s₁(N), cons₂(X, XS)) -> sel₂(N, activate₁(XS))
minus₂(X, 0) -> 0
minus₂(s₁(X), s₁(Y)) -> minus₂(X, Y)
quot₂(0, s₁(Y)) -> 0
quot₂(s₁(X), s₁(Y)) -> s₁(quot₂(minus₂(X, Y), s₁(Y)))
zWquot₂(XS, nil) -> nil
zWquot₂(nil, XS) -> nil
zWquot₂(cons₂(X, XS), cons₂(Y, YS)) -> cons₂(quot₂(X, Y), n__zWquot₂(activate₁(XS), activate₁(YS)))
from₁(X) -> n__from₁(X)
s₁(X) -> n__s₁(X)
zWquot₂(X1, X2) -> n__zWquot₂(X1, X2)
activate₁(n__from₁(X)) -> from₁(activate₁(X))
activate₁(n__s₁(X)) -> s₁(activate₁(X))
activate₁(n__zWquot₂(X1, X2)) -> zWquot₂(activate₁(X1), activate₁(X2))
activate₁(X) -> 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:

QUOT₂(s₁(X), s₁(Y)) -> QUOT₂(minus₂(X, Y), s₁(Y))
ZWQUOT₂(cons₂(X, XS), cons₂(Y, YS)) -> ACTIVATE₁(XS)
ACTIVATE₁(n__zWquot₂(X1, X2)) -> ZWQUOT₂(activate₁(X1), activate₁(X2))
MINUS₂(s₁(X), s₁(Y)) -> MINUS₂(X, Y)
QUOT₂(s₁(X), s₁(Y)) -> MINUS₂(X, Y)
ZWQUOT₂(cons₂(X, XS), cons₂(Y, YS)) -> QUOT₂(X, Y)
ACTIVATE₁(n__from₁(X)) -> FROM₁(activate₁(X))
SEL₂(s₁(N), cons₂(X, XS)) -> ACTIVATE₁(XS)
ACTIVATE₁(n__s₁(X)) -> S₁(activate₁(X))
ACTIVATE₁(n__zWquot₂(X1, X2)) -> ACTIVATE₁(X1)
SEL₂(s₁(N), cons₂(X, XS)) -> SEL₂(N, activate₁(XS))
ACTIVATE₁(n__zWquot₂(X1, X2)) -> ACTIVATE₁(X2)
QUOT₂(s₁(X), s₁(Y)) -> S₁(quot₂(minus₂(X, Y), s₁(Y)))
ZWQUOT₂(cons₂(X, XS), cons₂(Y, YS)) -> ACTIVATE₁(YS)
ACTIVATE₁(n__from₁(X)) -> ACTIVATE₁(X)
ACTIVATE₁(n__s₁(X)) -> ACTIVATE₁(X)

The TRS R consists of the following rules:

from₁(X) -> cons₂(X, n__from₁(n__s₁(X)))
sel₂(0, cons₂(X, XS)) -> X
sel₂(s₁(N), cons₂(X, XS)) -> sel₂(N, activate₁(XS))
minus₂(X, 0) -> 0
minus₂(s₁(X), s₁(Y)) -> minus₂(X, Y)
quot₂(0, s₁(Y)) -> 0
quot₂(s₁(X), s₁(Y)) -> s₁(quot₂(minus₂(X, Y), s₁(Y)))
zWquot₂(XS, nil) -> nil
zWquot₂(nil, XS) -> nil
zWquot₂(cons₂(X, XS), cons₂(Y, YS)) -> cons₂(quot₂(X, Y), n__zWquot₂(activate₁(XS), activate₁(YS)))
from₁(X) -> n__from₁(X)
s₁(X) -> n__s₁(X)
zWquot₂(X1, X2) -> n__zWquot₂(X1, X2)
activate₁(n__from₁(X)) -> from₁(activate₁(X))
activate₁(n__s₁(X)) -> s₁(activate₁(X))
activate₁(n__zWquot₂(X1, X2)) -> zWquot₂(activate₁(X1), activate₁(X2))
activate₁(X) -> 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:

