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(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(X)
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__length(X)) → length(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(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(X)
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__length(X)) → length(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:

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

LENGTH(cons(X, L)) → S(n__length(activate(L)))
ACTIVATE(n__length(X)) → LENGTH(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__s(X)) → S(X)
INF(X) → S(X)
EQ(n__s(X), n__s(Y)) → ACTIVATE(X)
EQ(n__s(X), n__s(Y)) → EQ(activate(X), activate(Y))
TAKE(s(X), cons(Y, L)) → ACTIVATE(L)
ACTIVATE(n__0) → 01
EQ(n__s(X), n__s(Y)) → ACTIVATE(Y)
ACTIVATE(n__take(X1, X2)) → TAKE(X1, X2)
ACTIVATE(n__inf(X)) → INF(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(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(X)
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__length(X)) → length(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 8 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ UsableRulesProof
          ↳ QDP

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

ACTIVATE(n__length(X)) → LENGTH(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(X1, 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(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(X)
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__length(X)) → length(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ UsableRulesProof
              ↳ QDP
                ↳ QDPSizeChangeProof
          ↳ QDP

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

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

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs: 



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ Narrowing

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(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(X)
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__length(X)) → length(X)
activate(X) → X

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

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing

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

EQ(n__s(n__inf(x0)), n__s(y1)) → EQ(inf(x0), activate(y1))
EQ(n__s(x0), n__s(y1)) → EQ(x0, activate(y1))
EQ(n__s(n__0), n__s(y1)) → EQ(0, activate(y1))
EQ(n__s(n__s(x0)), n__s(y1)) → EQ(s(x0), activate(y1))
EQ(n__s(n__length(x0)), n__s(y1)) → EQ(length(x0), activate(y1))
EQ(n__s(n__take(x0, x1)), n__s(y1)) → EQ(take(x0, 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(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(X)
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__length(X)) → length(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule EQ(n__s(n__inf(x0)), n__s(y1)) → EQ(inf(x0), activate(y1)) at position [0] we obtained the following new rules:

EQ(n__s(n__inf(x0)), n__s(y1)) → EQ(n__inf(x0), activate(y1))
EQ(n__s(n__inf(x0)), n__s(y1)) → EQ(cons(x0, n__inf(s(x0))), activate(y1))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ DependencyGraphProof

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

EQ(n__s(x0), n__s(y1)) → EQ(x0, activate(y1))
EQ(n__s(n__0), n__s(y1)) → EQ(0, activate(y1))
EQ(n__s(n__inf(x0)), n__s(y1)) → EQ(n__inf(x0), activate(y1))
EQ(n__s(n__s(x0)), n__s(y1)) → EQ(s(x0), activate(y1))
EQ(n__s(n__length(x0)), n__s(y1)) → EQ(length(x0), activate(y1))
EQ(n__s(n__take(x0, x1)), n__s(y1)) → EQ(take(x0, x1), activate(y1))
EQ(n__s(n__inf(x0)), n__s(y1)) → EQ(cons(x0, n__inf(s(x0))), 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(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(X)
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__length(X)) → length(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 2 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ DependencyGraphProof
                      ↳ QDP
                        ↳ Narrowing

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

EQ(n__s(x0), n__s(y1)) → EQ(x0, activate(y1))
EQ(n__s(n__0), n__s(y1)) → EQ(0, activate(y1))
EQ(n__s(n__s(x0)), n__s(y1)) → EQ(s(x0), activate(y1))
EQ(n__s(n__length(x0)), n__s(y1)) → EQ(length(x0), activate(y1))
EQ(n__s(n__take(x0, x1)), n__s(y1)) → EQ(take(x0, 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(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(X)
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__length(X)) → length(X)
activate(X) → X

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

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ DependencyGraphProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing

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

EQ(n__s(y0), n__s(n__take(x0, x1))) → EQ(y0, take(x0, x1))
EQ(n__s(y0), n__s(n__s(x0))) → EQ(y0, s(x0))
EQ(n__s(n__0), n__s(y1)) → EQ(0, activate(y1))
EQ(n__s(y0), n__s(n__0)) → EQ(y0, 0)
EQ(n__s(y0), n__s(x0)) → EQ(y0, x0)
EQ(n__s(n__s(x0)), n__s(y1)) → EQ(s(x0), activate(y1))
EQ(n__s(y0), n__s(n__inf(x0))) → EQ(y0, inf(x0))
EQ(n__s(y0), n__s(n__length(x0))) → EQ(y0, length(x0))
EQ(n__s(n__length(x0)), n__s(y1)) → EQ(length(x0), activate(y1))
EQ(n__s(n__take(x0, x1)), n__s(y1)) → EQ(take(x0, 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(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(X)
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__length(X)) → length(X)
activate(X) → X

