Termination w.r.t. Q proof of /tmp/tmpg6L6lq/Ex5_7_Luc97

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:

dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(n__dbl(activate(X)), n__dbls(activate(Y)))
sel(0, cons(X, Y)) → activate(X)
sel(s(X), cons(Y, Z)) → sel(activate(X), activate(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(n__sel(activate(X), activate(Z)), n__indx(activate(Y), activate(Z)))
from(X) → cons(activate(X), n__from(n__s(activate(X))))
dbl1(0) → 01
dbl1(s(X)) → s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) → activate(X)
sel1(s(X), cons(Y, Z)) → sel1(activate(X), activate(Z))
quote(0) → 01
quote(s(X)) → s1(quote(activate(X)))
quote(dbl(X)) → dbl1(X)
quote(sel(X, Y)) → sel1(X, Y)
s(X) → n__s(X)
dbl(X) → n__dbl(X)
dbls(X) → n__dbls(X)
sel(X1, X2) → n__sel(X1, X2)
indx(X1, X2) → n__indx(X1, X2)
from(X) → n__from(X)
activate(n__s(X)) → s(X)
activate(n__dbl(X)) → dbl(X)
activate(n__dbls(X)) → dbls(X)
activate(n__sel(X1, X2)) → sel(X1, X2)
activate(n__indx(X1, X2)) → indx(X1, X2)
activate(n__from(X)) → from(X)
activate(X) → X

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(n__dbl(activate(X)), n__dbls(activate(Y)))
sel(0, cons(X, Y)) → activate(X)
sel(s(X), cons(Y, Z)) → sel(activate(X), activate(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(n__sel(activate(X), activate(Z)), n__indx(activate(Y), activate(Z)))
from(X) → cons(activate(X), n__from(n__s(activate(X))))
dbl1(0) → 01
dbl1(s(X)) → s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) → activate(X)
sel1(s(X), cons(Y, Z)) → sel1(activate(X), activate(Z))
quote(0) → 01
quote(s(X)) → s1(quote(activate(X)))
quote(dbl(X)) → dbl1(X)
quote(sel(X, Y)) → sel1(X, Y)
s(X) → n__s(X)
dbl(X) → n__dbl(X)
dbls(X) → n__dbls(X)
sel(X1, X2) → n__sel(X1, X2)
indx(X1, X2) → n__indx(X1, X2)
from(X) → n__from(X)
activate(n__s(X)) → s(X)
activate(n__dbl(X)) → dbl(X)
activate(n__dbls(X)) → dbls(X)
activate(n__sel(X1, X2)) → sel(X1, X2)
activate(n__indx(X1, X2)) → indx(X1, X2)
activate(n__from(X)) → from(X)
activate(X) → X

Q is empty.

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

ACTIVATE(n__sel(X1, X2)) → SEL(X1, X2)
INDX(cons(X, Y), Z) → ACTIVATE(Z)
DBL1(s(X)) → DBL1(activate(X))
SEL(s(X), cons(Y, Z)) → ACTIVATE(Z)
SEL1(s(X), cons(Y, Z)) → SEL1(activate(X), activate(Z))
SEL1(s(X), cons(Y, Z)) → ACTIVATE(X)
SEL(0, cons(X, Y)) → ACTIVATE(X)
SEL1(0, cons(X, Y)) → ACTIVATE(X)
DBLS(cons(X, Y)) → ACTIVATE(Y)
SEL1(s(X), cons(Y, Z)) → ACTIVATE(Z)
DBL(s(X)) → S(n__s(n__dbl(activate(X))))
ACTIVATE(n__from(X)) → FROM(X)
QUOTE(dbl(X)) → DBL1(X)
QUOTE(sel(X, Y)) → SEL1(X, Y)
ACTIVATE(n__indx(X1, X2)) → INDX(X1, X2)
ACTIVATE(n__dbls(X)) → DBLS(X)
INDX(cons(X, Y), Z) → ACTIVATE(X)
ACTIVATE(n__dbl(X)) → DBL(X)
ACTIVATE(n__s(X)) → S(X)
QUOTE(s(X)) → ACTIVATE(X)
FROM(X) → ACTIVATE(X)
SEL(s(X), cons(Y, Z)) → SEL(activate(X), activate(Z))
SEL(s(X), cons(Y, Z)) → ACTIVATE(X)
DBLS(cons(X, Y)) → ACTIVATE(X)
DBL(s(X)) → ACTIVATE(X)
INDX(cons(X, Y), Z) → ACTIVATE(Y)
QUOTE(s(X)) → QUOTE(activate(X))
DBL1(s(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(n__dbl(activate(X)), n__dbls(activate(Y)))
sel(0, cons(X, Y)) → activate(X)
sel(s(X), cons(Y, Z)) → sel(activate(X), activate(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(n__sel(activate(X), activate(Z)), n__indx(activate(Y), activate(Z)))
from(X) → cons(activate(X), n__from(n__s(activate(X))))
dbl1(0) → 01
dbl1(s(X)) → s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) → activate(X)
sel1(s(X), cons(Y, Z)) → sel1(activate(X), activate(Z))
quote(0) → 01
quote(s(X)) → s1(quote(activate(X)))
quote(dbl(X)) → dbl1(X)
quote(sel(X, Y)) → sel1(X, Y)
s(X) → n__s(X)
dbl(X) → n__dbl(X)
dbls(X) → n__dbls(X)
sel(X1, X2) → n__sel(X1, X2)
indx(X1, X2) → n__indx(X1, X2)
from(X) → n__from(X)
activate(n__s(X)) → s(X)
activate(n__dbl(X)) → dbl(X)
activate(n__dbls(X)) → dbls(X)
activate(n__sel(X1, X2)) → sel(X1, X2)
activate(n__indx(X1, X2)) → indx(X1, X2)
activate(n__from(X)) → from(X)
activate(X) → X

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

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

ACTIVATE(n__sel(X1, X2)) → SEL(X1, X2)
INDX(cons(X, Y), Z) → ACTIVATE(Z)
DBL1(s(X)) → DBL1(activate(X))
SEL(s(X), cons(Y, Z)) → ACTIVATE(Z)
SEL1(s(X), cons(Y, Z)) → SEL1(activate(X), activate(Z))
SEL1(s(X), cons(Y, Z)) → ACTIVATE(X)
SEL(0, cons(X, Y)) → ACTIVATE(X)
SEL1(0, cons(X, Y)) → ACTIVATE(X)
DBLS(cons(X, Y)) → ACTIVATE(Y)
SEL1(s(X), cons(Y, Z)) → ACTIVATE(Z)
DBL(s(X)) → S(n__s(n__dbl(activate(X))))
ACTIVATE(n__from(X)) → FROM(X)
QUOTE(dbl(X)) → DBL1(X)
QUOTE(sel(X, Y)) → SEL1(X, Y)
ACTIVATE(n__indx(X1, X2)) → INDX(X1, X2)
ACTIVATE(n__dbls(X)) → DBLS(X)
INDX(cons(X, Y), Z) → ACTIVATE(X)
ACTIVATE(n__dbl(X)) → DBL(X)
ACTIVATE(n__s(X)) → S(X)
QUOTE(s(X)) → ACTIVATE(X)
FROM(X) → ACTIVATE(X)
SEL(s(X), cons(Y, Z)) → SEL(activate(X), activate(Z))
SEL(s(X), cons(Y, Z)) → ACTIVATE(X)
DBLS(cons(X, Y)) → ACTIVATE(X)
DBL(s(X)) → ACTIVATE(X)
INDX(cons(X, Y), Z) → ACTIVATE(Y)
QUOTE(s(X)) → QUOTE(activate(X))
DBL1(s(X)) → ACTIVATE(X)

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(n__dbl(activate(X)), n__dbls(activate(Y)))
sel(0, cons(X, Y)) → activate(X)
sel(s(X), cons(Y, Z)) → sel(activate(X), activate(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(n__sel(activate(X), activate(Z)), n__indx(activate(Y), activate(Z)))
from(X) → cons(activate(X), n__from(n__s(activate(X))))
dbl1(0) → 01
dbl1(s(X)) → s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) → activate(X)
sel1(s(X), cons(Y, Z)) → sel1(activate(X), activate(Z))
quote(0) → 01
quote(s(X)) → s1(quote(activate(X)))
quote(dbl(X)) → dbl1(X)
quote(sel(X, Y)) → sel1(X, Y)
s(X) → n__s(X)
dbl(X) → n__dbl(X)
dbls(X) → n__dbls(X)
sel(X1, X2) → n__sel(X1, X2)
indx(X1, X2) → n__indx(X1, X2)
from(X) → n__from(X)
activate(n__s(X)) → s(X)
activate(n__dbl(X)) → dbl(X)
activate(n__dbls(X)) → dbls(X)
activate(n__sel(X1, X2)) → sel(X1, X2)
activate(n__indx(X1, X2)) → indx(X1, X2)
activate(n__from(X)) → from(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 4 SCCs with 9 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ Narrowing

