Termination w.r.t. Q proof of /tmp/tmpxJ_Gh-/Ex5_DLMMU04

Termination w.r.t. Q of the following Term Rewriting System could not be shown:

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

pairNs → cons(0, n__incr(n__oddNs))
oddNs → incr(pairNs)
incr(cons(X, XS)) → cons(s(X), n__incr(activate(XS)))
take(0, XS) → nil
take(s(N), cons(X, XS)) → cons(X, n__take(N, activate(XS)))
zip(nil, XS) → nil
zip(X, nil) → nil
zip(cons(X, XS), cons(Y, YS)) → cons(pair(X, Y), n__zip(activate(XS), activate(YS)))
tail(cons(X, XS)) → activate(XS)
repItems(nil) → nil
repItems(cons(X, XS)) → cons(X, n__cons(X, n__repItems(activate(XS))))
incr(X) → n__incr(X)
oddNs → n__oddNs
take(X1, X2) → n__take(X1, X2)
zip(X1, X2) → n__zip(X1, X2)
cons(X1, X2) → n__cons(X1, X2)
repItems(X) → n__repItems(X)
activate(n__incr(X)) → incr(activate(X))
activate(n__oddNs) → oddNs
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__zip(X1, X2)) → zip(activate(X1), activate(X2))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__repItems(X)) → repItems(activate(X))
activate(X) → X

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

pairNs → cons(0, n__incr(n__oddNs))
oddNs → incr(pairNs)
incr(cons(X, XS)) → cons(s(X), n__incr(activate(XS)))
take(0, XS) → nil
take(s(N), cons(X, XS)) → cons(X, n__take(N, activate(XS)))
zip(nil, XS) → nil
zip(X, nil) → nil
zip(cons(X, XS), cons(Y, YS)) → cons(pair(X, Y), n__zip(activate(XS), activate(YS)))
tail(cons(X, XS)) → activate(XS)
repItems(nil) → nil
repItems(cons(X, XS)) → cons(X, n__cons(X, n__repItems(activate(XS))))
incr(X) → n__incr(X)
oddNs → n__oddNs
take(X1, X2) → n__take(X1, X2)
zip(X1, X2) → n__zip(X1, X2)
cons(X1, X2) → n__cons(X1, X2)
repItems(X) → n__repItems(X)
activate(n__incr(X)) → incr(activate(X))
activate(n__oddNs) → oddNs
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__zip(X1, X2)) → zip(activate(X1), activate(X2))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__repItems(X)) → repItems(activate(X))
activate(X) → X

Q is empty.

Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

TAKE(s(N), cons(X, XS)) → CONS(X, n__take(N, activate(XS)))
INCR(cons(X, XS)) → ACTIVATE(XS)
ACTIVATE(n__zip(X1, X2)) → ACTIVATE(X1)
ZIP(cons(X, XS), cons(Y, YS)) → ACTIVATE(YS)
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
ACTIVATE(n__zip(X1, X2)) → ZIP(activate(X1), activate(X2))
INCR(cons(X, XS)) → CONS(s(X), n__incr(activate(XS)))
PAIRNS → CONS(0, n__incr(n__oddNs))
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
REPITEMS(cons(X, XS)) → ACTIVATE(XS)
ACTIVATE(n__incr(X)) → ACTIVATE(X)
ACTIVATE(n__repItems(X)) → ACTIVATE(X)
REPITEMS(cons(X, XS)) → CONS(X, n__cons(X, n__repItems(activate(XS))))
TAIL(cons(X, XS)) → ACTIVATE(XS)
ZIP(cons(X, XS), cons(Y, YS)) → ACTIVATE(XS)
ACTIVATE(n__incr(X)) → INCR(activate(X))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__cons(X1, X2)) → CONS(activate(X1), X2)
ODDNS → INCR(pairNs)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
TAKE(s(N), cons(X, XS)) → ACTIVATE(XS)
ODDNS → PAIRNS
ZIP(cons(X, XS), cons(Y, YS)) → CONS(pair(X, Y), n__zip(activate(XS), activate(YS)))
ACTIVATE(n__repItems(X)) → REPITEMS(activate(X))
ACTIVATE(n__zip(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__oddNs) → ODDNS