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

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ DependencyGraphProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing

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

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

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(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(X)
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__length(X)) → length(X)
activate(X) → X

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

EQ(n__s(n__0), n__s(y0)) → EQ(n__0, activate(y0))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ DependencyGraphProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof

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

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

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(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(X)
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__length(X)) → length(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
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ DependencyGraphProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing

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

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

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(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(X)
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__length(X)) → length(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule EQ(n__s(n__take(x0, x1)), n__s(y1)) → EQ(take(x0, x1), activate(y1)) at position [0] we obtained the following new rules:

EQ(n__s(n__take(0, x0)), n__s(y2)) → EQ(nil, activate(y2))
EQ(n__s(n__take(x0, x1)), n__s(y2)) → EQ(n__take(x0, x1), activate(y2))
EQ(n__s(n__take(s(x0), cons(x1, x2))), n__s(y2)) → EQ(cons(activate(x1), n__take(activate(x0), activate(x2))), activate(y2))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ DependencyGraphProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ DependencyGraphProof

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

EQ(n__s(n__length(y0)), n__s(n__0)) → EQ(length(y0), 0)
EQ(n__s(n__length(y0)), n__s(x0)) → EQ(length(y0), x0)
EQ(n__s(y0), n__s(n__take(x0, x1))) → EQ(y0, take(x0, x1))
EQ(n__s(n__take(0, x0)), n__s(y2)) → EQ(nil, activate(y2))
EQ(n__s(y0), n__s(n__0)) → EQ(y0, 0)
EQ(n__s(n__s(x0)), n__s(y1)) → EQ(s(x0), activate(y1))
EQ(n__s(y0), n__s(n__inf(x0))) → EQ(y0, inf(x0))
EQ(n__s(n__length(y0)), n__s(n__take(x0, x1))) → EQ(length(y0), take(x0, x1))
EQ(n__s(y0), n__s(n__s(x0))) → EQ(y0, s(x0))
EQ(n__s(n__take(s(x0), cons(x1, x2))), n__s(y2)) → EQ(cons(activate(x1), n__take(activate(x0), activate(x2))), activate(y2))
EQ(n__s(y0), n__s(x0)) → EQ(y0, x0)
EQ(n__s(n__length(y0)), n__s(n__length(x0))) → EQ(length(y0), length(x0))
EQ(n__s(y0), n__s(n__length(x0))) → EQ(y0, length(x0))
EQ(n__s(n__length(y0)), n__s(n__inf(x0))) → EQ(length(y0), inf(x0))
EQ(n__s(n__length(y0)), n__s(n__s(x0))) → EQ(length(y0), s(x0))
EQ(n__s(n__take(x0, x1)), n__s(y2)) → EQ(n__take(x0, x1), activate(y2))

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(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(X)
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__length(X)) → length(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 3 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ DependencyGraphProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Narrowing

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

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

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(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(X)
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__length(X)) → length(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule EQ(n__s(n__s(x0)), n__s(y1)) → EQ(s(x0), activate(y1)) at position [0] we obtained the following new rules:

EQ(n__s(n__s(x0)), n__s(y1)) → EQ(n__s(x0), activate(y1))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ DependencyGraphProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing

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

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

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(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(X)
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__length(X)) → length(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule EQ(n__s(y0), n__s(n__take(x0, x1))) → EQ(y0, take(x0, x1)) at position [1] we obtained the following new rules:

EQ(n__s(y0), n__s(n__take(s(x0), cons(x1, x2)))) → EQ(y0, cons(activate(x1), n__take(activate(x0), activate(x2))))
EQ(n__s(y0), n__s(n__take(x0, x1))) → EQ(y0, n__take(x0, x1))
EQ(n__s(y0), n__s(n__take(0, x0))) → EQ(y0, nil)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ DependencyGraphProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ DependencyGraphProof