QUOT₂(s₁(X), s₁(Y)) -> QUOT₂(minus₂(X, Y), s₁(Y))
ZWQUOT₂(cons₂(X, XS), cons₂(Y, YS)) -> ACTIVATE₁(XS)
ACTIVATE₁(n__zWquot₂(X1, X2)) -> ZWQUOT₂(activate₁(X1), activate₁(X2))
MINUS₂(s₁(X), s₁(Y)) -> MINUS₂(X, Y)
QUOT₂(s₁(X), s₁(Y)) -> MINUS₂(X, Y)
ZWQUOT₂(cons₂(X, XS), cons₂(Y, YS)) -> QUOT₂(X, Y)
ACTIVATE₁(n__from₁(X)) -> FROM₁(activate₁(X))
SEL₂(s₁(N), cons₂(X, XS)) -> ACTIVATE₁(XS)
ACTIVATE₁(n__s₁(X)) -> S₁(activate₁(X))
ACTIVATE₁(n__zWquot₂(X1, X2)) -> ACTIVATE₁(X1)
SEL₂(s₁(N), cons₂(X, XS)) -> SEL₂(N, activate₁(XS))
ACTIVATE₁(n__zWquot₂(X1, X2)) -> ACTIVATE₁(X2)
QUOT₂(s₁(X), s₁(Y)) -> S₁(quot₂(minus₂(X, Y), s₁(Y)))
ZWQUOT₂(cons₂(X, XS), cons₂(Y, YS)) -> ACTIVATE₁(YS)
ACTIVATE₁(n__from₁(X)) -> ACTIVATE₁(X)
ACTIVATE₁(n__s₁(X)) -> ACTIVATE₁(X)

The TRS R consists of the following rules:

from₁(X) -> cons₂(X, n__from₁(n__s₁(X)))
sel₂(0, cons₂(X, XS)) -> X
sel₂(s₁(N), cons₂(X, XS)) -> sel₂(N, activate₁(XS))
minus₂(X, 0) -> 0
minus₂(s₁(X), s₁(Y)) -> minus₂(X, Y)
quot₂(0, s₁(Y)) -> 0
quot₂(s₁(X), s₁(Y)) -> s₁(quot₂(minus₂(X, Y), s₁(Y)))
zWquot₂(XS, nil) -> nil
zWquot₂(nil, XS) -> nil
zWquot₂(cons₂(X, XS), cons₂(Y, YS)) -> cons₂(quot₂(X, Y), n__zWquot₂(activate₁(XS), activate₁(YS)))
from₁(X) -> n__from₁(X)
s₁(X) -> n__s₁(X)
zWquot₂(X1, X2) -> n__zWquot₂(X1, X2)
activate₁(n__from₁(X)) -> from₁(activate₁(X))
activate₁(n__s₁(X)) -> s₁(activate₁(X))
activate₁(n__zWquot₂(X1, X2)) -> zWquot₂(activate₁(X1), activate₁(X2))
activate₁(X) -> X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 4 SCCs with 6 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

MINUS₂(s₁(X), s₁(Y)) -> MINUS₂(X, Y)

The TRS R consists of the following rules:

from₁(X) -> cons₂(X, n__from₁(n__s₁(X)))
sel₂(0, cons₂(X, XS)) -> X
sel₂(s₁(N), cons₂(X, XS)) -> sel₂(N, activate₁(XS))
minus₂(X, 0) -> 0
minus₂(s₁(X), s₁(Y)) -> minus₂(X, Y)
quot₂(0, s₁(Y)) -> 0
quot₂(s₁(X), s₁(Y)) -> s₁(quot₂(minus₂(X, Y), s₁(Y)))
zWquot₂(XS, nil) -> nil
zWquot₂(nil, XS) -> nil
zWquot₂(cons₂(X, XS), cons₂(Y, YS)) -> cons₂(quot₂(X, Y), n__zWquot₂(activate₁(XS), activate₁(YS)))
from₁(X) -> n__from₁(X)
s₁(X) -> n__s₁(X)
zWquot₂(X1, X2) -> n__zWquot₂(X1, X2)
activate₁(n__from₁(X)) -> from₁(activate₁(X))
activate₁(n__s₁(X)) -> s₁(activate₁(X))
activate₁(n__zWquot₂(X1, X2)) -> zWquot₂(activate₁(X1), activate₁(X2))
activate₁(X) -> 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 strictly oriented and are deleted.