The TRS R consists of the following rules:

pairNs → cons(0, n__incr(n__oddNs))
oddNs → incr(pairNs)
incr(cons(X, XS)) → cons(s(X), n__incr(activate(XS)))
take(0, XS) → nil
take(s(N), cons(X, XS)) → cons(X, n__take(N, activate(XS)))
zip(nil, XS) → nil
zip(X, nil) → nil
zip(cons(X, XS), cons(Y, YS)) → cons(pair(X, Y), n__zip(activate(XS), activate(YS)))
tail(cons(X, XS)) → activate(XS)
repItems(nil) → nil
repItems(cons(X, XS)) → cons(X, n__cons(X, n__repItems(activate(XS))))
incr(X) → n__incr(X)
oddNs → n__oddNs
take(X1, X2) → n__take(X1, X2)
zip(X1, X2) → n__zip(X1, X2)
cons(X1, X2) → n__cons(X1, X2)
repItems(X) → n__repItems(X)
activate(n__incr(X)) → incr(activate(X))
activate(n__oddNs) → oddNs
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__zip(X1, X2)) → zip(activate(X1), activate(X2))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__repItems(X)) → repItems(activate(X))
activate(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:

TAKE(s(N), cons(X, XS)) → CONS(X, n__take(N, activate(XS)))
INCR(cons(X, XS)) → ACTIVATE(XS)
ACTIVATE(n__zip(X1, X2)) → ACTIVATE(X1)
ZIP(cons(X, XS), cons(Y, YS)) → ACTIVATE(YS)
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
ACTIVATE(n__zip(X1, X2)) → ZIP(activate(X1), activate(X2))
INCR(cons(X, XS)) → CONS(s(X), n__incr(activate(XS)))
PAIRNS → CONS(0, n__incr(n__oddNs))
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
REPITEMS(cons(X, XS)) → ACTIVATE(XS)
ACTIVATE(n__incr(X)) → ACTIVATE(X)
ACTIVATE(n__repItems(X)) → ACTIVATE(X)
REPITEMS(cons(X, XS)) → CONS(X, n__cons(X, n__repItems(activate(XS))))
TAIL(cons(X, XS)) → ACTIVATE(XS)
ZIP(cons(X, XS), cons(Y, YS)) → ACTIVATE(XS)
ACTIVATE(n__incr(X)) → INCR(activate(X))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__cons(X1, X2)) → CONS(activate(X1), X2)
ODDNS → INCR(pairNs)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
TAKE(s(N), cons(X, XS)) → ACTIVATE(XS)
ODDNS → PAIRNS
ZIP(cons(X, XS), cons(Y, YS)) → CONS(pair(X, Y), n__zip(activate(XS), activate(YS)))
ACTIVATE(n__repItems(X)) → REPITEMS(activate(X))
ACTIVATE(n__zip(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__oddNs) → ODDNS

The TRS R consists of the following rules:

pairNs → cons(0, n__incr(n__oddNs))
oddNs → incr(pairNs)
incr(cons(X, XS)) → cons(s(X), n__incr(activate(XS)))
take(0, XS) → nil
take(s(N), cons(X, XS)) → cons(X, n__take(N, activate(XS)))
zip(nil, XS) → nil
zip(X, nil) → nil
zip(cons(X, XS), cons(Y, YS)) → cons(pair(X, Y), n__zip(activate(XS), activate(YS)))
tail(cons(X, XS)) → activate(XS)
repItems(nil) → nil
repItems(cons(X, XS)) → cons(X, n__cons(X, n__repItems(activate(XS))))
incr(X) → n__incr(X)
oddNs → n__oddNs
take(X1, X2) → n__take(X1, X2)
zip(X1, X2) → n__zip(X1, X2)
cons(X1, X2) → n__cons(X1, X2)
repItems(X) → n__repItems(X)
activate(n__incr(X)) → incr(activate(X))
activate(n__oddNs) → oddNs
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__zip(X1, X2)) → zip(activate(X1), activate(X2))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__repItems(X)) → repItems(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 8 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