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

EQ(n__s(n__length(y0)), n__s(x0)) → EQ(length(y0), x0)
EQ(n__s(n__length(y0)), n__s(n__0)) → EQ(length(y0), 0)
EQ(n__s(y0), n__s(n__take(s(x0), cons(x1, x2)))) → EQ(y0, cons(activate(x1), n__take(activate(x0), activate(x2))))
EQ(n__s(y0), n__s(n__0)) → EQ(y0, 0)
EQ(n__s(y0), n__s(n__inf(x0))) → EQ(y0, inf(x0))
EQ(n__s(n__length(y0)), n__s(n__take(x0, x1))) → EQ(length(y0), take(x0, x1))
EQ(n__s(n__s(x0)), n__s(y1)) → EQ(n__s(x0), activate(y1))
EQ(n__s(y0), n__s(n__take(x0, x1))) → EQ(y0, n__take(x0, x1))
EQ(n__s(y0), n__s(n__s(x0))) → EQ(y0, s(x0))
EQ(n__s(y0), n__s(x0)) → EQ(y0, x0)
EQ(n__s(n__length(y0)), n__s(n__length(x0))) → EQ(length(y0), length(x0))
EQ(n__s(y0), n__s(n__length(x0))) → EQ(y0, length(x0))
EQ(n__s(n__length(y0)), n__s(n__inf(x0))) → EQ(length(y0), inf(x0))
EQ(n__s(y0), n__s(n__take(0, x0))) → EQ(y0, nil)
EQ(n__s(n__length(y0)), n__s(n__s(x0))) → EQ(length(y0), s(x0))

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(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(X)
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__length(X)) → length(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 3 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ DependencyGraphProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ DependencyGraphProof
                                                          ↳ QDP
                                                            ↳ Narrowing

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

EQ(n__s(n__length(y0)), n__s(n__0)) → EQ(length(y0), 0)
EQ(n__s(n__length(y0)), n__s(x0)) → EQ(length(y0), x0)
EQ(n__s(y0), n__s(n__s(x0))) → EQ(y0, s(x0))
EQ(n__s(y0), n__s(n__0)) → EQ(y0, 0)
EQ(n__s(y0), n__s(x0)) → EQ(y0, x0)
EQ(n__s(n__length(y0)), n__s(n__length(x0))) → EQ(length(y0), length(x0))
EQ(n__s(y0), n__s(n__inf(x0))) → EQ(y0, inf(x0))
EQ(n__s(y0), n__s(n__length(x0))) → EQ(y0, length(x0))
EQ(n__s(n__length(y0)), n__s(n__inf(x0))) → EQ(length(y0), inf(x0))
EQ(n__s(n__length(y0)), n__s(n__s(x0))) → EQ(length(y0), s(x0))
EQ(n__s(n__length(y0)), n__s(n__take(x0, x1))) → EQ(length(y0), take(x0, x1))
EQ(n__s(n__s(x0)), n__s(y1)) → EQ(n__s(x0), 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(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(X)
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__length(X)) → length(X)
activate(X) → X

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

EQ(n__s(y0), n__s(n__inf(x0))) → EQ(y0, cons(x0, n__inf(s(x0))))
EQ(n__s(y0), n__s(n__inf(x0))) → EQ(y0, n__inf(x0))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ DependencyGraphProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ DependencyGraphProof
                                                          ↳ QDP
                                                            ↳ Narrowing
                                                              ↳ QDP
                                                                ↳ DependencyGraphProof

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

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

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(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(X)
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__length(X)) → length(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 2 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ DependencyGraphProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ DependencyGraphProof
                                                          ↳ QDP
                                                            ↳ Narrowing
                                                              ↳ QDP
                                                                ↳ DependencyGraphProof
                                                                  ↳ QDP
                                                                    ↳ Narrowing

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

EQ(n__s(n__length(y0)), n__s(n__0)) → EQ(length(y0), 0)
EQ(n__s(n__length(y0)), n__s(x0)) → EQ(length(y0), x0)
EQ(n__s(y0), n__s(n__s(x0))) → EQ(y0, s(x0))
EQ(n__s(y0), n__s(n__0)) → EQ(y0, 0)
EQ(n__s(y0), n__s(x0)) → EQ(y0, x0)
EQ(n__s(n__length(y0)), n__s(n__length(x0))) → EQ(length(y0), length(x0))
EQ(n__s(y0), n__s(n__length(x0))) → EQ(y0, length(x0))
EQ(n__s(n__length(y0)), n__s(n__inf(x0))) → EQ(length(y0), inf(x0))
EQ(n__s(n__length(y0)), n__s(n__s(x0))) → EQ(length(y0), s(x0))
EQ(n__s(n__length(y0)), n__s(n__take(x0, x1))) → EQ(length(y0), take(x0, x1))
EQ(n__s(n__s(x0)), n__s(y1)) → EQ(n__s(x0), 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(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(X)
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__length(X)) → length(X)
activate(X) → X

