Termination w.r.t. Q proof of /tmp/tmp3Xbc1z/Ex1_GL02a

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:

eq(n__0, n__0) → true
eq(n__s(X), n__s(Y)) → eq(activate(X), activate(Y))
eq(X, Y) → false
inf(X) → cons(X, n__inf(n__s(X)))
take(0, X) → nil
take(s(X), cons(Y, L)) → cons(activate(Y), n__take(activate(X), activate(L)))
length(nil) → 0
length(cons(X, L)) → s(n__length(activate(L)))
0 → n__0
s(X) → n__s(X)
inf(X) → n__inf(X)
take(X1, X2) → n__take(X1, X2)
length(X) → n__length(X)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__inf(X)) → inf(activate(X))
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__length(X)) → length(activate(X))
activate(X) → X

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

eq(n__0, n__0) → true
eq(n__s(X), n__s(Y)) → eq(activate(X), activate(Y))
eq(X, Y) → false
inf(X) → cons(X, n__inf(n__s(X)))
take(0, X) → nil
take(s(X), cons(Y, L)) → cons(activate(Y), n__take(activate(X), activate(L)))
length(nil) → 0
length(cons(X, L)) → s(n__length(activate(L)))
0 → n__0
s(X) → n__s(X)
inf(X) → n__inf(X)
take(X1, X2) → n__take(X1, X2)
length(X) → n__length(X)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__inf(X)) → inf(activate(X))
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__length(X)) → length(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:

ACTIVATE(n__inf(X)) → ACTIVATE(X)
LENGTH(cons(X, L)) → S(n__length(activate(L)))
TAKE(s(X), cons(Y, L)) → ACTIVATE(X)
LENGTH(cons(X, L)) → ACTIVATE(L)
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
TAKE(s(X), cons(Y, L)) → ACTIVATE(Y)
ACTIVATE(n__length(X)) → LENGTH(activate(X))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → S(X)
EQ(n__s(X), n__s(Y)) → ACTIVATE(X)
ACTIVATE(n__inf(X)) → INF(activate(X))
EQ(n__s(X), n__s(Y)) → EQ(activate(X), activate(Y))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
TAKE(s(X), cons(Y, L)) → ACTIVATE(L)
ACTIVATE(n__0) → 0¹
EQ(n__s(X), n__s(Y)) → ACTIVATE(Y)
ACTIVATE(n__length(X)) → ACTIVATE(X)
LENGTH(nil) → 0¹

The TRS R consists of the following rules:

eq(n__0, n__0) → true
eq(n__s(X), n__s(Y)) → eq(activate(X), activate(Y))
eq(X, Y) → false
inf(X) → cons(X, n__inf(n__s(X)))
take(0, X) → nil
take(s(X), cons(Y, L)) → cons(activate(Y), n__take(activate(X), activate(L)))
length(nil) → 0
length(cons(X, L)) → s(n__length(activate(L)))
0 → n__0
s(X) → n__s(X)
inf(X) → n__inf(X)
take(X1, X2) → n__take(X1, X2)
length(X) → n__length(X)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__inf(X)) → inf(activate(X))
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__length(X)) → length(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:

ACTIVATE(n__inf(X)) → ACTIVATE(X)
LENGTH(cons(X, L)) → S(n__length(activate(L)))
TAKE(s(X), cons(Y, L)) → ACTIVATE(X)
LENGTH(cons(X, L)) → ACTIVATE(L)
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
TAKE(s(X), cons(Y, L)) → ACTIVATE(Y)
ACTIVATE(n__length(X)) → LENGTH(activate(X))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__s(X)) → S(X)
EQ(n__s(X), n__s(Y)) → ACTIVATE(X)
ACTIVATE(n__inf(X)) → INF(activate(X))
EQ(n__s(X), n__s(Y)) → EQ(activate(X), activate(Y))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
TAKE(s(X), cons(Y, L)) → ACTIVATE(L)
ACTIVATE(n__0) → 0¹
EQ(n__s(X), n__s(Y)) → ACTIVATE(Y)
ACTIVATE(n__length(X)) → ACTIVATE(X)
LENGTH(nil) → 0¹

The TRS R consists of the following rules:

eq(n__0, n__0) → true
eq(n__s(X), n__s(Y)) → eq(activate(X), activate(Y))
eq(X, Y) → false
inf(X) → cons(X, n__inf(n__s(X)))
take(0, X) → nil
take(s(X), cons(Y, L)) → cons(activate(Y), n__take(activate(X), activate(L)))
length(nil) → 0
length(cons(X, L)) → s(n__length(activate(L)))
0 → n__0
s(X) → n__s(X)
inf(X) → n__inf(X)
take(X1, X2) → n__take(X1, X2)
length(X) → n__length(X)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__inf(X)) → inf(activate(X))
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__length(X)) → length(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 2 SCCs with 7 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ Narrowing

          ↳ QDP

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

ACTIVATE(n__inf(X)) → ACTIVATE(X)
TAKE(s(X), cons(Y, L)) → ACTIVATE(X)
LENGTH(cons(X, L)) → ACTIVATE(L)
TAKE(s(X), cons(Y, L)) → ACTIVATE(Y)
ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2))
ACTIVATE(n__length(X)) → LENGTH(activate(X))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__length(X)) → ACTIVATE(X)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
TAKE(s(X), cons(Y, L)) → ACTIVATE(L)

The TRS R consists of the following rules:

eq(n__0, n__0) → true
eq(n__s(X), n__s(Y)) → eq(activate(X), activate(Y))
eq(X, Y) → false
inf(X) → cons(X, n__inf(n__s(X)))
take(0, X) → nil
take(s(X), cons(Y, L)) → cons(activate(Y), n__take(activate(X), activate(L)))
length(nil) → 0
length(cons(X, L)) → s(n__length(activate(L)))
0 → n__0
s(X) → n__s(X)
inf(X) → n__inf(X)
take(X1, X2) → n__take(X1, X2)
length(X) → n__length(X)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__inf(X)) → inf(activate(X))
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__length(X)) → length(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ACTIVATE(n__take(X1, X2)) → TAKE(activate(X1), activate(X2)) at position [0] we obtained the following new rules:

ACTIVATE(n__take(n__length(x0), y1)) → TAKE(length(activate(x0)), activate(y1))
ACTIVATE(n__take(n__s(x0), y1)) → TAKE(s(x0), activate(y1))
ACTIVATE(n__take(n__inf(x0), y1)) → TAKE(inf(activate(x0)), activate(y1))
ACTIVATE(n__take(n__0, y1)) → TAKE(0, activate(y1))
ACTIVATE(n__take(n__take(x0, x1), y1)) → TAKE(take(activate(x0), activate(x1)), activate(y1))
ACTIVATE(n__take(x0, y1)) → TAKE(x0, activate(y1))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

          ↳ QDP

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

ACTIVATE(n__inf(X)) → ACTIVATE(X)
TAKE(s(X), cons(Y, L)) → ACTIVATE(X)
LENGTH(cons(X, L)) → ACTIVATE(L)
TAKE(s(X), cons(Y, L)) → ACTIVATE(Y)
ACTIVATE(n__take(x0, y1)) → TAKE(x0, activate(y1))
ACTIVATE(n__length(X)) → LENGTH(activate(X))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(n__s(x0), y1)) → TAKE(s(x0), activate(y1))
ACTIVATE(n__take(n__0, y1)) → TAKE(0, activate(y1))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
TAKE(s(X), cons(Y, L)) → ACTIVATE(L)
ACTIVATE(n__take(n__length(x0), y1)) → TAKE(length(activate(x0)), activate(y1))
ACTIVATE(n__take(n__inf(x0), y1)) → TAKE(inf(activate(x0)), activate(y1))
ACTIVATE(n__take(n__take(x0, x1), y1)) → TAKE(take(activate(x0), activate(x1)), activate(y1))
ACTIVATE(n__length(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

eq(n__0, n__0) → true
eq(n__s(X), n__s(Y)) → eq(activate(X), activate(Y))
eq(X, Y) → false
inf(X) → cons(X, n__inf(n__s(X)))
take(0, X) → nil
take(s(X), cons(Y, L)) → cons(activate(Y), n__take(activate(X), activate(L)))
length(nil) → 0
length(cons(X, L)) → s(n__length(activate(L)))
0 → n__0
s(X) → n__s(X)
inf(X) → n__inf(X)
take(X1, X2) → n__take(X1, X2)
length(X) → n__length(X)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__inf(X)) → inf(activate(X))
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__length(X)) → length(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ACTIVATE(n__length(X)) → LENGTH(activate(X)) at position [0] we obtained the following new rules:

ACTIVATE(n__length(n__take(x0, x1))) → LENGTH(take(activate(x0), activate(x1)))
ACTIVATE(n__length(n__length(x0))) → LENGTH(length(activate(x0)))
ACTIVATE(n__length(n__0)) → LENGTH(0)
ACTIVATE(n__length(x0)) → LENGTH(x0)
ACTIVATE(n__length(n__inf(x0))) → LENGTH(inf(activate(x0)))
ACTIVATE(n__length(n__s(x0))) → LENGTH(s(x0))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

          ↳ QDP

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

ACTIVATE(n__length(n__take(x0, x1))) → LENGTH(take(activate(x0), activate(x1)))
ACTIVATE(n__inf(X)) → ACTIVATE(X)
ACTIVATE(n__length(n__length(x0))) → LENGTH(length(activate(x0)))
ACTIVATE(n__length(x0)) → LENGTH(x0)
LENGTH(cons(X, L)) → ACTIVATE(L)
TAKE(s(X), cons(Y, L)) → ACTIVATE(X)
ACTIVATE(n__length(n__s(x0))) → LENGTH(s(x0))
TAKE(s(X), cons(Y, L)) → ACTIVATE(Y)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(x0, y1)) → TAKE(x0, activate(y1))
ACTIVATE(n__take(n__s(x0), y1)) → TAKE(s(x0), activate(y1))
ACTIVATE(n__take(n__0, y1)) → TAKE(0, activate(y1))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
TAKE(s(X), cons(Y, L)) → ACTIVATE(L)
ACTIVATE(n__length(n__inf(x0))) → LENGTH(inf(activate(x0)))
ACTIVATE(n__take(n__length(x0), y1)) → TAKE(length(activate(x0)), activate(y1))
ACTIVATE(n__length(n__0)) → LENGTH(0)
ACTIVATE(n__take(n__inf(x0), y1)) → TAKE(inf(activate(x0)), activate(y1))
ACTIVATE(n__length(X)) → ACTIVATE(X)
ACTIVATE(n__take(n__take(x0, x1), y1)) → TAKE(take(activate(x0), activate(x1)), activate(y1))

The TRS R consists of the following rules:

eq(n__0, n__0) → true
eq(n__s(X), n__s(Y)) → eq(activate(X), activate(Y))
eq(X, Y) → false
inf(X) → cons(X, n__inf(n__s(X)))
take(0, X) → nil
take(s(X), cons(Y, L)) → cons(activate(Y), n__take(activate(X), activate(L)))
length(nil) → 0
length(cons(X, L)) → s(n__length(activate(L)))
0 → n__0
s(X) → n__s(X)
inf(X) → n__inf(X)
take(X1, X2) → n__take(X1, X2)
length(X) → n__length(X)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__inf(X)) → inf(activate(X))
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__length(X)) → length(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ACTIVATE(n__take(n__length(x0), y1)) → TAKE(length(activate(x0)), activate(y1)) at position [1] we obtained the following new rules:

ACTIVATE(n__take(n__length(y0), n__s(x0))) → TAKE(length(activate(y0)), s(x0))
ACTIVATE(n__take(n__length(y0), n__inf(x0))) → TAKE(length(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(n__length(y0), n__length(x0))) → TAKE(length(activate(y0)), length(activate(x0)))
ACTIVATE(n__take(n__length(y0), x0)) → TAKE(length(activate(y0)), x0)
ACTIVATE(n__take(n__length(y0), n__0)) → TAKE(length(activate(y0)), 0)
ACTIVATE(n__take(n__length(y0), n__take(x0, x1))) → TAKE(length(activate(y0)), take(activate(x0), activate(x1)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

          ↳ QDP

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

ACTIVATE(n__length(n__take(x0, x1))) → LENGTH(take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__length(y0), n__s(x0))) → TAKE(length(activate(y0)), s(x0))
ACTIVATE(n__take(n__length(y0), n__inf(x0))) → TAKE(length(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(n__length(y0), n__length(x0))) → TAKE(length(activate(y0)), length(activate(x0)))
ACTIVATE(n__length(n__s(x0))) → LENGTH(s(x0))
TAKE(s(X), cons(Y, L)) → ACTIVATE(Y)
ACTIVATE(n__take(n__s(x0), y1)) → TAKE(s(x0), activate(y1))
ACTIVATE(n__take(n__length(y0), n__0)) → TAKE(length(activate(y0)), 0)
ACTIVATE(n__length(n__0)) → LENGTH(0)
ACTIVATE(n__take(n__length(y0), x0)) → TAKE(length(activate(y0)), x0)
ACTIVATE(n__inf(X)) → ACTIVATE(X)
ACTIVATE(n__length(n__length(x0))) → LENGTH(length(activate(x0)))
ACTIVATE(n__length(x0)) → LENGTH(x0)
LENGTH(cons(X, L)) → ACTIVATE(L)
TAKE(s(X), cons(Y, L)) → ACTIVATE(X)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(x0, y1)) → TAKE(x0, activate(y1))
ACTIVATE(n__take(n__0, y1)) → TAKE(0, activate(y1))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__take(n__length(y0), n__take(x0, x1))) → TAKE(length(activate(y0)), take(activate(x0), activate(x1)))
TAKE(s(X), cons(Y, L)) → ACTIVATE(L)
ACTIVATE(n__length(n__inf(x0))) → LENGTH(inf(activate(x0)))
ACTIVATE(n__take(n__inf(x0), y1)) → TAKE(inf(activate(x0)), activate(y1))
ACTIVATE(n__length(X)) → ACTIVATE(X)
ACTIVATE(n__take(n__take(x0, x1), y1)) → TAKE(take(activate(x0), activate(x1)), activate(y1))

The TRS R consists of the following rules:

eq(n__0, n__0) → true
eq(n__s(X), n__s(Y)) → eq(activate(X), activate(Y))
eq(X, Y) → false
inf(X) → cons(X, n__inf(n__s(X)))
take(0, X) → nil
take(s(X), cons(Y, L)) → cons(activate(Y), n__take(activate(X), activate(L)))
length(nil) → 0
length(cons(X, L)) → s(n__length(activate(L)))
0 → n__0
s(X) → n__s(X)
inf(X) → n__inf(X)
take(X1, X2) → n__take(X1, X2)
length(X) → n__length(X)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__inf(X)) → inf(activate(X))
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__length(X)) → length(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ACTIVATE(n__take(n__s(x0), y1)) → TAKE(s(x0), activate(y1)) at position [1] we obtained the following new rules:

ACTIVATE(n__take(n__s(y0), n__inf(x0))) → TAKE(s(y0), inf(activate(x0)))
ACTIVATE(n__take(n__s(y0), n__s(x0))) → TAKE(s(y0), s(x0))
ACTIVATE(n__take(n__s(y0), x0)) → TAKE(s(y0), x0)
ACTIVATE(n__take(n__s(y0), n__take(x0, x1))) → TAKE(s(y0), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__s(y0), n__length(x0))) → TAKE(s(y0), length(activate(x0)))
ACTIVATE(n__take(n__s(y0), n__0)) → TAKE(s(y0), 0)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

          ↳ QDP

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

ACTIVATE(n__take(n__length(y0), n__s(x0))) → TAKE(length(activate(y0)), s(x0))
ACTIVATE(n__length(n__take(x0, x1))) → LENGTH(take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__length(y0), n__inf(x0))) → TAKE(length(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(n__s(y0), n__0)) → TAKE(s(y0), 0)
ACTIVATE(n__take(n__length(y0), n__length(x0))) → TAKE(length(activate(y0)), length(activate(x0)))
ACTIVATE(n__length(n__s(x0))) → LENGTH(s(x0))
TAKE(s(X), cons(Y, L)) → ACTIVATE(Y)
ACTIVATE(n__take(n__length(y0), n__0)) → TAKE(length(activate(y0)), 0)
ACTIVATE(n__take(n__s(y0), x0)) → TAKE(s(y0), x0)
ACTIVATE(n__take(n__s(y0), n__inf(x0))) → TAKE(s(y0), inf(activate(x0)))
ACTIVATE(n__length(n__0)) → LENGTH(0)
ACTIVATE(n__take(n__s(y0), n__take(x0, x1))) → TAKE(s(y0), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__length(y0), x0)) → TAKE(length(activate(y0)), x0)
ACTIVATE(n__inf(X)) → ACTIVATE(X)
ACTIVATE(n__take(n__s(y0), n__s(x0))) → TAKE(s(y0), s(x0))
ACTIVATE(n__length(n__length(x0))) → LENGTH(length(activate(x0)))
ACTIVATE(n__length(x0)) → LENGTH(x0)
TAKE(s(X), cons(Y, L)) → ACTIVATE(X)
LENGTH(cons(X, L)) → ACTIVATE(L)
ACTIVATE(n__take(x0, y1)) → TAKE(x0, activate(y1))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(n__s(y0), n__length(x0))) → TAKE(s(y0), length(activate(x0)))
ACTIVATE(n__take(n__0, y1)) → TAKE(0, activate(y1))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
TAKE(s(X), cons(Y, L)) → ACTIVATE(L)
ACTIVATE(n__take(n__length(y0), n__take(x0, x1))) → TAKE(length(activate(y0)), take(activate(x0), activate(x1)))
ACTIVATE(n__length(n__inf(x0))) → LENGTH(inf(activate(x0)))
ACTIVATE(n__take(n__inf(x0), y1)) → TAKE(inf(activate(x0)), activate(y1))
ACTIVATE(n__take(n__take(x0, x1), y1)) → TAKE(take(activate(x0), activate(x1)), activate(y1))
ACTIVATE(n__length(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

eq(n__0, n__0) → true
eq(n__s(X), n__s(Y)) → eq(activate(X), activate(Y))
eq(X, Y) → false
inf(X) → cons(X, n__inf(n__s(X)))
take(0, X) → nil
take(s(X), cons(Y, L)) → cons(activate(Y), n__take(activate(X), activate(L)))
length(nil) → 0
length(cons(X, L)) → s(n__length(activate(L)))
0 → n__0
s(X) → n__s(X)
inf(X) → n__inf(X)
take(X1, X2) → n__take(X1, X2)
length(X) → n__length(X)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__inf(X)) → inf(activate(X))
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__length(X)) → length(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ACTIVATE(n__take(n__inf(x0), y1)) → TAKE(inf(activate(x0)), activate(y1)) at position [1] we obtained the following new rules:

ACTIVATE(n__take(n__inf(y0), n__take(x0, x1))) → TAKE(inf(activate(y0)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__inf(y0), n__length(x0))) → TAKE(inf(activate(y0)), length(activate(x0)))
ACTIVATE(n__take(n__inf(y0), n__0)) → TAKE(inf(activate(y0)), 0)
ACTIVATE(n__take(n__inf(y0), x0)) → TAKE(inf(activate(y0)), x0)
ACTIVATE(n__take(n__inf(y0), n__inf(x0))) → TAKE(inf(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(n__inf(y0), n__s(x0))) → TAKE(inf(activate(y0)), s(x0))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

          ↳ QDP

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

ACTIVATE(n__take(n__inf(y0), n__take(x0, x1))) → TAKE(inf(activate(y0)), take(activate(x0), activate(x1)))
ACTIVATE(n__length(n__take(x0, x1))) → LENGTH(take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__length(y0), n__s(x0))) → TAKE(length(activate(y0)), s(x0))
ACTIVATE(n__take(n__inf(y0), n__length(x0))) → TAKE(inf(activate(y0)), length(activate(x0)))
ACTIVATE(n__take(n__length(y0), n__inf(x0))) → TAKE(length(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(n__length(y0), n__length(x0))) → TAKE(length(activate(y0)), length(activate(x0)))
ACTIVATE(n__take(n__s(y0), n__0)) → TAKE(s(y0), 0)
ACTIVATE(n__length(n__s(x0))) → LENGTH(s(x0))
TAKE(s(X), cons(Y, L)) → ACTIVATE(Y)
ACTIVATE(n__take(n__inf(y0), x0)) → TAKE(inf(activate(y0)), x0)
ACTIVATE(n__take(n__length(y0), n__0)) → TAKE(length(activate(y0)), 0)
ACTIVATE(n__take(n__inf(y0), n__s(x0))) → TAKE(inf(activate(y0)), s(x0))
ACTIVATE(n__take(n__s(y0), x0)) → TAKE(s(y0), x0)
ACTIVATE(n__take(n__s(y0), n__inf(x0))) → TAKE(s(y0), inf(activate(x0)))
ACTIVATE(n__length(n__0)) → LENGTH(0)
ACTIVATE(n__take(n__length(y0), x0)) → TAKE(length(activate(y0)), x0)
ACTIVATE(n__take(n__s(y0), n__take(x0, x1))) → TAKE(s(y0), take(activate(x0), activate(x1)))
ACTIVATE(n__inf(X)) → ACTIVATE(X)
ACTIVATE(n__length(n__length(x0))) → LENGTH(length(activate(x0)))
ACTIVATE(n__take(n__s(y0), n__s(x0))) → TAKE(s(y0), s(x0))
ACTIVATE(n__take(n__inf(y0), n__0)) → TAKE(inf(activate(y0)), 0)
ACTIVATE(n__length(x0)) → LENGTH(x0)
LENGTH(cons(X, L)) → ACTIVATE(L)
TAKE(s(X), cons(Y, L)) → ACTIVATE(X)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(x0, y1)) → TAKE(x0, activate(y1))
ACTIVATE(n__take(n__0, y1)) → TAKE(0, activate(y1))
ACTIVATE(n__take(n__s(y0), n__length(x0))) → TAKE(s(y0), length(activate(x0)))
ACTIVATE(n__take(n__inf(y0), n__inf(x0))) → TAKE(inf(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__take(n__length(y0), n__take(x0, x1))) → TAKE(length(activate(y0)), take(activate(x0), activate(x1)))
TAKE(s(X), cons(Y, L)) → ACTIVATE(L)
ACTIVATE(n__length(n__inf(x0))) → LENGTH(inf(activate(x0)))
ACTIVATE(n__length(X)) → ACTIVATE(X)
ACTIVATE(n__take(n__take(x0, x1), y1)) → TAKE(take(activate(x0), activate(x1)), activate(y1))

The TRS R consists of the following rules:

eq(n__0, n__0) → true
eq(n__s(X), n__s(Y)) → eq(activate(X), activate(Y))
eq(X, Y) → false
inf(X) → cons(X, n__inf(n__s(X)))
take(0, X) → nil
take(s(X), cons(Y, L)) → cons(activate(Y), n__take(activate(X), activate(L)))
length(nil) → 0
length(cons(X, L)) → s(n__length(activate(L)))
0 → n__0
s(X) → n__s(X)
inf(X) → n__inf(X)
take(X1, X2) → n__take(X1, X2)
length(X) → n__length(X)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__inf(X)) → inf(activate(X))
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__length(X)) → length(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ACTIVATE(n__take(n__0, y1)) → TAKE(0, activate(y1)) at position [0] we obtained the following new rules:

ACTIVATE(n__take(n__0, y0)) → TAKE(n__0, activate(y0))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

          ↳ QDP

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

ACTIVATE(n__take(n__length(y0), n__s(x0))) → TAKE(length(activate(y0)), s(x0))
ACTIVATE(n__length(n__take(x0, x1))) → LENGTH(take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__inf(y0), n__take(x0, x1))) → TAKE(inf(activate(y0)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__length(y0), n__inf(x0))) → TAKE(length(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(n__inf(y0), n__length(x0))) → TAKE(inf(activate(y0)), length(activate(x0)))
ACTIVATE(n__take(n__s(y0), n__0)) → TAKE(s(y0), 0)
ACTIVATE(n__take(n__length(y0), n__length(x0))) → TAKE(length(activate(y0)), length(activate(x0)))
ACTIVATE(n__length(n__s(x0))) → LENGTH(s(x0))
TAKE(s(X), cons(Y, L)) → ACTIVATE(Y)
ACTIVATE(n__take(n__inf(y0), x0)) → TAKE(inf(activate(y0)), x0)
ACTIVATE(n__take(n__length(y0), n__0)) → TAKE(length(activate(y0)), 0)
ACTIVATE(n__take(n__inf(y0), n__s(x0))) → TAKE(inf(activate(y0)), s(x0))
ACTIVATE(n__take(n__s(y0), x0)) → TAKE(s(y0), x0)
ACTIVATE(n__take(n__s(y0), n__inf(x0))) → TAKE(s(y0), inf(activate(x0)))
ACTIVATE(n__length(n__0)) → LENGTH(0)
ACTIVATE(n__take(n__s(y0), n__take(x0, x1))) → TAKE(s(y0), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__length(y0), x0)) → TAKE(length(activate(y0)), x0)
ACTIVATE(n__inf(X)) → ACTIVATE(X)
ACTIVATE(n__take(n__s(y0), n__s(x0))) → TAKE(s(y0), s(x0))
ACTIVATE(n__length(n__length(x0))) → LENGTH(length(activate(x0)))
ACTIVATE(n__take(n__inf(y0), n__0)) → TAKE(inf(activate(y0)), 0)
ACTIVATE(n__take(n__0, y0)) → TAKE(n__0, activate(y0))
ACTIVATE(n__length(x0)) → LENGTH(x0)
TAKE(s(X), cons(Y, L)) → ACTIVATE(X)
LENGTH(cons(X, L)) → ACTIVATE(L)
ACTIVATE(n__take(x0, y1)) → TAKE(x0, activate(y1))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(n__s(y0), n__length(x0))) → TAKE(s(y0), length(activate(x0)))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__take(n__inf(y0), n__inf(x0))) → TAKE(inf(activate(y0)), inf(activate(x0)))
TAKE(s(X), cons(Y, L)) → ACTIVATE(L)
ACTIVATE(n__take(n__length(y0), n__take(x0, x1))) → TAKE(length(activate(y0)), take(activate(x0), activate(x1)))
ACTIVATE(n__length(n__inf(x0))) → LENGTH(inf(activate(x0)))
ACTIVATE(n__take(n__take(x0, x1), y1)) → TAKE(take(activate(x0), activate(x1)), activate(y1))
ACTIVATE(n__length(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

eq(n__0, n__0) → true
eq(n__s(X), n__s(Y)) → eq(activate(X), activate(Y))
eq(X, Y) → false
inf(X) → cons(X, n__inf(n__s(X)))
take(0, X) → nil
take(s(X), cons(Y, L)) → cons(activate(Y), n__take(activate(X), activate(L)))
length(nil) → 0
length(cons(X, L)) → s(n__length(activate(L)))
0 → n__0
s(X) → n__s(X)
inf(X) → n__inf(X)
take(X1, X2) → n__take(X1, X2)
length(X) → n__length(X)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__inf(X)) → inf(activate(X))
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__length(X)) → length(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