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

INCR(cons(X, XS)) → ACTIVATE(XS)
ACTIVATE(n__zip(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__incr(X)) → INCR(activate(X))
ZIP(cons(X, XS), cons(Y, YS)) → ACTIVATE(YS)
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__zip(X1, X2)) → ZIP(activate(X1), activate(X2))
ODDNS → INCR(pairNs)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
TAKE(s(N), cons(X, XS)) → ACTIVATE(XS)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__repItems(X)) → REPITEMS(activate(X))
REPITEMS(cons(X, XS)) → ACTIVATE(XS)
ACTIVATE(n__incr(X)) → ACTIVATE(X)
ACTIVATE(n__repItems(X)) → ACTIVATE(X)
ACTIVATE(n__zip(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__oddNs) → ODDNS
ZIP(cons(X, XS), cons(Y, YS)) → ACTIVATE(XS)

The TRS R consists of the following rules:

pairNs → cons(0, n__incr(n__oddNs))
oddNs → incr(pairNs)
incr(cons(X, XS)) → cons(s(X), n__incr(activate(XS)))
take(0, XS) → nil
take(s(N), cons(X, XS)) → cons(X, n__take(N, activate(XS)))
zip(nil, XS) → nil
zip(X, nil) → nil
zip(cons(X, XS), cons(Y, YS)) → cons(pair(X, Y), n__zip(activate(XS), activate(YS)))
tail(cons(X, XS)) → activate(XS)
repItems(nil) → nil
repItems(cons(X, XS)) → cons(X, n__cons(X, n__repItems(activate(XS))))
incr(X) → n__incr(X)
oddNs → n__oddNs
take(X1, X2) → n__take(X1, X2)
zip(X1, X2) → n__zip(X1, X2)
cons(X1, X2) → n__cons(X1, X2)
repItems(X) → n__repItems(X)
activate(n__incr(X)) → incr(activate(X))
activate(n__oddNs) → oddNs
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__zip(X1, X2)) → zip(activate(X1), activate(X2))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__repItems(X)) → repItems(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)

The remaining pairs can at least be oriented weakly.

INCR(cons(X, XS)) → ACTIVATE(XS)
ACTIVATE(n__zip(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__incr(X)) → INCR(activate(X))
ZIP(cons(X, XS), cons(Y, YS)) → ACTIVATE(YS)
ACTIVATE(n__zip(X1, X2)) → ZIP(activate(X1), activate(X2))
ODDNS → INCR(pairNs)
TAKE(s(N), cons(X, XS)) → ACTIVATE(XS)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__repItems(X)) → REPITEMS(activate(X))
REPITEMS(cons(X, XS)) → ACTIVATE(XS)
ACTIVATE(n__incr(X)) → ACTIVATE(X)
ACTIVATE(n__repItems(X)) → ACTIVATE(X)
ACTIVATE(n__zip(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__oddNs) → ODDNS
ZIP(cons(X, XS), cons(Y, YS)) → ACTIVATE(XS)

Used ordering: Polynomial interpretation [25,35]:

POL(zip(x₁, x₂)) = (2)x_1 + x_2
POL(n__incr(x₁)) = x_1
POL(oddNs) = 0
POL(n__cons(x₁, x₂)) = (4)x_1 + x_2
POL(n__repItems(x₁)) = (2)x_1
POL(activate(x₁)) = x_1
POL(take(x₁, x₂)) = 4 + (2)x_1 + x_2
POL(0) = 0
POL(repItems(x₁)) = (2)x_1
POL(REPITEMS(x₁)) = (2)x_1
POL(ODDNS) = 0
POL(cons(x₁, x₂)) = (4)x_1 + x_2
POL(TAKE(x₁, x₂)) = (2)x_1 + x_2
POL(incr(x₁)) = x_1
POL(n__take(x₁, x₂)) = 4 + (2)x_1 + x_2
POL(n__oddNs) = 0
POL(pair(x₁, x₂)) = 0
POL(s(x₁)) = x_1
POL(pairNs) = 0
POL(ZIP(x₁, x₂)) = (2)x_1 + x_2
POL(n__zip(x₁, x₂)) = (2)x_1 + x_2
POL(ACTIVATE(x₁)) = x_1
POL(INCR(x₁)) = x_1
POL(nil) = 0