MINUS₂(s₁(X), s₁(Y)) -> MINUS₂(X, Y)

The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
MINUS₂(x1, x2) = MINUS₁(x1)
s₁(x1) = s₁(x1)

Lexicographic Path Order [19].
Precedence:

trivial

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

from₁(X) -> cons₂(X, n__from₁(n__s₁(X)))
sel₂(0, cons₂(X, XS)) -> X
sel₂(s₁(N), cons₂(X, XS)) -> sel₂(N, activate₁(XS))
minus₂(X, 0) -> 0
minus₂(s₁(X), s₁(Y)) -> minus₂(X, Y)
quot₂(0, s₁(Y)) -> 0
quot₂(s₁(X), s₁(Y)) -> s₁(quot₂(minus₂(X, Y), s₁(Y)))
zWquot₂(XS, nil) -> nil
zWquot₂(nil, XS) -> nil
zWquot₂(cons₂(X, XS), cons₂(Y, YS)) -> cons₂(quot₂(X, Y), n__zWquot₂(activate₁(XS), activate₁(YS)))
from₁(X) -> n__from₁(X)
s₁(X) -> n__s₁(X)
zWquot₂(X1, X2) -> n__zWquot₂(X1, X2)
activate₁(n__from₁(X)) -> from₁(activate₁(X))
activate₁(n__s₁(X)) -> s₁(activate₁(X))
activate₁(n__zWquot₂(X1, X2)) -> zWquot₂(activate₁(X1), activate₁(X2))
activate₁(X) -> 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.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

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

QUOT₂(s₁(X), s₁(Y)) -> QUOT₂(minus₂(X, Y), s₁(Y))

The TRS R consists of the following rules:

from₁(X) -> cons₂(X, n__from₁(n__s₁(X)))
sel₂(0, cons₂(X, XS)) -> X
sel₂(s₁(N), cons₂(X, XS)) -> sel₂(N, activate₁(XS))
minus₂(X, 0) -> 0
minus₂(s₁(X), s₁(Y)) -> minus₂(X, Y)
quot₂(0, s₁(Y)) -> 0
quot₂(s₁(X), s₁(Y)) -> s₁(quot₂(minus₂(X, Y), s₁(Y)))
zWquot₂(XS, nil) -> nil
zWquot₂(nil, XS) -> nil
zWquot₂(cons₂(X, XS), cons₂(Y, YS)) -> cons₂(quot₂(X, Y), n__zWquot₂(activate₁(XS), activate₁(YS)))
from₁(X) -> n__from₁(X)
s₁(X) -> n__s₁(X)
zWquot₂(X1, X2) -> n__zWquot₂(X1, X2)
activate₁(n__from₁(X)) -> from₁(activate₁(X))
activate₁(n__s₁(X)) -> s₁(activate₁(X))
activate₁(n__zWquot₂(X1, X2)) -> zWquot₂(activate₁(X1), activate₁(X2))
activate₁(X) -> 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 strictly oriented and are deleted.