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

          ↳ QDP

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

ACTIVATE(n__take(n__inf(y0), n__take(x0, x1))) → TAKE(inf(activate(y0)), take(activate(x0), activate(x1)))
ACTIVATE(n__length(n__take(x0, x1))) → LENGTH(take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__length(y0), n__s(x0))) → TAKE(length(activate(y0)), s(x0))
ACTIVATE(n__take(n__inf(y0), n__length(x0))) → TAKE(inf(activate(y0)), length(activate(x0)))
ACTIVATE(n__take(n__length(y0), n__inf(x0))) → TAKE(length(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(n__s(y0), n__0)) → TAKE(s(y0), 0)
ACTIVATE(n__take(n__length(y0), n__length(x0))) → TAKE(length(activate(y0)), length(activate(x0)))
ACTIVATE(n__length(n__s(x0))) → LENGTH(s(x0))
TAKE(s(X), cons(Y, L)) → ACTIVATE(Y)
ACTIVATE(n__take(n__inf(y0), x0)) → TAKE(inf(activate(y0)), x0)
ACTIVATE(n__take(n__length(y0), n__0)) → TAKE(length(activate(y0)), 0)
ACTIVATE(n__take(n__inf(y0), n__s(x0))) → TAKE(inf(activate(y0)), s(x0))
ACTIVATE(n__take(n__s(y0), x0)) → TAKE(s(y0), x0)
ACTIVATE(n__take(n__s(y0), n__inf(x0))) → TAKE(s(y0), inf(activate(x0)))
ACTIVATE(n__length(n__0)) → LENGTH(0)
ACTIVATE(n__take(n__s(y0), n__take(x0, x1))) → TAKE(s(y0), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__length(y0), x0)) → TAKE(length(activate(y0)), x0)
ACTIVATE(n__inf(X)) → ACTIVATE(X)
ACTIVATE(n__length(n__length(x0))) → LENGTH(length(activate(x0)))
ACTIVATE(n__take(n__s(y0), n__s(x0))) → TAKE(s(y0), s(x0))
ACTIVATE(n__take(n__inf(y0), n__0)) → TAKE(inf(activate(y0)), 0)
ACTIVATE(n__length(x0)) → LENGTH(x0)
LENGTH(cons(X, L)) → ACTIVATE(L)
TAKE(s(X), cons(Y, L)) → ACTIVATE(X)
ACTIVATE(n__take(x0, y1)) → TAKE(x0, activate(y1))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(n__s(y0), n__length(x0))) → TAKE(s(y0), length(activate(x0)))
ACTIVATE(n__take(n__inf(y0), n__inf(x0))) → TAKE(inf(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
TAKE(s(X), cons(Y, L)) → ACTIVATE(L)
ACTIVATE(n__take(n__length(y0), n__take(x0, x1))) → TAKE(length(activate(y0)), take(activate(x0), activate(x1)))
ACTIVATE(n__length(n__inf(x0))) → LENGTH(inf(activate(x0)))
ACTIVATE(n__take(n__take(x0, x1), y1)) → TAKE(take(activate(x0), activate(x1)), activate(y1))
ACTIVATE(n__length(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

eq(n__0, n__0) → true
eq(n__s(X), n__s(Y)) → eq(activate(X), activate(Y))
eq(X, Y) → false
inf(X) → cons(X, n__inf(n__s(X)))
take(0, X) → nil
take(s(X), cons(Y, L)) → cons(activate(Y), n__take(activate(X), activate(L)))
length(nil) → 0
length(cons(X, L)) → s(n__length(activate(L)))
0 → n__0
s(X) → n__s(X)
inf(X) → n__inf(X)
take(X1, X2) → n__take(X1, X2)
length(X) → n__length(X)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__inf(X)) → inf(activate(X))
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__length(X)) → length(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ACTIVATE(n__take(n__take(x0, x1), y1)) → TAKE(take(activate(x0), activate(x1)), activate(y1)) at position [1] we obtained the following new rules:

ACTIVATE(n__take(n__take(y0, y1), n__inf(x0))) → TAKE(take(activate(y0), activate(y1)), inf(activate(x0)))
ACTIVATE(n__take(n__take(y0, y1), n__0)) → TAKE(take(activate(y0), activate(y1)), 0)
ACTIVATE(n__take(n__take(y0, y1), n__s(x0))) → TAKE(take(activate(y0), activate(y1)), s(x0))
ACTIVATE(n__take(n__take(y0, y1), n__length(x0))) → TAKE(take(activate(y0), activate(y1)), length(activate(x0)))
ACTIVATE(n__take(n__take(y0, y1), x0)) → TAKE(take(activate(y0), activate(y1)), x0)
ACTIVATE(n__take(n__take(y0, y1), n__take(x0, x1))) → TAKE(take(activate(y0), activate(y1)), take(activate(x0), activate(x1)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Narrowing

          ↳ QDP

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

ACTIVATE(n__take(n__take(y0, y1), n__inf(x0))) → TAKE(take(activate(y0), activate(y1)), inf(activate(x0)))
ACTIVATE(n__take(n__length(y0), n__s(x0))) → TAKE(length(activate(y0)), s(x0))
ACTIVATE(n__length(n__take(x0, x1))) → LENGTH(take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__inf(y0), n__take(x0, x1))) → TAKE(inf(activate(y0)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__length(y0), n__inf(x0))) → TAKE(length(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(n__inf(y0), n__length(x0))) → TAKE(inf(activate(y0)), length(activate(x0)))
ACTIVATE(n__take(n__take(y0, y1), n__0)) → TAKE(take(activate(y0), activate(y1)), 0)
ACTIVATE(n__take(n__length(y0), n__length(x0))) → TAKE(length(activate(y0)), length(activate(x0)))
ACTIVATE(n__take(n__s(y0), n__0)) → TAKE(s(y0), 0)
ACTIVATE(n__length(n__s(x0))) → LENGTH(s(x0))
TAKE(s(X), cons(Y, L)) → ACTIVATE(Y)
ACTIVATE(n__take(n__inf(y0), x0)) → TAKE(inf(activate(y0)), x0)
ACTIVATE(n__take(n__length(y0), n__0)) → TAKE(length(activate(y0)), 0)
ACTIVATE(n__take(n__inf(y0), n__s(x0))) → TAKE(inf(activate(y0)), s(x0))
ACTIVATE(n__take(n__s(y0), x0)) → TAKE(s(y0), x0)
ACTIVATE(n__take(n__s(y0), n__inf(x0))) → TAKE(s(y0), inf(activate(x0)))
ACTIVATE(n__length(n__0)) → LENGTH(0)
ACTIVATE(n__take(n__length(y0), x0)) → TAKE(length(activate(y0)), x0)
ACTIVATE(n__take(n__s(y0), n__take(x0, x1))) → TAKE(s(y0), take(activate(x0), activate(x1)))
ACTIVATE(n__inf(X)) → ACTIVATE(X)
ACTIVATE(n__take(n__s(y0), n__s(x0))) → TAKE(s(y0), s(x0))
ACTIVATE(n__length(n__length(x0))) → LENGTH(length(activate(x0)))
ACTIVATE(n__take(n__inf(y0), n__0)) → TAKE(inf(activate(y0)), 0)
ACTIVATE(n__length(x0)) → LENGTH(x0)
TAKE(s(X), cons(Y, L)) → ACTIVATE(X)
LENGTH(cons(X, L)) → ACTIVATE(L)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(x0, y1)) → TAKE(x0, activate(y1))
ACTIVATE(n__take(n__take(y0, y1), n__s(x0))) → TAKE(take(activate(y0), activate(y1)), s(x0))
ACTIVATE(n__take(n__s(y0), n__length(x0))) → TAKE(s(y0), length(activate(x0)))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__take(n__inf(y0), n__inf(x0))) → TAKE(inf(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(n__take(y0, y1), n__length(x0))) → TAKE(take(activate(y0), activate(y1)), length(activate(x0)))
ACTIVATE(n__take(n__length(y0), n__take(x0, x1))) → TAKE(length(activate(y0)), take(activate(x0), activate(x1)))
TAKE(s(X), cons(Y, L)) → ACTIVATE(L)
ACTIVATE(n__take(n__take(y0, y1), n__take(x0, x1))) → TAKE(take(activate(y0), activate(y1)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__take(y0, y1), x0)) → TAKE(take(activate(y0), activate(y1)), x0)
ACTIVATE(n__length(n__inf(x0))) → LENGTH(inf(activate(x0)))
ACTIVATE(n__length(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

eq(n__0, n__0) → true
eq(n__s(X), n__s(Y)) → eq(activate(X), activate(Y))
eq(X, Y) → false
inf(X) → cons(X, n__inf(n__s(X)))
take(0, X) → nil
take(s(X), cons(Y, L)) → cons(activate(Y), n__take(activate(X), activate(L)))
length(nil) → 0
length(cons(X, L)) → s(n__length(activate(L)))
0 → n__0
s(X) → n__s(X)
inf(X) → n__inf(X)
take(X1, X2) → n__take(X1, X2)
length(X) → n__length(X)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__inf(X)) → inf(activate(X))
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__length(X)) → length(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ACTIVATE(n__take(x0, y1)) → TAKE(x0, activate(y1)) at position [1] we obtained the following new rules:

ACTIVATE(n__take(y0, n__0)) → TAKE(y0, 0)
ACTIVATE(n__take(y0, n__inf(x0))) → TAKE(y0, inf(activate(x0)))
ACTIVATE(n__take(y0, x0)) → TAKE(y0, x0)
ACTIVATE(n__take(y0, n__length(x0))) → TAKE(y0, length(activate(x0)))
ACTIVATE(n__take(y0, n__take(x0, x1))) → TAKE(y0, take(activate(x0), activate(x1)))
ACTIVATE(n__take(y0, n__s(x0))) → TAKE(y0, s(x0))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ Narrowing

          ↳ QDP

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

ACTIVATE(n__take(n__inf(y0), n__take(x0, x1))) → TAKE(inf(activate(y0)), take(activate(x0), activate(x1)))
ACTIVATE(n__length(n__take(x0, x1))) → LENGTH(take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__length(y0), n__s(x0))) → TAKE(length(activate(y0)), s(x0))
ACTIVATE(n__take(n__take(y0, y1), n__inf(x0))) → TAKE(take(activate(y0), activate(y1)), inf(activate(x0)))
ACTIVATE(n__take(n__inf(y0), n__length(x0))) → TAKE(inf(activate(y0)), length(activate(x0)))
ACTIVATE(n__take(n__length(y0), n__inf(x0))) → TAKE(length(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(n__take(y0, y1), n__0)) → TAKE(take(activate(y0), activate(y1)), 0)
ACTIVATE(n__take(n__s(y0), n__0)) → TAKE(s(y0), 0)
ACTIVATE(n__take(n__length(y0), n__length(x0))) → TAKE(length(activate(y0)), length(activate(x0)))
ACTIVATE(n__length(n__s(x0))) → LENGTH(s(x0))
TAKE(s(X), cons(Y, L)) → ACTIVATE(Y)
ACTIVATE(n__take(n__inf(y0), x0)) → TAKE(inf(activate(y0)), x0)
ACTIVATE(n__take(n__length(y0), n__0)) → TAKE(length(activate(y0)), 0)
ACTIVATE(n__take(n__inf(y0), n__s(x0))) → TAKE(inf(activate(y0)), s(x0))
ACTIVATE(n__take(n__s(y0), x0)) → TAKE(s(y0), x0)
ACTIVATE(n__take(n__s(y0), n__inf(x0))) → TAKE(s(y0), inf(activate(x0)))
ACTIVATE(n__length(n__0)) → LENGTH(0)
ACTIVATE(n__take(n__s(y0), n__take(x0, x1))) → TAKE(s(y0), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__length(y0), x0)) → TAKE(length(activate(y0)), x0)
ACTIVATE(n__inf(X)) → ACTIVATE(X)
ACTIVATE(n__length(n__length(x0))) → LENGTH(length(activate(x0)))
ACTIVATE(n__take(n__s(y0), n__s(x0))) → TAKE(s(y0), s(x0))
ACTIVATE(n__take(n__inf(y0), n__0)) → TAKE(inf(activate(y0)), 0)
ACTIVATE(n__length(x0)) → LENGTH(x0)
LENGTH(cons(X, L)) → ACTIVATE(L)
TAKE(s(X), cons(Y, L)) → ACTIVATE(X)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(n__take(y0, y1), n__s(x0))) → TAKE(take(activate(y0), activate(y1)), s(x0))
ACTIVATE(n__take(n__s(y0), n__length(x0))) → TAKE(s(y0), length(activate(x0)))
ACTIVATE(n__take(y0, n__take(x0, x1))) → TAKE(y0, take(activate(x0), activate(x1)))
ACTIVATE(n__take(y0, n__s(x0))) → TAKE(y0, s(x0))
ACTIVATE(n__take(n__inf(y0), n__inf(x0))) → TAKE(inf(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
TAKE(s(X), cons(Y, L)) → ACTIVATE(L)
ACTIVATE(n__take(n__length(y0), n__take(x0, x1))) → TAKE(length(activate(y0)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__take(y0, y1), n__length(x0))) → TAKE(take(activate(y0), activate(y1)), length(activate(x0)))
ACTIVATE(n__take(y0, n__0)) → TAKE(y0, 0)
ACTIVATE(n__take(n__take(y0, y1), x0)) → TAKE(take(activate(y0), activate(y1)), x0)
ACTIVATE(n__take(n__take(y0, y1), n__take(x0, x1))) → TAKE(take(activate(y0), activate(y1)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(y0, n__length(x0))) → TAKE(y0, length(activate(x0)))
ACTIVATE(n__length(n__inf(x0))) → LENGTH(inf(activate(x0)))
ACTIVATE(n__take(y0, n__inf(x0))) → TAKE(y0, inf(activate(x0)))
ACTIVATE(n__take(y0, x0)) → TAKE(y0, x0)
ACTIVATE(n__length(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

eq(n__0, n__0) → true
eq(n__s(X), n__s(Y)) → eq(activate(X), activate(Y))
eq(X, Y) → false
inf(X) → cons(X, n__inf(n__s(X)))
take(0, X) → nil
take(s(X), cons(Y, L)) → cons(activate(Y), n__take(activate(X), activate(L)))
length(nil) → 0
length(cons(X, L)) → s(n__length(activate(L)))
0 → n__0
s(X) → n__s(X)
inf(X) → n__inf(X)
take(X1, X2) → n__take(X1, X2)
length(X) → n__length(X)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__inf(X)) → inf(activate(X))
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__length(X)) → length(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ACTIVATE(n__length(n__0)) → LENGTH(0) at position [0] we obtained the following new rules:

ACTIVATE(n__length(n__0)) → LENGTH(n__0)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

          ↳ QDP

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

ACTIVATE(n__take(n__take(y0, y1), n__inf(x0))) → TAKE(take(activate(y0), activate(y1)), inf(activate(x0)))
ACTIVATE(n__take(n__length(y0), n__s(x0))) → TAKE(length(activate(y0)), s(x0))
ACTIVATE(n__length(n__take(x0, x1))) → LENGTH(take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__inf(y0), n__take(x0, x1))) → TAKE(inf(activate(y0)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__length(y0), n__inf(x0))) → TAKE(length(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(n__inf(y0), n__length(x0))) → TAKE(inf(activate(y0)), length(activate(x0)))
ACTIVATE(n__take(n__take(y0, y1), n__0)) → TAKE(take(activate(y0), activate(y1)), 0)
ACTIVATE(n__take(n__length(y0), n__length(x0))) → TAKE(length(activate(y0)), length(activate(x0)))
ACTIVATE(n__take(n__s(y0), n__0)) → TAKE(s(y0), 0)
ACTIVATE(n__length(n__s(x0))) → LENGTH(s(x0))
TAKE(s(X), cons(Y, L)) → ACTIVATE(Y)
ACTIVATE(n__take(n__inf(y0), x0)) → TAKE(inf(activate(y0)), x0)
ACTIVATE(n__take(n__length(y0), n__0)) → TAKE(length(activate(y0)), 0)
ACTIVATE(n__take(n__inf(y0), n__s(x0))) → TAKE(inf(activate(y0)), s(x0))
ACTIVATE(n__take(n__s(y0), x0)) → TAKE(s(y0), x0)
ACTIVATE(n__take(n__s(y0), n__inf(x0))) → TAKE(s(y0), inf(activate(x0)))
ACTIVATE(n__take(n__length(y0), x0)) → TAKE(length(activate(y0)), x0)
ACTIVATE(n__take(n__s(y0), n__take(x0, x1))) → TAKE(s(y0), take(activate(x0), activate(x1)))
ACTIVATE(n__inf(X)) → ACTIVATE(X)
ACTIVATE(n__take(n__s(y0), n__s(x0))) → TAKE(s(y0), s(x0))
ACTIVATE(n__length(n__length(x0))) → LENGTH(length(activate(x0)))
ACTIVATE(n__take(n__inf(y0), n__0)) → TAKE(inf(activate(y0)), 0)
ACTIVATE(n__length(x0)) → LENGTH(x0)
TAKE(s(X), cons(Y, L)) → ACTIVATE(X)
LENGTH(cons(X, L)) → ACTIVATE(L)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(n__take(y0, y1), n__s(x0))) → TAKE(take(activate(y0), activate(y1)), s(x0))
ACTIVATE(n__take(n__s(y0), n__length(x0))) → TAKE(s(y0), length(activate(x0)))
ACTIVATE(n__take(y0, n__take(x0, x1))) → TAKE(y0, take(activate(x0), activate(x1)))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__take(n__inf(y0), n__inf(x0))) → TAKE(inf(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(y0, n__s(x0))) → TAKE(y0, s(x0))
ACTIVATE(n__take(n__take(y0, y1), n__length(x0))) → TAKE(take(activate(y0), activate(y1)), length(activate(x0)))
ACTIVATE(n__take(n__length(y0), n__take(x0, x1))) → TAKE(length(activate(y0)), take(activate(x0), activate(x1)))
TAKE(s(X), cons(Y, L)) → ACTIVATE(L)
ACTIVATE(n__take(y0, n__0)) → TAKE(y0, 0)
ACTIVATE(n__take(n__take(y0, y1), n__take(x0, x1))) → TAKE(take(activate(y0), activate(y1)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__take(y0, y1), x0)) → TAKE(take(activate(y0), activate(y1)), x0)
ACTIVATE(n__take(y0, n__length(x0))) → TAKE(y0, length(activate(x0)))
ACTIVATE(n__length(n__inf(x0))) → LENGTH(inf(activate(x0)))
ACTIVATE(n__take(y0, n__inf(x0))) → TAKE(y0, inf(activate(x0)))
ACTIVATE(n__take(y0, x0)) → TAKE(y0, x0)
ACTIVATE(n__length(n__0)) → LENGTH(n__0)
ACTIVATE(n__length(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

eq(n__0, n__0) → true
eq(n__s(X), n__s(Y)) → eq(activate(X), activate(Y))
eq(X, Y) → false
inf(X) → cons(X, n__inf(n__s(X)))
take(0, X) → nil
take(s(X), cons(Y, L)) → cons(activate(Y), n__take(activate(X), activate(L)))
length(nil) → 0
length(cons(X, L)) → s(n__length(activate(L)))
0 → n__0
s(X) → n__s(X)
inf(X) → n__inf(X)
take(X1, X2) → n__take(X1, X2)
length(X) → n__length(X)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__inf(X)) → inf(activate(X))
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__length(X)) → length(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

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

          ↳ QDP

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

ACTIVATE(n__take(n__take(y0, y1), n__inf(x0))) → TAKE(take(activate(y0), activate(y1)), inf(activate(x0)))
ACTIVATE(n__take(n__inf(y0), n__take(x0, x1))) → TAKE(inf(activate(y0)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__length(y0), n__s(x0))) → TAKE(length(activate(y0)), s(x0))
ACTIVATE(n__length(n__take(x0, x1))) → LENGTH(take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__inf(y0), n__length(x0))) → TAKE(inf(activate(y0)), length(activate(x0)))
ACTIVATE(n__take(n__length(y0), n__inf(x0))) → TAKE(length(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(n__take(y0, y1), n__0)) → TAKE(take(activate(y0), activate(y1)), 0)
ACTIVATE(n__take(n__length(y0), n__length(x0))) → TAKE(length(activate(y0)), length(activate(x0)))
ACTIVATE(n__take(n__s(y0), n__0)) → TAKE(s(y0), 0)
ACTIVATE(n__length(n__s(x0))) → LENGTH(s(x0))
TAKE(s(X), cons(Y, L)) → ACTIVATE(Y)
ACTIVATE(n__take(n__inf(y0), x0)) → TAKE(inf(activate(y0)), x0)
ACTIVATE(n__take(n__length(y0), n__0)) → TAKE(length(activate(y0)), 0)
ACTIVATE(n__take(n__inf(y0), n__s(x0))) → TAKE(inf(activate(y0)), s(x0))
ACTIVATE(n__take(n__s(y0), x0)) → TAKE(s(y0), x0)
ACTIVATE(n__take(n__s(y0), n__inf(x0))) → TAKE(s(y0), inf(activate(x0)))
ACTIVATE(n__take(n__length(y0), x0)) → TAKE(length(activate(y0)), x0)
ACTIVATE(n__take(n__s(y0), n__take(x0, x1))) → TAKE(s(y0), take(activate(x0), activate(x1)))
ACTIVATE(n__inf(X)) → ACTIVATE(X)
ACTIVATE(n__length(n__length(x0))) → LENGTH(length(activate(x0)))
ACTIVATE(n__take(n__s(y0), n__s(x0))) → TAKE(s(y0), s(x0))
ACTIVATE(n__take(n__inf(y0), n__0)) → TAKE(inf(activate(y0)), 0)
ACTIVATE(n__length(x0)) → LENGTH(x0)
LENGTH(cons(X, L)) → ACTIVATE(L)
TAKE(s(X), cons(Y, L)) → ACTIVATE(X)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(n__take(y0, y1), n__s(x0))) → TAKE(take(activate(y0), activate(y1)), s(x0))
ACTIVATE(n__take(n__s(y0), n__length(x0))) → TAKE(s(y0), length(activate(x0)))
ACTIVATE(n__take(y0, n__take(x0, x1))) → TAKE(y0, take(activate(x0), activate(x1)))
ACTIVATE(n__take(y0, n__s(x0))) → TAKE(y0, s(x0))
ACTIVATE(n__take(n__inf(y0), n__inf(x0))) → TAKE(inf(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__take(n__take(y0, y1), n__length(x0))) → TAKE(take(activate(y0), activate(y1)), length(activate(x0)))
TAKE(s(X), cons(Y, L)) → ACTIVATE(L)
ACTIVATE(n__take(n__length(y0), n__take(x0, x1))) → TAKE(length(activate(y0)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(y0, n__0)) → TAKE(y0, 0)
ACTIVATE(n__take(n__take(y0, y1), n__take(x0, x1))) → TAKE(take(activate(y0), activate(y1)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__take(y0, y1), x0)) → TAKE(take(activate(y0), activate(y1)), x0)
ACTIVATE(n__take(y0, n__length(x0))) → TAKE(y0, length(activate(x0)))
ACTIVATE(n__length(n__inf(x0))) → LENGTH(inf(activate(x0)))
ACTIVATE(n__take(y0, n__inf(x0))) → TAKE(y0, inf(activate(x0)))
ACTIVATE(n__take(y0, x0)) → TAKE(y0, x0)
ACTIVATE(n__length(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

eq(n__0, n__0) → true
eq(n__s(X), n__s(Y)) → eq(activate(X), activate(Y))
eq(X, Y) → false
inf(X) → cons(X, n__inf(n__s(X)))
take(0, X) → nil
take(s(X), cons(Y, L)) → cons(activate(Y), n__take(activate(X), activate(L)))
length(nil) → 0
length(cons(X, L)) → s(n__length(activate(L)))
0 → n__0
s(X) → n__s(X)
inf(X) → n__inf(X)
take(X1, X2) → n__take(X1, X2)
length(X) → n__length(X)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__inf(X)) → inf(activate(X))
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__length(X)) → length(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ACTIVATE(n__length(n__s(x0))) → LENGTH(s(x0)) at position [0] we obtained the following new rules:

ACTIVATE(n__length(n__s(x0))) → LENGTH(n__s(x0))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

          ↳ QDP

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

ACTIVATE(n__length(n__take(x0, x1))) → LENGTH(take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__length(y0), n__s(x0))) → TAKE(length(activate(y0)), s(x0))
ACTIVATE(n__take(n__inf(y0), n__take(x0, x1))) → TAKE(inf(activate(y0)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__take(y0, y1), n__inf(x0))) → TAKE(take(activate(y0), activate(y1)), inf(activate(x0)))
ACTIVATE(n__take(n__length(y0), n__inf(x0))) → TAKE(length(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(n__inf(y0), n__length(x0))) → TAKE(inf(activate(y0)), length(activate(x0)))
ACTIVATE(n__take(n__take(y0, y1), n__0)) → TAKE(take(activate(y0), activate(y1)), 0)
ACTIVATE(n__take(n__s(y0), n__0)) → TAKE(s(y0), 0)
ACTIVATE(n__take(n__length(y0), n__length(x0))) → TAKE(length(activate(y0)), length(activate(x0)))
TAKE(s(X), cons(Y, L)) → ACTIVATE(Y)
ACTIVATE(n__take(n__inf(y0), x0)) → TAKE(inf(activate(y0)), x0)
ACTIVATE(n__take(n__length(y0), n__0)) → TAKE(length(activate(y0)), 0)
ACTIVATE(n__take(n__inf(y0), n__s(x0))) → TAKE(inf(activate(y0)), s(x0))
ACTIVATE(n__take(n__s(y0), x0)) → TAKE(s(y0), x0)
ACTIVATE(n__take(n__s(y0), n__inf(x0))) → TAKE(s(y0), inf(activate(x0)))
ACTIVATE(n__take(n__s(y0), n__take(x0, x1))) → TAKE(s(y0), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__length(y0), x0)) → TAKE(length(activate(y0)), x0)
ACTIVATE(n__inf(X)) → ACTIVATE(X)
ACTIVATE(n__take(n__s(y0), n__s(x0))) → TAKE(s(y0), s(x0))
ACTIVATE(n__length(n__length(x0))) → LENGTH(length(activate(x0)))
ACTIVATE(n__take(n__inf(y0), n__0)) → TAKE(inf(activate(y0)), 0)
ACTIVATE(n__length(x0)) → LENGTH(x0)
TAKE(s(X), cons(Y, L)) → ACTIVATE(X)
LENGTH(cons(X, L)) → ACTIVATE(L)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(n__take(y0, y1), n__s(x0))) → TAKE(take(activate(y0), activate(y1)), s(x0))
ACTIVATE(n__take(n__s(y0), n__length(x0))) → TAKE(s(y0), length(activate(x0)))
ACTIVATE(n__take(y0, n__take(x0, x1))) → TAKE(y0, take(activate(x0), activate(x1)))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__take(n__inf(y0), n__inf(x0))) → TAKE(inf(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(y0, n__s(x0))) → TAKE(y0, s(x0))
ACTIVATE(n__length(n__s(x0))) → LENGTH(n__s(x0))
ACTIVATE(n__take(n__length(y0), n__take(x0, x1))) → TAKE(length(activate(y0)), take(activate(x0), activate(x1)))
TAKE(s(X), cons(Y, L)) → ACTIVATE(L)
ACTIVATE(n__take(n__take(y0, y1), n__length(x0))) → TAKE(take(activate(y0), activate(y1)), length(activate(x0)))
ACTIVATE(n__take(y0, n__0)) → TAKE(y0, 0)
ACTIVATE(n__take(n__take(y0, y1), x0)) → TAKE(take(activate(y0), activate(y1)), x0)
ACTIVATE(n__take(n__take(y0, y1), n__take(x0, x1))) → TAKE(take(activate(y0), activate(y1)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(y0, n__length(x0))) → TAKE(y0, length(activate(x0)))
ACTIVATE(n__length(n__inf(x0))) → LENGTH(inf(activate(x0)))
ACTIVATE(n__take(y0, n__inf(x0))) → TAKE(y0, inf(activate(x0)))
ACTIVATE(n__take(y0, x0)) → TAKE(y0, x0)
ACTIVATE(n__length(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

eq(n__0, n__0) → true
eq(n__s(X), n__s(Y)) → eq(activate(X), activate(Y))
eq(X, Y) → false
inf(X) → cons(X, n__inf(n__s(X)))
take(0, X) → nil
take(s(X), cons(Y, L)) → cons(activate(Y), n__take(activate(X), activate(L)))
length(nil) → 0
length(cons(X, L)) → s(n__length(activate(L)))
0 → n__0
s(X) → n__s(X)
inf(X) → n__inf(X)
take(X1, X2) → n__take(X1, X2)
length(X) → n__length(X)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__inf(X)) → inf(activate(X))
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__length(X)) → length(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