The value of delta used in the strict ordering is 4.
The following usable rules [17] were oriented:

zip(cons(X, XS), cons(Y, YS)) → cons(pair(X, Y), n__zip(activate(XS), activate(YS)))
zip(nil, XS) → nil
zip(X, nil) → nil
incr(X) → n__incr(X)
oddNs → n__oddNs
repItems(nil) → nil
repItems(cons(X, XS)) → cons(X, n__cons(X, n__repItems(activate(XS))))
cons(X1, X2) → n__cons(X1, X2)
repItems(X) → n__repItems(X)
take(X1, X2) → n__take(X1, X2)
zip(X1, X2) → n__zip(X1, X2)
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__zip(X1, X2)) → zip(activate(X1), activate(X2))
activate(n__incr(X)) → incr(activate(X))
activate(n__oddNs) → oddNs
activate(X) → X
activate(n__repItems(X)) → repItems(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
pairNs → cons(0, n__incr(n__oddNs))
incr(cons(X, XS)) → cons(s(X), n__incr(activate(XS)))
oddNs → incr(pairNs)
take(s(N), cons(X, XS)) → cons(X, n__take(N, activate(XS)))
take(0, XS) → nil

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

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

INCR(cons(X, XS)) → ACTIVATE(XS)
ACTIVATE(n__zip(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__incr(X)) → INCR(activate(X))
ZIP(cons(X, XS), cons(Y, YS)) → ACTIVATE(YS)
ACTIVATE(n__zip(X1, X2)) → ZIP(activate(X1), activate(X2))
ODDNS → INCR(pairNs)
TAKE(s(N), cons(X, XS)) → ACTIVATE(XS)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__repItems(X)) → REPITEMS(activate(X))
REPITEMS(cons(X, XS)) → ACTIVATE(XS)
ACTIVATE(n__incr(X)) → ACTIVATE(X)
ACTIVATE(n__repItems(X)) → ACTIVATE(X)
ACTIVATE(n__zip(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__oddNs) → ODDNS
ZIP(cons(X, XS), cons(Y, YS)) → ACTIVATE(XS)

The TRS R consists of the following rules:

pairNs → cons(0, n__incr(n__oddNs))
oddNs → incr(pairNs)
incr(cons(X, XS)) → cons(s(X), n__incr(activate(XS)))
take(0, XS) → nil
take(s(N), cons(X, XS)) → cons(X, n__take(N, activate(XS)))
zip(nil, XS) → nil
zip(X, nil) → nil
zip(cons(X, XS), cons(Y, YS)) → cons(pair(X, Y), n__zip(activate(XS), activate(YS)))
tail(cons(X, XS)) → activate(XS)
repItems(nil) → nil
repItems(cons(X, XS)) → cons(X, n__cons(X, n__repItems(activate(XS))))
incr(X) → n__incr(X)
oddNs → n__oddNs
take(X1, X2) → n__take(X1, X2)
zip(X1, X2) → n__zip(X1, X2)
cons(X1, X2) → n__cons(X1, X2)
repItems(X) → n__repItems(X)
activate(n__incr(X)) → incr(activate(X))
activate(n__oddNs) → oddNs
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__zip(X1, X2)) → zip(activate(X1), activate(X2))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__repItems(X)) → repItems(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] 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:

INCR(cons(X, XS)) → ACTIVATE(XS)
ACTIVATE(n__zip(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__incr(X)) → INCR(activate(X))
ZIP(cons(X, XS), cons(Y, YS)) → ACTIVATE(YS)
ACTIVATE(n__zip(X1, X2)) → ZIP(activate(X1), activate(X2))
ODDNS → INCR(pairNs)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__repItems(X)) → REPITEMS(activate(X))
REPITEMS(cons(X, XS)) → ACTIVATE(XS)
ACTIVATE(n__incr(X)) → ACTIVATE(X)
ACTIVATE(n__repItems(X)) → ACTIVATE(X)
ACTIVATE(n__zip(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__oddNs) → ODDNS
ZIP(cons(X, XS), cons(Y, YS)) → ACTIVATE(XS)