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

EQ(n__s(y0), n__s(n__s(x0))) → EQ(y0, n__s(x0))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ DependencyGraphProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ DependencyGraphProof
                                                          ↳ QDP
                                                            ↳ Narrowing
                                                              ↳ QDP
                                                                ↳ DependencyGraphProof
                                                                  ↳ QDP
                                                                    ↳ Narrowing
                                                                      ↳ QDP
                                                                        ↳ Narrowing

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

EQ(n__s(n__length(y0)), n__s(x0)) → EQ(length(y0), x0)
EQ(n__s(n__length(y0)), n__s(n__0)) → EQ(length(y0), 0)
EQ(n__s(y0), n__s(n__s(x0))) → EQ(y0, n__s(x0))
EQ(n__s(y0), n__s(x0)) → EQ(y0, x0)
EQ(n__s(y0), n__s(n__0)) → EQ(y0, 0)
EQ(n__s(n__length(y0)), n__s(n__length(x0))) → EQ(length(y0), length(x0))
EQ(n__s(y0), n__s(n__length(x0))) → EQ(y0, length(x0))
EQ(n__s(n__length(y0)), n__s(n__inf(x0))) → EQ(length(y0), inf(x0))
EQ(n__s(n__length(y0)), n__s(n__take(x0, x1))) → EQ(length(y0), take(x0, x1))
EQ(n__s(n__length(y0)), n__s(n__s(x0))) → EQ(length(y0), s(x0))
EQ(n__s(n__s(x0)), n__s(y1)) → EQ(n__s(x0), 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(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(X)
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__length(X)) → length(X)
activate(X) → X

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

EQ(n__s(y0), n__s(n__0)) → EQ(y0, n__0)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ DependencyGraphProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ DependencyGraphProof
                                                          ↳ QDP
                                                            ↳ Narrowing
                                                              ↳ QDP
                                                                ↳ DependencyGraphProof
                                                                  ↳ QDP
                                                                    ↳ Narrowing
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof

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

EQ(n__s(n__length(y0)), n__s(n__0)) → EQ(length(y0), 0)
EQ(n__s(n__length(y0)), n__s(x0)) → EQ(length(y0), x0)
EQ(n__s(y0), n__s(n__0)) → EQ(y0, n__0)
EQ(n__s(y0), n__s(x0)) → EQ(y0, x0)
EQ(n__s(y0), n__s(n__s(x0))) → EQ(y0, n__s(x0))
EQ(n__s(n__length(y0)), n__s(n__length(x0))) → EQ(length(y0), length(x0))
EQ(n__s(y0), n__s(n__length(x0))) → EQ(y0, length(x0))
EQ(n__s(n__length(y0)), n__s(n__inf(x0))) → EQ(length(y0), inf(x0))
EQ(n__s(n__length(y0)), n__s(n__s(x0))) → EQ(length(y0), s(x0))
EQ(n__s(n__length(y0)), n__s(n__take(x0, x1))) → EQ(length(y0), take(x0, x1))
EQ(n__s(n__s(x0)), n__s(y1)) → EQ(n__s(x0), 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(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(X)
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__length(X)) → length(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
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ DependencyGraphProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ DependencyGraphProof
                                                          ↳ QDP
                                                            ↳ Narrowing
                                                              ↳ QDP
                                                                ↳ DependencyGraphProof
                                                                  ↳ QDP
                                                                    ↳ Narrowing
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Narrowing

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

EQ(n__s(n__length(y0)), n__s(n__0)) → EQ(length(y0), 0)
EQ(n__s(n__length(y0)), n__s(x0)) → EQ(length(y0), x0)
EQ(n__s(y0), n__s(n__s(x0))) → EQ(y0, n__s(x0))
EQ(n__s(y0), n__s(x0)) → EQ(y0, x0)
EQ(n__s(n__length(y0)), n__s(n__length(x0))) → EQ(length(y0), length(x0))
EQ(n__s(y0), n__s(n__length(x0))) → EQ(y0, length(x0))
EQ(n__s(n__length(y0)), n__s(n__inf(x0))) → EQ(length(y0), inf(x0))
EQ(n__s(n__length(y0)), n__s(n__s(x0))) → EQ(length(y0), s(x0))
EQ(n__s(n__length(y0)), n__s(n__take(x0, x1))) → EQ(length(y0), take(x0, x1))
EQ(n__s(n__s(x0)), n__s(y1)) → EQ(n__s(x0), 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(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(X)
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__length(X)) → length(X)
activate(X) → X