QUOT₂(s₁(X), s₁(Y)) -> QUOT₂(minus₂(X, Y), s₁(Y))

The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
QUOT₂(x1, x2) = x1
s₁(x1) = s₁(x1)
minus₂(x1, x2) = minus₁(x1)
0 = 0
n__s₁(x1) = n__s₁(x1)

Lexicographic Path Order [19].
Precedence:

s₁ > [minus₁, 0] > n_{_s₁}

The following usable rules [14] were oriented:

minus₂(X, 0) -> 0
minus₂(s₁(X), s₁(Y)) -> minus₂(X, Y)
s₁(X) -> n__s₁(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

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

from₁(X) -> cons₂(X, n__from₁(n__s₁(X)))
sel₂(0, cons₂(X, XS)) -> X
sel₂(s₁(N), cons₂(X, XS)) -> sel₂(N, activate₁(XS))
minus₂(X, 0) -> 0
minus₂(s₁(X), s₁(Y)) -> minus₂(X, Y)
quot₂(0, s₁(Y)) -> 0
quot₂(s₁(X), s₁(Y)) -> s₁(quot₂(minus₂(X, Y), s₁(Y)))
zWquot₂(XS, nil) -> nil
zWquot₂(nil, XS) -> nil
zWquot₂(cons₂(X, XS), cons₂(Y, YS)) -> cons₂(quot₂(X, Y), n__zWquot₂(activate₁(XS), activate₁(YS)))
from₁(X) -> n__from₁(X)
s₁(X) -> n__s₁(X)
zWquot₂(X1, X2) -> n__zWquot₂(X1, X2)
activate₁(n__from₁(X)) -> from₁(activate₁(X))
activate₁(n__s₁(X)) -> s₁(activate₁(X))
activate₁(n__zWquot₂(X1, X2)) -> zWquot₂(activate₁(X1), activate₁(X2))
activate₁(X) -> 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.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

          ↳ QDP

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

ACTIVATE₁(n__zWquot₂(X1, X2)) -> ACTIVATE₁(X1)
ZWQUOT₂(cons₂(X, XS), cons₂(Y, YS)) -> ACTIVATE₁(XS)
ACTIVATE₁(n__zWquot₂(X1, X2)) -> ZWQUOT₂(activate₁(X1), activate₁(X2))
ACTIVATE₁(n__zWquot₂(X1, X2)) -> ACTIVATE₁(X2)
ZWQUOT₂(cons₂(X, XS), cons₂(Y, YS)) -> ACTIVATE₁(YS)
ACTIVATE₁(n__from₁(X)) -> ACTIVATE₁(X)
ACTIVATE₁(n__s₁(X)) -> ACTIVATE₁(X)

The TRS R consists of the following rules:

from₁(X) -> cons₂(X, n__from₁(n__s₁(X)))
sel₂(0, cons₂(X, XS)) -> X
sel₂(s₁(N), cons₂(X, XS)) -> sel₂(N, activate₁(XS))
minus₂(X, 0) -> 0
minus₂(s₁(X), s₁(Y)) -> minus₂(X, Y)
quot₂(0, s₁(Y)) -> 0
quot₂(s₁(X), s₁(Y)) -> s₁(quot₂(minus₂(X, Y), s₁(Y)))
zWquot₂(XS, nil) -> nil
zWquot₂(nil, XS) -> nil
zWquot₂(cons₂(X, XS), cons₂(Y, YS)) -> cons₂(quot₂(X, Y), n__zWquot₂(activate₁(XS), activate₁(YS)))
from₁(X) -> n__from₁(X)
s₁(X) -> n__s₁(X)
zWquot₂(X1, X2) -> n__zWquot₂(X1, X2)
activate₁(n__from₁(X)) -> from₁(activate₁(X))
activate₁(n__s₁(X)) -> s₁(activate₁(X))
activate₁(n__zWquot₂(X1, X2)) -> zWquot₂(activate₁(X1), activate₁(X2))
activate₁(X) -> 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 strictly oriented and are deleted.

ACTIVATE₁(n__zWquot₂(X1, X2)) -> ACTIVATE₁(X1)
ZWQUOT₂(cons₂(X, XS), cons₂(Y, YS)) -> ACTIVATE₁(XS)
ACTIVATE₁(n__zWquot₂(X1, X2)) -> ZWQUOT₂(activate₁(X1), activate₁(X2))
ACTIVATE₁(n__zWquot₂(X1, X2)) -> ACTIVATE₁(X2)
ZWQUOT₂(cons₂(X, XS), cons₂(Y, YS)) -> ACTIVATE₁(YS)

The remaining pairs can at least by weakly be oriented.