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Narrowing

          ↳ QDP

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

ACTIVATE(n__take(n__take(y0, y1), n__inf(x0))) → TAKE(take(activate(y0), activate(y1)), inf(activate(x0)))
ACTIVATE(n__take(n__length(y0), n__s(x0))) → TAKE(length(activate(y0)), s(x0))
ACTIVATE(n__length(n__take(x0, x1))) → LENGTH(take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__inf(y0), n__take(x0, x1))) → TAKE(inf(activate(y0)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__inf(y0), n__length(x0))) → TAKE(inf(activate(y0)), length(activate(x0)))
ACTIVATE(n__take(n__length(y0), n__inf(x0))) → TAKE(length(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(n__take(y0, y1), n__0)) → TAKE(take(activate(y0), activate(y1)), 0)
ACTIVATE(n__take(n__length(y0), n__length(x0))) → TAKE(length(activate(y0)), length(activate(x0)))
ACTIVATE(n__take(n__s(y0), n__0)) → TAKE(s(y0), 0)
TAKE(s(X), cons(Y, L)) → ACTIVATE(Y)
ACTIVATE(n__take(n__inf(y0), x0)) → TAKE(inf(activate(y0)), x0)
ACTIVATE(n__take(n__length(y0), n__0)) → TAKE(length(activate(y0)), 0)
ACTIVATE(n__take(n__inf(y0), n__s(x0))) → TAKE(inf(activate(y0)), s(x0))
ACTIVATE(n__take(n__s(y0), x0)) → TAKE(s(y0), x0)
ACTIVATE(n__take(n__s(y0), n__inf(x0))) → TAKE(s(y0), inf(activate(x0)))
ACTIVATE(n__take(n__length(y0), x0)) → TAKE(length(activate(y0)), x0)
ACTIVATE(n__take(n__s(y0), n__take(x0, x1))) → TAKE(s(y0), take(activate(x0), activate(x1)))
ACTIVATE(n__inf(X)) → ACTIVATE(X)
ACTIVATE(n__length(n__length(x0))) → LENGTH(length(activate(x0)))
ACTIVATE(n__take(n__s(y0), n__s(x0))) → TAKE(s(y0), s(x0))
ACTIVATE(n__take(n__inf(y0), n__0)) → TAKE(inf(activate(y0)), 0)
ACTIVATE(n__length(x0)) → LENGTH(x0)
TAKE(s(X), cons(Y, L)) → ACTIVATE(X)
LENGTH(cons(X, L)) → ACTIVATE(L)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(n__take(y0, y1), n__s(x0))) → TAKE(take(activate(y0), activate(y1)), s(x0))
ACTIVATE(n__take(n__s(y0), n__length(x0))) → TAKE(s(y0), length(activate(x0)))
ACTIVATE(n__take(y0, n__take(x0, x1))) → TAKE(y0, take(activate(x0), activate(x1)))
ACTIVATE(n__take(y0, n__s(x0))) → TAKE(y0, s(x0))
ACTIVATE(n__take(n__inf(y0), n__inf(x0))) → TAKE(inf(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__take(n__take(y0, y1), n__length(x0))) → TAKE(take(activate(y0), activate(y1)), length(activate(x0)))
TAKE(s(X), cons(Y, L)) → ACTIVATE(L)
ACTIVATE(n__take(n__length(y0), n__take(x0, x1))) → TAKE(length(activate(y0)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(y0, n__0)) → TAKE(y0, 0)
ACTIVATE(n__take(n__take(y0, y1), n__take(x0, x1))) → TAKE(take(activate(y0), activate(y1)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__take(y0, y1), x0)) → TAKE(take(activate(y0), activate(y1)), x0)
ACTIVATE(n__take(y0, n__length(x0))) → TAKE(y0, length(activate(x0)))
ACTIVATE(n__length(n__inf(x0))) → LENGTH(inf(activate(x0)))
ACTIVATE(n__take(y0, n__inf(x0))) → TAKE(y0, inf(activate(x0)))
ACTIVATE(n__take(y0, x0)) → TAKE(y0, x0)
ACTIVATE(n__length(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

eq(n__0, n__0) → true
eq(n__s(X), n__s(Y)) → eq(activate(X), activate(Y))
eq(X, Y) → false
inf(X) → cons(X, n__inf(n__s(X)))
take(0, X) → nil
take(s(X), cons(Y, L)) → cons(activate(Y), n__take(activate(X), activate(L)))
length(nil) → 0
length(cons(X, L)) → s(n__length(activate(L)))
0 → n__0
s(X) → n__s(X)
inf(X) → n__inf(X)
take(X1, X2) → n__take(X1, X2)
length(X) → n__length(X)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__inf(X)) → inf(activate(X))
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__length(X)) → length(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ACTIVATE(n__take(n__length(y0), n__s(x0))) → TAKE(length(activate(y0)), s(x0)) at position [1] we obtained the following new rules:

ACTIVATE(n__take(n__length(y0), n__s(x0))) → TAKE(length(activate(y0)), n__s(x0))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

          ↳ QDP

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

ACTIVATE(n__take(n__inf(y0), n__take(x0, x1))) → TAKE(inf(activate(y0)), take(activate(x0), activate(x1)))
ACTIVATE(n__length(n__take(x0, x1))) → LENGTH(take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__take(y0, y1), n__inf(x0))) → TAKE(take(activate(y0), activate(y1)), inf(activate(x0)))
ACTIVATE(n__take(n__length(y0), n__inf(x0))) → TAKE(length(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(n__inf(y0), n__length(x0))) → TAKE(inf(activate(y0)), length(activate(x0)))
ACTIVATE(n__take(n__take(y0, y1), n__0)) → TAKE(take(activate(y0), activate(y1)), 0)
ACTIVATE(n__take(n__s(y0), n__0)) → TAKE(s(y0), 0)
ACTIVATE(n__take(n__length(y0), n__length(x0))) → TAKE(length(activate(y0)), length(activate(x0)))
TAKE(s(X), cons(Y, L)) → ACTIVATE(Y)
ACTIVATE(n__take(n__inf(y0), x0)) → TAKE(inf(activate(y0)), x0)
ACTIVATE(n__take(n__length(y0), n__0)) → TAKE(length(activate(y0)), 0)
ACTIVATE(n__take(n__inf(y0), n__s(x0))) → TAKE(inf(activate(y0)), s(x0))
ACTIVATE(n__take(n__s(y0), x0)) → TAKE(s(y0), x0)
ACTIVATE(n__take(n__length(y0), n__s(x0))) → TAKE(length(activate(y0)), n__s(x0))
ACTIVATE(n__take(n__s(y0), n__inf(x0))) → TAKE(s(y0), inf(activate(x0)))
ACTIVATE(n__take(n__s(y0), n__take(x0, x1))) → TAKE(s(y0), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__length(y0), x0)) → TAKE(length(activate(y0)), x0)
ACTIVATE(n__inf(X)) → ACTIVATE(X)
ACTIVATE(n__take(n__s(y0), n__s(x0))) → TAKE(s(y0), s(x0))
ACTIVATE(n__length(n__length(x0))) → LENGTH(length(activate(x0)))
ACTIVATE(n__take(n__inf(y0), n__0)) → TAKE(inf(activate(y0)), 0)
ACTIVATE(n__length(x0)) → LENGTH(x0)
LENGTH(cons(X, L)) → ACTIVATE(L)
TAKE(s(X), cons(Y, L)) → ACTIVATE(X)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(n__take(y0, y1), n__s(x0))) → TAKE(take(activate(y0), activate(y1)), s(x0))
ACTIVATE(n__take(n__s(y0), n__length(x0))) → TAKE(s(y0), length(activate(x0)))
ACTIVATE(n__take(y0, n__take(x0, x1))) → TAKE(y0, take(activate(x0), activate(x1)))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__take(n__inf(y0), n__inf(x0))) → TAKE(inf(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(y0, n__s(x0))) → TAKE(y0, s(x0))
ACTIVATE(n__take(n__length(y0), n__take(x0, x1))) → TAKE(length(activate(y0)), take(activate(x0), activate(x1)))
TAKE(s(X), cons(Y, L)) → ACTIVATE(L)
ACTIVATE(n__take(n__take(y0, y1), n__length(x0))) → TAKE(take(activate(y0), activate(y1)), length(activate(x0)))
ACTIVATE(n__take(y0, n__0)) → TAKE(y0, 0)
ACTIVATE(n__take(n__take(y0, y1), x0)) → TAKE(take(activate(y0), activate(y1)), x0)
ACTIVATE(n__take(n__take(y0, y1), n__take(x0, x1))) → TAKE(take(activate(y0), activate(y1)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(y0, n__length(x0))) → TAKE(y0, length(activate(x0)))
ACTIVATE(n__length(n__inf(x0))) → LENGTH(inf(activate(x0)))
ACTIVATE(n__take(y0, n__inf(x0))) → TAKE(y0, inf(activate(x0)))
ACTIVATE(n__take(y0, x0)) → TAKE(y0, x0)
ACTIVATE(n__length(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

eq(n__0, n__0) → true
eq(n__s(X), n__s(Y)) → eq(activate(X), activate(Y))
eq(X, Y) → false
inf(X) → cons(X, n__inf(n__s(X)))
take(0, X) → nil
take(s(X), cons(Y, L)) → cons(activate(Y), n__take(activate(X), activate(L)))
length(nil) → 0
length(cons(X, L)) → s(n__length(activate(L)))
0 → n__0
s(X) → n__s(X)
inf(X) → n__inf(X)
take(X1, X2) → n__take(X1, X2)
length(X) → n__length(X)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__inf(X)) → inf(activate(X))
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__length(X)) → length(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

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Narrowing

          ↳ QDP

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

ACTIVATE(n__take(n__take(y0, y1), n__inf(x0))) → TAKE(take(activate(y0), activate(y1)), inf(activate(x0)))
ACTIVATE(n__length(n__take(x0, x1))) → LENGTH(take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__inf(y0), n__take(x0, x1))) → TAKE(inf(activate(y0)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__length(y0), n__inf(x0))) → TAKE(length(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(n__inf(y0), n__length(x0))) → TAKE(inf(activate(y0)), length(activate(x0)))
ACTIVATE(n__take(n__take(y0, y1), n__0)) → TAKE(take(activate(y0), activate(y1)), 0)
ACTIVATE(n__take(n__length(y0), n__length(x0))) → TAKE(length(activate(y0)), length(activate(x0)))
ACTIVATE(n__take(n__s(y0), n__0)) → TAKE(s(y0), 0)
TAKE(s(X), cons(Y, L)) → ACTIVATE(Y)
ACTIVATE(n__take(n__inf(y0), x0)) → TAKE(inf(activate(y0)), x0)
ACTIVATE(n__take(n__length(y0), n__0)) → TAKE(length(activate(y0)), 0)
ACTIVATE(n__take(n__inf(y0), n__s(x0))) → TAKE(inf(activate(y0)), s(x0))
ACTIVATE(n__take(n__s(y0), x0)) → TAKE(s(y0), x0)
ACTIVATE(n__take(n__s(y0), n__inf(x0))) → TAKE(s(y0), inf(activate(x0)))
ACTIVATE(n__take(n__length(y0), x0)) → TAKE(length(activate(y0)), x0)
ACTIVATE(n__take(n__s(y0), n__take(x0, x1))) → TAKE(s(y0), take(activate(x0), activate(x1)))
ACTIVATE(n__inf(X)) → ACTIVATE(X)
ACTIVATE(n__length(n__length(x0))) → LENGTH(length(activate(x0)))
ACTIVATE(n__take(n__s(y0), n__s(x0))) → TAKE(s(y0), s(x0))
ACTIVATE(n__take(n__inf(y0), n__0)) → TAKE(inf(activate(y0)), 0)
ACTIVATE(n__length(x0)) → LENGTH(x0)
TAKE(s(X), cons(Y, L)) → ACTIVATE(X)
LENGTH(cons(X, L)) → ACTIVATE(L)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(n__take(y0, y1), n__s(x0))) → TAKE(take(activate(y0), activate(y1)), s(x0))
ACTIVATE(n__take(n__s(y0), n__length(x0))) → TAKE(s(y0), length(activate(x0)))
ACTIVATE(n__take(y0, n__take(x0, x1))) → TAKE(y0, take(activate(x0), activate(x1)))
ACTIVATE(n__take(y0, n__s(x0))) → TAKE(y0, s(x0))
ACTIVATE(n__take(n__inf(y0), n__inf(x0))) → TAKE(inf(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__take(n__take(y0, y1), n__length(x0))) → TAKE(take(activate(y0), activate(y1)), length(activate(x0)))
TAKE(s(X), cons(Y, L)) → ACTIVATE(L)
ACTIVATE(n__take(n__length(y0), n__take(x0, x1))) → TAKE(length(activate(y0)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(y0, n__0)) → TAKE(y0, 0)
ACTIVATE(n__take(n__take(y0, y1), n__take(x0, x1))) → TAKE(take(activate(y0), activate(y1)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__take(y0, y1), x0)) → TAKE(take(activate(y0), activate(y1)), x0)
ACTIVATE(n__take(y0, n__length(x0))) → TAKE(y0, length(activate(x0)))
ACTIVATE(n__length(n__inf(x0))) → LENGTH(inf(activate(x0)))
ACTIVATE(n__take(y0, n__inf(x0))) → TAKE(y0, inf(activate(x0)))
ACTIVATE(n__take(y0, x0)) → TAKE(y0, x0)
ACTIVATE(n__length(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

eq(n__0, n__0) → true
eq(n__s(X), n__s(Y)) → eq(activate(X), activate(Y))
eq(X, Y) → false
inf(X) → cons(X, n__inf(n__s(X)))
take(0, X) → nil
take(s(X), cons(Y, L)) → cons(activate(Y), n__take(activate(X), activate(L)))
length(nil) → 0
length(cons(X, L)) → s(n__length(activate(L)))
0 → n__0
s(X) → n__s(X)
inf(X) → n__inf(X)
take(X1, X2) → n__take(X1, X2)
length(X) → n__length(X)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__inf(X)) → inf(activate(X))
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__length(X)) → length(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ACTIVATE(n__take(n__length(y0), n__0)) → TAKE(length(activate(y0)), 0) at position [1] we obtained the following new rules:

ACTIVATE(n__take(n__length(y0), n__0)) → TAKE(length(activate(y0)), n__0)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

          ↳ QDP

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

ACTIVATE(n__take(n__inf(y0), n__take(x0, x1))) → TAKE(inf(activate(y0)), take(activate(x0), activate(x1)))
ACTIVATE(n__length(n__take(x0, x1))) → LENGTH(take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__take(y0, y1), n__inf(x0))) → TAKE(take(activate(y0), activate(y1)), inf(activate(x0)))
ACTIVATE(n__take(n__inf(y0), n__length(x0))) → TAKE(inf(activate(y0)), length(activate(x0)))
ACTIVATE(n__take(n__length(y0), n__inf(x0))) → TAKE(length(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(n__take(y0, y1), n__0)) → TAKE(take(activate(y0), activate(y1)), 0)
ACTIVATE(n__take(n__s(y0), n__0)) → TAKE(s(y0), 0)
ACTIVATE(n__take(n__length(y0), n__length(x0))) → TAKE(length(activate(y0)), length(activate(x0)))
TAKE(s(X), cons(Y, L)) → ACTIVATE(Y)
ACTIVATE(n__take(n__inf(y0), x0)) → TAKE(inf(activate(y0)), x0)
ACTIVATE(n__take(n__inf(y0), n__s(x0))) → TAKE(inf(activate(y0)), s(x0))
ACTIVATE(n__take(n__s(y0), x0)) → TAKE(s(y0), x0)
ACTIVATE(n__take(n__s(y0), n__inf(x0))) → TAKE(s(y0), inf(activate(x0)))
ACTIVATE(n__take(n__s(y0), n__take(x0, x1))) → TAKE(s(y0), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__length(y0), x0)) → TAKE(length(activate(y0)), x0)
ACTIVATE(n__inf(X)) → ACTIVATE(X)
ACTIVATE(n__take(n__s(y0), n__s(x0))) → TAKE(s(y0), s(x0))
ACTIVATE(n__length(n__length(x0))) → LENGTH(length(activate(x0)))
ACTIVATE(n__take(n__inf(y0), n__0)) → TAKE(inf(activate(y0)), 0)
ACTIVATE(n__length(x0)) → LENGTH(x0)
ACTIVATE(n__take(n__length(y0), n__0)) → TAKE(length(activate(y0)), n__0)
LENGTH(cons(X, L)) → ACTIVATE(L)
TAKE(s(X), cons(Y, L)) → ACTIVATE(X)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(n__take(y0, y1), n__s(x0))) → TAKE(take(activate(y0), activate(y1)), s(x0))
ACTIVATE(n__take(n__s(y0), n__length(x0))) → TAKE(s(y0), length(activate(x0)))
ACTIVATE(n__take(y0, n__take(x0, x1))) → TAKE(y0, take(activate(x0), activate(x1)))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__take(n__inf(y0), n__inf(x0))) → TAKE(inf(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(y0, n__s(x0))) → TAKE(y0, s(x0))
ACTIVATE(n__take(n__length(y0), n__take(x0, x1))) → TAKE(length(activate(y0)), take(activate(x0), activate(x1)))
TAKE(s(X), cons(Y, L)) → ACTIVATE(L)
ACTIVATE(n__take(n__take(y0, y1), n__length(x0))) → TAKE(take(activate(y0), activate(y1)), length(activate(x0)))
ACTIVATE(n__take(y0, n__0)) → TAKE(y0, 0)
ACTIVATE(n__take(n__take(y0, y1), x0)) → TAKE(take(activate(y0), activate(y1)), x0)
ACTIVATE(n__take(n__take(y0, y1), n__take(x0, x1))) → TAKE(take(activate(y0), activate(y1)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(y0, n__length(x0))) → TAKE(y0, length(activate(x0)))
ACTIVATE(n__length(n__inf(x0))) → LENGTH(inf(activate(x0)))
ACTIVATE(n__take(y0, n__inf(x0))) → TAKE(y0, inf(activate(x0)))
ACTIVATE(n__take(y0, x0)) → TAKE(y0, x0)
ACTIVATE(n__length(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

eq(n__0, n__0) → true
eq(n__s(X), n__s(Y)) → eq(activate(X), activate(Y))
eq(X, Y) → false
inf(X) → cons(X, n__inf(n__s(X)))
take(0, X) → nil
take(s(X), cons(Y, L)) → cons(activate(Y), n__take(activate(X), activate(L)))
length(nil) → 0
length(cons(X, L)) → s(n__length(activate(L)))
0 → n__0
s(X) → n__s(X)
inf(X) → n__inf(X)
take(X1, X2) → n__take(X1, X2)
length(X) → n__length(X)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__inf(X)) → inf(activate(X))
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__length(X)) → length(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

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

          ↳ QDP

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

ACTIVATE(n__take(n__take(y0, y1), n__inf(x0))) → TAKE(take(activate(y0), activate(y1)), inf(activate(x0)))
ACTIVATE(n__length(n__take(x0, x1))) → LENGTH(take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__inf(y0), n__take(x0, x1))) → TAKE(inf(activate(y0)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__length(y0), n__inf(x0))) → TAKE(length(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(n__inf(y0), n__length(x0))) → TAKE(inf(activate(y0)), length(activate(x0)))
ACTIVATE(n__take(n__take(y0, y1), n__0)) → TAKE(take(activate(y0), activate(y1)), 0)
ACTIVATE(n__take(n__length(y0), n__length(x0))) → TAKE(length(activate(y0)), length(activate(x0)))
ACTIVATE(n__take(n__s(y0), n__0)) → TAKE(s(y0), 0)
TAKE(s(X), cons(Y, L)) → ACTIVATE(Y)
ACTIVATE(n__take(n__inf(y0), x0)) → TAKE(inf(activate(y0)), x0)
ACTIVATE(n__take(n__inf(y0), n__s(x0))) → TAKE(inf(activate(y0)), s(x0))
ACTIVATE(n__take(n__s(y0), x0)) → TAKE(s(y0), x0)
ACTIVATE(n__take(n__s(y0), n__inf(x0))) → TAKE(s(y0), inf(activate(x0)))
ACTIVATE(n__take(n__length(y0), x0)) → TAKE(length(activate(y0)), x0)
ACTIVATE(n__take(n__s(y0), n__take(x0, x1))) → TAKE(s(y0), take(activate(x0), activate(x1)))
ACTIVATE(n__inf(X)) → ACTIVATE(X)
ACTIVATE(n__length(n__length(x0))) → LENGTH(length(activate(x0)))
ACTIVATE(n__take(n__s(y0), n__s(x0))) → TAKE(s(y0), s(x0))
ACTIVATE(n__take(n__inf(y0), n__0)) → TAKE(inf(activate(y0)), 0)
ACTIVATE(n__length(x0)) → LENGTH(x0)
TAKE(s(X), cons(Y, L)) → ACTIVATE(X)
LENGTH(cons(X, L)) → ACTIVATE(L)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(n__take(y0, y1), n__s(x0))) → TAKE(take(activate(y0), activate(y1)), s(x0))
ACTIVATE(n__take(n__s(y0), n__length(x0))) → TAKE(s(y0), length(activate(x0)))
ACTIVATE(n__take(y0, n__take(x0, x1))) → TAKE(y0, take(activate(x0), activate(x1)))
ACTIVATE(n__take(y0, n__s(x0))) → TAKE(y0, s(x0))
ACTIVATE(n__take(n__inf(y0), n__inf(x0))) → TAKE(inf(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__take(n__take(y0, y1), n__length(x0))) → TAKE(take(activate(y0), activate(y1)), length(activate(x0)))
TAKE(s(X), cons(Y, L)) → ACTIVATE(L)
ACTIVATE(n__take(n__length(y0), n__take(x0, x1))) → TAKE(length(activate(y0)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(y0, n__0)) → TAKE(y0, 0)
ACTIVATE(n__take(n__take(y0, y1), n__take(x0, x1))) → TAKE(take(activate(y0), activate(y1)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__take(y0, y1), x0)) → TAKE(take(activate(y0), activate(y1)), x0)
ACTIVATE(n__take(y0, n__length(x0))) → TAKE(y0, length(activate(x0)))
ACTIVATE(n__length(n__inf(x0))) → LENGTH(inf(activate(x0)))
ACTIVATE(n__take(y0, n__inf(x0))) → TAKE(y0, inf(activate(x0)))
ACTIVATE(n__take(y0, x0)) → TAKE(y0, x0)
ACTIVATE(n__length(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

eq(n__0, n__0) → true
eq(n__s(X), n__s(Y)) → eq(activate(X), activate(Y))
eq(X, Y) → false
inf(X) → cons(X, n__inf(n__s(X)))
take(0, X) → nil
take(s(X), cons(Y, L)) → cons(activate(Y), n__take(activate(X), activate(L)))
length(nil) → 0
length(cons(X, L)) → s(n__length(activate(L)))
0 → n__0
s(X) → n__s(X)
inf(X) → n__inf(X)
take(X1, X2) → n__take(X1, X2)
length(X) → n__length(X)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__inf(X)) → inf(activate(X))
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__length(X)) → length(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ACTIVATE(n__take(n__s(y0), n__s(x0))) → TAKE(s(y0), s(x0)) at position [1] we obtained the following new rules:

ACTIVATE(n__take(n__s(y0), n__s(x0))) → TAKE(s(y0), n__s(x0))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

          ↳ QDP

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

ACTIVATE(n__take(n__inf(y0), n__take(x0, x1))) → TAKE(inf(activate(y0)), take(activate(x0), activate(x1)))
ACTIVATE(n__length(n__take(x0, x1))) → LENGTH(take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__take(y0, y1), n__inf(x0))) → TAKE(take(activate(y0), activate(y1)), inf(activate(x0)))
ACTIVATE(n__take(n__inf(y0), n__length(x0))) → TAKE(inf(activate(y0)), length(activate(x0)))
ACTIVATE(n__take(n__length(y0), n__inf(x0))) → TAKE(length(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(n__take(y0, y1), n__0)) → TAKE(take(activate(y0), activate(y1)), 0)
ACTIVATE(n__take(n__s(y0), n__0)) → TAKE(s(y0), 0)
ACTIVATE(n__take(n__length(y0), n__length(x0))) → TAKE(length(activate(y0)), length(activate(x0)))
TAKE(s(X), cons(Y, L)) → ACTIVATE(Y)
ACTIVATE(n__take(n__inf(y0), x0)) → TAKE(inf(activate(y0)), x0)
ACTIVATE(n__take(n__inf(y0), n__s(x0))) → TAKE(inf(activate(y0)), s(x0))
ACTIVATE(n__take(n__s(y0), x0)) → TAKE(s(y0), x0)
ACTIVATE(n__take(n__s(y0), n__inf(x0))) → TAKE(s(y0), inf(activate(x0)))
ACTIVATE(n__take(n__s(y0), n__take(x0, x1))) → TAKE(s(y0), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__length(y0), x0)) → TAKE(length(activate(y0)), x0)
ACTIVATE(n__inf(X)) → ACTIVATE(X)
ACTIVATE(n__length(n__length(x0))) → LENGTH(length(activate(x0)))
ACTIVATE(n__take(n__inf(y0), n__0)) → TAKE(inf(activate(y0)), 0)
ACTIVATE(n__length(x0)) → LENGTH(x0)
LENGTH(cons(X, L)) → ACTIVATE(L)
TAKE(s(X), cons(Y, L)) → ACTIVATE(X)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(n__take(y0, y1), n__s(x0))) → TAKE(take(activate(y0), activate(y1)), s(x0))
ACTIVATE(n__take(n__s(y0), n__length(x0))) → TAKE(s(y0), length(activate(x0)))
ACTIVATE(n__take(y0, n__take(x0, x1))) → TAKE(y0, take(activate(x0), activate(x1)))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__take(n__inf(y0), n__inf(x0))) → TAKE(inf(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(y0, n__s(x0))) → TAKE(y0, s(x0))
ACTIVATE(n__take(n__length(y0), n__take(x0, x1))) → TAKE(length(activate(y0)), take(activate(x0), activate(x1)))
TAKE(s(X), cons(Y, L)) → ACTIVATE(L)
ACTIVATE(n__take(n__take(y0, y1), n__length(x0))) → TAKE(take(activate(y0), activate(y1)), length(activate(x0)))
ACTIVATE(n__take(y0, n__0)) → TAKE(y0, 0)
ACTIVATE(n__take(n__take(y0, y1), x0)) → TAKE(take(activate(y0), activate(y1)), x0)
ACTIVATE(n__take(n__take(y0, y1), n__take(x0, x1))) → TAKE(take(activate(y0), activate(y1)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__s(y0), n__s(x0))) → TAKE(s(y0), n__s(x0))
ACTIVATE(n__take(y0, n__length(x0))) → TAKE(y0, length(activate(x0)))
ACTIVATE(n__length(n__inf(x0))) → LENGTH(inf(activate(x0)))
ACTIVATE(n__take(y0, n__inf(x0))) → TAKE(y0, inf(activate(x0)))
ACTIVATE(n__take(y0, x0)) → TAKE(y0, x0)
ACTIVATE(n__length(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

eq(n__0, n__0) → true
eq(n__s(X), n__s(Y)) → eq(activate(X), activate(Y))
eq(X, Y) → false
inf(X) → cons(X, n__inf(n__s(X)))
take(0, X) → nil
take(s(X), cons(Y, L)) → cons(activate(Y), n__take(activate(X), activate(L)))
length(nil) → 0
length(cons(X, L)) → s(n__length(activate(L)))
0 → n__0
s(X) → n__s(X)
inf(X) → n__inf(X)
take(X1, X2) → n__take(X1, X2)
length(X) → n__length(X)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__inf(X)) → inf(activate(X))
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__length(X)) → length(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

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

          ↳ QDP

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

ACTIVATE(n__take(n__take(y0, y1), n__inf(x0))) → TAKE(take(activate(y0), activate(y1)), inf(activate(x0)))
ACTIVATE(n__length(n__take(x0, x1))) → LENGTH(take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__inf(y0), n__take(x0, x1))) → TAKE(inf(activate(y0)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__length(y0), n__inf(x0))) → TAKE(length(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(n__inf(y0), n__length(x0))) → TAKE(inf(activate(y0)), length(activate(x0)))
ACTIVATE(n__take(n__take(y0, y1), n__0)) → TAKE(take(activate(y0), activate(y1)), 0)
ACTIVATE(n__take(n__length(y0), n__length(x0))) → TAKE(length(activate(y0)), length(activate(x0)))
ACTIVATE(n__take(n__s(y0), n__0)) → TAKE(s(y0), 0)
TAKE(s(X), cons(Y, L)) → ACTIVATE(Y)
ACTIVATE(n__take(n__inf(y0), x0)) → TAKE(inf(activate(y0)), x0)
ACTIVATE(n__take(n__inf(y0), n__s(x0))) → TAKE(inf(activate(y0)), s(x0))
ACTIVATE(n__take(n__s(y0), x0)) → TAKE(s(y0), x0)
ACTIVATE(n__take(n__s(y0), n__inf(x0))) → TAKE(s(y0), inf(activate(x0)))
ACTIVATE(n__take(n__length(y0), x0)) → TAKE(length(activate(y0)), x0)
ACTIVATE(n__take(n__s(y0), n__take(x0, x1))) → TAKE(s(y0), take(activate(x0), activate(x1)))
ACTIVATE(n__inf(X)) → ACTIVATE(X)
ACTIVATE(n__length(n__length(x0))) → LENGTH(length(activate(x0)))
ACTIVATE(n__take(n__inf(y0), n__0)) → TAKE(inf(activate(y0)), 0)
ACTIVATE(n__length(x0)) → LENGTH(x0)
TAKE(s(X), cons(Y, L)) → ACTIVATE(X)
LENGTH(cons(X, L)) → ACTIVATE(L)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(n__take(y0, y1), n__s(x0))) → TAKE(take(activate(y0), activate(y1)), s(x0))
ACTIVATE(n__take(n__s(y0), n__length(x0))) → TAKE(s(y0), length(activate(x0)))
ACTIVATE(n__take(y0, n__take(x0, x1))) → TAKE(y0, take(activate(x0), activate(x1)))
ACTIVATE(n__take(y0, n__s(x0))) → TAKE(y0, s(x0))
ACTIVATE(n__take(n__inf(y0), n__inf(x0))) → TAKE(inf(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__take(n__take(y0, y1), n__length(x0))) → TAKE(take(activate(y0), activate(y1)), length(activate(x0)))
TAKE(s(X), cons(Y, L)) → ACTIVATE(L)
ACTIVATE(n__take(n__length(y0), n__take(x0, x1))) → TAKE(length(activate(y0)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(y0, n__0)) → TAKE(y0, 0)
ACTIVATE(n__take(n__take(y0, y1), n__take(x0, x1))) → TAKE(take(activate(y0), activate(y1)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__take(y0, y1), x0)) → TAKE(take(activate(y0), activate(y1)), x0)
ACTIVATE(n__take(y0, n__length(x0))) → TAKE(y0, length(activate(x0)))
ACTIVATE(n__length(n__inf(x0))) → LENGTH(inf(activate(x0)))
ACTIVATE(n__take(y0, n__inf(x0))) → TAKE(y0, inf(activate(x0)))
ACTIVATE(n__take(y0, x0)) → TAKE(y0, x0)
ACTIVATE(n__length(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

eq(n__0, n__0) → true
eq(n__s(X), n__s(Y)) → eq(activate(X), activate(Y))
eq(X, Y) → false
inf(X) → cons(X, n__inf(n__s(X)))
take(0, X) → nil
take(s(X), cons(Y, L)) → cons(activate(Y), n__take(activate(X), activate(L)))
length(nil) → 0
length(cons(X, L)) → s(n__length(activate(L)))
0 → n__0
s(X) → n__s(X)
inf(X) → n__inf(X)
take(X1, X2) → n__take(X1, X2)
length(X) → n__length(X)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__inf(X)) → inf(activate(X))
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__length(X)) → length(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ACTIVATE(n__take(n__s(y0), n__0)) → TAKE(s(y0), 0) at position [1] we obtained the following new rules:

ACTIVATE(n__take(n__s(y0), n__0)) → TAKE(s(y0), n__0)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

          ↳ QDP

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

ACTIVATE(n__take(n__inf(y0), n__take(x0, x1))) → TAKE(inf(activate(y0)), take(activate(x0), activate(x1)))
ACTIVATE(n__length(n__take(x0, x1))) → LENGTH(take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__take(y0, y1), n__inf(x0))) → TAKE(take(activate(y0), activate(y1)), inf(activate(x0)))
ACTIVATE(n__take(n__inf(y0), n__length(x0))) → TAKE(inf(activate(y0)), length(activate(x0)))
ACTIVATE(n__take(n__length(y0), n__inf(x0))) → TAKE(length(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(n__s(y0), n__0)) → TAKE(s(y0), n__0)
ACTIVATE(n__take(n__take(y0, y1), n__0)) → TAKE(take(activate(y0), activate(y1)), 0)
ACTIVATE(n__take(n__length(y0), n__length(x0))) → TAKE(length(activate(y0)), length(activate(x0)))
TAKE(s(X), cons(Y, L)) → ACTIVATE(Y)
ACTIVATE(n__take(n__inf(y0), x0)) → TAKE(inf(activate(y0)), x0)
ACTIVATE(n__take(n__inf(y0), n__s(x0))) → TAKE(inf(activate(y0)), s(x0))
ACTIVATE(n__take(n__s(y0), x0)) → TAKE(s(y0), x0)
ACTIVATE(n__take(n__s(y0), n__inf(x0))) → TAKE(s(y0), inf(activate(x0)))
ACTIVATE(n__take(n__s(y0), n__take(x0, x1))) → TAKE(s(y0), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__length(y0), x0)) → TAKE(length(activate(y0)), x0)
ACTIVATE(n__inf(X)) → ACTIVATE(X)
ACTIVATE(n__length(n__length(x0))) → LENGTH(length(activate(x0)))
ACTIVATE(n__take(n__inf(y0), n__0)) → TAKE(inf(activate(y0)), 0)
ACTIVATE(n__length(x0)) → LENGTH(x0)
LENGTH(cons(X, L)) → ACTIVATE(L)
TAKE(s(X), cons(Y, L)) → ACTIVATE(X)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(n__take(y0, y1), n__s(x0))) → TAKE(take(activate(y0), activate(y1)), s(x0))
ACTIVATE(n__take(n__s(y0), n__length(x0))) → TAKE(s(y0), length(activate(x0)))
ACTIVATE(n__take(y0, n__take(x0, x1))) → TAKE(y0, take(activate(x0), activate(x1)))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__take(n__inf(y0), n__inf(x0))) → TAKE(inf(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(y0, n__s(x0))) → TAKE(y0, s(x0))
ACTIVATE(n__take(n__length(y0), n__take(x0, x1))) → TAKE(length(activate(y0)), take(activate(x0), activate(x1)))
TAKE(s(X), cons(Y, L)) → ACTIVATE(L)
ACTIVATE(n__take(n__take(y0, y1), n__length(x0))) → TAKE(take(activate(y0), activate(y1)), length(activate(x0)))
ACTIVATE(n__take(y0, n__0)) → TAKE(y0, 0)
ACTIVATE(n__take(n__take(y0, y1), x0)) → TAKE(take(activate(y0), activate(y1)), x0)
ACTIVATE(n__take(n__take(y0, y1), n__take(x0, x1))) → TAKE(take(activate(y0), activate(y1)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(y0, n__length(x0))) → TAKE(y0, length(activate(x0)))
ACTIVATE(n__length(n__inf(x0))) → LENGTH(inf(activate(x0)))
ACTIVATE(n__take(y0, n__inf(x0))) → TAKE(y0, inf(activate(x0)))
ACTIVATE(n__take(y0, x0)) → TAKE(y0, x0)
ACTIVATE(n__length(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

eq(n__0, n__0) → true
eq(n__s(X), n__s(Y)) → eq(activate(X), activate(Y))
eq(X, Y) → false
inf(X) → cons(X, n__inf(n__s(X)))
take(0, X) → nil
take(s(X), cons(Y, L)) → cons(activate(Y), n__take(activate(X), activate(L)))
length(nil) → 0
length(cons(X, L)) → s(n__length(activate(L)))
0 → n__0
s(X) → n__s(X)
inf(X) → n__inf(X)
take(X1, X2) → n__take(X1, X2)
length(X) → n__length(X)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__inf(X)) → inf(activate(X))
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__length(X)) → length(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

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

          ↳ QDP

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

ACTIVATE(n__take(n__take(y0, y1), n__inf(x0))) → TAKE(take(activate(y0), activate(y1)), inf(activate(x0)))
ACTIVATE(n__length(n__take(x0, x1))) → LENGTH(take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__inf(y0), n__take(x0, x1))) → TAKE(inf(activate(y0)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__length(y0), n__inf(x0))) → TAKE(length(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(n__inf(y0), n__length(x0))) → TAKE(inf(activate(y0)), length(activate(x0)))
ACTIVATE(n__take(n__take(y0, y1), n__0)) → TAKE(take(activate(y0), activate(y1)), 0)
ACTIVATE(n__take(n__length(y0), n__length(x0))) → TAKE(length(activate(y0)), length(activate(x0)))
TAKE(s(X), cons(Y, L)) → ACTIVATE(Y)
ACTIVATE(n__take(n__inf(y0), x0)) → TAKE(inf(activate(y0)), x0)
ACTIVATE(n__take(n__inf(y0), n__s(x0))) → TAKE(inf(activate(y0)), s(x0))
ACTIVATE(n__take(n__s(y0), x0)) → TAKE(s(y0), x0)
ACTIVATE(n__take(n__s(y0), n__inf(x0))) → TAKE(s(y0), inf(activate(x0)))
ACTIVATE(n__take(n__length(y0), x0)) → TAKE(length(activate(y0)), x0)
ACTIVATE(n__take(n__s(y0), n__take(x0, x1))) → TAKE(s(y0), take(activate(x0), activate(x1)))
ACTIVATE(n__inf(X)) → ACTIVATE(X)
ACTIVATE(n__length(n__length(x0))) → LENGTH(length(activate(x0)))
ACTIVATE(n__take(n__inf(y0), n__0)) → TAKE(inf(activate(y0)), 0)
ACTIVATE(n__length(x0)) → LENGTH(x0)
TAKE(s(X), cons(Y, L)) → ACTIVATE(X)
LENGTH(cons(X, L)) → ACTIVATE(L)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(n__take(y0, y1), n__s(x0))) → TAKE(take(activate(y0), activate(y1)), s(x0))
ACTIVATE(n__take(n__s(y0), n__length(x0))) → TAKE(s(y0), length(activate(x0)))
ACTIVATE(n__take(y0, n__take(x0, x1))) → TAKE(y0, take(activate(x0), activate(x1)))
ACTIVATE(n__take(y0, n__s(x0))) → TAKE(y0, s(x0))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__take(n__inf(y0), n__inf(x0))) → TAKE(inf(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(n__take(y0, y1), n__length(x0))) → TAKE(take(activate(y0), activate(y1)), length(activate(x0)))
TAKE(s(X), cons(Y, L)) → ACTIVATE(L)
ACTIVATE(n__take(n__length(y0), n__take(x0, x1))) → TAKE(length(activate(y0)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(y0, n__0)) → TAKE(y0, 0)
ACTIVATE(n__take(n__take(y0, y1), n__take(x0, x1))) → TAKE(take(activate(y0), activate(y1)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__take(y0, y1), x0)) → TAKE(take(activate(y0), activate(y1)), x0)
ACTIVATE(n__take(y0, n__length(x0))) → TAKE(y0, length(activate(x0)))
ACTIVATE(n__length(n__inf(x0))) → LENGTH(inf(activate(x0)))
ACTIVATE(n__take(y0, n__inf(x0))) → TAKE(y0, inf(activate(x0)))
ACTIVATE(n__take(y0, x0)) → TAKE(y0, x0)
ACTIVATE(n__length(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

eq(n__0, n__0) → true
eq(n__s(X), n__s(Y)) → eq(activate(X), activate(Y))
eq(X, Y) → false
inf(X) → cons(X, n__inf(n__s(X)))
take(0, X) → nil
take(s(X), cons(Y, L)) → cons(activate(Y), n__take(activate(X), activate(L)))
length(nil) → 0
length(cons(X, L)) → s(n__length(activate(L)))
0 → n__0
s(X) → n__s(X)
inf(X) → n__inf(X)
take(X1, X2) → n__take(X1, X2)
length(X) → n__length(X)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__inf(X)) → inf(activate(X))
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__length(X)) → length(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ACTIVATE(n__take(n__inf(y0), n__0)) → TAKE(inf(activate(y0)), 0) at position [1] we obtained the following new rules:

ACTIVATE(n__take(n__inf(y0), n__0)) → TAKE(inf(activate(y0)), n__0)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

          ↳ QDP

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

ACTIVATE(n__take(n__inf(y0), n__take(x0, x1))) → TAKE(inf(activate(y0)), take(activate(x0), activate(x1)))
ACTIVATE(n__length(n__take(x0, x1))) → LENGTH(take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__take(y0, y1), n__inf(x0))) → TAKE(take(activate(y0), activate(y1)), inf(activate(x0)))
ACTIVATE(n__take(n__inf(y0), n__length(x0))) → TAKE(inf(activate(y0)), length(activate(x0)))
ACTIVATE(n__take(n__length(y0), n__inf(x0))) → TAKE(length(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(n__take(y0, y1), n__0)) → TAKE(take(activate(y0), activate(y1)), 0)
ACTIVATE(n__take(n__length(y0), n__length(x0))) → TAKE(length(activate(y0)), length(activate(x0)))
TAKE(s(X), cons(Y, L)) → ACTIVATE(Y)
ACTIVATE(n__take(n__inf(y0), x0)) → TAKE(inf(activate(y0)), x0)
ACTIVATE(n__take(n__inf(y0), n__0)) → TAKE(inf(activate(y0)), n__0)
ACTIVATE(n__take(n__inf(y0), n__s(x0))) → TAKE(inf(activate(y0)), s(x0))
ACTIVATE(n__take(n__s(y0), x0)) → TAKE(s(y0), x0)
ACTIVATE(n__take(n__s(y0), n__inf(x0))) → TAKE(s(y0), inf(activate(x0)))
ACTIVATE(n__take(n__s(y0), n__take(x0, x1))) → TAKE(s(y0), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__length(y0), x0)) → TAKE(length(activate(y0)), x0)
ACTIVATE(n__inf(X)) → ACTIVATE(X)
ACTIVATE(n__length(n__length(x0))) → LENGTH(length(activate(x0)))
ACTIVATE(n__length(x0)) → LENGTH(x0)
LENGTH(cons(X, L)) → ACTIVATE(L)
TAKE(s(X), cons(Y, L)) → ACTIVATE(X)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(n__take(y0, y1), n__s(x0))) → TAKE(take(activate(y0), activate(y1)), s(x0))
ACTIVATE(n__take(n__s(y0), n__length(x0))) → TAKE(s(y0), length(activate(x0)))
ACTIVATE(n__take(y0, n__take(x0, x1))) → TAKE(y0, take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__inf(y0), n__inf(x0))) → TAKE(inf(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__take(y0, n__s(x0))) → TAKE(y0, s(x0))
ACTIVATE(n__take(n__length(y0), n__take(x0, x1))) → TAKE(length(activate(y0)), take(activate(x0), activate(x1)))
TAKE(s(X), cons(Y, L)) → ACTIVATE(L)
ACTIVATE(n__take(n__take(y0, y1), n__length(x0))) → TAKE(take(activate(y0), activate(y1)), length(activate(x0)))
ACTIVATE(n__take(y0, n__0)) → TAKE(y0, 0)
ACTIVATE(n__take(n__take(y0, y1), x0)) → TAKE(take(activate(y0), activate(y1)), x0)
ACTIVATE(n__take(n__take(y0, y1), n__take(x0, x1))) → TAKE(take(activate(y0), activate(y1)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(y0, n__length(x0))) → TAKE(y0, length(activate(x0)))
ACTIVATE(n__length(n__inf(x0))) → LENGTH(inf(activate(x0)))
ACTIVATE(n__take(y0, n__inf(x0))) → TAKE(y0, inf(activate(x0)))
ACTIVATE(n__take(y0, x0)) → TAKE(y0, x0)
ACTIVATE(n__length(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

eq(n__0, n__0) → true
eq(n__s(X), n__s(Y)) → eq(activate(X), activate(Y))
eq(X, Y) → false
inf(X) → cons(X, n__inf(n__s(X)))
take(0, X) → nil
take(s(X), cons(Y, L)) → cons(activate(Y), n__take(activate(X), activate(L)))
length(nil) → 0
length(cons(X, L)) → s(n__length(activate(L)))
0 → n__0
s(X) → n__s(X)
inf(X) → n__inf(X)
take(X1, X2) → n__take(X1, X2)
length(X) → n__length(X)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__inf(X)) → inf(activate(X))
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__length(X)) → length(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

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Narrowing

          ↳ QDP

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

ACTIVATE(n__take(n__take(y0, y1), n__inf(x0))) → TAKE(take(activate(y0), activate(y1)), inf(activate(x0)))
ACTIVATE(n__length(n__take(x0, x1))) → LENGTH(take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__inf(y0), n__take(x0, x1))) → TAKE(inf(activate(y0)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__length(y0), n__inf(x0))) → TAKE(length(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(n__inf(y0), n__length(x0))) → TAKE(inf(activate(y0)), length(activate(x0)))
ACTIVATE(n__take(n__take(y0, y1), n__0)) → TAKE(take(activate(y0), activate(y1)), 0)
ACTIVATE(n__take(n__length(y0), n__length(x0))) → TAKE(length(activate(y0)), length(activate(x0)))
TAKE(s(X), cons(Y, L)) → ACTIVATE(Y)
ACTIVATE(n__take(n__inf(y0), x0)) → TAKE(inf(activate(y0)), x0)
ACTIVATE(n__take(n__inf(y0), n__s(x0))) → TAKE(inf(activate(y0)), s(x0))
ACTIVATE(n__take(n__s(y0), x0)) → TAKE(s(y0), x0)
ACTIVATE(n__take(n__s(y0), n__inf(x0))) → TAKE(s(y0), inf(activate(x0)))
ACTIVATE(n__take(n__length(y0), x0)) → TAKE(length(activate(y0)), x0)
ACTIVATE(n__take(n__s(y0), n__take(x0, x1))) → TAKE(s(y0), take(activate(x0), activate(x1)))
ACTIVATE(n__inf(X)) → ACTIVATE(X)
ACTIVATE(n__length(n__length(x0))) → LENGTH(length(activate(x0)))
ACTIVATE(n__length(x0)) → LENGTH(x0)
TAKE(s(X), cons(Y, L)) → ACTIVATE(X)
LENGTH(cons(X, L)) → ACTIVATE(L)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(n__take(y0, y1), n__s(x0))) → TAKE(take(activate(y0), activate(y1)), s(x0))
ACTIVATE(n__take(n__s(y0), n__length(x0))) → TAKE(s(y0), length(activate(x0)))
ACTIVATE(n__take(y0, n__take(x0, x1))) → TAKE(y0, take(activate(x0), activate(x1)))
ACTIVATE(n__take(y0, n__s(x0))) → TAKE(y0, s(x0))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__take(n__inf(y0), n__inf(x0))) → TAKE(inf(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(n__take(y0, y1), n__length(x0))) → TAKE(take(activate(y0), activate(y1)), length(activate(x0)))
TAKE(s(X), cons(Y, L)) → ACTIVATE(L)
ACTIVATE(n__take(n__length(y0), n__take(x0, x1))) → TAKE(length(activate(y0)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(y0, n__0)) → TAKE(y0, 0)
ACTIVATE(n__take(n__take(y0, y1), n__take(x0, x1))) → TAKE(take(activate(y0), activate(y1)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__take(y0, y1), x0)) → TAKE(take(activate(y0), activate(y1)), x0)
ACTIVATE(n__take(y0, n__length(x0))) → TAKE(y0, length(activate(x0)))
ACTIVATE(n__length(n__inf(x0))) → LENGTH(inf(activate(x0)))
ACTIVATE(n__take(y0, n__inf(x0))) → TAKE(y0, inf(activate(x0)))
ACTIVATE(n__take(y0, x0)) → TAKE(y0, x0)
ACTIVATE(n__length(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

eq(n__0, n__0) → true
eq(n__s(X), n__s(Y)) → eq(activate(X), activate(Y))
eq(X, Y) → false
inf(X) → cons(X, n__inf(n__s(X)))
take(0, X) → nil
take(s(X), cons(Y, L)) → cons(activate(Y), n__take(activate(X), activate(L)))
length(nil) → 0
length(cons(X, L)) → s(n__length(activate(L)))
0 → n__0
s(X) → n__s(X)
inf(X) → n__inf(X)
take(X1, X2) → n__take(X1, X2)
length(X) → n__length(X)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__inf(X)) → inf(activate(X))
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__length(X)) → length(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ACTIVATE(n__take(n__inf(y0), n__s(x0))) → TAKE(inf(activate(y0)), s(x0)) at position [1] we obtained the following new rules:

ACTIVATE(n__take(n__inf(y0), n__s(x0))) → TAKE(inf(activate(y0)), n__s(x0))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Narrowing

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

          ↳ QDP

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

ACTIVATE(n__take(n__inf(y0), n__take(x0, x1))) → TAKE(inf(activate(y0)), take(activate(x0), activate(x1)))
ACTIVATE(n__length(n__take(x0, x1))) → LENGTH(take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__take(y0, y1), n__inf(x0))) → TAKE(take(activate(y0), activate(y1)), inf(activate(x0)))
ACTIVATE(n__take(n__inf(y0), n__length(x0))) → TAKE(inf(activate(y0)), length(activate(x0)))
ACTIVATE(n__take(n__length(y0), n__inf(x0))) → TAKE(length(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(n__take(y0, y1), n__0)) → TAKE(take(activate(y0), activate(y1)), 0)
ACTIVATE(n__take(n__length(y0), n__length(x0))) → TAKE(length(activate(y0)), length(activate(x0)))
TAKE(s(X), cons(Y, L)) → ACTIVATE(Y)
ACTIVATE(n__take(n__inf(y0), x0)) → TAKE(inf(activate(y0)), x0)
ACTIVATE(n__take(n__inf(y0), n__s(x0))) → TAKE(inf(activate(y0)), n__s(x0))
ACTIVATE(n__take(n__s(y0), x0)) → TAKE(s(y0), x0)
ACTIVATE(n__take(n__s(y0), n__inf(x0))) → TAKE(s(y0), inf(activate(x0)))
ACTIVATE(n__take(n__s(y0), n__take(x0, x1))) → TAKE(s(y0), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__length(y0), x0)) → TAKE(length(activate(y0)), x0)
ACTIVATE(n__inf(X)) → ACTIVATE(X)
ACTIVATE(n__length(n__length(x0))) → LENGTH(length(activate(x0)))
ACTIVATE(n__length(x0)) → LENGTH(x0)
LENGTH(cons(X, L)) → ACTIVATE(L)
TAKE(s(X), cons(Y, L)) → ACTIVATE(X)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(n__take(y0, y1), n__s(x0))) → TAKE(take(activate(y0), activate(y1)), s(x0))
ACTIVATE(n__take(n__s(y0), n__length(x0))) → TAKE(s(y0), length(activate(x0)))
ACTIVATE(n__take(y0, n__take(x0, x1))) → TAKE(y0, take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__inf(y0), n__inf(x0))) → TAKE(inf(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__take(y0, n__s(x0))) → TAKE(y0, s(x0))
ACTIVATE(n__take(n__length(y0), n__take(x0, x1))) → TAKE(length(activate(y0)), take(activate(x0), activate(x1)))
TAKE(s(X), cons(Y, L)) → ACTIVATE(L)
ACTIVATE(n__take(n__take(y0, y1), n__length(x0))) → TAKE(take(activate(y0), activate(y1)), length(activate(x0)))
ACTIVATE(n__take(y0, n__0)) → TAKE(y0, 0)
ACTIVATE(n__take(n__take(y0, y1), x0)) → TAKE(take(activate(y0), activate(y1)), x0)
ACTIVATE(n__take(n__take(y0, y1), n__take(x0, x1))) → TAKE(take(activate(y0), activate(y1)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(y0, n__length(x0))) → TAKE(y0, length(activate(x0)))
ACTIVATE(n__length(n__inf(x0))) → LENGTH(inf(activate(x0)))
ACTIVATE(n__take(y0, n__inf(x0))) → TAKE(y0, inf(activate(x0)))
ACTIVATE(n__take(y0, x0)) → TAKE(y0, x0)
ACTIVATE(n__length(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

eq(n__0, n__0) → true
eq(n__s(X), n__s(Y)) → eq(activate(X), activate(Y))
eq(X, Y) → false
inf(X) → cons(X, n__inf(n__s(X)))
take(0, X) → nil
take(s(X), cons(Y, L)) → cons(activate(Y), n__take(activate(X), activate(L)))
length(nil) → 0
length(cons(X, L)) → s(n__length(activate(L)))
0 → n__0
s(X) → n__s(X)
inf(X) → n__inf(X)
take(X1, X2) → n__take(X1, X2)
length(X) → n__length(X)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__inf(X)) → inf(activate(X))
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__length(X)) → length(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

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Narrowing

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Narrowing

          ↳ QDP

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

ACTIVATE(n__length(n__take(x0, x1))) → LENGTH(take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__inf(y0), n__take(x0, x1))) → TAKE(inf(activate(y0)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__take(y0, y1), n__inf(x0))) → TAKE(take(activate(y0), activate(y1)), inf(activate(x0)))
ACTIVATE(n__take(n__length(y0), n__inf(x0))) → TAKE(length(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(n__inf(y0), n__length(x0))) → TAKE(inf(activate(y0)), length(activate(x0)))
ACTIVATE(n__take(n__take(y0, y1), n__0)) → TAKE(take(activate(y0), activate(y1)), 0)
ACTIVATE(n__take(n__length(y0), n__length(x0))) → TAKE(length(activate(y0)), length(activate(x0)))
TAKE(s(X), cons(Y, L)) → ACTIVATE(Y)
ACTIVATE(n__take(n__inf(y0), x0)) → TAKE(inf(activate(y0)), x0)
ACTIVATE(n__take(n__s(y0), x0)) → TAKE(s(y0), x0)
ACTIVATE(n__take(n__s(y0), n__inf(x0))) → TAKE(s(y0), inf(activate(x0)))
ACTIVATE(n__take(n__length(y0), x0)) → TAKE(length(activate(y0)), x0)
ACTIVATE(n__take(n__s(y0), n__take(x0, x1))) → TAKE(s(y0), take(activate(x0), activate(x1)))
ACTIVATE(n__inf(X)) → ACTIVATE(X)
ACTIVATE(n__length(n__length(x0))) → LENGTH(length(activate(x0)))
ACTIVATE(n__length(x0)) → LENGTH(x0)
TAKE(s(X), cons(Y, L)) → ACTIVATE(X)
LENGTH(cons(X, L)) → ACTIVATE(L)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(n__take(y0, y1), n__s(x0))) → TAKE(take(activate(y0), activate(y1)), s(x0))
ACTIVATE(n__take(n__s(y0), n__length(x0))) → TAKE(s(y0), length(activate(x0)))
ACTIVATE(n__take(y0, n__take(x0, x1))) → TAKE(y0, take(activate(x0), activate(x1)))
ACTIVATE(n__take(y0, n__s(x0))) → TAKE(y0, s(x0))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__take(n__inf(y0), n__inf(x0))) → TAKE(inf(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(n__take(y0, y1), n__length(x0))) → TAKE(take(activate(y0), activate(y1)), length(activate(x0)))
TAKE(s(X), cons(Y, L)) → ACTIVATE(L)
ACTIVATE(n__take(n__length(y0), n__take(x0, x1))) → TAKE(length(activate(y0)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(y0, n__0)) → TAKE(y0, 0)
ACTIVATE(n__take(n__take(y0, y1), n__take(x0, x1))) → TAKE(take(activate(y0), activate(y1)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__take(y0, y1), x0)) → TAKE(take(activate(y0), activate(y1)), x0)
ACTIVATE(n__take(y0, n__length(x0))) → TAKE(y0, length(activate(x0)))
ACTIVATE(n__length(n__inf(x0))) → LENGTH(inf(activate(x0)))
ACTIVATE(n__take(y0, n__inf(x0))) → TAKE(y0, inf(activate(x0)))
ACTIVATE(n__take(y0, x0)) → TAKE(y0, x0)
ACTIVATE(n__length(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

eq(n__0, n__0) → true
eq(n__s(X), n__s(Y)) → eq(activate(X), activate(Y))
eq(X, Y) → false
inf(X) → cons(X, n__inf(n__s(X)))
take(0, X) → nil
take(s(X), cons(Y, L)) → cons(activate(Y), n__take(activate(X), activate(L)))
length(nil) → 0
length(cons(X, L)) → s(n__length(activate(L)))
0 → n__0
s(X) → n__s(X)
inf(X) → n__inf(X)
take(X1, X2) → n__take(X1, X2)
length(X) → n__length(X)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__inf(X)) → inf(activate(X))
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__length(X)) → length(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ACTIVATE(n__take(n__take(y0, y1), n__0)) → TAKE(take(activate(y0), activate(y1)), 0) at position [1] we obtained the following new rules:

ACTIVATE(n__take(n__take(y0, y1), n__0)) → TAKE(take(activate(y0), activate(y1)), n__0)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Narrowing