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ACTIVATE(n__sel(X1, X2)) → SEL(X1, X2)
INDX(cons(X, Y), Z) → ACTIVATE(Z)
INDX(cons(X, Y), Z) → ACTIVATE(X)
ACTIVATE(n__dbl(X)) → DBL(X)
SEL(s(X), cons(Y, Z)) → ACTIVATE(Z)
FROM(X) → ACTIVATE(X)
SEL(0, cons(X, Y)) → ACTIVATE(X)
SEL(s(X), cons(Y, Z)) → SEL(activate(X), activate(Z))
DBLS(cons(X, Y)) → ACTIVATE(Y)
SEL(s(X), cons(Y, Z)) → ACTIVATE(X)
DBLS(cons(X, Y)) → ACTIVATE(X)
DBL(s(X)) → ACTIVATE(X)
ACTIVATE(n__from(X)) → FROM(X)
INDX(cons(X, Y), Z) → ACTIVATE(Y)
ACTIVATE(n__indx(X1, X2)) → INDX(X1, X2)
ACTIVATE(n__dbls(X)) → DBLS(X)

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(n__dbl(activate(X)), n__dbls(activate(Y)))
sel(0, cons(X, Y)) → activate(X)
sel(s(X), cons(Y, Z)) → sel(activate(X), activate(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(n__sel(activate(X), activate(Z)), n__indx(activate(Y), activate(Z)))
from(X) → cons(activate(X), n__from(n__s(activate(X))))
dbl1(0) → 01
dbl1(s(X)) → s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) → activate(X)
sel1(s(X), cons(Y, Z)) → sel1(activate(X), activate(Z))
quote(0) → 01
quote(s(X)) → s1(quote(activate(X)))
quote(dbl(X)) → dbl1(X)
quote(sel(X, Y)) → sel1(X, Y)
s(X) → n__s(X)
dbl(X) → n__dbl(X)
dbls(X) → n__dbls(X)
sel(X1, X2) → n__sel(X1, X2)
indx(X1, X2) → n__indx(X1, X2)
from(X) → n__from(X)
activate(n__s(X)) → s(X)
activate(n__dbl(X)) → dbl(X)
activate(n__dbls(X)) → dbls(X)
activate(n__sel(X1, X2)) → sel(X1, X2)
activate(n__indx(X1, X2)) → indx(X1, X2)
activate(n__from(X)) → from(X)
activate(X) → X

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

SEL(s(n__sel(x0, x1)), cons(y1, y2)) → SEL(sel(x0, x1), activate(y2))
SEL(s(n__dbls(x0)), cons(y1, y2)) → SEL(dbls(x0), activate(y2))
SEL(s(n__s(x0)), cons(y1, y2)) → SEL(s(x0), activate(y2))
SEL(s(n__dbl(x0)), cons(y1, y2)) → SEL(dbl(x0), activate(y2))
SEL(s(n__indx(x0, x1)), cons(y1, y2)) → SEL(indx(x0, x1), activate(y2))
SEL(s(n__from(x0)), cons(y1, y2)) → SEL(from(x0), activate(y2))
SEL(s(x0), cons(y1, y2)) → SEL(x0, activate(y2))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ACTIVATE(n__sel(X1, X2)) → SEL(X1, X2)
INDX(cons(X, Y), Z) → ACTIVATE(Z)
SEL(s(n__dbls(x0)), cons(y1, y2)) → SEL(dbls(x0), activate(y2))
INDX(cons(X, Y), Z) → ACTIVATE(X)
ACTIVATE(n__dbl(X)) → DBL(X)
SEL(s(X), cons(Y, Z)) → ACTIVATE(Z)
SEL(s(n__indx(x0, x1)), cons(y1, y2)) → SEL(indx(x0, x1), activate(y2))
FROM(X) → ACTIVATE(X)
SEL(0, cons(X, Y)) → ACTIVATE(X)
SEL(s(n__sel(x0, x1)), cons(y1, y2)) → SEL(sel(x0, x1), activate(y2))
DBLS(cons(X, Y)) → ACTIVATE(Y)
DBLS(cons(X, Y)) → ACTIVATE(X)
SEL(s(X), cons(Y, Z)) → ACTIVATE(X)
SEL(s(n__s(x0)), cons(y1, y2)) → SEL(s(x0), activate(y2))
DBL(s(X)) → ACTIVATE(X)
SEL(s(n__dbl(x0)), cons(y1, y2)) → SEL(dbl(x0), activate(y2))
INDX(cons(X, Y), Z) → ACTIVATE(Y)
ACTIVATE(n__from(X)) → FROM(X)
SEL(s(x0), cons(y1, y2)) → SEL(x0, activate(y2))
SEL(s(n__from(x0)), cons(y1, y2)) → SEL(from(x0), activate(y2))
ACTIVATE(n__indx(X1, X2)) → INDX(X1, X2)
ACTIVATE(n__dbls(X)) → DBLS(X)

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(n__dbl(activate(X)), n__dbls(activate(Y)))
sel(0, cons(X, Y)) → activate(X)
sel(s(X), cons(Y, Z)) → sel(activate(X), activate(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(n__sel(activate(X), activate(Z)), n__indx(activate(Y), activate(Z)))
from(X) → cons(activate(X), n__from(n__s(activate(X))))
dbl1(0) → 01
dbl1(s(X)) → s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) → activate(X)
sel1(s(X), cons(Y, Z)) → sel1(activate(X), activate(Z))
quote(0) → 01
quote(s(X)) → s1(quote(activate(X)))
quote(dbl(X)) → dbl1(X)
quote(sel(X, Y)) → sel1(X, Y)
s(X) → n__s(X)
dbl(X) → n__dbl(X)
dbls(X) → n__dbls(X)
sel(X1, X2) → n__sel(X1, X2)
indx(X1, X2) → n__indx(X1, X2)
from(X) → n__from(X)
activate(n__s(X)) → s(X)
activate(n__dbl(X)) → dbl(X)
activate(n__dbls(X)) → dbls(X)
activate(n__sel(X1, X2)) → sel(X1, X2)
activate(n__indx(X1, X2)) → indx(X1, X2)
activate(n__from(X)) → from(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule SEL(s(n__sel(x0, x1)), cons(y1, y2)) → SEL(sel(x0, x1), activate(y2)) at position [1] we obtained the following new rules:

SEL(s(n__sel(y0, y1)), cons(y2, n__dbls(x0))) → SEL(sel(y0, y1), dbls(x0))
SEL(s(n__sel(y0, y1)), cons(y2, n__s(x0))) → SEL(sel(y0, y1), s(x0))
SEL(s(n__sel(y0, y1)), cons(y2, n__dbl(x0))) → SEL(sel(y0, y1), dbl(x0))
SEL(s(n__sel(y0, y1)), cons(y2, n__from(x0))) → SEL(sel(y0, y1), from(x0))
SEL(s(n__sel(y0, y1)), cons(y2, n__indx(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
SEL(s(n__sel(y0, y1)), cons(y2, n__sel(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ACTIVATE(n__sel(X1, X2)) → SEL(X1, X2)
INDX(cons(X, Y), Z) → ACTIVATE(Z)
SEL(s(X), cons(Y, Z)) → ACTIVATE(Z)
SEL(s(n__sel(y0, y1)), cons(y2, n__sel(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))
SEL(0, cons(X, Y)) → ACTIVATE(X)
SEL(s(n__sel(y0, y1)), cons(y2, n__dbls(x0))) → SEL(sel(y0, y1), dbls(x0))
DBLS(cons(X, Y)) → ACTIVATE(Y)
SEL(s(n__s(x0)), cons(y1, y2)) → SEL(s(x0), activate(y2))
SEL(s(n__dbl(x0)), cons(y1, y2)) → SEL(dbl(x0), activate(y2))
SEL(s(n__sel(y0, y1)), cons(y2, n__from(x0))) → SEL(sel(y0, y1), from(x0))
SEL(s(n__sel(y0, y1)), cons(y2, n__indx(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
ACTIVATE(n__from(X)) → FROM(X)
SEL(s(n__from(x0)), cons(y1, y2)) → SEL(from(x0), activate(y2))
SEL(s(n__sel(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
ACTIVATE(n__indx(X1, X2)) → INDX(X1, X2)
ACTIVATE(n__dbls(X)) → DBLS(X)
SEL(s(n__dbls(x0)), cons(y1, y2)) → SEL(dbls(x0), activate(y2))
INDX(cons(X, Y), Z) → ACTIVATE(X)
ACTIVATE(n__dbl(X)) → DBL(X)
SEL(s(n__sel(y0, y1)), cons(y2, n__s(x0))) → SEL(sel(y0, y1), s(x0))
SEL(s(n__sel(y0, y1)), cons(y2, n__dbl(x0))) → SEL(sel(y0, y1), dbl(x0))
SEL(s(n__indx(x0, x1)), cons(y1, y2)) → SEL(indx(x0, x1), activate(y2))
FROM(X) → ACTIVATE(X)
DBLS(cons(X, Y)) → ACTIVATE(X)
SEL(s(X), cons(Y, Z)) → ACTIVATE(X)
DBL(s(X)) → ACTIVATE(X)
INDX(cons(X, Y), Z) → ACTIVATE(Y)
SEL(s(x0), cons(y1, y2)) → SEL(x0, activate(y2))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(n__dbl(activate(X)), n__dbls(activate(Y)))
sel(0, cons(X, Y)) → activate(X)
sel(s(X), cons(Y, Z)) → sel(activate(X), activate(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(n__sel(activate(X), activate(Z)), n__indx(activate(Y), activate(Z)))
from(X) → cons(activate(X), n__from(n__s(activate(X))))
dbl1(0) → 01
dbl1(s(X)) → s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) → activate(X)
sel1(s(X), cons(Y, Z)) → sel1(activate(X), activate(Z))
quote(0) → 01
quote(s(X)) → s1(quote(activate(X)))
quote(dbl(X)) → dbl1(X)
quote(sel(X, Y)) → sel1(X, Y)
s(X) → n__s(X)
dbl(X) → n__dbl(X)
dbls(X) → n__dbls(X)
sel(X1, X2) → n__sel(X1, X2)
indx(X1, X2) → n__indx(X1, X2)
from(X) → n__from(X)
activate(n__s(X)) → s(X)
activate(n__dbl(X)) → dbl(X)
activate(n__dbls(X)) → dbls(X)
activate(n__sel(X1, X2)) → sel(X1, X2)
activate(n__indx(X1, X2)) → indx(X1, X2)
activate(n__from(X)) → from(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule SEL(s(n__dbls(x0)), cons(y1, y2)) → SEL(dbls(x0), activate(y2)) at position [0] we obtained the following new rules:

SEL(s(n__dbls(cons(x0, x1))), cons(y1, y2)) → SEL(cons(n__dbl(activate(x0)), n__dbls(activate(x1))), activate(y2))
SEL(s(n__dbls(nil)), cons(y1, y2)) → SEL(nil, activate(y2))
SEL(s(n__dbls(x0)), cons(y1, y2)) → SEL(n__dbls(x0), activate(y2))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ACTIVATE(n__sel(X1, X2)) → SEL(X1, X2)
INDX(cons(X, Y), Z) → ACTIVATE(Z)
SEL(s(X), cons(Y, Z)) → ACTIVATE(Z)
SEL(s(n__sel(y0, y1)), cons(y2, n__sel(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))
SEL(0, cons(X, Y)) → ACTIVATE(X)
SEL(s(n__sel(y0, y1)), cons(y2, n__dbls(x0))) → SEL(sel(y0, y1), dbls(x0))
DBLS(cons(X, Y)) → ACTIVATE(Y)
SEL(s(n__s(x0)), cons(y1, y2)) → SEL(s(x0), activate(y2))
SEL(s(n__dbl(x0)), cons(y1, y2)) → SEL(dbl(x0), activate(y2))
SEL(s(n__sel(y0, y1)), cons(y2, n__from(x0))) → SEL(sel(y0, y1), from(x0))
ACTIVATE(n__from(X)) → FROM(X)
SEL(s(n__sel(y0, y1)), cons(y2, n__indx(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
SEL(s(n__from(x0)), cons(y1, y2)) → SEL(from(x0), activate(y2))
SEL(s(n__sel(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
ACTIVATE(n__indx(X1, X2)) → INDX(X1, X2)
ACTIVATE(n__dbls(X)) → DBLS(X)
INDX(cons(X, Y), Z) → ACTIVATE(X)
ACTIVATE(n__dbl(X)) → DBL(X)
SEL(s(n__sel(y0, y1)), cons(y2, n__s(x0))) → SEL(sel(y0, y1), s(x0))
SEL(s(n__sel(y0, y1)), cons(y2, n__dbl(x0))) → SEL(sel(y0, y1), dbl(x0))
SEL(s(n__indx(x0, x1)), cons(y1, y2)) → SEL(indx(x0, x1), activate(y2))
SEL(s(n__dbls(nil)), cons(y1, y2)) → SEL(nil, activate(y2))
FROM(X) → ACTIVATE(X)
SEL(s(n__dbls(x0)), cons(y1, y2)) → SEL(n__dbls(x0), activate(y2))
SEL(s(X), cons(Y, Z)) → ACTIVATE(X)
DBLS(cons(X, Y)) → ACTIVATE(X)
DBL(s(X)) → ACTIVATE(X)
INDX(cons(X, Y), Z) → ACTIVATE(Y)
SEL(s(x0), cons(y1, y2)) → SEL(x0, activate(y2))
SEL(s(n__dbls(cons(x0, x1))), cons(y1, y2)) → SEL(cons(n__dbl(activate(x0)), n__dbls(activate(x1))), activate(y2))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(n__dbl(activate(X)), n__dbls(activate(Y)))
sel(0, cons(X, Y)) → activate(X)
sel(s(X), cons(Y, Z)) → sel(activate(X), activate(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(n__sel(activate(X), activate(Z)), n__indx(activate(Y), activate(Z)))
from(X) → cons(activate(X), n__from(n__s(activate(X))))
dbl1(0) → 01
dbl1(s(X)) → s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) → activate(X)
sel1(s(X), cons(Y, Z)) → sel1(activate(X), activate(Z))
quote(0) → 01
quote(s(X)) → s1(quote(activate(X)))
quote(dbl(X)) → dbl1(X)
quote(sel(X, Y)) → sel1(X, Y)
s(X) → n__s(X)
dbl(X) → n__dbl(X)
dbls(X) → n__dbls(X)
sel(X1, X2) → n__sel(X1, X2)
indx(X1, X2) → n__indx(X1, X2)
from(X) → n__from(X)
activate(n__s(X)) → s(X)
activate(n__dbl(X)) → dbl(X)
activate(n__dbls(X)) → dbls(X)
activate(n__sel(X1, X2)) → sel(X1, X2)
activate(n__indx(X1, X2)) → indx(X1, X2)
activate(n__from(X)) → from(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

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ Narrowing

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ACTIVATE(n__sel(X1, X2)) → SEL(X1, X2)
INDX(cons(X, Y), Z) → ACTIVATE(Z)
SEL(s(X), cons(Y, Z)) → ACTIVATE(Z)
SEL(s(n__sel(y0, y1)), cons(y2, n__sel(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))
SEL(0, cons(X, Y)) → ACTIVATE(X)
SEL(s(n__sel(y0, y1)), cons(y2, n__dbls(x0))) → SEL(sel(y0, y1), dbls(x0))
DBLS(cons(X, Y)) → ACTIVATE(Y)
SEL(s(n__s(x0)), cons(y1, y2)) → SEL(s(x0), activate(y2))
SEL(s(n__dbl(x0)), cons(y1, y2)) → SEL(dbl(x0), activate(y2))
SEL(s(n__sel(y0, y1)), cons(y2, n__from(x0))) → SEL(sel(y0, y1), from(x0))
ACTIVATE(n__from(X)) → FROM(X)
SEL(s(n__sel(y0, y1)), cons(y2, n__indx(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
SEL(s(n__from(x0)), cons(y1, y2)) → SEL(from(x0), activate(y2))
SEL(s(n__sel(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
ACTIVATE(n__indx(X1, X2)) → INDX(X1, X2)
ACTIVATE(n__dbls(X)) → DBLS(X)
INDX(cons(X, Y), Z) → ACTIVATE(X)
ACTIVATE(n__dbl(X)) → DBL(X)
SEL(s(n__sel(y0, y1)), cons(y2, n__s(x0))) → SEL(sel(y0, y1), s(x0))
SEL(s(n__sel(y0, y1)), cons(y2, n__dbl(x0))) → SEL(sel(y0, y1), dbl(x0))
SEL(s(n__indx(x0, x1)), cons(y1, y2)) → SEL(indx(x0, x1), activate(y2))
FROM(X) → ACTIVATE(X)
SEL(s(X), cons(Y, Z)) → ACTIVATE(X)
DBLS(cons(X, Y)) → ACTIVATE(X)
DBL(s(X)) → ACTIVATE(X)
INDX(cons(X, Y), Z) → ACTIVATE(Y)
SEL(s(x0), cons(y1, y2)) → SEL(x0, activate(y2))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(n__dbl(activate(X)), n__dbls(activate(Y)))
sel(0, cons(X, Y)) → activate(X)
sel(s(X), cons(Y, Z)) → sel(activate(X), activate(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(n__sel(activate(X), activate(Z)), n__indx(activate(Y), activate(Z)))
from(X) → cons(activate(X), n__from(n__s(activate(X))))
dbl1(0) → 01
dbl1(s(X)) → s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) → activate(X)
sel1(s(X), cons(Y, Z)) → sel1(activate(X), activate(Z))
quote(0) → 01
quote(s(X)) → s1(quote(activate(X)))
quote(dbl(X)) → dbl1(X)
quote(sel(X, Y)) → sel1(X, Y)
s(X) → n__s(X)
dbl(X) → n__dbl(X)
dbls(X) → n__dbls(X)
sel(X1, X2) → n__sel(X1, X2)
indx(X1, X2) → n__indx(X1, X2)
from(X) → n__from(X)
activate(n__s(X)) → s(X)
activate(n__dbl(X)) → dbl(X)
activate(n__dbls(X)) → dbls(X)
activate(n__sel(X1, X2)) → sel(X1, X2)
activate(n__indx(X1, X2)) → indx(X1, X2)
activate(n__from(X)) → from(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule SEL(s(n__s(x0)), cons(y1, y2)) → SEL(s(x0), activate(y2)) at position [1] we obtained the following new rules:

SEL(s(n__s(y0)), cons(y1, n__from(x0))) → SEL(s(y0), from(x0))
SEL(s(n__s(y0)), cons(y1, n__s(x0))) → SEL(s(y0), s(x0))
SEL(s(n__s(y0)), cons(y1, x0)) → SEL(s(y0), x0)
SEL(s(n__s(y0)), cons(y1, n__sel(x0, x1))) → SEL(s(y0), sel(x0, x1))
SEL(s(n__s(y0)), cons(y1, n__dbl(x0))) → SEL(s(y0), dbl(x0))
SEL(s(n__s(y0)), cons(y1, n__dbls(x0))) → SEL(s(y0), dbls(x0))
SEL(s(n__s(y0)), cons(y1, n__indx(x0, x1))) → SEL(s(y0), indx(x0, x1))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ACTIVATE(n__sel(X1, X2)) → SEL(X1, X2)
INDX(cons(X, Y), Z) → ACTIVATE(Z)
SEL(s(n__s(y0)), cons(y1, n__s(x0))) → SEL(s(y0), s(x0))
SEL(s(X), cons(Y, Z)) → ACTIVATE(Z)
SEL(s(n__s(y0)), cons(y1, n__sel(x0, x1))) → SEL(s(y0), sel(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, n__sel(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))
SEL(s(n__s(y0)), cons(y1, n__dbl(x0))) → SEL(s(y0), dbl(x0))
SEL(0, cons(X, Y)) → ACTIVATE(X)
SEL(s(n__sel(y0, y1)), cons(y2, n__dbls(x0))) → SEL(sel(y0, y1), dbls(x0))
DBLS(cons(X, Y)) → ACTIVATE(Y)
SEL(s(n__dbl(x0)), cons(y1, y2)) → SEL(dbl(x0), activate(y2))
SEL(s(n__sel(y0, y1)), cons(y2, n__from(x0))) → SEL(sel(y0, y1), from(x0))
SEL(s(n__sel(y0, y1)), cons(y2, n__indx(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
ACTIVATE(n__from(X)) → FROM(X)
SEL(s(n__from(x0)), cons(y1, y2)) → SEL(from(x0), activate(y2))
SEL(s(n__sel(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
ACTIVATE(n__indx(X1, X2)) → INDX(X1, X2)
ACTIVATE(n__dbls(X)) → DBLS(X)
SEL(s(n__s(y0)), cons(y1, n__from(x0))) → SEL(s(y0), from(x0))
INDX(cons(X, Y), Z) → ACTIVATE(X)
ACTIVATE(n__dbl(X)) → DBL(X)
SEL(s(n__sel(y0, y1)), cons(y2, n__s(x0))) → SEL(sel(y0, y1), s(x0))
SEL(s(n__sel(y0, y1)), cons(y2, n__dbl(x0))) → SEL(sel(y0, y1), dbl(x0))
SEL(s(n__indx(x0, x1)), cons(y1, y2)) → SEL(indx(x0, x1), activate(y2))
FROM(X) → ACTIVATE(X)
SEL(s(n__s(y0)), cons(y1, n__dbls(x0))) → SEL(s(y0), dbls(x0))
DBLS(cons(X, Y)) → ACTIVATE(X)
SEL(s(X), cons(Y, Z)) → ACTIVATE(X)
DBL(s(X)) → ACTIVATE(X)
SEL(s(n__s(y0)), cons(y1, x0)) → SEL(s(y0), x0)
INDX(cons(X, Y), Z) → ACTIVATE(Y)
SEL(s(x0), cons(y1, y2)) → SEL(x0, activate(y2))
SEL(s(n__s(y0)), cons(y1, n__indx(x0, x1))) → SEL(s(y0), indx(x0, x1))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(n__dbl(activate(X)), n__dbls(activate(Y)))
sel(0, cons(X, Y)) → activate(X)
sel(s(X), cons(Y, Z)) → sel(activate(X), activate(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(n__sel(activate(X), activate(Z)), n__indx(activate(Y), activate(Z)))
from(X) → cons(activate(X), n__from(n__s(activate(X))))
dbl1(0) → 01
dbl1(s(X)) → s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) → activate(X)
sel1(s(X), cons(Y, Z)) → sel1(activate(X), activate(Z))
quote(0) → 01
quote(s(X)) → s1(quote(activate(X)))
quote(dbl(X)) → dbl1(X)
quote(sel(X, Y)) → sel1(X, Y)
s(X) → n__s(X)
dbl(X) → n__dbl(X)
dbls(X) → n__dbls(X)
sel(X1, X2) → n__sel(X1, X2)
indx(X1, X2) → n__indx(X1, X2)
from(X) → n__from(X)
activate(n__s(X)) → s(X)
activate(n__dbl(X)) → dbl(X)
activate(n__dbls(X)) → dbls(X)
activate(n__sel(X1, X2)) → sel(X1, X2)
activate(n__indx(X1, X2)) → indx(X1, X2)
activate(n__from(X)) → from(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule SEL(s(n__dbl(x0)), cons(y1, y2)) → SEL(dbl(x0), activate(y2)) at position [1] we obtained the following new rules:

SEL(s(n__dbl(y0)), cons(y1, n__dbls(x0))) → SEL(dbl(y0), dbls(x0))
SEL(s(n__dbl(y0)), cons(y1, x0)) → SEL(dbl(y0), x0)
SEL(s(n__dbl(y0)), cons(y1, n__sel(x0, x1))) → SEL(dbl(y0), sel(x0, x1))
SEL(s(n__dbl(y0)), cons(y1, n__s(x0))) → SEL(dbl(y0), s(x0))
SEL(s(n__dbl(y0)), cons(y1, n__dbl(x0))) → SEL(dbl(y0), dbl(x0))
SEL(s(n__dbl(y0)), cons(y1, n__indx(x0, x1))) → SEL(dbl(y0), indx(x0, x1))
SEL(s(n__dbl(y0)), cons(y1, n__from(x0))) → SEL(dbl(y0), from(x0))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ Narrowing

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ACTIVATE(n__sel(X1, X2)) → SEL(X1, X2)
INDX(cons(X, Y), Z) → ACTIVATE(Z)
SEL(s(X), cons(Y, Z)) → ACTIVATE(Z)
SEL(s(n__s(y0)), cons(y1, n__s(x0))) → SEL(s(y0), s(x0))
SEL(s(n__s(y0)), cons(y1, n__sel(x0, x1))) → SEL(s(y0), sel(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, n__sel(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))
SEL(s(n__dbl(y0)), cons(y1, n__s(x0))) → SEL(dbl(y0), s(x0))
SEL(0, cons(X, Y)) → ACTIVATE(X)
SEL(s(n__s(y0)), cons(y1, n__dbl(x0))) → SEL(s(y0), dbl(x0))
SEL(s(n__dbl(y0)), cons(y1, n__indx(x0, x1))) → SEL(dbl(y0), indx(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, n__dbls(x0))) → SEL(sel(y0, y1), dbls(x0))
DBLS(cons(X, Y)) → ACTIVATE(Y)
SEL(s(n__dbl(y0)), cons(y1, n__sel(x0, x1))) → SEL(dbl(y0), sel(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, n__from(x0))) → SEL(sel(y0, y1), from(x0))
ACTIVATE(n__from(X)) → FROM(X)
SEL(s(n__sel(y0, y1)), cons(y2, n__indx(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
SEL(s(n__from(x0)), cons(y1, y2)) → SEL(from(x0), activate(y2))
SEL(s(n__sel(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
ACTIVATE(n__indx(X1, X2)) → INDX(X1, X2)
SEL(s(n__dbl(y0)), cons(y1, n__dbl(x0))) → SEL(dbl(y0), dbl(x0))
ACTIVATE(n__dbls(X)) → DBLS(X)
SEL(s(n__dbl(y0)), cons(y1, n__from(x0))) → SEL(dbl(y0), from(x0))
SEL(s(n__dbl(y0)), cons(y1, n__dbls(x0))) → SEL(dbl(y0), dbls(x0))
SEL(s(n__s(y0)), cons(y1, n__from(x0))) → SEL(s(y0), from(x0))
SEL(s(n__dbl(y0)), cons(y1, x0)) → SEL(dbl(y0), x0)
INDX(cons(X, Y), Z) → ACTIVATE(X)
ACTIVATE(n__dbl(X)) → DBL(X)
SEL(s(n__sel(y0, y1)), cons(y2, n__s(x0))) → SEL(sel(y0, y1), s(x0))
SEL(s(n__sel(y0, y1)), cons(y2, n__dbl(x0))) → SEL(sel(y0, y1), dbl(x0))
SEL(s(n__indx(x0, x1)), cons(y1, y2)) → SEL(indx(x0, x1), activate(y2))
FROM(X) → ACTIVATE(X)
SEL(s(n__s(y0)), cons(y1, n__dbls(x0))) → SEL(s(y0), dbls(x0))
SEL(s(X), cons(Y, Z)) → ACTIVATE(X)
DBLS(cons(X, Y)) → ACTIVATE(X)
DBL(s(X)) → ACTIVATE(X)
SEL(s(n__s(y0)), cons(y1, x0)) → SEL(s(y0), x0)
INDX(cons(X, Y), Z) → ACTIVATE(Y)
SEL(s(x0), cons(y1, y2)) → SEL(x0, activate(y2))
SEL(s(n__s(y0)), cons(y1, n__indx(x0, x1))) → SEL(s(y0), indx(x0, x1))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(n__dbl(activate(X)), n__dbls(activate(Y)))
sel(0, cons(X, Y)) → activate(X)
sel(s(X), cons(Y, Z)) → sel(activate(X), activate(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(n__sel(activate(X), activate(Z)), n__indx(activate(Y), activate(Z)))
from(X) → cons(activate(X), n__from(n__s(activate(X))))
dbl1(0) → 01
dbl1(s(X)) → s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) → activate(X)
sel1(s(X), cons(Y, Z)) → sel1(activate(X), activate(Z))
quote(0) → 01
quote(s(X)) → s1(quote(activate(X)))
quote(dbl(X)) → dbl1(X)
quote(sel(X, Y)) → sel1(X, Y)
s(X) → n__s(X)
dbl(X) → n__dbl(X)
dbls(X) → n__dbls(X)
sel(X1, X2) → n__sel(X1, X2)
indx(X1, X2) → n__indx(X1, X2)
from(X) → n__from(X)
activate(n__s(X)) → s(X)
activate(n__dbl(X)) → dbl(X)
activate(n__dbls(X)) → dbls(X)
activate(n__sel(X1, X2)) → sel(X1, X2)
activate(n__indx(X1, X2)) → indx(X1, X2)
activate(n__from(X)) → from(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule SEL(s(n__indx(x0, x1)), cons(y1, y2)) → SEL(indx(x0, x1), activate(y2)) at position [0] we obtained the following new rules:

SEL(s(n__indx(x0, x1)), cons(y2, y3)) → SEL(n__indx(x0, x1), activate(y3))
SEL(s(n__indx(cons(x0, x1), x2)), cons(y2, y3)) → SEL(cons(n__sel(activate(x0), activate(x2)), n__indx(activate(x1), activate(x2))), activate(y3))
SEL(s(n__indx(nil, x0)), cons(y2, y3)) → SEL(nil, activate(y3))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ACTIVATE(n__sel(X1, X2)) → SEL(X1, X2)
INDX(cons(X, Y), Z) → ACTIVATE(Z)
SEL(s(n__s(y0)), cons(y1, n__s(x0))) → SEL(s(y0), s(x0))
SEL(s(X), cons(Y, Z)) → ACTIVATE(Z)
SEL(s(n__s(y0)), cons(y1, n__sel(x0, x1))) → SEL(s(y0), sel(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, n__sel(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))
SEL(s(n__s(y0)), cons(y1, n__dbl(x0))) → SEL(s(y0), dbl(x0))
SEL(0, cons(X, Y)) → ACTIVATE(X)
SEL(s(n__dbl(y0)), cons(y1, n__s(x0))) → SEL(dbl(y0), s(x0))
SEL(s(n__dbl(y0)), cons(y1, n__indx(x0, x1))) → SEL(dbl(y0), indx(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, n__dbls(x0))) → SEL(sel(y0, y1), dbls(x0))
DBLS(cons(X, Y)) → ACTIVATE(Y)
SEL(s(n__sel(y0, y1)), cons(y2, n__from(x0))) → SEL(sel(y0, y1), from(x0))
SEL(s(n__dbl(y0)), cons(y1, n__sel(x0, x1))) → SEL(dbl(y0), sel(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, n__indx(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
ACTIVATE(n__from(X)) → FROM(X)
SEL(s(n__from(x0)), cons(y1, y2)) → SEL(from(x0), activate(y2))
SEL(s(n__sel(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
ACTIVATE(n__indx(X1, X2)) → INDX(X1, X2)
SEL(s(n__indx(nil, x0)), cons(y2, y3)) → SEL(nil, activate(y3))
ACTIVATE(n__dbls(X)) → DBLS(X)
SEL(s(n__dbl(y0)), cons(y1, n__dbl(x0))) → SEL(dbl(y0), dbl(x0))
SEL(s(n__dbl(y0)), cons(y1, n__from(x0))) → SEL(dbl(y0), from(x0))
SEL(s(n__s(y0)), cons(y1, n__from(x0))) → SEL(s(y0), from(x0))
SEL(s(n__dbl(y0)), cons(y1, n__dbls(x0))) → SEL(dbl(y0), dbls(x0))
INDX(cons(X, Y), Z) → ACTIVATE(X)
SEL(s(n__dbl(y0)), cons(y1, x0)) → SEL(dbl(y0), x0)
ACTIVATE(n__dbl(X)) → DBL(X)
SEL(s(n__sel(y0, y1)), cons(y2, n__s(x0))) → SEL(sel(y0, y1), s(x0))
SEL(s(n__sel(y0, y1)), cons(y2, n__dbl(x0))) → SEL(sel(y0, y1), dbl(x0))
SEL(s(n__indx(x0, x1)), cons(y2, y3)) → SEL(n__indx(x0, x1), activate(y3))
FROM(X) → ACTIVATE(X)
SEL(s(n__indx(cons(x0, x1), x2)), cons(y2, y3)) → SEL(cons(n__sel(activate(x0), activate(x2)), n__indx(activate(x1), activate(x2))), activate(y3))
SEL(s(n__s(y0)), cons(y1, n__dbls(x0))) → SEL(s(y0), dbls(x0))
DBLS(cons(X, Y)) → ACTIVATE(X)
SEL(s(X), cons(Y, Z)) → ACTIVATE(X)
DBL(s(X)) → ACTIVATE(X)
SEL(s(n__s(y0)), cons(y1, x0)) → SEL(s(y0), x0)
INDX(cons(X, Y), Z) → ACTIVATE(Y)
SEL(s(x0), cons(y1, y2)) → SEL(x0, activate(y2))
SEL(s(n__s(y0)), cons(y1, n__indx(x0, x1))) → SEL(s(y0), indx(x0, x1))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(n__dbl(activate(X)), n__dbls(activate(Y)))
sel(0, cons(X, Y)) → activate(X)
sel(s(X), cons(Y, Z)) → sel(activate(X), activate(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(n__sel(activate(X), activate(Z)), n__indx(activate(Y), activate(Z)))
from(X) → cons(activate(X), n__from(n__s(activate(X))))
dbl1(0) → 01
dbl1(s(X)) → s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) → activate(X)
sel1(s(X), cons(Y, Z)) → sel1(activate(X), activate(Z))
quote(0) → 01
quote(s(X)) → s1(quote(activate(X)))
quote(dbl(X)) → dbl1(X)
quote(sel(X, Y)) → sel1(X, Y)
s(X) → n__s(X)
dbl(X) → n__dbl(X)
dbls(X) → n__dbls(X)
sel(X1, X2) → n__sel(X1, X2)
indx(X1, X2) → n__indx(X1, X2)
from(X) → n__from(X)
activate(n__s(X)) → s(X)
activate(n__dbl(X)) → dbl(X)
activate(n__dbls(X)) → dbls(X)
activate(n__sel(X1, X2)) → sel(X1, X2)
activate(n__indx(X1, X2)) → indx(X1, X2)
activate(n__from(X)) → from(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

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ACTIVATE(n__sel(X1, X2)) → SEL(X1, X2)
INDX(cons(X, Y), Z) → ACTIVATE(Z)
SEL(s(X), cons(Y, Z)) → ACTIVATE(Z)
SEL(s(n__s(y0)), cons(y1, n__s(x0))) → SEL(s(y0), s(x0))
SEL(s(n__s(y0)), cons(y1, n__sel(x0, x1))) → SEL(s(y0), sel(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, n__sel(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))
SEL(s(n__dbl(y0)), cons(y1, n__s(x0))) → SEL(dbl(y0), s(x0))
SEL(s(n__s(y0)), cons(y1, n__dbl(x0))) → SEL(s(y0), dbl(x0))
SEL(0, cons(X, Y)) → ACTIVATE(X)
SEL(s(n__dbl(y0)), cons(y1, n__indx(x0, x1))) → SEL(dbl(y0), indx(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, n__dbls(x0))) → SEL(sel(y0, y1), dbls(x0))
DBLS(cons(X, Y)) → ACTIVATE(Y)
SEL(s(n__dbl(y0)), cons(y1, n__sel(x0, x1))) → SEL(dbl(y0), sel(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, n__from(x0))) → SEL(sel(y0, y1), from(x0))
ACTIVATE(n__from(X)) → FROM(X)
SEL(s(n__sel(y0, y1)), cons(y2, n__indx(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
SEL(s(n__from(x0)), cons(y1, y2)) → SEL(from(x0), activate(y2))
SEL(s(n__sel(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
ACTIVATE(n__indx(X1, X2)) → INDX(X1, X2)
SEL(s(n__dbl(y0)), cons(y1, n__dbl(x0))) → SEL(dbl(y0), dbl(x0))
ACTIVATE(n__dbls(X)) → DBLS(X)
SEL(s(n__dbl(y0)), cons(y1, n__from(x0))) → SEL(dbl(y0), from(x0))
SEL(s(n__dbl(y0)), cons(y1, n__dbls(x0))) → SEL(dbl(y0), dbls(x0))
SEL(s(n__s(y0)), cons(y1, n__from(x0))) → SEL(s(y0), from(x0))
SEL(s(n__dbl(y0)), cons(y1, x0)) → SEL(dbl(y0), x0)
INDX(cons(X, Y), Z) → ACTIVATE(X)
ACTIVATE(n__dbl(X)) → DBL(X)
SEL(s(n__sel(y0, y1)), cons(y2, n__s(x0))) → SEL(sel(y0, y1), s(x0))
SEL(s(n__sel(y0, y1)), cons(y2, n__dbl(x0))) → SEL(sel(y0, y1), dbl(x0))
FROM(X) → ACTIVATE(X)
SEL(s(n__s(y0)), cons(y1, n__dbls(x0))) → SEL(s(y0), dbls(x0))
SEL(s(X), cons(Y, Z)) → ACTIVATE(X)
DBLS(cons(X, Y)) → ACTIVATE(X)
DBL(s(X)) → ACTIVATE(X)
SEL(s(n__s(y0)), cons(y1, x0)) → SEL(s(y0), x0)
INDX(cons(X, Y), Z) → ACTIVATE(Y)
SEL(s(x0), cons(y1, y2)) → SEL(x0, activate(y2))
SEL(s(n__s(y0)), cons(y1, n__indx(x0, x1))) → SEL(s(y0), indx(x0, x1))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(n__dbl(activate(X)), n__dbls(activate(Y)))
sel(0, cons(X, Y)) → activate(X)
sel(s(X), cons(Y, Z)) → sel(activate(X), activate(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(n__sel(activate(X), activate(Z)), n__indx(activate(Y), activate(Z)))
from(X) → cons(activate(X), n__from(n__s(activate(X))))
dbl1(0) → 01
dbl1(s(X)) → s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) → activate(X)
sel1(s(X), cons(Y, Z)) → sel1(activate(X), activate(Z))
quote(0) → 01
quote(s(X)) → s1(quote(activate(X)))
quote(dbl(X)) → dbl1(X)
quote(sel(X, Y)) → sel1(X, Y)
s(X) → n__s(X)
dbl(X) → n__dbl(X)
dbls(X) → n__dbls(X)
sel(X1, X2) → n__sel(X1, X2)
indx(X1, X2) → n__indx(X1, X2)
from(X) → n__from(X)
activate(n__s(X)) → s(X)
activate(n__dbl(X)) → dbl(X)
activate(n__dbls(X)) → dbls(X)
activate(n__sel(X1, X2)) → sel(X1, X2)
activate(n__indx(X1, X2)) → indx(X1, X2)
activate(n__from(X)) → from(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule SEL(s(n__from(x0)), cons(y1, y2)) → SEL(from(x0), activate(y2)) at position [0] we obtained the following new rules:

SEL(s(n__from(x0)), cons(y1, y2)) → SEL(cons(activate(x0), n__from(n__s(activate(x0)))), activate(y2))
SEL(s(n__from(x0)), cons(y1, y2)) → SEL(n__from(x0), activate(y2))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ACTIVATE(n__sel(X1, X2)) → SEL(X1, X2)
INDX(cons(X, Y), Z) → ACTIVATE(Z)
SEL(s(n__s(y0)), cons(y1, n__s(x0))) → SEL(s(y0), s(x0))
SEL(s(X), cons(Y, Z)) → ACTIVATE(Z)
SEL(s(n__s(y0)), cons(y1, n__sel(x0, x1))) → SEL(s(y0), sel(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, n__sel(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))
SEL(0, cons(X, Y)) → ACTIVATE(X)
SEL(s(n__s(y0)), cons(y1, n__dbl(x0))) → SEL(s(y0), dbl(x0))
SEL(s(n__dbl(y0)), cons(y1, n__s(x0))) → SEL(dbl(y0), s(x0))
SEL(s(n__dbl(y0)), cons(y1, n__indx(x0, x1))) → SEL(dbl(y0), indx(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, n__dbls(x0))) → SEL(sel(y0, y1), dbls(x0))
DBLS(cons(X, Y)) → ACTIVATE(Y)
SEL(s(n__sel(y0, y1)), cons(y2, n__from(x0))) → SEL(sel(y0, y1), from(x0))
SEL(s(n__dbl(y0)), cons(y1, n__sel(x0, x1))) → SEL(dbl(y0), sel(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, n__indx(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
ACTIVATE(n__from(X)) → FROM(X)
SEL(s(n__sel(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
ACTIVATE(n__indx(X1, X2)) → INDX(X1, X2)
SEL(s(n__from(x0)), cons(y1, y2)) → SEL(cons(activate(x0), n__from(n__s(activate(x0)))), activate(y2))
ACTIVATE(n__dbls(X)) → DBLS(X)
SEL(s(n__dbl(y0)), cons(y1, n__dbl(x0))) → SEL(dbl(y0), dbl(x0))
SEL(s(n__dbl(y0)), cons(y1, n__from(x0))) → SEL(dbl(y0), from(x0))
SEL(s(n__s(y0)), cons(y1, n__from(x0))) → SEL(s(y0), from(x0))
SEL(s(n__dbl(y0)), cons(y1, n__dbls(x0))) → SEL(dbl(y0), dbls(x0))
INDX(cons(X, Y), Z) → ACTIVATE(X)
SEL(s(n__dbl(y0)), cons(y1, x0)) → SEL(dbl(y0), x0)
ACTIVATE(n__dbl(X)) → DBL(X)
SEL(s(n__sel(y0, y1)), cons(y2, n__s(x0))) → SEL(sel(y0, y1), s(x0))
SEL(s(n__sel(y0, y1)), cons(y2, n__dbl(x0))) → SEL(sel(y0, y1), dbl(x0))
FROM(X) → ACTIVATE(X)
SEL(s(n__s(y0)), cons(y1, n__dbls(x0))) → SEL(s(y0), dbls(x0))
DBLS(cons(X, Y)) → ACTIVATE(X)
SEL(s(X), cons(Y, Z)) → ACTIVATE(X)
DBL(s(X)) → ACTIVATE(X)
SEL(s(n__s(y0)), cons(y1, x0)) → SEL(s(y0), x0)
INDX(cons(X, Y), Z) → ACTIVATE(Y)
SEL(s(x0), cons(y1, y2)) → SEL(x0, activate(y2))
SEL(s(n__from(x0)), cons(y1, y2)) → SEL(n__from(x0), activate(y2))
SEL(s(n__s(y0)), cons(y1, n__indx(x0, x1))) → SEL(s(y0), indx(x0, x1))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(n__dbl(activate(X)), n__dbls(activate(Y)))
sel(0, cons(X, Y)) → activate(X)
sel(s(X), cons(Y, Z)) → sel(activate(X), activate(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(n__sel(activate(X), activate(Z)), n__indx(activate(Y), activate(Z)))
from(X) → cons(activate(X), n__from(n__s(activate(X))))
dbl1(0) → 01
dbl1(s(X)) → s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) → activate(X)
sel1(s(X), cons(Y, Z)) → sel1(activate(X), activate(Z))
quote(0) → 01
quote(s(X)) → s1(quote(activate(X)))
quote(dbl(X)) → dbl1(X)
quote(sel(X, Y)) → sel1(X, Y)
s(X) → n__s(X)
dbl(X) → n__dbl(X)
dbls(X) → n__dbls(X)
sel(X1, X2) → n__sel(X1, X2)
indx(X1, X2) → n__indx(X1, X2)
from(X) → n__from(X)
activate(n__s(X)) → s(X)
activate(n__dbl(X)) → dbl(X)
activate(n__dbls(X)) → dbls(X)
activate(n__sel(X1, X2)) → sel(X1, X2)
activate(n__indx(X1, X2)) → indx(X1, X2)
activate(n__from(X)) → from(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

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Narrowing

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ACTIVATE(n__sel(X1, X2)) → SEL(X1, X2)
INDX(cons(X, Y), Z) → ACTIVATE(Z)
SEL(s(X), cons(Y, Z)) → ACTIVATE(Z)
SEL(s(n__s(y0)), cons(y1, n__s(x0))) → SEL(s(y0), s(x0))
SEL(s(n__s(y0)), cons(y1, n__sel(x0, x1))) → SEL(s(y0), sel(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, n__sel(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))
SEL(s(n__dbl(y0)), cons(y1, n__s(x0))) → SEL(dbl(y0), s(x0))
SEL(s(n__s(y0)), cons(y1, n__dbl(x0))) → SEL(s(y0), dbl(x0))
SEL(0, cons(X, Y)) → ACTIVATE(X)
SEL(s(n__dbl(y0)), cons(y1, n__indx(x0, x1))) → SEL(dbl(y0), indx(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, n__dbls(x0))) → SEL(sel(y0, y1), dbls(x0))
DBLS(cons(X, Y)) → ACTIVATE(Y)
SEL(s(n__dbl(y0)), cons(y1, n__sel(x0, x1))) → SEL(dbl(y0), sel(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, n__from(x0))) → SEL(sel(y0, y1), from(x0))
ACTIVATE(n__from(X)) → FROM(X)
SEL(s(n__sel(y0, y1)), cons(y2, n__indx(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
ACTIVATE(n__indx(X1, X2)) → INDX(X1, X2)
SEL(s(n__dbl(y0)), cons(y1, n__dbl(x0))) → SEL(dbl(y0), dbl(x0))
ACTIVATE(n__dbls(X)) → DBLS(X)
SEL(s(n__dbl(y0)), cons(y1, n__from(x0))) → SEL(dbl(y0), from(x0))
SEL(s(n__dbl(y0)), cons(y1, n__dbls(x0))) → SEL(dbl(y0), dbls(x0))
SEL(s(n__s(y0)), cons(y1, n__from(x0))) → SEL(s(y0), from(x0))
SEL(s(n__dbl(y0)), cons(y1, x0)) → SEL(dbl(y0), x0)
INDX(cons(X, Y), Z) → ACTIVATE(X)
ACTIVATE(n__dbl(X)) → DBL(X)
SEL(s(n__sel(y0, y1)), cons(y2, n__s(x0))) → SEL(sel(y0, y1), s(x0))
SEL(s(n__sel(y0, y1)), cons(y2, n__dbl(x0))) → SEL(sel(y0, y1), dbl(x0))
FROM(X) → ACTIVATE(X)
SEL(s(n__s(y0)), cons(y1, n__dbls(x0))) → SEL(s(y0), dbls(x0))
SEL(s(X), cons(Y, Z)) → ACTIVATE(X)
DBLS(cons(X, Y)) → ACTIVATE(X)
DBL(s(X)) → ACTIVATE(X)
SEL(s(n__s(y0)), cons(y1, x0)) → SEL(s(y0), x0)
INDX(cons(X, Y), Z) → ACTIVATE(Y)
SEL(s(x0), cons(y1, y2)) → SEL(x0, activate(y2))
SEL(s(n__s(y0)), cons(y1, n__indx(x0, x1))) → SEL(s(y0), indx(x0, x1))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(n__dbl(activate(X)), n__dbls(activate(Y)))
sel(0, cons(X, Y)) → activate(X)
sel(s(X), cons(Y, Z)) → sel(activate(X), activate(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(n__sel(activate(X), activate(Z)), n__indx(activate(Y), activate(Z)))
from(X) → cons(activate(X), n__from(n__s(activate(X))))
dbl1(0) → 01
dbl1(s(X)) → s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) → activate(X)
sel1(s(X), cons(Y, Z)) → sel1(activate(X), activate(Z))
quote(0) → 01
quote(s(X)) → s1(quote(activate(X)))
quote(dbl(X)) → dbl1(X)
quote(sel(X, Y)) → sel1(X, Y)
s(X) → n__s(X)
dbl(X) → n__dbl(X)
dbls(X) → n__dbls(X)
sel(X1, X2) → n__sel(X1, X2)
indx(X1, X2) → n__indx(X1, X2)
from(X) → n__from(X)
activate(n__s(X)) → s(X)
activate(n__dbl(X)) → dbl(X)
activate(n__dbls(X)) → dbls(X)
activate(n__sel(X1, X2)) → sel(X1, X2)
activate(n__indx(X1, X2)) → indx(X1, X2)
activate(n__from(X)) → from(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule SEL(s(x0), cons(y1, y2)) → SEL(x0, activate(y2)) at position [1] we obtained the following new rules:

SEL(s(y0), cons(y1, n__s(x0))) → SEL(y0, s(x0))
SEL(s(y0), cons(y1, n__dbl(x0))) → SEL(y0, dbl(x0))
SEL(s(y0), cons(y1, n__dbls(x0))) → SEL(y0, dbls(x0))
SEL(s(y0), cons(y1, n__sel(x0, x1))) → SEL(y0, sel(x0, x1))
SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(y0), cons(y1, n__indx(x0, x1))) → SEL(y0, indx(x0, x1))
SEL(s(y0), cons(y1, n__from(x0))) → SEL(y0, from(x0))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ACTIVATE(n__sel(X1, X2)) → SEL(X1, X2)
INDX(cons(X, Y), Z) → ACTIVATE(Z)
SEL(s(n__s(y0)), cons(y1, n__s(x0))) → SEL(s(y0), s(x0))
SEL(s(X), cons(Y, Z)) → ACTIVATE(Z)
SEL(s(y0), cons(y1, n__sel(x0, x1))) → SEL(y0, sel(x0, x1))
SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(n__s(y0)), cons(y1, n__sel(x0, x1))) → SEL(s(y0), sel(x0, x1))
SEL(s(y0), cons(y1, n__indx(x0, x1))) → SEL(y0, indx(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, n__sel(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))
SEL(0, cons(X, Y)) → ACTIVATE(X)
SEL(s(n__s(y0)), cons(y1, n__dbl(x0))) → SEL(s(y0), dbl(x0))
SEL(s(n__dbl(y0)), cons(y1, n__s(x0))) → SEL(dbl(y0), s(x0))
SEL(s(y0), cons(y1, n__from(x0))) → SEL(y0, from(x0))
SEL(s(n__dbl(y0)), cons(y1, n__indx(x0, x1))) → SEL(dbl(y0), indx(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, n__dbls(x0))) → SEL(sel(y0, y1), dbls(x0))
SEL(s(y0), cons(y1, n__dbl(x0))) → SEL(y0, dbl(x0))
DBLS(cons(X, Y)) → ACTIVATE(Y)
SEL(s(n__sel(y0, y1)), cons(y2, n__from(x0))) → SEL(sel(y0, y1), from(x0))
SEL(s(n__dbl(y0)), cons(y1, n__sel(x0, x1))) → SEL(dbl(y0), sel(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, n__indx(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
ACTIVATE(n__from(X)) → FROM(X)
SEL(s(n__sel(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
ACTIVATE(n__indx(X1, X2)) → INDX(X1, X2)
ACTIVATE(n__dbls(X)) → DBLS(X)
SEL(s(n__dbl(y0)), cons(y1, n__dbl(x0))) → SEL(dbl(y0), dbl(x0))
SEL(s(n__dbl(y0)), cons(y1, n__from(x0))) → SEL(dbl(y0), from(x0))
SEL(s(y0), cons(y1, n__s(x0))) → SEL(y0, s(x0))
SEL(s(n__s(y0)), cons(y1, n__from(x0))) → SEL(s(y0), from(x0))
SEL(s(n__dbl(y0)), cons(y1, n__dbls(x0))) → SEL(dbl(y0), dbls(x0))
INDX(cons(X, Y), Z) → ACTIVATE(X)
SEL(s(n__dbl(y0)), cons(y1, x0)) → SEL(dbl(y0), x0)
ACTIVATE(n__dbl(X)) → DBL(X)
SEL(s(n__sel(y0, y1)), cons(y2, n__s(x0))) → SEL(sel(y0, y1), s(x0))
SEL(s(y0), cons(y1, n__dbls(x0))) → SEL(y0, dbls(x0))
SEL(s(n__sel(y0, y1)), cons(y2, n__dbl(x0))) → SEL(sel(y0, y1), dbl(x0))
FROM(X) → ACTIVATE(X)
SEL(s(n__s(y0)), cons(y1, n__dbls(x0))) → SEL(s(y0), dbls(x0))
DBLS(cons(X, Y)) → ACTIVATE(X)
SEL(s(X), cons(Y, Z)) → ACTIVATE(X)
DBL(s(X)) → ACTIVATE(X)
SEL(s(n__s(y0)), cons(y1, x0)) → SEL(s(y0), x0)
INDX(cons(X, Y), Z) → ACTIVATE(Y)
SEL(s(n__s(y0)), cons(y1, n__indx(x0, x1))) → SEL(s(y0), indx(x0, x1))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(n__dbl(activate(X)), n__dbls(activate(Y)))
sel(0, cons(X, Y)) → activate(X)
sel(s(X), cons(Y, Z)) → sel(activate(X), activate(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(n__sel(activate(X), activate(Z)), n__indx(activate(Y), activate(Z)))
from(X) → cons(activate(X), n__from(n__s(activate(X))))
dbl1(0) → 01
dbl1(s(X)) → s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) → activate(X)
sel1(s(X), cons(Y, Z)) → sel1(activate(X), activate(Z))
quote(0) → 01
quote(s(X)) → s1(quote(activate(X)))
quote(dbl(X)) → dbl1(X)
quote(sel(X, Y)) → sel1(X, Y)
s(X) → n__s(X)
dbl(X) → n__dbl(X)
dbls(X) → n__dbls(X)
sel(X1, X2) → n__sel(X1, X2)
indx(X1, X2) → n__indx(X1, X2)
from(X) → n__from(X)
activate(n__s(X)) → s(X)
activate(n__dbl(X)) → dbl(X)
activate(n__dbls(X)) → dbls(X)
activate(n__sel(X1, X2)) → sel(X1, X2)
activate(n__indx(X1, X2)) → indx(X1, X2)
activate(n__from(X)) → from(X)
activate(X) → X

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule SEL(s(n__sel(y0, y1)), cons(y2, n__s(x0))) → SEL(sel(y0, y1), s(x0)) at position [1] we obtained the following new rules:

SEL(s(n__sel(y0, y1)), cons(y2, n__s(x0))) → SEL(sel(y0, y1), n__s(x0))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ 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