The TRS R consists of the following rules:

pairNs → cons(0, n__incr(n__oddNs))
oddNs → incr(pairNs)
incr(cons(X, XS)) → cons(s(X), n__incr(activate(XS)))
take(0, XS) → nil
take(s(N), cons(X, XS)) → cons(X, n__take(N, activate(XS)))
zip(nil, XS) → nil
zip(X, nil) → nil
zip(cons(X, XS), cons(Y, YS)) → cons(pair(X, Y), n__zip(activate(XS), activate(YS)))
tail(cons(X, XS)) → activate(XS)
repItems(nil) → nil
repItems(cons(X, XS)) → cons(X, n__cons(X, n__repItems(activate(XS))))
incr(X) → n__incr(X)
oddNs → n__oddNs
take(X1, X2) → n__take(X1, X2)
zip(X1, X2) → n__zip(X1, X2)
cons(X1, X2) → n__cons(X1, X2)
repItems(X) → n__repItems(X)
activate(n__incr(X)) → incr(activate(X))
activate(n__oddNs) → oddNs
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__zip(X1, X2)) → zip(activate(X1), activate(X2))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__repItems(X)) → repItems(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

ACTIVATE(n__zip(X1, X2)) → ACTIVATE(X1)
ZIP(cons(X, XS), cons(Y, YS)) → ACTIVATE(YS)
ACTIVATE(n__zip(X1, X2)) → ZIP(activate(X1), activate(X2))
ACTIVATE(n__zip(X1, X2)) → ACTIVATE(X2)
ZIP(cons(X, XS), cons(Y, YS)) → ACTIVATE(XS)

The remaining pairs can at least be oriented weakly.

INCR(cons(X, XS)) → ACTIVATE(XS)
ACTIVATE(n__incr(X)) → INCR(activate(X))
ODDNS → INCR(pairNs)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__repItems(X)) → REPITEMS(activate(X))
REPITEMS(cons(X, XS)) → ACTIVATE(XS)
ACTIVATE(n__incr(X)) → ACTIVATE(X)
ACTIVATE(n__repItems(X)) → ACTIVATE(X)
ACTIVATE(n__oddNs) → ODDNS

Used ordering: Polynomial interpretation [25,35]:

POL(zip(x₁, x₂)) = 1 + (2)x_1 + x_2
POL(n__incr(x₁)) = x_1
POL(oddNs) = 0
POL(n__cons(x₁, x₂)) = (4)x_1 + x_2
POL(activate(x₁)) = x_1
POL(n__repItems(x₁)) = (4)x_1
POL(take(x₁, x₂)) = x_2
POL(0) = 0
POL(repItems(x₁)) = (4)x_1
POL(REPITEMS(x₁)) = 1 + (4)x_1
POL(ODDNS) = 1
POL(cons(x₁, x₂)) = (4)x_1 + x_2
POL(incr(x₁)) = x_1
POL(n__oddNs) = 0
POL(pair(x₁, x₂)) = (2)x_1 + x_2
POL(n__take(x₁, x₂)) = x_2
POL(pairNs) = 0
POL(ZIP(x₁, x₂)) = 2 + (2)x_1 + (2)x_2
POL(s(x₁)) = 0
POL(n__zip(x₁, x₂)) = 1 + (2)x_1 + x_2
POL(ACTIVATE(x₁)) = 1 + (2)x_1
POL(INCR(x₁)) = 1 + (2)x_1
POL(nil) = 0

The value of delta used in the strict ordering is 1.
The following usable rules [17] were oriented:

zip(cons(X, XS), cons(Y, YS)) → cons(pair(X, Y), n__zip(activate(XS), activate(YS)))
zip(nil, XS) → nil
zip(X, nil) → nil
incr(X) → n__incr(X)
oddNs → n__oddNs
repItems(nil) → nil
repItems(cons(X, XS)) → cons(X, n__cons(X, n__repItems(activate(XS))))
cons(X1, X2) → n__cons(X1, X2)
repItems(X) → n__repItems(X)
take(X1, X2) → n__take(X1, X2)
zip(X1, X2) → n__zip(X1, X2)
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__zip(X1, X2)) → zip(activate(X1), activate(X2))
activate(n__incr(X)) → incr(activate(X))
activate(n__oddNs) → oddNs
activate(X) → X
activate(n__repItems(X)) → repItems(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
pairNs → cons(0, n__incr(n__oddNs))
incr(cons(X, XS)) → cons(s(X), n__incr(activate(XS)))
oddNs → incr(pairNs)
take(s(N), cons(X, XS)) → cons(X, n__take(N, activate(XS)))
take(0, XS) → nil