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Narrowing

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

          ↳ QDP

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

ACTIVATE(n__take(n__take(y0, y1), n__inf(x0))) → TAKE(take(activate(y0), activate(y1)), inf(activate(x0)))
ACTIVATE(n__take(n__inf(y0), n__take(x0, x1))) → TAKE(inf(activate(y0)), take(activate(x0), activate(x1)))
ACTIVATE(n__length(n__take(x0, x1))) → LENGTH(take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__inf(y0), n__length(x0))) → TAKE(inf(activate(y0)), length(activate(x0)))
ACTIVATE(n__take(n__length(y0), n__inf(x0))) → TAKE(length(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(n__length(y0), n__length(x0))) → TAKE(length(activate(y0)), length(activate(x0)))
TAKE(s(X), cons(Y, L)) → ACTIVATE(Y)
ACTIVATE(n__take(n__inf(y0), x0)) → TAKE(inf(activate(y0)), x0)
ACTIVATE(n__take(n__s(y0), x0)) → TAKE(s(y0), x0)
ACTIVATE(n__take(n__s(y0), n__inf(x0))) → TAKE(s(y0), inf(activate(x0)))
ACTIVATE(n__take(n__s(y0), n__take(x0, x1))) → TAKE(s(y0), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__length(y0), x0)) → TAKE(length(activate(y0)), x0)
ACTIVATE(n__inf(X)) → ACTIVATE(X)
ACTIVATE(n__length(n__length(x0))) → LENGTH(length(activate(x0)))
ACTIVATE(n__length(x0)) → LENGTH(x0)
LENGTH(cons(X, L)) → ACTIVATE(L)
TAKE(s(X), cons(Y, L)) → ACTIVATE(X)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(n__take(y0, y1), n__s(x0))) → TAKE(take(activate(y0), activate(y1)), s(x0))
ACTIVATE(n__take(n__s(y0), n__length(x0))) → TAKE(s(y0), length(activate(x0)))
ACTIVATE(n__take(y0, n__take(x0, x1))) → TAKE(y0, take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__inf(y0), n__inf(x0))) → TAKE(inf(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__take(y0, n__s(x0))) → TAKE(y0, s(x0))
ACTIVATE(n__take(n__length(y0), n__take(x0, x1))) → TAKE(length(activate(y0)), take(activate(x0), activate(x1)))
TAKE(s(X), cons(Y, L)) → ACTIVATE(L)
ACTIVATE(n__take(n__take(y0, y1), n__length(x0))) → TAKE(take(activate(y0), activate(y1)), length(activate(x0)))
ACTIVATE(n__take(y0, n__0)) → TAKE(y0, 0)
ACTIVATE(n__take(n__take(y0, y1), n__0)) → TAKE(take(activate(y0), activate(y1)), n__0)
ACTIVATE(n__take(n__take(y0, y1), x0)) → TAKE(take(activate(y0), activate(y1)), x0)
ACTIVATE(n__take(n__take(y0, y1), n__take(x0, x1))) → TAKE(take(activate(y0), activate(y1)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(y0, n__length(x0))) → TAKE(y0, length(activate(x0)))
ACTIVATE(n__length(n__inf(x0))) → LENGTH(inf(activate(x0)))
ACTIVATE(n__take(y0, n__inf(x0))) → TAKE(y0, inf(activate(x0)))
ACTIVATE(n__take(y0, x0)) → TAKE(y0, x0)
ACTIVATE(n__length(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

eq(n__0, n__0) → true
eq(n__s(X), n__s(Y)) → eq(activate(X), activate(Y))
eq(X, Y) → false
inf(X) → cons(X, n__inf(n__s(X)))
take(0, X) → nil
take(s(X), cons(Y, L)) → cons(activate(Y), n__take(activate(X), activate(L)))
length(nil) → 0
length(cons(X, L)) → s(n__length(activate(L)))
0 → n__0
s(X) → n__s(X)
inf(X) → n__inf(X)
take(X1, X2) → n__take(X1, X2)
length(X) → n__length(X)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__inf(X)) → inf(activate(X))
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__length(X)) → length(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

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Narrowing

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Narrowing

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

          ↳ QDP

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

ACTIVATE(n__length(n__take(x0, x1))) → LENGTH(take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__inf(y0), n__take(x0, x1))) → TAKE(inf(activate(y0)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__take(y0, y1), n__inf(x0))) → TAKE(take(activate(y0), activate(y1)), inf(activate(x0)))
ACTIVATE(n__take(n__length(y0), n__inf(x0))) → TAKE(length(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(n__inf(y0), n__length(x0))) → TAKE(inf(activate(y0)), length(activate(x0)))
ACTIVATE(n__take(n__length(y0), n__length(x0))) → TAKE(length(activate(y0)), length(activate(x0)))
TAKE(s(X), cons(Y, L)) → ACTIVATE(Y)
ACTIVATE(n__take(n__inf(y0), x0)) → TAKE(inf(activate(y0)), x0)
ACTIVATE(n__take(n__s(y0), x0)) → TAKE(s(y0), x0)
ACTIVATE(n__take(n__s(y0), n__inf(x0))) → TAKE(s(y0), inf(activate(x0)))
ACTIVATE(n__take(n__length(y0), x0)) → TAKE(length(activate(y0)), x0)
ACTIVATE(n__take(n__s(y0), n__take(x0, x1))) → TAKE(s(y0), take(activate(x0), activate(x1)))
ACTIVATE(n__inf(X)) → ACTIVATE(X)
ACTIVATE(n__length(n__length(x0))) → LENGTH(length(activate(x0)))
ACTIVATE(n__length(x0)) → LENGTH(x0)
TAKE(s(X), cons(Y, L)) → ACTIVATE(X)
LENGTH(cons(X, L)) → ACTIVATE(L)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(n__take(y0, y1), n__s(x0))) → TAKE(take(activate(y0), activate(y1)), s(x0))
ACTIVATE(n__take(n__s(y0), n__length(x0))) → TAKE(s(y0), length(activate(x0)))
ACTIVATE(n__take(y0, n__take(x0, x1))) → TAKE(y0, take(activate(x0), activate(x1)))
ACTIVATE(n__take(y0, n__s(x0))) → TAKE(y0, s(x0))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__take(n__inf(y0), n__inf(x0))) → TAKE(inf(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(n__take(y0, y1), n__length(x0))) → TAKE(take(activate(y0), activate(y1)), length(activate(x0)))
TAKE(s(X), cons(Y, L)) → ACTIVATE(L)
ACTIVATE(n__take(n__length(y0), n__take(x0, x1))) → TAKE(length(activate(y0)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(y0, n__0)) → TAKE(y0, 0)
ACTIVATE(n__take(n__take(y0, y1), n__take(x0, x1))) → TAKE(take(activate(y0), activate(y1)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__take(y0, y1), x0)) → TAKE(take(activate(y0), activate(y1)), x0)
ACTIVATE(n__take(y0, n__length(x0))) → TAKE(y0, length(activate(x0)))
ACTIVATE(n__length(n__inf(x0))) → LENGTH(inf(activate(x0)))
ACTIVATE(n__take(y0, n__inf(x0))) → TAKE(y0, inf(activate(x0)))
ACTIVATE(n__take(y0, x0)) → TAKE(y0, x0)
ACTIVATE(n__length(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

eq(n__0, n__0) → true
eq(n__s(X), n__s(Y)) → eq(activate(X), activate(Y))
eq(X, Y) → false
inf(X) → cons(X, n__inf(n__s(X)))
take(0, X) → nil
take(s(X), cons(Y, L)) → cons(activate(Y), n__take(activate(X), activate(L)))
length(nil) → 0
length(cons(X, L)) → s(n__length(activate(L)))
0 → n__0
s(X) → n__s(X)
inf(X) → n__inf(X)
take(X1, X2) → n__take(X1, X2)
length(X) → n__length(X)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__inf(X)) → inf(activate(X))
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__length(X)) → length(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ACTIVATE(n__take(n__take(y0, y1), n__s(x0))) → TAKE(take(activate(y0), activate(y1)), s(x0)) at position [1] we obtained the following new rules:

ACTIVATE(n__take(n__take(y0, y1), n__s(x0))) → TAKE(take(activate(y0), activate(y1)), n__s(x0))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Narrowing

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Narrowing

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

          ↳ QDP

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

ACTIVATE(n__take(n__take(y0, y1), n__inf(x0))) → TAKE(take(activate(y0), activate(y1)), inf(activate(x0)))
ACTIVATE(n__take(n__inf(y0), n__take(x0, x1))) → TAKE(inf(activate(y0)), take(activate(x0), activate(x1)))
ACTIVATE(n__length(n__take(x0, x1))) → LENGTH(take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__inf(y0), n__length(x0))) → TAKE(inf(activate(y0)), length(activate(x0)))
ACTIVATE(n__take(n__length(y0), n__inf(x0))) → TAKE(length(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(n__length(y0), n__length(x0))) → TAKE(length(activate(y0)), length(activate(x0)))
TAKE(s(X), cons(Y, L)) → ACTIVATE(Y)
ACTIVATE(n__take(n__inf(y0), x0)) → TAKE(inf(activate(y0)), x0)
ACTIVATE(n__take(n__s(y0), x0)) → TAKE(s(y0), x0)
ACTIVATE(n__take(n__s(y0), n__inf(x0))) → TAKE(s(y0), inf(activate(x0)))
ACTIVATE(n__take(n__s(y0), n__take(x0, x1))) → TAKE(s(y0), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__length(y0), x0)) → TAKE(length(activate(y0)), x0)
ACTIVATE(n__inf(X)) → ACTIVATE(X)
ACTIVATE(n__length(n__length(x0))) → LENGTH(length(activate(x0)))
ACTIVATE(n__length(x0)) → LENGTH(x0)
LENGTH(cons(X, L)) → ACTIVATE(L)
TAKE(s(X), cons(Y, L)) → ACTIVATE(X)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(n__s(y0), n__length(x0))) → TAKE(s(y0), length(activate(x0)))
ACTIVATE(n__take(y0, n__take(x0, x1))) → TAKE(y0, take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__inf(y0), n__inf(x0))) → TAKE(inf(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__take(y0, n__s(x0))) → TAKE(y0, s(x0))
ACTIVATE(n__take(n__length(y0), n__take(x0, x1))) → TAKE(length(activate(y0)), take(activate(x0), activate(x1)))
TAKE(s(X), cons(Y, L)) → ACTIVATE(L)
ACTIVATE(n__take(n__take(y0, y1), n__length(x0))) → TAKE(take(activate(y0), activate(y1)), length(activate(x0)))
ACTIVATE(n__take(y0, n__0)) → TAKE(y0, 0)
ACTIVATE(n__take(n__take(y0, y1), x0)) → TAKE(take(activate(y0), activate(y1)), x0)
ACTIVATE(n__take(n__take(y0, y1), n__take(x0, x1))) → TAKE(take(activate(y0), activate(y1)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(y0, n__length(x0))) → TAKE(y0, length(activate(x0)))
ACTIVATE(n__length(n__inf(x0))) → LENGTH(inf(activate(x0)))
ACTIVATE(n__take(n__take(y0, y1), n__s(x0))) → TAKE(take(activate(y0), activate(y1)), n__s(x0))
ACTIVATE(n__take(y0, n__inf(x0))) → TAKE(y0, inf(activate(x0)))
ACTIVATE(n__take(y0, x0)) → TAKE(y0, x0)
ACTIVATE(n__length(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

eq(n__0, n__0) → true
eq(n__s(X), n__s(Y)) → eq(activate(X), activate(Y))
eq(X, Y) → false
inf(X) → cons(X, n__inf(n__s(X)))
take(0, X) → nil
take(s(X), cons(Y, L)) → cons(activate(Y), n__take(activate(X), activate(L)))
length(nil) → 0
length(cons(X, L)) → s(n__length(activate(L)))
0 → n__0
s(X) → n__s(X)
inf(X) → n__inf(X)
take(X1, X2) → n__take(X1, X2)
length(X) → n__length(X)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__inf(X)) → inf(activate(X))
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__length(X)) → length(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

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Narrowing

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Narrowing

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

          ↳ QDP

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

ACTIVATE(n__length(n__take(x0, x1))) → LENGTH(take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__inf(y0), n__take(x0, x1))) → TAKE(inf(activate(y0)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__take(y0, y1), n__inf(x0))) → TAKE(take(activate(y0), activate(y1)), inf(activate(x0)))
ACTIVATE(n__take(n__length(y0), n__inf(x0))) → TAKE(length(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(n__inf(y0), n__length(x0))) → TAKE(inf(activate(y0)), length(activate(x0)))
ACTIVATE(n__take(n__length(y0), n__length(x0))) → TAKE(length(activate(y0)), length(activate(x0)))
TAKE(s(X), cons(Y, L)) → ACTIVATE(Y)
ACTIVATE(n__take(n__inf(y0), x0)) → TAKE(inf(activate(y0)), x0)
ACTIVATE(n__take(n__s(y0), x0)) → TAKE(s(y0), x0)
ACTIVATE(n__take(n__s(y0), n__inf(x0))) → TAKE(s(y0), inf(activate(x0)))
ACTIVATE(n__take(n__length(y0), x0)) → TAKE(length(activate(y0)), x0)
ACTIVATE(n__take(n__s(y0), n__take(x0, x1))) → TAKE(s(y0), take(activate(x0), activate(x1)))
ACTIVATE(n__inf(X)) → ACTIVATE(X)
ACTIVATE(n__length(n__length(x0))) → LENGTH(length(activate(x0)))
ACTIVATE(n__length(x0)) → LENGTH(x0)
TAKE(s(X), cons(Y, L)) → ACTIVATE(X)
LENGTH(cons(X, L)) → ACTIVATE(L)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(n__s(y0), n__length(x0))) → TAKE(s(y0), length(activate(x0)))
ACTIVATE(n__take(y0, n__take(x0, x1))) → TAKE(y0, take(activate(x0), activate(x1)))
ACTIVATE(n__take(y0, n__s(x0))) → TAKE(y0, s(x0))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__take(n__inf(y0), n__inf(x0))) → TAKE(inf(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(n__take(y0, y1), n__length(x0))) → TAKE(take(activate(y0), activate(y1)), length(activate(x0)))
TAKE(s(X), cons(Y, L)) → ACTIVATE(L)
ACTIVATE(n__take(n__length(y0), n__take(x0, x1))) → TAKE(length(activate(y0)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(y0, n__0)) → TAKE(y0, 0)
ACTIVATE(n__take(n__take(y0, y1), n__take(x0, x1))) → TAKE(take(activate(y0), activate(y1)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__take(y0, y1), x0)) → TAKE(take(activate(y0), activate(y1)), x0)
ACTIVATE(n__take(y0, n__length(x0))) → TAKE(y0, length(activate(x0)))
ACTIVATE(n__length(n__inf(x0))) → LENGTH(inf(activate(x0)))
ACTIVATE(n__take(y0, n__inf(x0))) → TAKE(y0, inf(activate(x0)))
ACTIVATE(n__take(y0, x0)) → TAKE(y0, x0)
ACTIVATE(n__length(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

eq(n__0, n__0) → true
eq(n__s(X), n__s(Y)) → eq(activate(X), activate(Y))
eq(X, Y) → false
inf(X) → cons(X, n__inf(n__s(X)))
take(0, X) → nil
take(s(X), cons(Y, L)) → cons(activate(Y), n__take(activate(X), activate(L)))
length(nil) → 0
length(cons(X, L)) → s(n__length(activate(L)))
0 → n__0
s(X) → n__s(X)
inf(X) → n__inf(X)
take(X1, X2) → n__take(X1, X2)
length(X) → n__length(X)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__inf(X)) → inf(activate(X))
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__length(X)) → length(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ACTIVATE(n__take(y0, n__0)) → TAKE(y0, 0) at position [1] we obtained the following new rules:

ACTIVATE(n__take(y0, n__0)) → TAKE(y0, n__0)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Narrowing

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Narrowing

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

          ↳ QDP

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

ACTIVATE(n__take(n__take(y0, y1), n__inf(x0))) → TAKE(take(activate(y0), activate(y1)), inf(activate(x0)))
ACTIVATE(n__take(n__inf(y0), n__take(x0, x1))) → TAKE(inf(activate(y0)), take(activate(x0), activate(x1)))
ACTIVATE(n__length(n__take(x0, x1))) → LENGTH(take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__inf(y0), n__length(x0))) → TAKE(inf(activate(y0)), length(activate(x0)))
ACTIVATE(n__take(n__length(y0), n__inf(x0))) → TAKE(length(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(n__length(y0), n__length(x0))) → TAKE(length(activate(y0)), length(activate(x0)))
TAKE(s(X), cons(Y, L)) → ACTIVATE(Y)
ACTIVATE(n__take(n__inf(y0), x0)) → TAKE(inf(activate(y0)), x0)
ACTIVATE(n__take(n__s(y0), x0)) → TAKE(s(y0), x0)
ACTIVATE(n__take(n__s(y0), n__inf(x0))) → TAKE(s(y0), inf(activate(x0)))
ACTIVATE(n__take(n__s(y0), n__take(x0, x1))) → TAKE(s(y0), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__length(y0), x0)) → TAKE(length(activate(y0)), x0)
ACTIVATE(n__inf(X)) → ACTIVATE(X)
ACTIVATE(n__length(n__length(x0))) → LENGTH(length(activate(x0)))
ACTIVATE(n__length(x0)) → LENGTH(x0)
LENGTH(cons(X, L)) → ACTIVATE(L)
TAKE(s(X), cons(Y, L)) → ACTIVATE(X)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(n__s(y0), n__length(x0))) → TAKE(s(y0), length(activate(x0)))
ACTIVATE(n__take(y0, n__take(x0, x1))) → TAKE(y0, take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__inf(y0), n__inf(x0))) → TAKE(inf(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__take(y0, n__s(x0))) → TAKE(y0, s(x0))
ACTIVATE(n__take(n__length(y0), n__take(x0, x1))) → TAKE(length(activate(y0)), take(activate(x0), activate(x1)))
TAKE(s(X), cons(Y, L)) → ACTIVATE(L)
ACTIVATE(n__take(n__take(y0, y1), n__length(x0))) → TAKE(take(activate(y0), activate(y1)), length(activate(x0)))
ACTIVATE(n__take(n__take(y0, y1), x0)) → TAKE(take(activate(y0), activate(y1)), x0)
ACTIVATE(n__take(n__take(y0, y1), n__take(x0, x1))) → TAKE(take(activate(y0), activate(y1)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(y0, n__length(x0))) → TAKE(y0, length(activate(x0)))
ACTIVATE(n__length(n__inf(x0))) → LENGTH(inf(activate(x0)))
ACTIVATE(n__take(y0, n__0)) → TAKE(y0, n__0)
ACTIVATE(n__take(y0, n__inf(x0))) → TAKE(y0, inf(activate(x0)))
ACTIVATE(n__take(y0, x0)) → TAKE(y0, x0)
ACTIVATE(n__length(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

eq(n__0, n__0) → true
eq(n__s(X), n__s(Y)) → eq(activate(X), activate(Y))
eq(X, Y) → false
inf(X) → cons(X, n__inf(n__s(X)))
take(0, X) → nil
take(s(X), cons(Y, L)) → cons(activate(Y), n__take(activate(X), activate(L)))
length(nil) → 0
length(cons(X, L)) → s(n__length(activate(L)))
0 → n__0
s(X) → n__s(X)
inf(X) → n__inf(X)
take(X1, X2) → n__take(X1, X2)
length(X) → n__length(X)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__inf(X)) → inf(activate(X))
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__length(X)) → length(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

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Narrowing

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Narrowing

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

          ↳ QDP

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

ACTIVATE(n__length(n__take(x0, x1))) → LENGTH(take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__inf(y0), n__take(x0, x1))) → TAKE(inf(activate(y0)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__take(y0, y1), n__inf(x0))) → TAKE(take(activate(y0), activate(y1)), inf(activate(x0)))
ACTIVATE(n__take(n__length(y0), n__inf(x0))) → TAKE(length(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(n__inf(y0), n__length(x0))) → TAKE(inf(activate(y0)), length(activate(x0)))
ACTIVATE(n__take(n__length(y0), n__length(x0))) → TAKE(length(activate(y0)), length(activate(x0)))
TAKE(s(X), cons(Y, L)) → ACTIVATE(Y)
ACTIVATE(n__take(n__inf(y0), x0)) → TAKE(inf(activate(y0)), x0)
ACTIVATE(n__take(n__s(y0), x0)) → TAKE(s(y0), x0)
ACTIVATE(n__take(n__s(y0), n__inf(x0))) → TAKE(s(y0), inf(activate(x0)))
ACTIVATE(n__take(n__length(y0), x0)) → TAKE(length(activate(y0)), x0)
ACTIVATE(n__take(n__s(y0), n__take(x0, x1))) → TAKE(s(y0), take(activate(x0), activate(x1)))
ACTIVATE(n__inf(X)) → ACTIVATE(X)
ACTIVATE(n__length(n__length(x0))) → LENGTH(length(activate(x0)))
ACTIVATE(n__length(x0)) → LENGTH(x0)
TAKE(s(X), cons(Y, L)) → ACTIVATE(X)
LENGTH(cons(X, L)) → ACTIVATE(L)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(n__s(y0), n__length(x0))) → TAKE(s(y0), length(activate(x0)))
ACTIVATE(n__take(y0, n__take(x0, x1))) → TAKE(y0, take(activate(x0), activate(x1)))
ACTIVATE(n__take(y0, n__s(x0))) → TAKE(y0, s(x0))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__take(n__inf(y0), n__inf(x0))) → TAKE(inf(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(n__take(y0, y1), n__length(x0))) → TAKE(take(activate(y0), activate(y1)), length(activate(x0)))
TAKE(s(X), cons(Y, L)) → ACTIVATE(L)
ACTIVATE(n__take(n__length(y0), n__take(x0, x1))) → TAKE(length(activate(y0)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__take(y0, y1), n__take(x0, x1))) → TAKE(take(activate(y0), activate(y1)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__take(y0, y1), x0)) → TAKE(take(activate(y0), activate(y1)), x0)
ACTIVATE(n__take(y0, n__length(x0))) → TAKE(y0, length(activate(x0)))
ACTIVATE(n__length(n__inf(x0))) → LENGTH(inf(activate(x0)))
ACTIVATE(n__take(y0, n__inf(x0))) → TAKE(y0, inf(activate(x0)))
ACTIVATE(n__take(y0, x0)) → TAKE(y0, x0)
ACTIVATE(n__length(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

eq(n__0, n__0) → true
eq(n__s(X), n__s(Y)) → eq(activate(X), activate(Y))
eq(X, Y) → false
inf(X) → cons(X, n__inf(n__s(X)))
take(0, X) → nil
take(s(X), cons(Y, L)) → cons(activate(Y), n__take(activate(X), activate(L)))
length(nil) → 0
length(cons(X, L)) → s(n__length(activate(L)))
0 → n__0
s(X) → n__s(X)
inf(X) → n__inf(X)
take(X1, X2) → n__take(X1, X2)
length(X) → n__length(X)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__inf(X)) → inf(activate(X))
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__length(X)) → length(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule ACTIVATE(n__take(y0, n__s(x0))) → TAKE(y0, s(x0)) at position [1] we obtained the following new rules:

ACTIVATE(n__take(y0, n__s(x0))) → TAKE(y0, n__s(x0))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Narrowing

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Narrowing

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

          ↳ QDP

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

ACTIVATE(n__take(n__take(y0, y1), n__inf(x0))) → TAKE(take(activate(y0), activate(y1)), inf(activate(x0)))
ACTIVATE(n__take(n__inf(y0), n__take(x0, x1))) → TAKE(inf(activate(y0)), take(activate(x0), activate(x1)))
ACTIVATE(n__length(n__take(x0, x1))) → LENGTH(take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__inf(y0), n__length(x0))) → TAKE(inf(activate(y0)), length(activate(x0)))
ACTIVATE(n__take(n__length(y0), n__inf(x0))) → TAKE(length(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(n__length(y0), n__length(x0))) → TAKE(length(activate(y0)), length(activate(x0)))
TAKE(s(X), cons(Y, L)) → ACTIVATE(Y)
ACTIVATE(n__take(n__inf(y0), x0)) → TAKE(inf(activate(y0)), x0)
ACTIVATE(n__take(n__s(y0), x0)) → TAKE(s(y0), x0)
ACTIVATE(n__take(n__s(y0), n__inf(x0))) → TAKE(s(y0), inf(activate(x0)))
ACTIVATE(n__take(n__s(y0), n__take(x0, x1))) → TAKE(s(y0), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__length(y0), x0)) → TAKE(length(activate(y0)), x0)
ACTIVATE(n__inf(X)) → ACTIVATE(X)
ACTIVATE(n__length(n__length(x0))) → LENGTH(length(activate(x0)))
ACTIVATE(n__length(x0)) → LENGTH(x0)
LENGTH(cons(X, L)) → ACTIVATE(L)
TAKE(s(X), cons(Y, L)) → ACTIVATE(X)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(y0, n__s(x0))) → TAKE(y0, n__s(x0))
ACTIVATE(n__take(n__s(y0), n__length(x0))) → TAKE(s(y0), length(activate(x0)))
ACTIVATE(n__take(y0, n__take(x0, x1))) → TAKE(y0, take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__inf(y0), n__inf(x0))) → TAKE(inf(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__take(n__length(y0), n__take(x0, x1))) → TAKE(length(activate(y0)), take(activate(x0), activate(x1)))
TAKE(s(X), cons(Y, L)) → ACTIVATE(L)
ACTIVATE(n__take(n__take(y0, y1), n__length(x0))) → TAKE(take(activate(y0), activate(y1)), length(activate(x0)))
ACTIVATE(n__take(n__take(y0, y1), x0)) → TAKE(take(activate(y0), activate(y1)), x0)
ACTIVATE(n__take(n__take(y0, y1), n__take(x0, x1))) → TAKE(take(activate(y0), activate(y1)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(y0, n__length(x0))) → TAKE(y0, length(activate(x0)))
ACTIVATE(n__length(n__inf(x0))) → LENGTH(inf(activate(x0)))
ACTIVATE(n__take(y0, n__inf(x0))) → TAKE(y0, inf(activate(x0)))
ACTIVATE(n__take(y0, x0)) → TAKE(y0, x0)
ACTIVATE(n__length(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

eq(n__0, n__0) → true
eq(n__s(X), n__s(Y)) → eq(activate(X), activate(Y))
eq(X, Y) → false
inf(X) → cons(X, n__inf(n__s(X)))
take(0, X) → nil
take(s(X), cons(Y, L)) → cons(activate(Y), n__take(activate(X), activate(L)))
length(nil) → 0
length(cons(X, L)) → s(n__length(activate(L)))
0 → n__0
s(X) → n__s(X)
inf(X) → n__inf(X)
take(X1, X2) → n__take(X1, X2)
length(X) → n__length(X)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__inf(X)) → inf(activate(X))
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__length(X)) → length(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

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Narrowing

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Narrowing

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ SemLabProof

                                                                                                                                                ↳ SemLabProof2

          ↳ QDP

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

ACTIVATE(n__length(n__take(x0, x1))) → LENGTH(take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__inf(y0), n__take(x0, x1))) → TAKE(inf(activate(y0)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__take(y0, y1), n__inf(x0))) → TAKE(take(activate(y0), activate(y1)), inf(activate(x0)))
ACTIVATE(n__take(n__length(y0), n__inf(x0))) → TAKE(length(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(n__inf(y0), n__length(x0))) → TAKE(inf(activate(y0)), length(activate(x0)))
ACTIVATE(n__take(n__length(y0), n__length(x0))) → TAKE(length(activate(y0)), length(activate(x0)))
TAKE(s(X), cons(Y, L)) → ACTIVATE(Y)
ACTIVATE(n__take(n__inf(y0), x0)) → TAKE(inf(activate(y0)), x0)
ACTIVATE(n__take(n__s(y0), x0)) → TAKE(s(y0), x0)
ACTIVATE(n__take(n__s(y0), n__inf(x0))) → TAKE(s(y0), inf(activate(x0)))
ACTIVATE(n__take(n__length(y0), x0)) → TAKE(length(activate(y0)), x0)
ACTIVATE(n__take(n__s(y0), n__take(x0, x1))) → TAKE(s(y0), take(activate(x0), activate(x1)))
ACTIVATE(n__inf(X)) → ACTIVATE(X)
ACTIVATE(n__length(n__length(x0))) → LENGTH(length(activate(x0)))
ACTIVATE(n__length(x0)) → LENGTH(x0)
TAKE(s(X), cons(Y, L)) → ACTIVATE(X)
LENGTH(cons(X, L)) → ACTIVATE(L)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(n__s(y0), n__length(x0))) → TAKE(s(y0), length(activate(x0)))
ACTIVATE(n__take(y0, n__take(x0, x1))) → TAKE(y0, take(activate(x0), activate(x1)))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__take(n__inf(y0), n__inf(x0))) → TAKE(inf(activate(y0)), inf(activate(x0)))
ACTIVATE(n__take(n__take(y0, y1), n__length(x0))) → TAKE(take(activate(y0), activate(y1)), length(activate(x0)))
TAKE(s(X), cons(Y, L)) → ACTIVATE(L)
ACTIVATE(n__take(n__length(y0), n__take(x0, x1))) → TAKE(length(activate(y0)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__take(y0, y1), n__take(x0, x1))) → TAKE(take(activate(y0), activate(y1)), take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__take(y0, y1), x0)) → TAKE(take(activate(y0), activate(y1)), x0)
ACTIVATE(n__take(y0, n__length(x0))) → TAKE(y0, length(activate(x0)))
ACTIVATE(n__length(n__inf(x0))) → LENGTH(inf(activate(x0)))
ACTIVATE(n__take(y0, n__inf(x0))) → TAKE(y0, inf(activate(x0)))
ACTIVATE(n__take(y0, x0)) → TAKE(y0, x0)
ACTIVATE(n__length(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

eq(n__0, n__0) → true
eq(n__s(X), n__s(Y)) → eq(activate(X), activate(Y))
eq(X, Y) → false
inf(X) → cons(X, n__inf(n__s(X)))
take(0, X) → nil
take(s(X), cons(Y, L)) → cons(activate(Y), n__take(activate(X), activate(L)))
length(nil) → 0
length(cons(X, L)) → s(n__length(activate(L)))
0 → n__0
s(X) → n__s(X)
inf(X) → n__inf(X)
take(X1, X2) → n__take(X1, X2)
length(X) → n__length(X)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__inf(X)) → inf(activate(X))
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__length(X)) → length(activate(X))
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We found the following model for the rules of the TRS R. Interpretation over the domain with elements from 0 to 1.LENGTH: 0
eq: 0
n__length: 1
true: 0
n__inf: 0
activate: x₀
n__s: 1
take: 0
0: 1
TAKE: 0
inf: 0
cons: 0
n__0: 1
n__take: 0
false: 0
s: 1
length: 1
ACTIVATE: 0
nil: 0
By semantic labelling [33] we obtain the following labelled TRS:Q DP problem:
The TRS P consists of the following rules:

TAKE.1-0(s.1(X), cons.0-1(Y, L)) → ACTIVATE.1(X)
ACTIVATE.0(n__take.0-0(n__take.0-1(y0, y1), n__take.0-1(x0, x1))) → TAKE.0-0(take.0-1(activate.0(y0), activate.1(y1)), take.0-1(activate.0(x0), activate.1(x1)))
ACTIVATE.1(n__length.1(X)) → ACTIVATE.1(X)
ACTIVATE.0(n__take.0-1(n__take.0-1(y0, y1), x0)) → TAKE.0-1(take.0-1(activate.0(y0), activate.1(y1)), x0)
ACTIVATE.0(n__take.1-0(y0, x0)) → TAKE.1-0(y0, x0)
ACTIVATE.0(n__take.1-0(n__s.0(y0), n__take.1-0(x0, x1))) → TAKE.1-0(s.0(y0), take.1-0(activate.1(x0), activate.0(x1)))
ACTIVATE.0(n__take.1-0(n__length.1(y0), n__inf.1(x0))) → TAKE.1-0(length.1(activate.1(y0)), inf.1(activate.1(x0)))
ACTIVATE.0(n__take.1-1(y0, n__length.0(x0))) → TAKE.1-1(y0, length.0(activate.0(x0)))
ACTIVATE.0(n__take.1-0(X1, X2)) → ACTIVATE.1(X1)
ACTIVATE.0(n__take.0-0(y0, n__take.1-1(x0, x1))) → TAKE.0-0(y0, take.1-1(activate.1(x0), activate.1(x1)))
ACTIVATE.0(n__take.1-0(n__length.1(y0), n__take.1-0(x0, x1))) → TAKE.1-0(length.1(activate.1(y0)), take.1-0(activate.1(x0), activate.0(x1)))
ACTIVATE.0(n__take.1-0(n__length.0(y0), n__take.1-1(x0, x1))) → TAKE.1-0(length.0(activate.0(y0)), take.1-1(activate.1(x0), activate.1(x1)))
ACTIVATE.0(n__take.1-1(n__s.0(y0), n__length.1(x0))) → TAKE.1-1(s.0(y0), length.1(activate.1(x0)))
ACTIVATE.0(n__take.0-1(n__take.0-1(y0, y1), n__length.1(x0))) → TAKE.0-1(take.0-1(activate.0(y0), activate.1(y1)), length.1(activate.1(x0)))
TAKE.1-0(s.0(X), cons.1-0(Y, L)) → ACTIVATE.0(L)
LENGTH.0(cons.1-0(X, L)) → ACTIVATE.0(L)
ACTIVATE.0(n__take.1-0(n__length.1(y0), x0)) → TAKE.1-0(length.1(activate.1(y0)), x0)
ACTIVATE.1(n__length.0(n__inf.0(x0))) → LENGTH.0(inf.0(activate.0(x0)))
ACTIVATE.0(n__take.0-1(n__take.1-0(y0, y1), n__length.0(x0))) → TAKE.0-1(take.1-0(activate.1(y0), activate.0(y1)), length.0(activate.0(x0)))
ACTIVATE.0(n__take.0-0(n__inf.1(y0), n__take.1-1(x0, x1))) → TAKE.0-0(inf.1(activate.1(y0)), take.1-1(activate.1(x0), activate.1(x1)))
ACTIVATE.0(n__take.0-0(n__take.1-1(y0, y1), n__inf.1(x0))) → TAKE.0-0(take.1-1(activate.1(y0), activate.1(y1)), inf.1(activate.1(x0)))
ACTIVATE.0(n__take.0-0(n__take.1-1(y0, y1), n__take.0-0(x0, x1))) → TAKE.0-0(take.1-1(activate.1(y0), activate.1(y1)), take.0-0(activate.0(x0), activate.0(x1)))
ACTIVATE.0(n__take.0-0(n__take.1-0(y0, y1), n__take.0-1(x0, x1))) → TAKE.0-0(take.1-0(activate.1(y0), activate.0(y1)), take.0-1(activate.0(x0), activate.1(x1)))
ACTIVATE.0(n__take.0-0(n__take.0-1(y0, y1), n__take.1-0(x0, x1))) → TAKE.0-0(take.0-1(activate.0(y0), activate.1(y1)), take.1-0(activate.1(x0), activate.0(x1)))
ACTIVATE.0(n__take.0-0(n__take.0-0(y0, y1), n__take.1-1(x0, x1))) → TAKE.0-0(take.0-0(activate.0(y0), activate.0(y1)), take.1-1(activate.1(x0), activate.1(x1)))
ACTIVATE.0(n__take.0-0(n__take.1-1(y0, y1), n__take.0-1(x0, x1))) → TAKE.0-0(take.1-1(activate.1(y0), activate.1(y1)), take.0-1(activate.0(x0), activate.1(x1)))
ACTIVATE.0(n__take.0-0(n__take.0-1(y0, y1), n__take.1-1(x0, x1))) → TAKE.0-0(take.0-1(activate.0(y0), activate.1(y1)), take.1-1(activate.1(x0), activate.1(x1)))
ACTIVATE.1(n__length.0(n__take.1-1(x0, x1))) → LENGTH.0(take.1-1(activate.1(x0), activate.1(x1)))
ACTIVATE.0(n__take.1-0(n__s.0(y0), n__take.0-1(x0, x1))) → TAKE.1-0(s.0(y0), take.0-1(activate.0(x0), activate.1(x1)))
TAKE.1-0(s.0(X), cons.1-1(Y, L)) → ACTIVATE.1(L)
TAKE.1-0(s.1(X), cons.1-0(Y, L)) → ACTIVATE.0(L)
ACTIVATE.0(n__take.0-0(n__inf.0(y0), x0)) → TAKE.0-0(inf.0(activate.0(y0)), x0)
LENGTH.0(cons.1-1(X, L)) → ACTIVATE.1(L)
ACTIVATE.0(n__take.0-0(y0, n__inf.0(x0))) → TAKE.0-0(y0, inf.0(activate.0(x0)))
ACTIVATE.0(n__take.1-1(y0, n__length.1(x0))) → TAKE.1-1(y0, length.1(activate.1(x0)))
ACTIVATE.0(n__take.0-0(n__take.0-0(y0, y1), x0)) → TAKE.0-0(take.0-0(activate.0(y0), activate.0(y1)), x0)
ACTIVATE.0(n__take.0-0(n__take.1-0(y0, y1), x0)) → TAKE.0-0(take.1-0(activate.1(y0), activate.0(y1)), x0)
ACTIVATE.0(n__take.1-0(n__length.1(y0), n__take.0-1(x0, x1))) → TAKE.1-0(length.1(activate.1(y0)), take.0-1(activate.0(x0), activate.1(x1)))
ACTIVATE.0(n__take.1-0(X1, X2)) → ACTIVATE.0(X2)
ACTIVATE.1(n__length.1(n__length.1(x0))) → LENGTH.1(length.1(activate.1(x0)))
ACTIVATE.0(n__take.0-0(y0, n__take.1-0(x0, x1))) → TAKE.0-0(y0, take.1-0(activate.1(x0), activate.0(x1)))
TAKE.1-0(s.1(X), cons.0-1(Y, L)) → ACTIVATE.0(Y)
ACTIVATE.0(n__take.1-0(n__length.0(y0), n__inf.1(x0))) → TAKE.1-0(length.0(activate.0(y0)), inf.1(activate.1(x0)))
ACTIVATE.0(n__take.1-0(n__length.1(y0), n__inf.0(x0))) → TAKE.1-0(length.1(activate.1(y0)), inf.0(activate.0(x0)))
ACTIVATE.0(n__take.0-1(n__take.0-0(y0, y1), x0)) → TAKE.0-1(take.0-0(activate.0(y0), activate.0(y1)), x0)
TAKE.1-0(s.1(X), cons.1-0(Y, L)) → ACTIVATE.1(X)
ACTIVATE.0(n__take.1-0(n__s.0(y0), n__inf.0(x0))) → TAKE.1-0(s.0(y0), inf.0(activate.0(x0)))
ACTIVATE.0(n__take.0-1(X1, X2)) → ACTIVATE.1(X2)
ACTIVATE.0(n__take.0-0(y0, n__take.0-0(x0, x1))) → TAKE.0-0(y0, take.0-0(activate.0(x0), activate.0(x1)))
ACTIVATE.0(n__take.0-0(n__take.1-1(y0, y1), x0)) → TAKE.0-0(take.1-1(activate.1(y0), activate.1(y1)), x0)
ACTIVATE.1(n__length.1(n__length.0(x0))) → LENGTH.1(length.0(activate.0(x0)))
ACTIVATE.0(n__take.0-1(n__take.1-1(y0, y1), n__length.1(x0))) → TAKE.0-1(take.1-1(activate.1(y0), activate.1(y1)), length.1(activate.1(x0)))
ACTIVATE.0(n__take.0-0(n__inf.0(y0), n__take.1-0(x0, x1))) → TAKE.0-0(inf.0(activate.0(y0)), take.1-0(activate.1(x0), activate.0(x1)))
ACTIVATE.0(n__take.0-0(n__take.1-0(y0, y1), n__inf.0(x0))) → TAKE.0-0(take.1-0(activate.1(y0), activate.0(y1)), inf.0(activate.0(x0)))
ACTIVATE.0(n__take.1-1(n__s.0(y0), x0)) → TAKE.1-1(s.0(y0), x0)
TAKE.1-0(s.1(X), cons.0-1(Y, L)) → ACTIVATE.1(L)
ACTIVATE.0(n__take.1-0(n__length.0(y0), n__inf.0(x0))) → TAKE.1-0(length.0(activate.0(y0)), inf.0(activate.0(x0)))
ACTIVATE.0(n__take.0-1(n__inf.1(y0), x0)) → TAKE.0-1(inf.1(activate.1(y0)), x0)
TAKE.1-0(s.0(X), cons.0-0(Y, L)) → ACTIVATE.0(L)
ACTIVATE.0(n__take.1-1(n__s.1(y0), x0)) → TAKE.1-1(s.1(y0), x0)
ACTIVATE.0(n__take.1-1(n__s.1(y0), n__length.0(x0))) → TAKE.1-1(s.1(y0), length.0(activate.0(x0)))
ACTIVATE.0(n__take.1-1(n__s.0(y0), n__length.0(x0))) → TAKE.1-1(s.0(y0), length.0(activate.0(x0)))
ACTIVATE.0(n__take.1-0(n__s.0(y0), n__take.1-1(x0, x1))) → TAKE.1-0(s.0(y0), take.1-1(activate.1(x0), activate.1(x1)))
TAKE.1-0(s.0(X), cons.1-1(Y, L)) → ACTIVATE.0(X)
ACTIVATE.0(n__take.0-0(n__take.1-0(y0, y1), n__take.1-0(x0, x1))) → TAKE.0-0(take.1-0(activate.1(y0), activate.0(y1)), take.1-0(activate.1(x0), activate.0(x1)))
ACTIVATE.0(n__take.0-1(n__inf.0(y0), n__length.0(x0))) → TAKE.0-1(inf.0(activate.0(y0)), length.0(activate.0(x0)))
ACTIVATE.1(n__length.0(x0)) → LENGTH.0(x0)
ACTIVATE.0(n__take.0-0(n__take.1-1(y0, y1), n__take.1-1(x0, x1))) → TAKE.0-0(take.1-1(activate.1(y0), activate.1(y1)), take.1-1(activate.1(x0), activate.1(x1)))
ACTIVATE.1(n__length.0(X)) → ACTIVATE.0(X)
ACTIVATE.0(n__take.1-0(y0, n__take.0-0(x0, x1))) → TAKE.1-0(y0, take.0-0(activate.0(x0), activate.0(x1)))
ACTIVATE.0(n__take.0-1(n__take.0-0(y0, y1), n__length.1(x0))) → TAKE.0-1(take.0-0(activate.0(y0), activate.0(y1)), length.1(activate.1(x0)))
ACTIVATE.0(n__take.0-1(n__take.0-1(y0, y1), n__length.0(x0))) → TAKE.0-1(take.0-1(activate.0(y0), activate.1(y1)), length.0(activate.0(x0)))
ACTIVATE.0(n__take.1-0(n__s.1(y0), n__inf.0(x0))) → TAKE.1-0(s.1(y0), inf.0(activate.0(x0)))
ACTIVATE.0(n__take.0-1(y0, x0)) → TAKE.0-1(y0, x0)
ACTIVATE.1(n__length.0(n__inf.1(x0))) → LENGTH.0(inf.1(activate.1(x0)))
TAKE.1-0(s.1(X), cons.1-1(Y, L)) → ACTIVATE.1(L)
TAKE.1-0(s.1(X), cons.1-1(Y, L)) → ACTIVATE.1(Y)
TAKE.1-0(s.1(X), cons.0-0(Y, L)) → ACTIVATE.0(L)
ACTIVATE.0(n__take.1-0(n__s.1(y0), n__inf.1(x0))) → TAKE.1-0(s.1(y0), inf.1(activate.1(x0)))
ACTIVATE.0(n__take.0-1(X1, X2)) → ACTIVATE.0(X1)
ACTIVATE.0(n__take.1-0(n__length.1(y0), n__take.1-1(x0, x1))) → TAKE.1-0(length.1(activate.1(y0)), take.1-1(activate.1(x0), activate.1(x1)))
TAKE.1-0(s.1(X), cons.1-1(Y, L)) → ACTIVATE.1(X)
TAKE.1-0(s.1(X), cons.0-0(Y, L)) → ACTIVATE.1(X)
ACTIVATE.0(n__take.0-0(n__inf.0(y0), n__take.1-1(x0, x1))) → TAKE.0-0(inf.0(activate.0(y0)), take.1-1(activate.1(x0), activate.1(x1)))
ACTIVATE.0(n__take.0-0(n__inf.1(y0), n__take.1-0(x0, x1))) → TAKE.0-0(inf.1(activate.1(y0)), take.1-0(activate.1(x0), activate.0(x1)))
ACTIVATE.0(n__take.0-0(n__take.1-0(y0, y1), n__inf.1(x0))) → TAKE.0-0(take.1-0(activate.1(y0), activate.0(y1)), inf.1(activate.1(x0)))
ACTIVATE.0(n__take.0-0(n__take.1-1(y0, y1), n__inf.0(x0))) → TAKE.0-0(take.1-1(activate.1(y0), activate.1(y1)), inf.0(activate.0(x0)))
TAKE.1-0(s.0(X), cons.0-0(Y, L)) → ACTIVATE.0(X)
TAKE.1-0(s.0(X), cons.1-0(Y, L)) → ACTIVATE.0(X)
ACTIVATE.0(n__take.0-0(n__take.1-0(y0, y1), n__take.0-0(x0, x1))) → TAKE.0-0(take.1-0(activate.1(y0), activate.0(y1)), take.0-0(activate.0(x0), activate.0(x1)))
ACTIVATE.0(n__take.0-0(n__take.0-0(y0, y1), n__take.1-0(x0, x1))) → TAKE.0-0(take.0-0(activate.0(y0), activate.0(y1)), take.1-0(activate.1(x0), activate.0(x1)))
ACTIVATE.0(n__take.0-0(y0, n__take.0-1(x0, x1))) → TAKE.0-0(y0, take.0-1(activate.0(x0), activate.1(x1)))
ACTIVATE.0(n__take.0-0(y0, x0)) → TAKE.0-0(y0, x0)
ACTIVATE.1(n__length.0(n__take.1-0(x0, x1))) → LENGTH.0(take.1-0(activate.1(x0), activate.0(x1)))
ACTIVATE.0(n__take.1-1(n__length.1(y0), x0)) → TAKE.1-1(length.1(activate.1(y0)), x0)
ACTIVATE.0(n__take.1-1(n__s.1(y0), n__length.1(x0))) → TAKE.1-1(s.1(y0), length.1(activate.1(x0)))
ACTIVATE.0(n__take.0-0(n__inf.0(y0), n__take.0-0(x0, x1))) → TAKE.0-0(inf.0(activate.0(y0)), take.0-0(activate.0(x0), activate.0(x1)))
ACTIVATE.0(n__take.0-0(n__take.0-0(y0, y1), n__inf.0(x0))) → TAKE.0-0(take.0-0(activate.0(y0), activate.0(y1)), inf.0(activate.0(x0)))
ACTIVATE.0(n__take.0-1(n__take.1-1(y0, y1), x0)) → TAKE.0-1(take.1-1(activate.1(y0), activate.1(y1)), x0)
ACTIVATE.1(n__length.0(n__take.0-0(x0, x1))) → LENGTH.0(take.0-0(activate.0(x0), activate.0(x1)))
ACTIVATE.0(n__take.0-0(n__inf.1(y0), n__inf.1(x0))) → TAKE.0-0(inf.1(activate.1(y0)), inf.1(activate.1(x0)))
ACTIVATE.0(n__take.0-0(n__take.0-1(y0, y1), x0)) → TAKE.0-0(take.0-1(activate.0(y0), activate.1(y1)), x0)
TAKE.1-0(s.0(X), cons.1-0(Y, L)) → ACTIVATE.1(Y)
ACTIVATE.0(n__take.0-0(n__take.0-0(y0, y1), n__take.0-0(x0, x1))) → TAKE.0-0(take.0-0(activate.0(y0), activate.0(y1)), take.0-0(activate.0(x0), activate.0(x1)))
ACTIVATE.0(n__take.0-0(X1, X2)) → ACTIVATE.0(X2)
ACTIVATE.0(n__take.1-0(y0, n__take.0-1(x0, x1))) → TAKE.1-0(y0, take.0-1(activate.0(x0), activate.1(x1)))
ACTIVATE.0(n__take.1-1(n__length.1(y0), n__length.0(x0))) → TAKE.1-1(length.1(activate.1(y0)), length.0(activate.0(x0)))
ACTIVATE.0(n__take.1-1(n__length.0(y0), n__length.1(x0))) → TAKE.1-1(length.0(activate.0(y0)), length.1(activate.1(x0)))
ACTIVATE.0(n__take.1-1(X1, X2)) → ACTIVATE.1(X1)
ACTIVATE.0(n__take.1-1(y0, x0)) → TAKE.1-1(y0, x0)
LENGTH.0(cons.0-1(X, L)) → ACTIVATE.1(L)
ACTIVATE.0(n__take.1-0(n__s.0(y0), x0)) → TAKE.1-0(s.0(y0), x0)
ACTIVATE.0(n__take.1-0(n__length.0(y0), n__take.0-0(x0, x1))) → TAKE.1-0(length.0(activate.0(y0)), take.0-0(activate.0(x0), activate.0(x1)))
ACTIVATE.0(n__inf.0(X)) → ACTIVATE.0(X)
ACTIVATE.0(n__take.1-0(n__s.0(y0), n__take.0-0(x0, x1))) → TAKE.1-0(s.0(y0), take.0-0(activate.0(x0), activate.0(x1)))
ACTIVATE.0(n__take.1-0(n__s.1(y0), n__take.1-0(x0, x1))) → TAKE.1-0(s.1(y0), take.1-0(activate.1(x0), activate.0(x1)))
ACTIVATE.0(n__take.1-0(n__s.1(y0), n__take.0-1(x0, x1))) → TAKE.1-0(s.1(y0), take.0-1(activate.0(x0), activate.1(x1)))
LENGTH.0(cons.0-0(X, L)) → ACTIVATE.0(L)
ACTIVATE.0(n__take.1-1(n__length.0(y0), n__length.0(x0))) → TAKE.1-1(length.0(activate.0(y0)), length.0(activate.0(x0)))
ACTIVATE.0(n__take.0-0(y0, n__inf.1(x0))) → TAKE.0-0(y0, inf.1(activate.1(x0)))
ACTIVATE.1(n__length.0(n__take.0-1(x0, x1))) → LENGTH.0(take.0-1(activate.0(x0), activate.1(x1)))
ACTIVATE.0(n__take.1-0(n__s.1(y0), x0)) → TAKE.1-0(s.1(y0), x0)
ACTIVATE.0(n__take.1-0(n__s.1(y0), n__take.1-1(x0, x1))) → TAKE.1-0(s.1(y0), take.1-1(activate.1(x0), activate.1(x1)))
ACTIVATE.0(n__take.0-0(n__take.0-1(y0, y1), n__take.0-0(x0, x1))) → TAKE.0-0(take.0-1(activate.0(y0), activate.1(y1)), take.0-0(activate.0(x0), activate.0(x1)))
ACTIVATE.0(n__take.0-0(n__take.0-0(y0, y1), n__take.0-1(x0, x1))) → TAKE.0-0(take.0-0(activate.0(y0), activate.0(y1)), take.0-1(activate.0(x0), activate.1(x1)))
ACTIVATE.0(n__take.0-1(n__inf.0(y0), x0)) → TAKE.0-1(inf.0(activate.0(y0)), x0)
ACTIVATE.0(n__take.0-1(y0, n__length.0(x0))) → TAKE.0-1(y0, length.0(activate.0(x0)))
ACTIVATE.0(n__take.1-1(X1, X2)) → ACTIVATE.1(X2)
ACTIVATE.0(n__take.0-0(n__inf.1(y0), n__take.0-1(x0, x1))) → TAKE.0-0(inf.1(activate.1(y0)), take.0-1(activate.0(x0), activate.1(x1)))
ACTIVATE.0(n__take.0-0(n__take.0-1(y0, y1), n__inf.1(x0))) → TAKE.0-0(take.0-1(activate.0(y0), activate.1(y1)), inf.1(activate.1(x0)))
ACTIVATE.0(n__take.1-0(y0, n__inf.1(x0))) → TAKE.1-0(y0, inf.1(activate.1(x0)))
ACTIVATE.0(n__inf.1(X)) → ACTIVATE.1(X)
TAKE.1-0(s.0(X), cons.0-1(Y, L)) → ACTIVATE.1(L)
ACTIVATE.0(n__take.1-0(y0, n__inf.0(x0))) → TAKE.1-0(y0, inf.0(activate.0(x0)))
ACTIVATE.0(n__take.1-0(y0, n__take.1-0(x0, x1))) → TAKE.1-0(y0, take.1-0(activate.1(x0), activate.0(x1)))
ACTIVATE.0(n__take.1-0(n__s.1(y0), n__take.0-0(x0, x1))) → TAKE.1-0(s.1(y0), take.0-0(activate.0(x0), activate.0(x1)))
ACTIVATE.0(n__take.0-1(n__inf.1(y0), n__length.1(x0))) → TAKE.0-1(inf.1(activate.1(y0)), length.1(activate.1(x0)))
ACTIVATE.0(n__take.1-1(n__length.1(y0), n__length.1(x0))) → TAKE.1-1(length.1(activate.1(y0)), length.1(activate.1(x0)))
ACTIVATE.0(n__take.0-0(n__inf.0(y0), n__take.0-1(x0, x1))) → TAKE.0-0(inf.0(activate.0(y0)), take.0-1(activate.0(x0), activate.1(x1)))
ACTIVATE.0(n__take.0-0(n__inf.1(y0), n__take.0-0(x0, x1))) → TAKE.0-0(inf.1(activate.1(y0)), take.0-0(activate.0(x0), activate.0(x1)))
ACTIVATE.0(n__take.0-0(n__take.0-0(y0, y1), n__inf.1(x0))) → TAKE.0-0(take.0-0(activate.0(y0), activate.0(y1)), inf.1(activate.1(x0)))
ACTIVATE.0(n__take.0-0(n__take.0-1(y0, y1), n__inf.0(x0))) → TAKE.0-0(take.0-1(activate.0(y0), activate.1(y1)), inf.0(activate.0(x0)))
ACTIVATE.0(n__take.0-1(n__take.1-0(y0, y1), n__length.1(x0))) → TAKE.0-1(take.1-0(activate.1(y0), activate.0(y1)), length.1(activate.1(x0)))
ACTIVATE.0(n__take.0-1(n__take.1-1(y0, y1), n__length.0(x0))) → TAKE.0-1(take.1-1(activate.1(y0), activate.1(y1)), length.0(activate.0(x0)))
TAKE.1-0(s.1(X), cons.0-0(Y, L)) → ACTIVATE.0(Y)
TAKE.1-0(s.0(X), cons.0-1(Y, L)) → ACTIVATE.0(Y)
ACTIVATE.0(n__take.1-1(n__length.0(y0), x0)) → TAKE.1-1(length.0(activate.0(y0)), x0)
ACTIVATE.0(n__take.0-0(X1, X2)) → ACTIVATE.0(X1)
ACTIVATE.0(n__take.0-1(n__inf.0(y0), n__length.1(x0))) → TAKE.0-1(inf.0(activate.0(y0)), length.1(activate.1(x0)))
ACTIVATE.0(n__take.0-1(n__inf.1(y0), n__length.0(x0))) → TAKE.0-1(inf.1(activate.1(y0)), length.0(activate.0(x0)))
ACTIVATE.0(n__take.0-1(n__take.0-0(y0, y1), n__length.0(x0))) → TAKE.0-1(take.0-0(activate.0(y0), activate.0(y1)), length.0(activate.0(x0)))
ACTIVATE.0(n__take.0-1(n__take.1-0(y0, y1), x0)) → TAKE.0-1(take.1-0(activate.1(y0), activate.0(y1)), x0)
ACTIVATE.0(n__take.0-1(y0, n__length.1(x0))) → TAKE.0-1(y0, length.1(activate.1(x0)))
ACTIVATE.0(n__take.1-0(n__s.0(y0), n__inf.1(x0))) → TAKE.1-0(s.0(y0), inf.1(activate.1(x0)))
ACTIVATE.1(n__length.1(x0)) → LENGTH.1(x0)
ACTIVATE.0(n__take.1-0(n__length.1(y0), n__take.0-0(x0, x1))) → TAKE.1-0(length.1(activate.1(y0)), take.0-0(activate.0(x0), activate.0(x1)))
ACTIVATE.0(n__take.1-0(n__length.0(y0), n__take.0-1(x0, x1))) → TAKE.1-0(length.0(activate.0(y0)), take.0-1(activate.0(x0), activate.1(x1)))
TAKE.1-0(s.1(X), cons.1-0(Y, L)) → ACTIVATE.1(Y)
TAKE.1-0(s.0(X), cons.1-1(Y, L)) → ACTIVATE.1(Y)
ACTIVATE.0(n__take.0-0(n__take.1-1(y0, y1), n__take.1-0(x0, x1))) → TAKE.0-0(take.1-1(activate.1(y0), activate.1(y1)), take.1-0(activate.1(x0), activate.0(x1)))
ACTIVATE.0(n__take.0-0(n__take.1-0(y0, y1), n__take.1-1(x0, x1))) → TAKE.0-0(take.1-0(activate.1(y0), activate.0(y1)), take.1-1(activate.1(x0), activate.1(x1)))
ACTIVATE.0(n__take.0-0(n__inf.1(y0), x0)) → TAKE.0-0(inf.1(activate.1(y0)), x0)
ACTIVATE.0(n__take.1-0(n__length.0(y0), n__take.1-0(x0, x1))) → TAKE.1-0(length.0(activate.0(y0)), take.1-0(activate.1(x0), activate.0(x1)))
ACTIVATE.0(n__take.1-0(y0, n__take.1-1(x0, x1))) → TAKE.1-0(y0, take.1-1(activate.1(x0), activate.1(x1)))
ACTIVATE.0(n__take.0-0(n__inf.0(y0), n__inf.0(x0))) → TAKE.0-0(inf.0(activate.0(y0)), inf.0(activate.0(x0)))
ACTIVATE.0(n__take.0-0(n__inf.0(y0), n__inf.1(x0))) → TAKE.0-0(inf.0(activate.0(y0)), inf.1(activate.1(x0)))
ACTIVATE.0(n__take.0-0(n__inf.1(y0), n__inf.0(x0))) → TAKE.0-0(inf.1(activate.1(y0)), inf.0(activate.0(x0)))
TAKE.1-0(s.0(X), cons.0-0(Y, L)) → ACTIVATE.0(Y)
ACTIVATE.0(n__take.1-0(n__length.0(y0), x0)) → TAKE.1-0(length.0(activate.0(y0)), x0)
TAKE.1-0(s.0(X), cons.0-1(Y, L)) → ACTIVATE.0(X)