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

EQ(n__s(n__length(y0)), n__s(n__0)) → EQ(length(y0), n__0)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ DependencyGraphProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ DependencyGraphProof
                                                          ↳ QDP
                                                            ↳ Narrowing
                                                              ↳ QDP
                                                                ↳ DependencyGraphProof
                                                                  ↳ QDP
                                                                    ↳ Narrowing
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Narrowing
                                                                                  ↳ QDP
                                                                                    ↳ DependencyGraphProof

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

EQ(n__s(n__length(y0)), n__s(x0)) → EQ(length(y0), x0)
EQ(n__s(y0), n__s(x0)) → EQ(y0, x0)
EQ(n__s(y0), n__s(n__s(x0))) → EQ(y0, n__s(x0))
EQ(n__s(n__length(y0)), n__s(n__0)) → EQ(length(y0), n__0)
EQ(n__s(n__length(y0)), n__s(n__length(x0))) → EQ(length(y0), length(x0))
EQ(n__s(y0), n__s(n__length(x0))) → EQ(y0, length(x0))
EQ(n__s(n__length(y0)), n__s(n__inf(x0))) → EQ(length(y0), inf(x0))
EQ(n__s(n__length(y0)), n__s(n__take(x0, x1))) → EQ(length(y0), take(x0, x1))
EQ(n__s(n__length(y0)), n__s(n__s(x0))) → EQ(length(y0), s(x0))
EQ(n__s(n__s(x0)), n__s(y1)) → EQ(n__s(x0), 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(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(X)
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__length(X)) → length(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
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ DependencyGraphProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ DependencyGraphProof
                                                          ↳ QDP
                                                            ↳ Narrowing
                                                              ↳ QDP
                                                                ↳ DependencyGraphProof
                                                                  ↳ QDP
                                                                    ↳ Narrowing
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Narrowing
                                                                                  ↳ QDP
                                                                                    ↳ DependencyGraphProof
                                                                                      ↳ QDP
                                                                                        ↳ Narrowing

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

EQ(n__s(n__length(y0)), n__s(x0)) → EQ(length(y0), x0)
EQ(n__s(y0), n__s(n__s(x0))) → EQ(y0, n__s(x0))
EQ(n__s(y0), n__s(x0)) → EQ(y0, x0)
EQ(n__s(n__length(y0)), n__s(n__length(x0))) → EQ(length(y0), length(x0))
EQ(n__s(y0), n__s(n__length(x0))) → EQ(y0, length(x0))
EQ(n__s(n__length(y0)), n__s(n__inf(x0))) → EQ(length(y0), inf(x0))
EQ(n__s(n__length(y0)), n__s(n__s(x0))) → EQ(length(y0), s(x0))
EQ(n__s(n__length(y0)), n__s(n__take(x0, x1))) → EQ(length(y0), take(x0, x1))
EQ(n__s(n__s(x0)), n__s(y1)) → EQ(n__s(x0), 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(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(X)
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__length(X)) → length(X)
activate(X) → X

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

EQ(n__s(n__length(y0)), n__s(n__inf(x0))) → EQ(length(y0), cons(x0, n__inf(s(x0))))
EQ(n__s(n__length(y0)), n__s(n__inf(x0))) → EQ(length(y0), n__inf(x0))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ DependencyGraphProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ DependencyGraphProof
                                                          ↳ QDP
                                                            ↳ Narrowing
                                                              ↳ QDP
                                                                ↳ DependencyGraphProof
                                                                  ↳ QDP
                                                                    ↳ Narrowing
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Narrowing
                                                                                  ↳ QDP
                                                                                    ↳ DependencyGraphProof
                                                                                      ↳ QDP
                                                                                        ↳ Narrowing
                                                                                          ↳ QDP
                                                                                            ↳ DependencyGraphProof