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

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

INCR(cons(X, XS)) → ACTIVATE(XS)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__repItems(X)) → REPITEMS(activate(X))
REPITEMS(cons(X, XS)) → ACTIVATE(XS)
ACTIVATE(n__incr(X)) → INCR(activate(X))
ACTIVATE(n__incr(X)) → ACTIVATE(X)
ACTIVATE(n__repItems(X)) → ACTIVATE(X)
ODDNS → INCR(pairNs)
ACTIVATE(n__oddNs) → ODDNS

The TRS R consists of the following rules:

pairNs → cons(0, n__incr(n__oddNs))
oddNs → incr(pairNs)
incr(cons(X, XS)) → cons(s(X), n__incr(activate(XS)))
take(0, XS) → nil
take(s(N), cons(X, XS)) → cons(X, n__take(N, activate(XS)))
zip(nil, XS) → nil
zip(X, nil) → nil
zip(cons(X, XS), cons(Y, YS)) → cons(pair(X, Y), n__zip(activate(XS), activate(YS)))
tail(cons(X, XS)) → activate(XS)
repItems(nil) → nil
repItems(cons(X, XS)) → cons(X, n__cons(X, n__repItems(activate(XS))))
incr(X) → n__incr(X)
oddNs → n__oddNs
take(X1, X2) → n__take(X1, X2)
zip(X1, X2) → n__zip(X1, X2)
cons(X1, X2) → n__cons(X1, X2)
repItems(X) → n__repItems(X)
activate(n__incr(X)) → incr(activate(X))
activate(n__oddNs) → oddNs
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__zip(X1, X2)) → zip(activate(X1), activate(X2))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__repItems(X)) → repItems(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

REPITEMS(cons(X, XS)) → ACTIVATE(XS)
ACTIVATE(n__repItems(X)) → ACTIVATE(X)

The remaining pairs can at least be oriented weakly.

INCR(cons(X, XS)) → ACTIVATE(XS)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__repItems(X)) → REPITEMS(activate(X))
ACTIVATE(n__incr(X)) → INCR(activate(X))
ACTIVATE(n__incr(X)) → ACTIVATE(X)
ODDNS → INCR(pairNs)
ACTIVATE(n__oddNs) → ODDNS

Used ordering: Polynomial interpretation [25,35]:

POL(zip(x₁, x₂)) = 1
POL(n__incr(x₁)) = (2)x_1
POL(oddNs) = 0
POL(n__cons(x₁, x₂)) = x_1 + x_2
POL(n__repItems(x₁)) = 1 + (2)x_1
POL(activate(x₁)) = x_1
POL(take(x₁, x₂)) = 1 + (4)x_1 + x_2
POL(0) = 0
POL(repItems(x₁)) = 1 + (2)x_1
POL(REPITEMS(x₁)) = 1 + x_1
POL(ODDNS) = 0
POL(cons(x₁, x₂)) = x_1 + x_2
POL(incr(x₁)) = (2)x_1
POL(pair(x₁, x₂)) = 0
POL(n__oddNs) = 0
POL(n__take(x₁, x₂)) = 1 + (4)x_1 + x_2
POL(pairNs) = 0
POL(s(x₁)) = (2)x_1
POL(n__zip(x₁, x₂)) = 1
POL(ACTIVATE(x₁)) = x_1
POL(INCR(x₁)) = x_1
POL(nil) = 1