The TRS R consists of the following rules:

take.1-0(s.0(X), cons.0-1(Y, L)) → cons.0-0(activate.0(Y), n__take.0-1(activate.0(X), activate.1(L)))
activate.1(n__s.1(X)) → s.1(X)
take.1-0(s.0(X), cons.1-0(Y, L)) → cons.1-0(activate.1(Y), n__take.0-0(activate.0(X), activate.0(L)))
eq.1-0(X, Y) → false.
take.1-1(X1, X2) → n__take.1-1(X1, X2)
activate.1(n__0.) → 0.
s.1(X) → n__s.1(X)
inf.0(X) → n__inf.0(X)
eq.1-1(n__s.1(X), n__s.0(Y)) → eq.1-0(activate.1(X), activate.0(Y))
inf.1(X) → cons.1-0(X, n__inf.1(n__s.1(X)))
take.1-0(X1, X2) → n__take.1-0(X1, X2)
inf.1(X) → n__inf.1(X)
activate.0(n__take.0-1(X1, X2)) → take.0-1(activate.0(X1), activate.1(X2))
length.0(cons.1-0(X, L)) → s.1(n__length.0(activate.0(L)))
length.0(X) → n__length.0(X)
take.1-0(s.1(X), cons.0-1(Y, L)) → cons.0-0(activate.0(Y), n__take.1-1(activate.1(X), activate.1(L)))
length.1(X) → n__length.1(X)
take.0-1(X1, X2) → n__take.0-1(X1, X2)
take.1-0(s.1(X), cons.1-1(Y, L)) → cons.1-0(activate.1(Y), n__take.1-1(activate.1(X), activate.1(L)))
activate.0(X) → X
eq.1-1(n__s.0(X), n__s.0(Y)) → eq.0-0(activate.0(X), activate.0(Y))
take.1-0(s.1(X), cons.1-0(Y, L)) → cons.1-0(activate.1(Y), n__take.1-0(activate.1(X), activate.0(L)))
activate.1(n__length.0(X)) → length.0(activate.0(X))
activate.1(X) → X
activate.0(n__take.1-1(X1, X2)) → take.1-1(activate.1(X1), activate.1(X2))
0. → n__0.
s.0(X) → n__s.0(X)
length.0(cons.1-1(X, L)) → s.1(n__length.1(activate.1(L)))
eq.0-1(X, Y) → false.
activate.1(n__length.1(X)) → length.1(activate.1(X))
activate.0(n__inf.1(X)) → inf.1(activate.1(X))
activate.1(n__s.0(X)) → s.0(X)
take.1-0(s.0(X), cons.0-0(Y, L)) → cons.0-0(activate.0(Y), n__take.0-0(activate.0(X), activate.0(L)))
eq.1-1(n__0., n__0.) → true.
take.1-0(s.1(X), cons.0-0(Y, L)) → cons.0-0(activate.0(Y), n__take.1-0(activate.1(X), activate.0(L)))
length.0(cons.0-0(X, L)) → s.1(n__length.0(activate.0(L)))
eq.1-1(n__s.1(X), n__s.1(Y)) → eq.1-1(activate.1(X), activate.1(Y))
eq.1-1(X, Y) → false.
take.1-0(s.0(X), cons.1-1(Y, L)) → cons.1-0(activate.1(Y), n__take.0-1(activate.0(X), activate.1(L)))
take.1-1(0., X) → nil.
take.0-0(X1, X2) → n__take.0-0(X1, X2)
length.0(cons.0-1(X, L)) → s.1(n__length.1(activate.1(L)))
activate.0(n__take.1-0(X1, X2)) → take.1-0(activate.1(X1), activate.0(X2))
activate.0(n__take.0-0(X1, X2)) → take.0-0(activate.0(X1), activate.0(X2))
eq.1-1(n__s.0(X), n__s.1(Y)) → eq.0-1(activate.0(X), activate.1(Y))
length.0(nil.) → 0.
inf.0(X) → cons.0-0(X, n__inf.1(n__s.0(X)))
eq.0-0(X, Y) → false.
activate.0(n__inf.0(X)) → inf.0(activate.0(X))
take.1-0(0., X) → nil.

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Narrowing

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Narrowing

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ SemLabProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                ↳ SemLabProof2

          ↳ QDP

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

TAKE.1-0(s.1(X), cons.0-1(Y, L)) → ACTIVATE.1(X)
ACTIVATE.0(n__take.0-0(n__take.0-1(y0, y1), n__take.0-1(x0, x1))) → TAKE.0-0(take.0-1(activate.0(y0), activate.1(y1)), take.0-1(activate.0(x0), activate.1(x1)))
ACTIVATE.1(n__length.1(X)) → ACTIVATE.1(X)
ACTIVATE.0(n__take.0-1(n__take.0-1(y0, y1), x0)) → TAKE.0-1(take.0-1(activate.0(y0), activate.1(y1)), x0)
ACTIVATE.0(n__take.1-0(y0, x0)) → TAKE.1-0(y0, x0)
ACTIVATE.0(n__take.1-0(n__s.0(y0), n__take.1-0(x0, x1))) → TAKE.1-0(s.0(y0), take.1-0(activate.1(x0), activate.0(x1)))
ACTIVATE.0(n__take.1-0(n__length.1(y0), n__inf.1(x0))) → TAKE.1-0(length.1(activate.1(y0)), inf.1(activate.1(x0)))
ACTIVATE.0(n__take.1-1(y0, n__length.0(x0))) → TAKE.1-1(y0, length.0(activate.0(x0)))
ACTIVATE.0(n__take.1-0(X1, X2)) → ACTIVATE.1(X1)
ACTIVATE.0(n__take.0-0(y0, n__take.1-1(x0, x1))) → TAKE.0-0(y0, take.1-1(activate.1(x0), activate.1(x1)))
ACTIVATE.0(n__take.1-0(n__length.1(y0), n__take.1-0(x0, x1))) → TAKE.1-0(length.1(activate.1(y0)), take.1-0(activate.1(x0), activate.0(x1)))
ACTIVATE.0(n__take.1-0(n__length.0(y0), n__take.1-1(x0, x1))) → TAKE.1-0(length.0(activate.0(y0)), take.1-1(activate.1(x0), activate.1(x1)))
ACTIVATE.0(n__take.1-1(n__s.0(y0), n__length.1(x0))) → TAKE.1-1(s.0(y0), length.1(activate.1(x0)))
ACTIVATE.0(n__take.0-1(n__take.0-1(y0, y1), n__length.1(x0))) → TAKE.0-1(take.0-1(activate.0(y0), activate.1(y1)), length.1(activate.1(x0)))
TAKE.1-0(s.0(X), cons.1-0(Y, L)) → ACTIVATE.0(L)
LENGTH.0(cons.1-0(X, L)) → ACTIVATE.0(L)
ACTIVATE.0(n__take.1-0(n__length.1(y0), x0)) → TAKE.1-0(length.1(activate.1(y0)), x0)
ACTIVATE.1(n__length.0(n__inf.0(x0))) → LENGTH.0(inf.0(activate.0(x0)))
ACTIVATE.0(n__take.0-1(n__take.1-0(y0, y1), n__length.0(x0))) → TAKE.0-1(take.1-0(activate.1(y0), activate.0(y1)), length.0(activate.0(x0)))
ACTIVATE.0(n__take.0-0(n__inf.1(y0), n__take.1-1(x0, x1))) → TAKE.0-0(inf.1(activate.1(y0)), take.1-1(activate.1(x0), activate.1(x1)))
ACTIVATE.0(n__take.0-0(n__take.1-1(y0, y1), n__inf.1(x0))) → TAKE.0-0(take.1-1(activate.1(y0), activate.1(y1)), inf.1(activate.1(x0)))
ACTIVATE.0(n__take.0-0(n__take.1-1(y0, y1), n__take.0-0(x0, x1))) → TAKE.0-0(take.1-1(activate.1(y0), activate.1(y1)), take.0-0(activate.0(x0), activate.0(x1)))
ACTIVATE.0(n__take.0-0(n__take.1-0(y0, y1), n__take.0-1(x0, x1))) → TAKE.0-0(take.1-0(activate.1(y0), activate.0(y1)), take.0-1(activate.0(x0), activate.1(x1)))
ACTIVATE.0(n__take.0-0(n__take.0-1(y0, y1), n__take.1-0(x0, x1))) → TAKE.0-0(take.0-1(activate.0(y0), activate.1(y1)), take.1-0(activate.1(x0), activate.0(x1)))
ACTIVATE.0(n__take.0-0(n__take.0-0(y0, y1), n__take.1-1(x0, x1))) → TAKE.0-0(take.0-0(activate.0(y0), activate.0(y1)), take.1-1(activate.1(x0), activate.1(x1)))
ACTIVATE.0(n__take.0-0(n__take.1-1(y0, y1), n__take.0-1(x0, x1))) → TAKE.0-0(take.1-1(activate.1(y0), activate.1(y1)), take.0-1(activate.0(x0), activate.1(x1)))
ACTIVATE.0(n__take.0-0(n__take.0-1(y0, y1), n__take.1-1(x0, x1))) → TAKE.0-0(take.0-1(activate.0(y0), activate.1(y1)), take.1-1(activate.1(x0), activate.1(x1)))
ACTIVATE.1(n__length.0(n__take.1-1(x0, x1))) → LENGTH.0(take.1-1(activate.1(x0), activate.1(x1)))
ACTIVATE.0(n__take.1-0(n__s.0(y0), n__take.0-1(x0, x1))) → TAKE.1-0(s.0(y0), take.0-1(activate.0(x0), activate.1(x1)))
TAKE.1-0(s.0(X), cons.1-1(Y, L)) → ACTIVATE.1(L)
TAKE.1-0(s.1(X), cons.1-0(Y, L)) → ACTIVATE.0(L)
ACTIVATE.0(n__take.0-0(n__inf.0(y0), x0)) → TAKE.0-0(inf.0(activate.0(y0)), x0)
LENGTH.0(cons.1-1(X, L)) → ACTIVATE.1(L)
ACTIVATE.0(n__take.0-0(y0, n__inf.0(x0))) → TAKE.0-0(y0, inf.0(activate.0(x0)))
ACTIVATE.0(n__take.1-1(y0, n__length.1(x0))) → TAKE.1-1(y0, length.1(activate.1(x0)))
ACTIVATE.0(n__take.0-0(n__take.0-0(y0, y1), x0)) → TAKE.0-0(take.0-0(activate.0(y0), activate.0(y1)), x0)
ACTIVATE.0(n__take.0-0(n__take.1-0(y0, y1), x0)) → TAKE.0-0(take.1-0(activate.1(y0), activate.0(y1)), x0)
ACTIVATE.0(n__take.1-0(n__length.1(y0), n__take.0-1(x0, x1))) → TAKE.1-0(length.1(activate.1(y0)), take.0-1(activate.0(x0), activate.1(x1)))
ACTIVATE.0(n__take.1-0(X1, X2)) → ACTIVATE.0(X2)
ACTIVATE.1(n__length.1(n__length.1(x0))) → LENGTH.1(length.1(activate.1(x0)))
ACTIVATE.0(n__take.0-0(y0, n__take.1-0(x0, x1))) → TAKE.0-0(y0, take.1-0(activate.1(x0), activate.0(x1)))
TAKE.1-0(s.1(X), cons.0-1(Y, L)) → ACTIVATE.0(Y)
ACTIVATE.0(n__take.1-0(n__length.0(y0), n__inf.1(x0))) → TAKE.1-0(length.0(activate.0(y0)), inf.1(activate.1(x0)))
ACTIVATE.0(n__take.1-0(n__length.1(y0), n__inf.0(x0))) → TAKE.1-0(length.1(activate.1(y0)), inf.0(activate.0(x0)))
ACTIVATE.0(n__take.0-1(n__take.0-0(y0, y1), x0)) → TAKE.0-1(take.0-0(activate.0(y0), activate.0(y1)), x0)
TAKE.1-0(s.1(X), cons.1-0(Y, L)) → ACTIVATE.1(X)
ACTIVATE.0(n__take.1-0(n__s.0(y0), n__inf.0(x0))) → TAKE.1-0(s.0(y0), inf.0(activate.0(x0)))
ACTIVATE.0(n__take.0-1(X1, X2)) → ACTIVATE.1(X2)
ACTIVATE.0(n__take.0-0(y0, n__take.0-0(x0, x1))) → TAKE.0-0(y0, take.0-0(activate.0(x0), activate.0(x1)))
ACTIVATE.0(n__take.0-0(n__take.1-1(y0, y1), x0)) → TAKE.0-0(take.1-1(activate.1(y0), activate.1(y1)), x0)
ACTIVATE.1(n__length.1(n__length.0(x0))) → LENGTH.1(length.0(activate.0(x0)))
ACTIVATE.0(n__take.0-1(n__take.1-1(y0, y1), n__length.1(x0))) → TAKE.0-1(take.1-1(activate.1(y0), activate.1(y1)), length.1(activate.1(x0)))
ACTIVATE.0(n__take.0-0(n__inf.0(y0), n__take.1-0(x0, x1))) → TAKE.0-0(inf.0(activate.0(y0)), take.1-0(activate.1(x0), activate.0(x1)))
ACTIVATE.0(n__take.0-0(n__take.1-0(y0, y1), n__inf.0(x0))) → TAKE.0-0(take.1-0(activate.1(y0), activate.0(y1)), inf.0(activate.0(x0)))
ACTIVATE.0(n__take.1-1(n__s.0(y0), x0)) → TAKE.1-1(s.0(y0), x0)
TAKE.1-0(s.1(X), cons.0-1(Y, L)) → ACTIVATE.1(L)
ACTIVATE.0(n__take.1-0(n__length.0(y0), n__inf.0(x0))) → TAKE.1-0(length.0(activate.0(y0)), inf.0(activate.0(x0)))
ACTIVATE.0(n__take.0-1(n__inf.1(y0), x0)) → TAKE.0-1(inf.1(activate.1(y0)), x0)
TAKE.1-0(s.0(X), cons.0-0(Y, L)) → ACTIVATE.0(L)
ACTIVATE.0(n__take.1-1(n__s.1(y0), x0)) → TAKE.1-1(s.1(y0), x0)
ACTIVATE.0(n__take.1-1(n__s.1(y0), n__length.0(x0))) → TAKE.1-1(s.1(y0), length.0(activate.0(x0)))
ACTIVATE.0(n__take.1-1(n__s.0(y0), n__length.0(x0))) → TAKE.1-1(s.0(y0), length.0(activate.0(x0)))
ACTIVATE.0(n__take.1-0(n__s.0(y0), n__take.1-1(x0, x1))) → TAKE.1-0(s.0(y0), take.1-1(activate.1(x0), activate.1(x1)))
TAKE.1-0(s.0(X), cons.1-1(Y, L)) → ACTIVATE.0(X)
ACTIVATE.0(n__take.0-0(n__take.1-0(y0, y1), n__take.1-0(x0, x1))) → TAKE.0-0(take.1-0(activate.1(y0), activate.0(y1)), take.1-0(activate.1(x0), activate.0(x1)))
ACTIVATE.0(n__take.0-1(n__inf.0(y0), n__length.0(x0))) → TAKE.0-1(inf.0(activate.0(y0)), length.0(activate.0(x0)))
ACTIVATE.1(n__length.0(x0)) → LENGTH.0(x0)
ACTIVATE.0(n__take.0-0(n__take.1-1(y0, y1), n__take.1-1(x0, x1))) → TAKE.0-0(take.1-1(activate.1(y0), activate.1(y1)), take.1-1(activate.1(x0), activate.1(x1)))
ACTIVATE.1(n__length.0(X)) → ACTIVATE.0(X)
ACTIVATE.0(n__take.1-0(y0, n__take.0-0(x0, x1))) → TAKE.1-0(y0, take.0-0(activate.0(x0), activate.0(x1)))
ACTIVATE.0(n__take.0-1(n__take.0-0(y0, y1), n__length.1(x0))) → TAKE.0-1(take.0-0(activate.0(y0), activate.0(y1)), length.1(activate.1(x0)))
ACTIVATE.0(n__take.0-1(n__take.0-1(y0, y1), n__length.0(x0))) → TAKE.0-1(take.0-1(activate.0(y0), activate.1(y1)), length.0(activate.0(x0)))
ACTIVATE.0(n__take.1-0(n__s.1(y0), n__inf.0(x0))) → TAKE.1-0(s.1(y0), inf.0(activate.0(x0)))
ACTIVATE.0(n__take.0-1(y0, x0)) → TAKE.0-1(y0, x0)
ACTIVATE.1(n__length.0(n__inf.1(x0))) → LENGTH.0(inf.1(activate.1(x0)))
TAKE.1-0(s.1(X), cons.1-1(Y, L)) → ACTIVATE.1(L)
TAKE.1-0(s.1(X), cons.1-1(Y, L)) → ACTIVATE.1(Y)
TAKE.1-0(s.1(X), cons.0-0(Y, L)) → ACTIVATE.0(L)
ACTIVATE.0(n__take.1-0(n__s.1(y0), n__inf.1(x0))) → TAKE.1-0(s.1(y0), inf.1(activate.1(x0)))
ACTIVATE.0(n__take.0-1(X1, X2)) → ACTIVATE.0(X1)
ACTIVATE.0(n__take.1-0(n__length.1(y0), n__take.1-1(x0, x1))) → TAKE.1-0(length.1(activate.1(y0)), take.1-1(activate.1(x0), activate.1(x1)))
TAKE.1-0(s.1(X), cons.1-1(Y, L)) → ACTIVATE.1(X)
TAKE.1-0(s.1(X), cons.0-0(Y, L)) → ACTIVATE.1(X)
ACTIVATE.0(n__take.0-0(n__inf.0(y0), n__take.1-1(x0, x1))) → TAKE.0-0(inf.0(activate.0(y0)), take.1-1(activate.1(x0), activate.1(x1)))
ACTIVATE.0(n__take.0-0(n__inf.1(y0), n__take.1-0(x0, x1))) → TAKE.0-0(inf.1(activate.1(y0)), take.1-0(activate.1(x0), activate.0(x1)))
ACTIVATE.0(n__take.0-0(n__take.1-0(y0, y1), n__inf.1(x0))) → TAKE.0-0(take.1-0(activate.1(y0), activate.0(y1)), inf.1(activate.1(x0)))
ACTIVATE.0(n__take.0-0(n__take.1-1(y0, y1), n__inf.0(x0))) → TAKE.0-0(take.1-1(activate.1(y0), activate.1(y1)), inf.0(activate.0(x0)))
TAKE.1-0(s.0(X), cons.0-0(Y, L)) → ACTIVATE.0(X)
TAKE.1-0(s.0(X), cons.1-0(Y, L)) → ACTIVATE.0(X)
ACTIVATE.0(n__take.0-0(n__take.1-0(y0, y1), n__take.0-0(x0, x1))) → TAKE.0-0(take.1-0(activate.1(y0), activate.0(y1)), take.0-0(activate.0(x0), activate.0(x1)))
ACTIVATE.0(n__take.0-0(n__take.0-0(y0, y1), n__take.1-0(x0, x1))) → TAKE.0-0(take.0-0(activate.0(y0), activate.0(y1)), take.1-0(activate.1(x0), activate.0(x1)))
ACTIVATE.0(n__take.0-0(y0, n__take.0-1(x0, x1))) → TAKE.0-0(y0, take.0-1(activate.0(x0), activate.1(x1)))
ACTIVATE.0(n__take.0-0(y0, x0)) → TAKE.0-0(y0, x0)
ACTIVATE.1(n__length.0(n__take.1-0(x0, x1))) → LENGTH.0(take.1-0(activate.1(x0), activate.0(x1)))
ACTIVATE.0(n__take.1-1(n__length.1(y0), x0)) → TAKE.1-1(length.1(activate.1(y0)), x0)
ACTIVATE.0(n__take.1-1(n__s.1(y0), n__length.1(x0))) → TAKE.1-1(s.1(y0), length.1(activate.1(x0)))
ACTIVATE.0(n__take.0-0(n__inf.0(y0), n__take.0-0(x0, x1))) → TAKE.0-0(inf.0(activate.0(y0)), take.0-0(activate.0(x0), activate.0(x1)))
ACTIVATE.0(n__take.0-0(n__take.0-0(y0, y1), n__inf.0(x0))) → TAKE.0-0(take.0-0(activate.0(y0), activate.0(y1)), inf.0(activate.0(x0)))
ACTIVATE.0(n__take.0-1(n__take.1-1(y0, y1), x0)) → TAKE.0-1(take.1-1(activate.1(y0), activate.1(y1)), x0)
ACTIVATE.1(n__length.0(n__take.0-0(x0, x1))) → LENGTH.0(take.0-0(activate.0(x0), activate.0(x1)))
ACTIVATE.0(n__take.0-0(n__inf.1(y0), n__inf.1(x0))) → TAKE.0-0(inf.1(activate.1(y0)), inf.1(activate.1(x0)))
ACTIVATE.0(n__take.0-0(n__take.0-1(y0, y1), x0)) → TAKE.0-0(take.0-1(activate.0(y0), activate.1(y1)), x0)
TAKE.1-0(s.0(X), cons.1-0(Y, L)) → ACTIVATE.1(Y)
ACTIVATE.0(n__take.0-0(n__take.0-0(y0, y1), n__take.0-0(x0, x1))) → TAKE.0-0(take.0-0(activate.0(y0), activate.0(y1)), take.0-0(activate.0(x0), activate.0(x1)))
ACTIVATE.0(n__take.0-0(X1, X2)) → ACTIVATE.0(X2)
ACTIVATE.0(n__take.1-0(y0, n__take.0-1(x0, x1))) → TAKE.1-0(y0, take.0-1(activate.0(x0), activate.1(x1)))
ACTIVATE.0(n__take.1-1(n__length.1(y0), n__length.0(x0))) → TAKE.1-1(length.1(activate.1(y0)), length.0(activate.0(x0)))
ACTIVATE.0(n__take.1-1(n__length.0(y0), n__length.1(x0))) → TAKE.1-1(length.0(activate.0(y0)), length.1(activate.1(x0)))
ACTIVATE.0(n__take.1-1(X1, X2)) → ACTIVATE.1(X1)
ACTIVATE.0(n__take.1-1(y0, x0)) → TAKE.1-1(y0, x0)
LENGTH.0(cons.0-1(X, L)) → ACTIVATE.1(L)
ACTIVATE.0(n__take.1-0(n__s.0(y0), x0)) → TAKE.1-0(s.0(y0), x0)
ACTIVATE.0(n__take.1-0(n__length.0(y0), n__take.0-0(x0, x1))) → TAKE.1-0(length.0(activate.0(y0)), take.0-0(activate.0(x0), activate.0(x1)))
ACTIVATE.0(n__inf.0(X)) → ACTIVATE.0(X)
ACTIVATE.0(n__take.1-0(n__s.0(y0), n__take.0-0(x0, x1))) → TAKE.1-0(s.0(y0), take.0-0(activate.0(x0), activate.0(x1)))
ACTIVATE.0(n__take.1-0(n__s.1(y0), n__take.1-0(x0, x1))) → TAKE.1-0(s.1(y0), take.1-0(activate.1(x0), activate.0(x1)))
ACTIVATE.0(n__take.1-0(n__s.1(y0), n__take.0-1(x0, x1))) → TAKE.1-0(s.1(y0), take.0-1(activate.0(x0), activate.1(x1)))
LENGTH.0(cons.0-0(X, L)) → ACTIVATE.0(L)
ACTIVATE.0(n__take.1-1(n__length.0(y0), n__length.0(x0))) → TAKE.1-1(length.0(activate.0(y0)), length.0(activate.0(x0)))
ACTIVATE.0(n__take.0-0(y0, n__inf.1(x0))) → TAKE.0-0(y0, inf.1(activate.1(x0)))
ACTIVATE.1(n__length.0(n__take.0-1(x0, x1))) → LENGTH.0(take.0-1(activate.0(x0), activate.1(x1)))
ACTIVATE.0(n__take.1-0(n__s.1(y0), x0)) → TAKE.1-0(s.1(y0), x0)
ACTIVATE.0(n__take.1-0(n__s.1(y0), n__take.1-1(x0, x1))) → TAKE.1-0(s.1(y0), take.1-1(activate.1(x0), activate.1(x1)))
ACTIVATE.0(n__take.0-0(n__take.0-1(y0, y1), n__take.0-0(x0, x1))) → TAKE.0-0(take.0-1(activate.0(y0), activate.1(y1)), take.0-0(activate.0(x0), activate.0(x1)))
ACTIVATE.0(n__take.0-0(n__take.0-0(y0, y1), n__take.0-1(x0, x1))) → TAKE.0-0(take.0-0(activate.0(y0), activate.0(y1)), take.0-1(activate.0(x0), activate.1(x1)))
ACTIVATE.0(n__take.0-1(n__inf.0(y0), x0)) → TAKE.0-1(inf.0(activate.0(y0)), x0)
ACTIVATE.0(n__take.0-1(y0, n__length.0(x0))) → TAKE.0-1(y0, length.0(activate.0(x0)))
ACTIVATE.0(n__take.1-1(X1, X2)) → ACTIVATE.1(X2)
ACTIVATE.0(n__take.0-0(n__inf.1(y0), n__take.0-1(x0, x1))) → TAKE.0-0(inf.1(activate.1(y0)), take.0-1(activate.0(x0), activate.1(x1)))
ACTIVATE.0(n__take.0-0(n__take.0-1(y0, y1), n__inf.1(x0))) → TAKE.0-0(take.0-1(activate.0(y0), activate.1(y1)), inf.1(activate.1(x0)))
ACTIVATE.0(n__take.1-0(y0, n__inf.1(x0))) → TAKE.1-0(y0, inf.1(activate.1(x0)))
ACTIVATE.0(n__inf.1(X)) → ACTIVATE.1(X)
TAKE.1-0(s.0(X), cons.0-1(Y, L)) → ACTIVATE.1(L)
ACTIVATE.0(n__take.1-0(y0, n__inf.0(x0))) → TAKE.1-0(y0, inf.0(activate.0(x0)))
ACTIVATE.0(n__take.1-0(y0, n__take.1-0(x0, x1))) → TAKE.1-0(y0, take.1-0(activate.1(x0), activate.0(x1)))
ACTIVATE.0(n__take.1-0(n__s.1(y0), n__take.0-0(x0, x1))) → TAKE.1-0(s.1(y0), take.0-0(activate.0(x0), activate.0(x1)))
ACTIVATE.0(n__take.0-1(n__inf.1(y0), n__length.1(x0))) → TAKE.0-1(inf.1(activate.1(y0)), length.1(activate.1(x0)))
ACTIVATE.0(n__take.1-1(n__length.1(y0), n__length.1(x0))) → TAKE.1-1(length.1(activate.1(y0)), length.1(activate.1(x0)))
ACTIVATE.0(n__take.0-0(n__inf.0(y0), n__take.0-1(x0, x1))) → TAKE.0-0(inf.0(activate.0(y0)), take.0-1(activate.0(x0), activate.1(x1)))
ACTIVATE.0(n__take.0-0(n__inf.1(y0), n__take.0-0(x0, x1))) → TAKE.0-0(inf.1(activate.1(y0)), take.0-0(activate.0(x0), activate.0(x1)))
ACTIVATE.0(n__take.0-0(n__take.0-0(y0, y1), n__inf.1(x0))) → TAKE.0-0(take.0-0(activate.0(y0), activate.0(y1)), inf.1(activate.1(x0)))
ACTIVATE.0(n__take.0-0(n__take.0-1(y0, y1), n__inf.0(x0))) → TAKE.0-0(take.0-1(activate.0(y0), activate.1(y1)), inf.0(activate.0(x0)))
ACTIVATE.0(n__take.0-1(n__take.1-0(y0, y1), n__length.1(x0))) → TAKE.0-1(take.1-0(activate.1(y0), activate.0(y1)), length.1(activate.1(x0)))
ACTIVATE.0(n__take.0-1(n__take.1-1(y0, y1), n__length.0(x0))) → TAKE.0-1(take.1-1(activate.1(y0), activate.1(y1)), length.0(activate.0(x0)))
TAKE.1-0(s.1(X), cons.0-0(Y, L)) → ACTIVATE.0(Y)
TAKE.1-0(s.0(X), cons.0-1(Y, L)) → ACTIVATE.0(Y)
ACTIVATE.0(n__take.1-1(n__length.0(y0), x0)) → TAKE.1-1(length.0(activate.0(y0)), x0)
ACTIVATE.0(n__take.0-0(X1, X2)) → ACTIVATE.0(X1)
ACTIVATE.0(n__take.0-1(n__inf.0(y0), n__length.1(x0))) → TAKE.0-1(inf.0(activate.0(y0)), length.1(activate.1(x0)))
ACTIVATE.0(n__take.0-1(n__inf.1(y0), n__length.0(x0))) → TAKE.0-1(inf.1(activate.1(y0)), length.0(activate.0(x0)))
ACTIVATE.0(n__take.0-1(n__take.0-0(y0, y1), n__length.0(x0))) → TAKE.0-1(take.0-0(activate.0(y0), activate.0(y1)), length.0(activate.0(x0)))
ACTIVATE.0(n__take.0-1(n__take.1-0(y0, y1), x0)) → TAKE.0-1(take.1-0(activate.1(y0), activate.0(y1)), x0)
ACTIVATE.0(n__take.0-1(y0, n__length.1(x0))) → TAKE.0-1(y0, length.1(activate.1(x0)))
ACTIVATE.0(n__take.1-0(n__s.0(y0), n__inf.1(x0))) → TAKE.1-0(s.0(y0), inf.1(activate.1(x0)))
ACTIVATE.1(n__length.1(x0)) → LENGTH.1(x0)
ACTIVATE.0(n__take.1-0(n__length.1(y0), n__take.0-0(x0, x1))) → TAKE.1-0(length.1(activate.1(y0)), take.0-0(activate.0(x0), activate.0(x1)))
ACTIVATE.0(n__take.1-0(n__length.0(y0), n__take.0-1(x0, x1))) → TAKE.1-0(length.0(activate.0(y0)), take.0-1(activate.0(x0), activate.1(x1)))
TAKE.1-0(s.1(X), cons.1-0(Y, L)) → ACTIVATE.1(Y)
TAKE.1-0(s.0(X), cons.1-1(Y, L)) → ACTIVATE.1(Y)
ACTIVATE.0(n__take.0-0(n__take.1-1(y0, y1), n__take.1-0(x0, x1))) → TAKE.0-0(take.1-1(activate.1(y0), activate.1(y1)), take.1-0(activate.1(x0), activate.0(x1)))
ACTIVATE.0(n__take.0-0(n__take.1-0(y0, y1), n__take.1-1(x0, x1))) → TAKE.0-0(take.1-0(activate.1(y0), activate.0(y1)), take.1-1(activate.1(x0), activate.1(x1)))
ACTIVATE.0(n__take.0-0(n__inf.1(y0), x0)) → TAKE.0-0(inf.1(activate.1(y0)), x0)
ACTIVATE.0(n__take.1-0(n__length.0(y0), n__take.1-0(x0, x1))) → TAKE.1-0(length.0(activate.0(y0)), take.1-0(activate.1(x0), activate.0(x1)))
ACTIVATE.0(n__take.1-0(y0, n__take.1-1(x0, x1))) → TAKE.1-0(y0, take.1-1(activate.1(x0), activate.1(x1)))
ACTIVATE.0(n__take.0-0(n__inf.0(y0), n__inf.0(x0))) → TAKE.0-0(inf.0(activate.0(y0)), inf.0(activate.0(x0)))
ACTIVATE.0(n__take.0-0(n__inf.0(y0), n__inf.1(x0))) → TAKE.0-0(inf.0(activate.0(y0)), inf.1(activate.1(x0)))
ACTIVATE.0(n__take.0-0(n__inf.1(y0), n__inf.0(x0))) → TAKE.0-0(inf.1(activate.1(y0)), inf.0(activate.0(x0)))
TAKE.1-0(s.0(X), cons.0-0(Y, L)) → ACTIVATE.0(Y)
ACTIVATE.0(n__take.1-0(n__length.0(y0), x0)) → TAKE.1-0(length.0(activate.0(y0)), x0)
TAKE.1-0(s.0(X), cons.0-1(Y, L)) → ACTIVATE.0(X)