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

EQ(n__s(n__length(y0)), n__s(x0)) → EQ(length(y0), x0)
EQ(n__s(n__length(y0)), n__s(n__inf(x0))) → EQ(length(y0), n__inf(x0))
EQ(n__s(y0), n__s(x0)) → EQ(y0, x0)
EQ(n__s(y0), n__s(n__s(x0))) → EQ(y0, n__s(x0))
EQ(n__s(n__length(y0)), n__s(n__inf(x0))) → EQ(length(y0), cons(x0, n__inf(s(x0))))
EQ(n__s(n__length(y0)), n__s(n__length(x0))) → EQ(length(y0), length(x0))
EQ(n__s(y0), n__s(n__length(x0))) → EQ(y0, length(x0))
EQ(n__s(n__length(y0)), n__s(n__take(x0, x1))) → EQ(length(y0), take(x0, x1))
EQ(n__s(n__length(y0)), n__s(n__s(x0))) → EQ(length(y0), s(x0))
EQ(n__s(n__s(x0)), n__s(y1)) → EQ(n__s(x0), 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(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(X)
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__length(X)) → length(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 2 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ DependencyGraphProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ DependencyGraphProof
                                                          ↳ QDP
                                                            ↳ Narrowing
                                                              ↳ QDP
                                                                ↳ DependencyGraphProof
                                                                  ↳ QDP
                                                                    ↳ Narrowing
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Narrowing
                                                                                  ↳ QDP
                                                                                    ↳ DependencyGraphProof
                                                                                      ↳ QDP
                                                                                        ↳ Narrowing
                                                                                          ↳ QDP
                                                                                            ↳ DependencyGraphProof
                                                                                              ↳ QDP
                                                                                                ↳ Narrowing

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

EQ(n__s(n__length(y0)), n__s(x0)) → EQ(length(y0), x0)
EQ(n__s(y0), n__s(n__s(x0))) → EQ(y0, n__s(x0))
EQ(n__s(y0), n__s(x0)) → EQ(y0, x0)
EQ(n__s(n__length(y0)), n__s(n__length(x0))) → EQ(length(y0), length(x0))
EQ(n__s(y0), n__s(n__length(x0))) → EQ(y0, length(x0))
EQ(n__s(n__length(y0)), n__s(n__s(x0))) → EQ(length(y0), s(x0))
EQ(n__s(n__length(y0)), n__s(n__take(x0, x1))) → EQ(length(y0), take(x0, x1))
EQ(n__s(n__s(x0)), n__s(y1)) → EQ(n__s(x0), 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(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(X)
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__length(X)) → length(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule EQ(n__s(n__length(y0)), n__s(n__take(x0, x1))) → EQ(length(y0), take(x0, x1)) at position [1] we obtained the following new rules:

EQ(n__s(n__length(y0)), n__s(n__take(x0, x1))) → EQ(length(y0), n__take(x0, x1))
EQ(n__s(n__length(y0)), n__s(n__take(s(x0), cons(x1, x2)))) → EQ(length(y0), cons(activate(x1), n__take(activate(x0), activate(x2))))
EQ(n__s(n__length(y0)), n__s(n__take(0, x0))) → EQ(length(y0), nil)



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ DependencyGraphProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ DependencyGraphProof
                                                          ↳ QDP
                                                            ↳ Narrowing
                                                              ↳ QDP
                                                                ↳ DependencyGraphProof
                                                                  ↳ QDP
                                                                    ↳ Narrowing
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Narrowing
                                                                                  ↳ QDP
                                                                                    ↳ DependencyGraphProof
                                                                                      ↳ QDP
                                                                                        ↳ Narrowing
                                                                                          ↳ QDP
                                                                                            ↳ DependencyGraphProof
                                                                                              ↳ QDP
                                                                                                ↳ Narrowing
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof

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

EQ(n__s(n__length(y0)), n__s(x0)) → EQ(length(y0), x0)
EQ(n__s(y0), n__s(x0)) → EQ(y0, x0)
EQ(n__s(y0), n__s(n__s(x0))) → EQ(y0, n__s(x0))
EQ(n__s(n__length(y0)), n__s(n__length(x0))) → EQ(length(y0), length(x0))
EQ(n__s(n__length(y0)), n__s(n__take(x0, x1))) → EQ(length(y0), n__take(x0, x1))
EQ(n__s(y0), n__s(n__length(x0))) → EQ(y0, length(x0))
EQ(n__s(n__length(y0)), n__s(n__take(s(x0), cons(x1, x2)))) → EQ(length(y0), cons(activate(x1), n__take(activate(x0), activate(x2))))
EQ(n__s(n__length(y0)), n__s(n__s(x0))) → EQ(length(y0), s(x0))
EQ(n__s(n__length(y0)), n__s(n__take(0, x0))) → EQ(length(y0), nil)
EQ(n__s(n__s(x0)), n__s(y1)) → EQ(n__s(x0), 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(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(X)
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__length(X)) → length(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 3 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ DependencyGraphProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ DependencyGraphProof
                                                          ↳ QDP
                                                            ↳ Narrowing
                                                              ↳ QDP
                                                                ↳ DependencyGraphProof
                                                                  ↳ QDP
                                                                    ↳ Narrowing
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Narrowing
                                                                                  ↳ QDP
                                                                                    ↳ DependencyGraphProof
                                                                                      ↳ QDP
                                                                                        ↳ Narrowing
                                                                                          ↳ QDP
                                                                                            ↳ DependencyGraphProof
                                                                                              ↳ QDP
                                                                                                ↳ Narrowing
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Narrowing