ACTIVATE₁(n__from₁(X)) -> ACTIVATE₁(X)
ACTIVATE₁(n__s₁(X)) -> ACTIVATE₁(X)

Used ordering: Combined order from the following AFS and order.
ACTIVATE₁(x1) = ACTIVATE₁(x1)
n__zWquot₂(x1, x2) = n__zWquot₂(x1, x2)
ZWQUOT₂(x1, x2) = ZWQUOT₂(x1, x2)
cons₂(x1, x2) = x2
activate₁(x1) = x1
n__from₁(x1) = x1
n__s₁(x1) = x1
from₁(x1) = x1
s₁(x1) = x1
zWquot₂(x1, x2) = zWquot₂(x1, x2)
nil = nil
quot₂(x1, x2) = x2
0 = 0
minus₂(x1, x2) = minus₁(x2)

Lexicographic Path Order [19].
Precedence:

[n_{_zWquot₂}, zWquot₂, nil] > ZWQUOT₂ > ACTIVATE₁ > 0
minus₁ > 0

The following usable rules [14] were oriented:

activate₁(n__from₁(X)) -> from₁(activate₁(X))
activate₁(n__s₁(X)) -> s₁(activate₁(X))
activate₁(n__zWquot₂(X1, X2)) -> zWquot₂(activate₁(X1), activate₁(X2))
activate₁(X) -> X
zWquot₂(XS, nil) -> nil
zWquot₂(nil, XS) -> nil
zWquot₂(cons₂(X, XS), cons₂(Y, YS)) -> cons₂(quot₂(X, Y), n__zWquot₂(activate₁(XS), activate₁(YS)))
zWquot₂(X1, X2) -> n__zWquot₂(X1, X2)
quot₂(0, s₁(Y)) -> 0
quot₂(s₁(X), s₁(Y)) -> s₁(quot₂(minus₂(X, Y), s₁(Y)))
minus₂(X, 0) -> 0
minus₂(s₁(X), s₁(Y)) -> minus₂(X, Y)
s₁(X) -> n__s₁(X)
from₁(X) -> cons₂(X, n__from₁(n__s₁(X)))
from₁(X) -> n__from₁(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

          ↳ QDP

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

ACTIVATE₁(n__from₁(X)) -> ACTIVATE₁(X)
ACTIVATE₁(n__s₁(X)) -> ACTIVATE₁(X)

The TRS R consists of the following rules:

from₁(X) -> cons₂(X, n__from₁(n__s₁(X)))
sel₂(0, cons₂(X, XS)) -> X
sel₂(s₁(N), cons₂(X, XS)) -> sel₂(N, activate₁(XS))
minus₂(X, 0) -> 0
minus₂(s₁(X), s₁(Y)) -> minus₂(X, Y)
quot₂(0, s₁(Y)) -> 0
quot₂(s₁(X), s₁(Y)) -> s₁(quot₂(minus₂(X, Y), s₁(Y)))
zWquot₂(XS, nil) -> nil
zWquot₂(nil, XS) -> nil
zWquot₂(cons₂(X, XS), cons₂(Y, YS)) -> cons₂(quot₂(X, Y), n__zWquot₂(activate₁(XS), activate₁(YS)))
from₁(X) -> n__from₁(X)
s₁(X) -> n__s₁(X)
zWquot₂(X1, X2) -> n__zWquot₂(X1, X2)
activate₁(n__from₁(X)) -> from₁(activate₁(X))
activate₁(n__s₁(X)) -> s₁(activate₁(X))
activate₁(n__zWquot₂(X1, X2)) -> zWquot₂(activate₁(X1), activate₁(X2))
activate₁(X) -> 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 strictly oriented and are deleted.

ACTIVATE₁(n__s₁(X)) -> ACTIVATE₁(X)

The remaining pairs can at least by weakly be oriented.