The TRS R consists of the following rules:

take.1-0(s.0(X), cons.0-1(Y, L)) → cons.0-0(activate.0(Y), n__take.0-1(activate.0(X), activate.1(L)))
activate.1(n__s.1(X)) → s.1(X)
take.1-0(s.0(X), cons.1-0(Y, L)) → cons.1-0(activate.1(Y), n__take.0-0(activate.0(X), activate.0(L)))
eq.1-0(X, Y) → false.
take.1-1(X1, X2) → n__take.1-1(X1, X2)
activate.1(n__0.) → 0.
s.1(X) → n__s.1(X)
inf.0(X) → n__inf.0(X)
eq.1-1(n__s.1(X), n__s.0(Y)) → eq.1-0(activate.1(X), activate.0(Y))
inf.1(X) → cons.1-0(X, n__inf.1(n__s.1(X)))
take.1-0(X1, X2) → n__take.1-0(X1, X2)
inf.1(X) → n__inf.1(X)
activate.0(n__take.0-1(X1, X2)) → take.0-1(activate.0(X1), activate.1(X2))
length.0(cons.1-0(X, L)) → s.1(n__length.0(activate.0(L)))
length.0(X) → n__length.0(X)
take.1-0(s.1(X), cons.0-1(Y, L)) → cons.0-0(activate.0(Y), n__take.1-1(activate.1(X), activate.1(L)))
length.1(X) → n__length.1(X)
take.0-1(X1, X2) → n__take.0-1(X1, X2)
take.1-0(s.1(X), cons.1-1(Y, L)) → cons.1-0(activate.1(Y), n__take.1-1(activate.1(X), activate.1(L)))
activate.0(X) → X
eq.1-1(n__s.0(X), n__s.0(Y)) → eq.0-0(activate.0(X), activate.0(Y))
take.1-0(s.1(X), cons.1-0(Y, L)) → cons.1-0(activate.1(Y), n__take.1-0(activate.1(X), activate.0(L)))
activate.1(n__length.0(X)) → length.0(activate.0(X))
activate.1(X) → X
activate.0(n__take.1-1(X1, X2)) → take.1-1(activate.1(X1), activate.1(X2))
0. → n__0.
s.0(X) → n__s.0(X)
length.0(cons.1-1(X, L)) → s.1(n__length.1(activate.1(L)))
eq.0-1(X, Y) → false.
activate.1(n__length.1(X)) → length.1(activate.1(X))
activate.0(n__inf.1(X)) → inf.1(activate.1(X))
activate.1(n__s.0(X)) → s.0(X)
take.1-0(s.0(X), cons.0-0(Y, L)) → cons.0-0(activate.0(Y), n__take.0-0(activate.0(X), activate.0(L)))
eq.1-1(n__0., n__0.) → true.
take.1-0(s.1(X), cons.0-0(Y, L)) → cons.0-0(activate.0(Y), n__take.1-0(activate.1(X), activate.0(L)))
length.0(cons.0-0(X, L)) → s.1(n__length.0(activate.0(L)))
eq.1-1(n__s.1(X), n__s.1(Y)) → eq.1-1(activate.1(X), activate.1(Y))
eq.1-1(X, Y) → false.
take.1-0(s.0(X), cons.1-1(Y, L)) → cons.1-0(activate.1(Y), n__take.0-1(activate.0(X), activate.1(L)))
take.1-1(0., X) → nil.
take.0-0(X1, X2) → n__take.0-0(X1, X2)
length.0(cons.0-1(X, L)) → s.1(n__length.1(activate.1(L)))
activate.0(n__take.1-0(X1, X2)) → take.1-0(activate.1(X1), activate.0(X2))
activate.0(n__take.0-0(X1, X2)) → take.0-0(activate.0(X1), activate.0(X2))
eq.1-1(n__s.0(X), n__s.1(Y)) → eq.0-1(activate.0(X), activate.1(Y))
length.0(nil.) → 0.
inf.0(X) → cons.0-0(X, n__inf.1(n__s.0(X)))
eq.0-0(X, Y) → false.
activate.0(n__inf.0(X)) → inf.0(activate.0(X))
take.1-0(0., X) → nil.

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 88 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Narrowing

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Narrowing

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ SemLabProof

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                                      ↳ QDP

                                                                                                                                                ↳ SemLabProof2

          ↳ QDP

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

TAKE.1-0(s.0(X), cons.1-0(Y, L)) → ACTIVATE.0(X)
TAKE.1-0(s.1(X), cons.0-1(Y, L)) → ACTIVATE.1(X)
ACTIVATE.1(n__length.0(n__take.1-0(x0, x1))) → LENGTH.0(take.1-0(activate.1(x0), activate.0(x1)))
ACTIVATE.1(n__length.1(X)) → ACTIVATE.1(X)
ACTIVATE.0(n__take.1-0(y0, x0)) → TAKE.1-0(y0, x0)
ACTIVATE.0(n__take.1-0(n__s.0(y0), n__take.1-0(x0, x1))) → TAKE.1-0(s.0(y0), take.1-0(activate.1(x0), activate.0(x1)))
ACTIVATE.0(n__take.1-0(n__length.1(y0), n__inf.1(x0))) → TAKE.1-0(length.1(activate.1(y0)), inf.1(activate.1(x0)))
ACTIVATE.0(n__take.1-0(X1, X2)) → ACTIVATE.1(X1)
ACTIVATE.1(n__length.0(n__take.0-0(x0, x1))) → LENGTH.0(take.0-0(activate.0(x0), activate.0(x1)))
ACTIVATE.0(n__take.1-0(n__length.1(y0), n__take.1-0(x0, x1))) → TAKE.1-0(length.1(activate.1(y0)), take.1-0(activate.1(x0), activate.0(x1)))
ACTIVATE.0(n__take.1-0(n__length.0(y0), n__take.1-1(x0, x1))) → TAKE.1-0(length.0(activate.0(y0)), take.1-1(activate.1(x0), activate.1(x1)))
TAKE.1-0(s.0(X), cons.1-0(Y, L)) → ACTIVATE.1(Y)
ACTIVATE.0(n__take.0-0(X1, X2)) → ACTIVATE.0(X2)
ACTIVATE.0(n__take.1-0(y0, n__take.0-1(x0, x1))) → TAKE.1-0(y0, take.0-1(activate.0(x0), activate.1(x1)))
TAKE.1-0(s.0(X), cons.1-0(Y, L)) → ACTIVATE.0(L)
LENGTH.0(cons.1-0(X, L)) → ACTIVATE.0(L)
ACTIVATE.0(n__take.1-0(n__length.1(y0), x0)) → TAKE.1-0(length.1(activate.1(y0)), x0)
ACTIVATE.1(n__length.0(n__inf.0(x0))) → LENGTH.0(inf.0(activate.0(x0)))
ACTIVATE.0(n__take.1-1(X1, X2)) → ACTIVATE.1(X1)
LENGTH.0(cons.0-1(X, L)) → ACTIVATE.1(L)
ACTIVATE.0(n__take.1-0(n__s.0(y0), x0)) → TAKE.1-0(s.0(y0), x0)
ACTIVATE.0(n__take.1-0(n__s.0(y0), n__take.0-1(x0, x1))) → TAKE.1-0(s.0(y0), take.0-1(activate.0(x0), activate.1(x1)))
ACTIVATE.1(n__length.0(n__take.1-1(x0, x1))) → LENGTH.0(take.1-1(activate.1(x0), activate.1(x1)))
ACTIVATE.0(n__inf.0(X)) → ACTIVATE.0(X)
ACTIVATE.0(n__take.1-0(n__length.0(y0), n__take.0-0(x0, x1))) → TAKE.1-0(length.0(activate.0(y0)), take.0-0(activate.0(x0), activate.0(x1)))
ACTIVATE.0(n__take.1-0(n__s.1(y0), n__take.1-0(x0, x1))) → TAKE.1-0(s.1(y0), take.1-0(activate.1(x0), activate.0(x1)))
ACTIVATE.0(n__take.1-0(n__s.0(y0), n__take.0-0(x0, x1))) → TAKE.1-0(s.0(y0), take.0-0(activate.0(x0), activate.0(x1)))
TAKE.1-0(s.0(X), cons.1-1(Y, L)) → ACTIVATE.1(L)
ACTIVATE.0(n__take.1-0(n__s.1(y0), n__take.0-1(x0, x1))) → TAKE.1-0(s.1(y0), take.0-1(activate.0(x0), activate.1(x1)))
TAKE.1-0(s.1(X), cons.1-0(Y, L)) → ACTIVATE.0(L)
LENGTH.0(cons.0-0(X, L)) → ACTIVATE.0(L)
LENGTH.0(cons.1-1(X, L)) → ACTIVATE.1(L)
ACTIVATE.0(n__take.1-0(n__length.1(y0), n__take.0-1(x0, x1))) → TAKE.1-0(length.1(activate.1(y0)), take.0-1(activate.0(x0), activate.1(x1)))
ACTIVATE.0(n__take.1-0(X1, X2)) → ACTIVATE.0(X2)
TAKE.1-0(s.1(X), cons.0-1(Y, L)) → ACTIVATE.0(Y)
ACTIVATE.1(n__length.0(n__take.0-1(x0, x1))) → LENGTH.0(take.0-1(activate.0(x0), activate.1(x1)))
ACTIVATE.0(n__take.1-0(n__length.0(y0), n__inf.1(x0))) → TAKE.1-0(length.0(activate.0(y0)), inf.1(activate.1(x0)))
ACTIVATE.0(n__take.1-0(n__length.1(y0), n__inf.0(x0))) → TAKE.1-0(length.1(activate.1(y0)), inf.0(activate.0(x0)))
ACTIVATE.0(n__take.1-0(n__s.0(y0), n__inf.0(x0))) → TAKE.1-0(s.0(y0), inf.0(activate.0(x0)))
ACTIVATE.0(n__take.0-1(X1, X2)) → ACTIVATE.1(X2)
TAKE.1-0(s.1(X), cons.1-0(Y, L)) → ACTIVATE.1(X)
ACTIVATE.0(n__take.1-0(n__s.1(y0), x0)) → TAKE.1-0(s.1(y0), x0)
ACTIVATE.0(n__take.1-0(n__s.1(y0), n__take.1-1(x0, x1))) → TAKE.1-0(s.1(y0), take.1-1(activate.1(x0), activate.1(x1)))
ACTIVATE.0(n__take.1-1(X1, X2)) → ACTIVATE.1(X2)
ACTIVATE.0(n__take.1-0(y0, n__inf.1(x0))) → TAKE.1-0(y0, inf.1(activate.1(x0)))
ACTIVATE.0(n__inf.1(X)) → ACTIVATE.1(X)
TAKE.1-0(s.1(X), cons.0-1(Y, L)) → ACTIVATE.1(L)
TAKE.1-0(s.0(X), cons.0-1(Y, L)) → ACTIVATE.1(L)
ACTIVATE.0(n__take.1-0(y0, n__inf.0(x0))) → TAKE.1-0(y0, inf.0(activate.0(x0)))
ACTIVATE.0(n__take.1-0(y0, n__take.1-0(x0, x1))) → TAKE.1-0(y0, take.1-0(activate.1(x0), activate.0(x1)))
ACTIVATE.0(n__take.1-0(n__length.0(y0), n__inf.0(x0))) → TAKE.1-0(length.0(activate.0(y0)), inf.0(activate.0(x0)))
ACTIVATE.0(n__take.1-0(n__s.1(y0), n__take.0-0(x0, x1))) → TAKE.1-0(s.1(y0), take.0-0(activate.0(x0), activate.0(x1)))
TAKE.1-0(s.0(X), cons.0-0(Y, L)) → ACTIVATE.0(L)
ACTIVATE.0(n__take.1-0(n__s.0(y0), n__take.1-1(x0, x1))) → TAKE.1-0(s.0(y0), take.1-1(activate.1(x0), activate.1(x1)))
TAKE.1-0(s.0(X), cons.1-1(Y, L)) → ACTIVATE.0(X)
ACTIVATE.1(n__length.0(x0)) → LENGTH.0(x0)
TAKE.1-0(s.0(X), cons.0-1(Y, L)) → ACTIVATE.0(Y)
TAKE.1-0(s.1(X), cons.0-0(Y, L)) → ACTIVATE.0(Y)
ACTIVATE.1(n__length.0(X)) → ACTIVATE.0(X)
ACTIVATE.0(n__take.0-0(X1, X2)) → ACTIVATE.0(X1)
ACTIVATE.0(n__take.1-0(y0, n__take.0-0(x0, x1))) → TAKE.1-0(y0, take.0-0(activate.0(x0), activate.0(x1)))
ACTIVATE.0(n__take.1-0(n__s.1(y0), n__inf.0(x0))) → TAKE.1-0(s.1(y0), inf.0(activate.0(x0)))
ACTIVATE.1(n__length.0(n__inf.1(x0))) → LENGTH.0(inf.1(activate.1(x0)))
TAKE.1-0(s.1(X), cons.1-1(Y, L)) → ACTIVATE.1(L)
TAKE.1-0(s.1(X), cons.1-1(Y, L)) → ACTIVATE.1(Y)
ACTIVATE.0(n__take.1-0(n__s.1(y0), n__inf.1(x0))) → TAKE.1-0(s.1(y0), inf.1(activate.1(x0)))
TAKE.1-0(s.1(X), cons.0-0(Y, L)) → ACTIVATE.0(L)
ACTIVATE.0(n__take.0-1(X1, X2)) → ACTIVATE.0(X1)
ACTIVATE.0(n__take.1-0(n__s.0(y0), n__inf.1(x0))) → TAKE.1-0(s.0(y0), inf.1(activate.1(x0)))
ACTIVATE.0(n__take.1-0(n__length.1(y0), n__take.1-1(x0, x1))) → TAKE.1-0(length.1(activate.1(y0)), take.1-1(activate.1(x0), activate.1(x1)))
ACTIVATE.0(n__take.1-0(n__length.1(y0), n__take.0-0(x0, x1))) → TAKE.1-0(length.1(activate.1(y0)), take.0-0(activate.0(x0), activate.0(x1)))
ACTIVATE.0(n__take.1-0(n__length.0(y0), n__take.0-1(x0, x1))) → TAKE.1-0(length.0(activate.0(y0)), take.0-1(activate.0(x0), activate.1(x1)))
TAKE.1-0(s.0(X), cons.1-1(Y, L)) → ACTIVATE.1(Y)
TAKE.1-0(s.1(X), cons.1-0(Y, L)) → ACTIVATE.1(Y)
TAKE.1-0(s.1(X), cons.1-1(Y, L)) → ACTIVATE.1(X)
ACTIVATE.0(n__take.1-0(n__length.0(y0), n__take.1-0(x0, x1))) → TAKE.1-0(length.0(activate.0(y0)), take.1-0(activate.1(x0), activate.0(x1)))
ACTIVATE.0(n__take.1-0(y0, n__take.1-1(x0, x1))) → TAKE.1-0(y0, take.1-1(activate.1(x0), activate.1(x1)))
TAKE.1-0(s.1(X), cons.0-0(Y, L)) → ACTIVATE.1(X)
TAKE.1-0(s.0(X), cons.0-0(Y, L)) → ACTIVATE.0(Y)
ACTIVATE.0(n__take.1-0(n__length.0(y0), x0)) → TAKE.1-0(length.0(activate.0(y0)), x0)
TAKE.1-0(s.0(X), cons.0-0(Y, L)) → ACTIVATE.0(X)
TAKE.1-0(s.0(X), cons.0-1(Y, L)) → ACTIVATE.0(X)

The TRS R consists of the following rules:

take.1-0(s.0(X), cons.0-1(Y, L)) → cons.0-0(activate.0(Y), n__take.0-1(activate.0(X), activate.1(L)))
activate.1(n__s.1(X)) → s.1(X)
take.1-0(s.0(X), cons.1-0(Y, L)) → cons.1-0(activate.1(Y), n__take.0-0(activate.0(X), activate.0(L)))
eq.1-0(X, Y) → false.
take.1-1(X1, X2) → n__take.1-1(X1, X2)
activate.1(n__0.) → 0.
s.1(X) → n__s.1(X)
inf.0(X) → n__inf.0(X)
eq.1-1(n__s.1(X), n__s.0(Y)) → eq.1-0(activate.1(X), activate.0(Y))
inf.1(X) → cons.1-0(X, n__inf.1(n__s.1(X)))
take.1-0(X1, X2) → n__take.1-0(X1, X2)
inf.1(X) → n__inf.1(X)
activate.0(n__take.0-1(X1, X2)) → take.0-1(activate.0(X1), activate.1(X2))
length.0(cons.1-0(X, L)) → s.1(n__length.0(activate.0(L)))
length.0(X) → n__length.0(X)
take.1-0(s.1(X), cons.0-1(Y, L)) → cons.0-0(activate.0(Y), n__take.1-1(activate.1(X), activate.1(L)))
length.1(X) → n__length.1(X)
take.0-1(X1, X2) → n__take.0-1(X1, X2)
take.1-0(s.1(X), cons.1-1(Y, L)) → cons.1-0(activate.1(Y), n__take.1-1(activate.1(X), activate.1(L)))
activate.0(X) → X
eq.1-1(n__s.0(X), n__s.0(Y)) → eq.0-0(activate.0(X), activate.0(Y))
take.1-0(s.1(X), cons.1-0(Y, L)) → cons.1-0(activate.1(Y), n__take.1-0(activate.1(X), activate.0(L)))
activate.1(n__length.0(X)) → length.0(activate.0(X))
activate.1(X) → X
activate.0(n__take.1-1(X1, X2)) → take.1-1(activate.1(X1), activate.1(X2))
0. → n__0.
s.0(X) → n__s.0(X)
length.0(cons.1-1(X, L)) → s.1(n__length.1(activate.1(L)))
eq.0-1(X, Y) → false.
activate.1(n__length.1(X)) → length.1(activate.1(X))
activate.0(n__inf.1(X)) → inf.1(activate.1(X))
activate.1(n__s.0(X)) → s.0(X)
take.1-0(s.0(X), cons.0-0(Y, L)) → cons.0-0(activate.0(Y), n__take.0-0(activate.0(X), activate.0(L)))
eq.1-1(n__0., n__0.) → true.
take.1-0(s.1(X), cons.0-0(Y, L)) → cons.0-0(activate.0(Y), n__take.1-0(activate.1(X), activate.0(L)))
length.0(cons.0-0(X, L)) → s.1(n__length.0(activate.0(L)))
eq.1-1(n__s.1(X), n__s.1(Y)) → eq.1-1(activate.1(X), activate.1(Y))
eq.1-1(X, Y) → false.
take.1-0(s.0(X), cons.1-1(Y, L)) → cons.1-0(activate.1(Y), n__take.0-1(activate.0(X), activate.1(L)))
take.1-1(0., X) → nil.
take.0-0(X1, X2) → n__take.0-0(X1, X2)
length.0(cons.0-1(X, L)) → s.1(n__length.1(activate.1(L)))
activate.0(n__take.1-0(X1, X2)) → take.1-0(activate.1(X1), activate.0(X2))
activate.0(n__take.0-0(X1, X2)) → take.0-0(activate.0(X1), activate.0(X2))
eq.1-1(n__s.0(X), n__s.1(Y)) → eq.0-1(activate.0(X), activate.1(Y))
length.0(nil.) → 0.
inf.0(X) → cons.0-0(X, n__inf.1(n__s.0(X)))
eq.0-0(X, Y) → false.
activate.0(n__inf.0(X)) → inf.0(activate.0(X))
take.1-0(0., X) → nil.

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
As can be seen after transforming the QDP problem by semantic labelling [33] and then some rule deleting processors, only certain labelled rules and pairs can be used. Hence, we only have to consider all unlabelled pairs and rules (without the decreasing rules for quasi-models).

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ Narrowing

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ QDP

                                        ↳ Narrowing

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Narrowing

                                                                  ↳ QDP

                                                                    ↳ DependencyGraphProof

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

                                                                                        ↳ Narrowing

                                                                                          ↳ QDP

                                                                                            ↳ DependencyGraphProof

                                                                                              ↳ QDP

                                                                                                ↳ Narrowing

                                                                                                  ↳ QDP

                                                                                                    ↳ DependencyGraphProof

                                                                                                      ↳ QDP

                                                                                                        ↳ Narrowing

                                                                                                          ↳ QDP

                                                                                                            ↳ DependencyGraphProof

                                                                                                              ↳ QDP

                                                                                                                ↳ Narrowing

                                                                                                                  ↳ QDP

                                                                                                                    ↳ DependencyGraphProof

                                                                                                                      ↳ QDP

                                                                                                                        ↳ Narrowing

                                                                                                                          ↳ QDP

                                                                                                                            ↳ DependencyGraphProof

                                                                                                                              ↳ QDP

                                                                                                                                ↳ Narrowing

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ DependencyGraphProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Narrowing

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ DependencyGraphProof

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ SemLabProof

                                                                                                                                                ↳ SemLabProof2

                                                                                                                                                  ↳ QDP

          ↳ QDP

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

ACTIVATE(n__inf(X)) → ACTIVATE(X)
ACTIVATE(n__length(n__take(x0, x1))) → LENGTH(take(activate(x0), activate(x1)))
ACTIVATE(n__take(n__length(y0), n__inf(x0))) → TAKE(length(activate(y0)), inf(activate(x0)))
ACTIVATE(n__length(x0)) → LENGTH(x0)
TAKE(s(X), cons(Y, L)) → ACTIVATE(X)
LENGTH(cons(X, L)) → ACTIVATE(L)
TAKE(s(X), cons(Y, L)) → ACTIVATE(Y)
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__take(y0, n__take(x0, x1))) → TAKE(y0, take(activate(x0), activate(x1)))
ACTIVATE(n__take(X1, X2)) → ACTIVATE(X2)
ACTIVATE(n__take(n__length(y0), n__take(x0, x1))) → TAKE(length(activate(y0)), take(activate(x0), activate(x1)))
TAKE(s(X), cons(Y, L)) → ACTIVATE(L)
ACTIVATE(n__take(n__s(y0), x0)) → TAKE(s(y0), x0)
ACTIVATE(n__length(n__inf(x0))) → LENGTH(inf(activate(x0)))
ACTIVATE(n__take(y0, n__inf(x0))) → TAKE(y0, inf(activate(x0)))
ACTIVATE(n__take(n__s(y0), n__inf(x0))) → TAKE(s(y0), inf(activate(x0)))
ACTIVATE(n__take(y0, x0)) → TAKE(y0, x0)
ACTIVATE(n__take(n__length(y0), x0)) → TAKE(length(activate(y0)), x0)
ACTIVATE(n__take(n__s(y0), n__take(x0, x1))) → TAKE(s(y0), take(activate(x0), activate(x1)))
ACTIVATE(n__length(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

eq(n__0, n__0) → true
eq(n__s(X), n__s(Y)) → eq(activate(X), activate(Y))
eq(X, Y) → false
inf(X) → cons(X, n__inf(n__s(X)))
take(0, X) → nil
take(s(X), cons(Y, L)) → cons(activate(Y), n__take(activate(X), activate(L)))
length(nil) → 0
length(cons(X, L)) → s(n__length(activate(L)))
0 → n__0
s(X) → n__s(X)
inf(X) → n__inf(X)
take(X1, X2) → n__take(X1, X2)
length(X) → n__length(X)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__inf(X)) → inf(activate(X))
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__length(X)) → length(activate(X))
activate(X) → X

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

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

EQ(n__s(X), n__s(Y)) → EQ(activate(X), activate(Y))

The TRS R consists of the following rules:

eq(n__0, n__0) → true
eq(n__s(X), n__s(Y)) → eq(activate(X), activate(Y))
eq(X, Y) → false
inf(X) → cons(X, n__inf(n__s(X)))
take(0, X) → nil
take(s(X), cons(Y, L)) → cons(activate(Y), n__take(activate(X), activate(L)))
length(nil) → 0
length(cons(X, L)) → s(n__length(activate(L)))
0 → n__0
s(X) → n__s(X)
inf(X) → n__inf(X)
take(X1, X2) → n__take(X1, X2)
length(X) → n__length(X)
activate(n__0) → 0
activate(n__s(X)) → s(X)
activate(n__inf(X)) → inf(activate(X))
activate(n__take(X1, X2)) → take(activate(X1), activate(X2))
activate(n__length(X)) → length(activate(X))
activate(X) → X

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