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

EQ(n__s(n__length(y0)), n__s(x0)) → EQ(length(y0), x0)
EQ(n__s(y0), n__s(n__s(x0))) → EQ(y0, n__s(x0))
EQ(n__s(y0), n__s(x0)) → EQ(y0, x0)
EQ(n__s(n__length(y0)), n__s(n__length(x0))) → EQ(length(y0), length(x0))
EQ(n__s(y0), n__s(n__length(x0))) → EQ(y0, length(x0))
EQ(n__s(n__length(y0)), n__s(n__s(x0))) → EQ(length(y0), s(x0))
EQ(n__s(n__s(x0)), n__s(y1)) → EQ(n__s(x0), 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(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(X)
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__length(X)) → length(X)
activate(X) → X

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

EQ(n__s(n__length(y0)), n__s(n__s(x0))) → EQ(length(y0), n__s(x0))



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ DependencyGraphProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ DependencyGraphProof
                                                          ↳ QDP
                                                            ↳ Narrowing
                                                              ↳ QDP
                                                                ↳ DependencyGraphProof
                                                                  ↳ QDP
                                                                    ↳ Narrowing
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Narrowing
                                                                                  ↳ QDP
                                                                                    ↳ DependencyGraphProof
                                                                                      ↳ QDP
                                                                                        ↳ Narrowing
                                                                                          ↳ QDP
                                                                                            ↳ DependencyGraphProof
                                                                                              ↳ QDP
                                                                                                ↳ Narrowing
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Narrowing
                                                                                                          ↳ QDP
                                                                                                            ↳ QDPOrderProof

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

EQ(n__s(n__length(y0)), n__s(x0)) → EQ(length(y0), x0)
EQ(n__s(y0), n__s(x0)) → EQ(y0, x0)
EQ(n__s(y0), n__s(n__s(x0))) → EQ(y0, n__s(x0))
EQ(n__s(n__length(y0)), n__s(n__length(x0))) → EQ(length(y0), length(x0))
EQ(n__s(n__length(y0)), n__s(n__s(x0))) → EQ(length(y0), n__s(x0))
EQ(n__s(y0), n__s(n__length(x0))) → EQ(y0, length(x0))
EQ(n__s(n__s(x0)), n__s(y1)) → EQ(n__s(x0), 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(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(X)
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__length(X)) → length(X)
activate(X) → X

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


The following pairs can be oriented strictly and are deleted.


EQ(n__s(y0), n__s(x0)) → EQ(y0, x0)
EQ(n__s(y0), n__s(n__s(x0))) → EQ(y0, n__s(x0))
EQ(n__s(y0), n__s(n__length(x0))) → EQ(y0, length(x0))
EQ(n__s(n__s(x0)), n__s(y1)) → EQ(n__s(x0), activate(y1))
The remaining pairs can at least be oriented weakly.

EQ(n__s(n__length(y0)), n__s(x0)) → EQ(length(y0), x0)
EQ(n__s(n__length(y0)), n__s(n__length(x0))) → EQ(length(y0), length(x0))
EQ(n__s(n__length(y0)), n__s(n__s(x0))) → EQ(length(y0), n__s(x0))
Used ordering: Polynomial interpretation [25]:

POL(0) = 1   
POL(EQ(x1, x2)) = x1   
POL(activate(x1)) = 1 + x1   
POL(cons(x1, x2)) = 0   
POL(inf(x1)) = 0   
POL(length(x1)) = 1   
POL(n__0) = 0   
POL(n__inf(x1)) = 0   
POL(n__length(x1)) = 0   
POL(n__s(x1)) = 1 + x1   
POL(n__take(x1, x2)) = 1 + x1 + x2   
POL(nil) = 0   
POL(s(x1)) = 1 + x1   
POL(take(x1, x2)) = 1 + x1 + x2   