The value of delta used in the strict ordering is 1.
The following usable rules [17] were oriented:

zip(cons(X, XS), cons(Y, YS)) → cons(pair(X, Y), n__zip(activate(XS), activate(YS)))
zip(nil, XS) → nil
zip(X, nil) → nil
incr(X) → n__incr(X)
oddNs → n__oddNs
repItems(nil) → nil
repItems(cons(X, XS)) → cons(X, n__cons(X, n__repItems(activate(XS))))
cons(X1, X2) → n__cons(X1, X2)
repItems(X) → n__repItems(X)
take(X1, X2) → n__take(X1, X2)
zip(X1, X2) → n__zip(X1, X2)
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__zip(X1, X2)) → zip(activate(X1), activate(X2))
activate(n__incr(X)) → incr(activate(X))
activate(n__oddNs) → oddNs
activate(X) → X
activate(n__repItems(X)) → repItems(activate(X))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
pairNs → cons(0, n__incr(n__oddNs))
incr(cons(X, XS)) → cons(s(X), n__incr(activate(XS)))
oddNs → incr(pairNs)
take(s(N), cons(X, XS)) → cons(X, n__take(N, activate(XS)))
take(0, XS) → nil

↳ 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:

INCR(cons(X, XS)) → ACTIVATE(XS)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__repItems(X)) → REPITEMS(activate(X))
ACTIVATE(n__incr(X)) → INCR(activate(X))
ACTIVATE(n__incr(X)) → ACTIVATE(X)
ODDNS → INCR(pairNs)
ACTIVATE(n__oddNs) → ODDNS

The TRS R consists of the following rules:

pairNs → cons(0, n__incr(n__oddNs))
oddNs → incr(pairNs)
incr(cons(X, XS)) → cons(s(X), n__incr(activate(XS)))
take(0, XS) → nil
take(s(N), cons(X, XS)) → cons(X, n__take(N, activate(XS)))
zip(nil, XS) → nil
zip(X, nil) → nil
zip(cons(X, XS), cons(Y, YS)) → cons(pair(X, Y), n__zip(activate(XS), activate(YS)))
tail(cons(X, XS)) → activate(XS)
repItems(nil) → nil
repItems(cons(X, XS)) → cons(X, n__cons(X, n__repItems(activate(XS))))
incr(X) → n__incr(X)
oddNs → n__oddNs
take(X1, X2) → n__take(X1, X2)
zip(X1, X2) → n__zip(X1, X2)
cons(X1, X2) → n__cons(X1, X2)
repItems(X) → n__repItems(X)
activate(n__incr(X)) → incr(activate(X))
activate(n__oddNs) → oddNs
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__zip(X1, X2)) → zip(activate(X1), activate(X2))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__repItems(X)) → repItems(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

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

INCR(cons(X, XS)) → ACTIVATE(XS)
ACTIVATE(n__cons(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__incr(X)) → INCR(activate(X))
ACTIVATE(n__incr(X)) → ACTIVATE(X)
ODDNS → INCR(pairNs)
ACTIVATE(n__oddNs) → ODDNS

The TRS R consists of the following rules:

pairNs → cons(0, n__incr(n__oddNs))
oddNs → incr(pairNs)
incr(cons(X, XS)) → cons(s(X), n__incr(activate(XS)))
take(0, XS) → nil
take(s(N), cons(X, XS)) → cons(X, n__take(N, activate(XS)))
zip(nil, XS) → nil
zip(X, nil) → nil
zip(cons(X, XS), cons(Y, YS)) → cons(pair(X, Y), n__zip(activate(XS), activate(YS)))
tail(cons(X, XS)) → activate(XS)
repItems(nil) → nil
repItems(cons(X, XS)) → cons(X, n__cons(X, n__repItems(activate(XS))))
incr(X) → n__incr(X)
oddNs → n__oddNs
take(X1, X2) → n__take(X1, X2)
zip(X1, X2) → n__zip(X1, X2)
cons(X1, X2) → n__cons(X1, X2)
repItems(X) → n__repItems(X)
activate(n__incr(X)) → incr(activate(X))
activate(n__oddNs) → oddNs
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__zip(X1, X2)) → zip(activate(X1), activate(X2))
activate(n__cons(X1, X2)) → cons(activate(X1), X2)
activate(n__repItems(X)) → repItems(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.