ACTIVATE₁(n__from₁(X)) -> ACTIVATE₁(X)

Used ordering: Combined order from the following AFS and order.
ACTIVATE₁(x1) = ACTIVATE₁(x1)
n__from₁(x1) = x1
n__s₁(x1) = n__s₁(x1)

Lexicographic Path Order [19].
Precedence:

trivial

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

          ↳ QDP

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

ACTIVATE₁(n__from₁(X)) -> ACTIVATE₁(X)

The TRS R consists of the following rules:

from₁(X) -> cons₂(X, n__from₁(n__s₁(X)))
sel₂(0, cons₂(X, XS)) -> X
sel₂(s₁(N), cons₂(X, XS)) -> sel₂(N, activate₁(XS))
minus₂(X, 0) -> 0
minus₂(s₁(X), s₁(Y)) -> minus₂(X, Y)
quot₂(0, s₁(Y)) -> 0
quot₂(s₁(X), s₁(Y)) -> s₁(quot₂(minus₂(X, Y), s₁(Y)))
zWquot₂(XS, nil) -> nil
zWquot₂(nil, XS) -> nil
zWquot₂(cons₂(X, XS), cons₂(Y, YS)) -> cons₂(quot₂(X, Y), n__zWquot₂(activate₁(XS), activate₁(YS)))
from₁(X) -> n__from₁(X)
s₁(X) -> n__s₁(X)
zWquot₂(X1, X2) -> n__zWquot₂(X1, X2)
activate₁(n__from₁(X)) -> from₁(activate₁(X))
activate₁(n__s₁(X)) -> s₁(activate₁(X))
activate₁(n__zWquot₂(X1, X2)) -> zWquot₂(activate₁(X1), activate₁(X2))
activate₁(X) -> 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 strictly oriented and are deleted.

ACTIVATE₁(n__from₁(X)) -> ACTIVATE₁(X)

The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
ACTIVATE₁(x1) = ACTIVATE₁(x1)
n__from₁(x1) = n__from₁(x1)

Lexicographic Path Order [19].
Precedence:

n_{_from₁} > ACTIVATE₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ PisEmptyProof

          ↳ QDP

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

from₁(X) -> cons₂(X, n__from₁(n__s₁(X)))
sel₂(0, cons₂(X, XS)) -> X
sel₂(s₁(N), cons₂(X, XS)) -> sel₂(N, activate₁(XS))
minus₂(X, 0) -> 0
minus₂(s₁(X), s₁(Y)) -> minus₂(X, Y)
quot₂(0, s₁(Y)) -> 0
quot₂(s₁(X), s₁(Y)) -> s₁(quot₂(minus₂(X, Y), s₁(Y)))
zWquot₂(XS, nil) -> nil
zWquot₂(nil, XS) -> nil
zWquot₂(cons₂(X, XS), cons₂(Y, YS)) -> cons₂(quot₂(X, Y), n__zWquot₂(activate₁(XS), activate₁(YS)))
from₁(X) -> n__from₁(X)
s₁(X) -> n__s₁(X)
zWquot₂(X1, X2) -> n__zWquot₂(X1, X2)
activate₁(n__from₁(X)) -> from₁(activate₁(X))
activate₁(n__s₁(X)) -> s₁(activate₁(X))
activate₁(n__zWquot₂(X1, X2)) -> zWquot₂(activate₁(X1), activate₁(X2))
activate₁(X) -> 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.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

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

SEL₂(s₁(N), cons₂(X, XS)) -> SEL₂(N, activate₁(XS))