The following usable rules [17] were oriented:

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ DependencyGraphProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ DependencyGraphProof
                                                          ↳ QDP
                                                            ↳ Narrowing
                                                              ↳ QDP
                                                                ↳ DependencyGraphProof
                                                                  ↳ QDP
                                                                    ↳ Narrowing
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Narrowing
                                                                                  ↳ QDP
                                                                                    ↳ DependencyGraphProof
                                                                                      ↳ QDP
                                                                                        ↳ Narrowing
                                                                                          ↳ QDP
                                                                                            ↳ DependencyGraphProof
                                                                                              ↳ QDP
                                                                                                ↳ Narrowing
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Narrowing
                                                                                                          ↳ QDP
                                                                                                            ↳ QDPOrderProof
                                                                                                              ↳ QDP
                                                                                                                ↳ QDPOrderProof

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

EQ(n__s(n__length(y0)), n__s(x0)) → EQ(length(y0), x0)
EQ(n__s(n__length(y0)), n__s(n__length(x0))) → EQ(length(y0), length(x0))
EQ(n__s(n__length(y0)), n__s(n__s(x0))) → EQ(length(y0), n__s(x0))

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(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(X)
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__length(X)) → length(X)
activate(X) → X

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


The following pairs can be oriented strictly and are deleted.


EQ(n__s(n__length(y0)), n__s(x0)) → EQ(length(y0), x0)
EQ(n__s(n__length(y0)), n__s(n__s(x0))) → EQ(length(y0), n__s(x0))
The remaining pairs can at least be oriented weakly.

EQ(n__s(n__length(y0)), n__s(n__length(x0))) → EQ(length(y0), length(x0))
Used ordering: Polynomial interpretation [25]:

POL(0) = 1   
POL(EQ(x1, x2)) = x1 + x2   
POL(activate(x1)) = 1 + x1   
POL(cons(x1, x2)) = 0   
POL(inf(x1)) = 0   
POL(length(x1)) = 1   
POL(n__0) = 0   
POL(n__inf(x1)) = 0   
POL(n__length(x1)) = 0   
POL(n__s(x1)) = 1 + x1   
POL(n__take(x1, x2)) = 1 + x1 + x2   
POL(nil) = 0   
POL(s(x1)) = 1 + x1   
POL(take(x1, x2)) = 1 + x1 + x2   

The following usable rules [17] were oriented:

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



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ Narrowing
              ↳ QDP
                ↳ Narrowing
                  ↳ QDP
                    ↳ DependencyGraphProof
                      ↳ QDP
                        ↳ Narrowing
                          ↳ QDP
                            ↳ Narrowing
                              ↳ QDP
                                ↳ Narrowing
                                  ↳ QDP
                                    ↳ DependencyGraphProof
                                      ↳ QDP
                                        ↳ Narrowing
                                          ↳ QDP
                                            ↳ DependencyGraphProof
                                              ↳ QDP
                                                ↳ Narrowing
                                                  ↳ QDP
                                                    ↳ Narrowing
                                                      ↳ QDP
                                                        ↳ DependencyGraphProof
                                                          ↳ QDP
                                                            ↳ Narrowing
                                                              ↳ QDP
                                                                ↳ DependencyGraphProof
                                                                  ↳ QDP
                                                                    ↳ Narrowing
                                                                      ↳ QDP
                                                                        ↳ Narrowing
                                                                          ↳ QDP
                                                                            ↳ DependencyGraphProof
                                                                              ↳ QDP
                                                                                ↳ Narrowing
                                                                                  ↳ QDP
                                                                                    ↳ DependencyGraphProof
                                                                                      ↳ QDP
                                                                                        ↳ Narrowing
                                                                                          ↳ QDP
                                                                                            ↳ DependencyGraphProof
                                                                                              ↳ QDP
                                                                                                ↳ Narrowing
                                                                                                  ↳ QDP
                                                                                                    ↳ DependencyGraphProof
                                                                                                      ↳ QDP
                                                                                                        ↳ Narrowing
                                                                                                          ↳ QDP
                                                                                                            ↳ QDPOrderProof
                                                                                                              ↳ QDP
                                                                                                                ↳ QDPOrderProof
                                                                                                                  ↳ QDP

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

EQ(n__s(n__length(y0)), n__s(n__length(x0))) → EQ(length(y0), length(x0))

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(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(X)
activate(n__take(X1, X2)) → take(X1, X2)
activate(n__length(X)) → length(X)
activate(X) → X

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