The TRS R consists of the following rules:

from₁(X) -> cons₂(X, n__from₁(n__s₁(X)))
sel₂(0, cons₂(X, XS)) -> X
sel₂(s₁(N), cons₂(X, XS)) -> sel₂(N, activate₁(XS))
minus₂(X, 0) -> 0
minus₂(s₁(X), s₁(Y)) -> minus₂(X, Y)
quot₂(0, s₁(Y)) -> 0
quot₂(s₁(X), s₁(Y)) -> s₁(quot₂(minus₂(X, Y), s₁(Y)))
zWquot₂(XS, nil) -> nil
zWquot₂(nil, XS) -> nil
zWquot₂(cons₂(X, XS), cons₂(Y, YS)) -> cons₂(quot₂(X, Y), n__zWquot₂(activate₁(XS), activate₁(YS)))
from₁(X) -> n__from₁(X)
s₁(X) -> n__s₁(X)
zWquot₂(X1, X2) -> n__zWquot₂(X1, X2)
activate₁(n__from₁(X)) -> from₁(activate₁(X))
activate₁(n__s₁(X)) -> s₁(activate₁(X))
activate₁(n__zWquot₂(X1, X2)) -> zWquot₂(activate₁(X1), activate₁(X2))
activate₁(X) -> 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 strictly oriented and are deleted.

SEL₂(s₁(N), cons₂(X, XS)) -> SEL₂(N, activate₁(XS))

The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
SEL₂(x1, x2) = x1
s₁(x1) = s₁(x1)
cons₂(x1, x2) = cons
activate₁(x1) = x1
n__from₁(x1) = n__from₁(x1)
from₁(x1) = from₁(x1)
n__s₁(x1) = n__s₁(x1)
n__zWquot₂(x1, x2) = x2
zWquot₂(x1, x2) = x2
nil = nil
quot₂(x1, x2) = quot₂(x1, x2)
0 = 0
minus₂(x1, x2) = x1

Lexicographic Path Order [19].
Precedence:

[cons, n_{_from₁}, from_1] > quot₂ > [s₁, n_{_s₁}, nil, 0]

The following usable rules [14] were oriented:

activate₁(n__from₁(X)) -> from₁(activate₁(X))
activate₁(n__s₁(X)) -> s₁(activate₁(X))
activate₁(n__zWquot₂(X1, X2)) -> zWquot₂(activate₁(X1), activate₁(X2))
activate₁(X) -> X
zWquot₂(XS, nil) -> nil
zWquot₂(nil, XS) -> nil
zWquot₂(cons₂(X, XS), cons₂(Y, YS)) -> cons₂(quot₂(X, Y), n__zWquot₂(activate₁(XS), activate₁(YS)))
zWquot₂(X1, X2) -> n__zWquot₂(X1, X2)
quot₂(0, s₁(Y)) -> 0
quot₂(s₁(X), s₁(Y)) -> s₁(quot₂(minus₂(X, Y), s₁(Y)))
minus₂(X, 0) -> 0
minus₂(s₁(X), s₁(Y)) -> minus₂(X, Y)
s₁(X) -> n__s₁(X)
from₁(X) -> cons₂(X, n__from₁(n__s₁(X)))
from₁(X) -> n__from₁(X)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ PisEmptyProof

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

from₁(X) -> cons₂(X, n__from₁(n__s₁(X)))
sel₂(0, cons₂(X, XS)) -> X
sel₂(s₁(N), cons₂(X, XS)) -> sel₂(N, activate₁(XS))
minus₂(X, 0) -> 0
minus₂(s₁(X), s₁(Y)) -> minus₂(X, Y)
quot₂(0, s₁(Y)) -> 0
quot₂(s₁(X), s₁(Y)) -> s₁(quot₂(minus₂(X, Y), s₁(Y)))
zWquot₂(XS, nil) -> nil
zWquot₂(nil, XS) -> nil
zWquot₂(cons₂(X, XS), cons₂(Y, YS)) -> cons₂(quot₂(X, Y), n__zWquot₂(activate₁(XS), activate₁(YS)))
from₁(X) -> n__from₁(X)
s₁(X) -> n__s₁(X)
zWquot₂(X1, X2) -> n__zWquot₂(X1, X2)
activate₁(n__from₁(X)) -> from₁(activate₁(X))
activate₁(n__s₁(X)) -> s₁(activate₁(X))
activate₁(n__zWquot₂(X1, X2)) -> zWquot₂(activate₁(X1), activate₁(X2))
activate₁(X) -> 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.