          ↳ QDP

          ↳ QDP

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

ACTIVATE(n__sel(X1, X2)) → SEL(X1, X2)
INDX(cons(X, Y), Z) → ACTIVATE(Z)
SEL(s(X), cons(Y, Z)) → ACTIVATE(Z)
SEL(s(n__s(y0)), cons(y1, n__s(x0))) → SEL(s(y0), s(x0))
SEL(s(y0), cons(y1, n__sel(x0, x1))) → SEL(y0, sel(x0, x1))
SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(n__s(y0)), cons(y1, n__sel(x0, x1))) → SEL(s(y0), sel(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, n__sel(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))
SEL(s(y0), cons(y1, n__indx(x0, x1))) → SEL(y0, indx(x0, x1))
SEL(s(n__dbl(y0)), cons(y1, n__s(x0))) → SEL(dbl(y0), s(x0))
SEL(s(n__s(y0)), cons(y1, n__dbl(x0))) → SEL(s(y0), dbl(x0))
SEL(0, cons(X, Y)) → ACTIVATE(X)
SEL(s(y0), cons(y1, n__from(x0))) → SEL(y0, from(x0))
SEL(s(n__dbl(y0)), cons(y1, n__indx(x0, x1))) → SEL(dbl(y0), indx(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, n__dbls(x0))) → SEL(sel(y0, y1), dbls(x0))
DBLS(cons(X, Y)) → ACTIVATE(Y)
SEL(s(y0), cons(y1, n__dbl(x0))) → SEL(y0, dbl(x0))
SEL(s(n__dbl(y0)), cons(y1, n__sel(x0, x1))) → SEL(dbl(y0), sel(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, n__from(x0))) → SEL(sel(y0, y1), from(x0))
SEL(s(n__sel(y0, y1)), cons(y2, n__s(x0))) → SEL(sel(y0, y1), n__s(x0))
ACTIVATE(n__from(X)) → FROM(X)
SEL(s(n__sel(y0, y1)), cons(y2, n__indx(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
ACTIVATE(n__indx(X1, X2)) → INDX(X1, X2)
SEL(s(n__dbl(y0)), cons(y1, n__dbl(x0))) → SEL(dbl(y0), dbl(x0))
ACTIVATE(n__dbls(X)) → DBLS(X)
SEL(s(n__dbl(y0)), cons(y1, n__from(x0))) → SEL(dbl(y0), from(x0))
SEL(s(y0), cons(y1, n__s(x0))) → SEL(y0, s(x0))
SEL(s(n__dbl(y0)), cons(y1, n__dbls(x0))) → SEL(dbl(y0), dbls(x0))
SEL(s(n__s(y0)), cons(y1, n__from(x0))) → SEL(s(y0), from(x0))
SEL(s(n__dbl(y0)), cons(y1, x0)) → SEL(dbl(y0), x0)
INDX(cons(X, Y), Z) → ACTIVATE(X)
ACTIVATE(n__dbl(X)) → DBL(X)
SEL(s(n__sel(y0, y1)), cons(y2, n__dbl(x0))) → SEL(sel(y0, y1), dbl(x0))
SEL(s(y0), cons(y1, n__dbls(x0))) → SEL(y0, dbls(x0))
FROM(X) → ACTIVATE(X)
SEL(s(n__s(y0)), cons(y1, n__dbls(x0))) → SEL(s(y0), dbls(x0))
SEL(s(X), cons(Y, Z)) → ACTIVATE(X)
DBLS(cons(X, Y)) → ACTIVATE(X)
DBL(s(X)) → ACTIVATE(X)
SEL(s(n__s(y0)), cons(y1, x0)) → SEL(s(y0), x0)
INDX(cons(X, Y), Z) → ACTIVATE(Y)
SEL(s(n__s(y0)), cons(y1, n__indx(x0, x1))) → SEL(s(y0), indx(x0, x1))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(n__dbl(activate(X)), n__dbls(activate(Y)))
sel(0, cons(X, Y)) → activate(X)
sel(s(X), cons(Y, Z)) → sel(activate(X), activate(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(n__sel(activate(X), activate(Z)), n__indx(activate(Y), activate(Z)))
from(X) → cons(activate(X), n__from(n__s(activate(X))))
dbl1(0) → 01
dbl1(s(X)) → s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) → activate(X)
sel1(s(X), cons(Y, Z)) → sel1(activate(X), activate(Z))
quote(0) → 01
quote(s(X)) → s1(quote(activate(X)))
quote(dbl(X)) → dbl1(X)
quote(sel(X, Y)) → sel1(X, Y)
s(X) → n__s(X)
dbl(X) → n__dbl(X)
dbls(X) → n__dbls(X)
sel(X1, X2) → n__sel(X1, X2)
indx(X1, X2) → n__indx(X1, X2)
from(X) → n__from(X)
activate(n__s(X)) → s(X)
activate(n__dbl(X)) → dbl(X)
activate(n__dbls(X)) → dbls(X)
activate(n__sel(X1, X2)) → sel(X1, X2)
activate(n__indx(X1, X2)) → indx(X1, X2)
activate(n__from(X)) → from(X)
activate(X) → X

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ Narrowing

                              ↳ QDP

                                ↳ Narrowing

                                  ↳ QDP

                                    ↳ Narrowing

                                      ↳ QDP

                                        ↳ DependencyGraphProof

                                          ↳ QDP

                                            ↳ Narrowing

                                              ↳ QDP

                                                ↳ DependencyGraphProof

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ DependencyGraphProof

                                                              ↳ QDP

                                                                ↳ Narrowing

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ACTIVATE(n__sel(X1, X2)) → SEL(X1, X2)
INDX(cons(X, Y), Z) → ACTIVATE(Z)
SEL(s(X), cons(Y, Z)) → ACTIVATE(Z)
SEL(s(n__s(y0)), cons(y1, n__s(x0))) → SEL(s(y0), s(x0))
SEL(s(y0), cons(y1, n__sel(x0, x1))) → SEL(y0, sel(x0, x1))
SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(n__s(y0)), cons(y1, n__sel(x0, x1))) → SEL(s(y0), sel(x0, x1))
SEL(s(y0), cons(y1, n__indx(x0, x1))) → SEL(y0, indx(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, n__sel(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))
SEL(s(n__dbl(y0)), cons(y1, n__s(x0))) → SEL(dbl(y0), s(x0))
SEL(0, cons(X, Y)) → ACTIVATE(X)
SEL(s(n__s(y0)), cons(y1, n__dbl(x0))) → SEL(s(y0), dbl(x0))
SEL(s(y0), cons(y1, n__from(x0))) → SEL(y0, from(x0))
SEL(s(n__dbl(y0)), cons(y1, n__indx(x0, x1))) → SEL(dbl(y0), indx(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, n__dbls(x0))) → SEL(sel(y0, y1), dbls(x0))
DBLS(cons(X, Y)) → ACTIVATE(Y)
SEL(s(y0), cons(y1, n__dbl(x0))) → SEL(y0, dbl(x0))
SEL(s(n__dbl(y0)), cons(y1, n__sel(x0, x1))) → SEL(dbl(y0), sel(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, n__from(x0))) → SEL(sel(y0, y1), from(x0))
ACTIVATE(n__from(X)) → FROM(X)
SEL(s(n__sel(y0, y1)), cons(y2, n__indx(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
ACTIVATE(n__indx(X1, X2)) → INDX(X1, X2)
SEL(s(n__dbl(y0)), cons(y1, n__dbl(x0))) → SEL(dbl(y0), dbl(x0))
ACTIVATE(n__dbls(X)) → DBLS(X)
SEL(s(n__dbl(y0)), cons(y1, n__from(x0))) → SEL(dbl(y0), from(x0))
SEL(s(y0), cons(y1, n__s(x0))) → SEL(y0, s(x0))
SEL(s(n__dbl(y0)), cons(y1, n__dbls(x0))) → SEL(dbl(y0), dbls(x0))
SEL(s(n__s(y0)), cons(y1, n__from(x0))) → SEL(s(y0), from(x0))
SEL(s(n__dbl(y0)), cons(y1, x0)) → SEL(dbl(y0), x0)
INDX(cons(X, Y), Z) → ACTIVATE(X)
ACTIVATE(n__dbl(X)) → DBL(X)
SEL(s(y0), cons(y1, n__dbls(x0))) → SEL(y0, dbls(x0))
SEL(s(n__sel(y0, y1)), cons(y2, n__dbl(x0))) → SEL(sel(y0, y1), dbl(x0))
FROM(X) → ACTIVATE(X)
SEL(s(n__s(y0)), cons(y1, n__dbls(x0))) → SEL(s(y0), dbls(x0))
SEL(s(X), cons(Y, Z)) → ACTIVATE(X)
DBLS(cons(X, Y)) → ACTIVATE(X)
DBL(s(X)) → ACTIVATE(X)
SEL(s(n__s(y0)), cons(y1, x0)) → SEL(s(y0), x0)
INDX(cons(X, Y), Z) → ACTIVATE(Y)
SEL(s(n__s(y0)), cons(y1, n__indx(x0, x1))) → SEL(s(y0), indx(x0, x1))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(n__dbl(activate(X)), n__dbls(activate(Y)))
sel(0, cons(X, Y)) → activate(X)
sel(s(X), cons(Y, Z)) → sel(activate(X), activate(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(n__sel(activate(X), activate(Z)), n__indx(activate(Y), activate(Z)))
from(X) → cons(activate(X), n__from(n__s(activate(X))))
dbl1(0) → 01
dbl1(s(X)) → s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) → activate(X)
sel1(s(X), cons(Y, Z)) → sel1(activate(X), activate(Z))
quote(0) → 01
quote(s(X)) → s1(quote(activate(X)))
quote(dbl(X)) → dbl1(X)
quote(sel(X, Y)) → sel1(X, Y)
s(X) → n__s(X)
dbl(X) → n__dbl(X)
dbls(X) → n__dbls(X)
sel(X1, X2) → n__sel(X1, X2)
indx(X1, X2) → n__indx(X1, X2)
from(X) → n__from(X)
activate(n__s(X)) → s(X)
activate(n__dbl(X)) → dbl(X)
activate(n__dbls(X)) → dbls(X)
activate(n__sel(X1, X2)) → sel(X1, X2)
activate(n__indx(X1, X2)) → indx(X1, X2)
activate(n__from(X)) → from(X)
activate(X) → X

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

SEL(s(n__s(y0)), cons(y1, n__s(x0))) → SEL(s(y0), n__s(x0))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ 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

          ↳ QDP

          ↳ QDP

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

ACTIVATE(n__sel(X1, X2)) → SEL(X1, X2)
INDX(cons(X, Y), Z) → ACTIVATE(Z)
SEL(s(X), cons(Y, Z)) → ACTIVATE(Z)
SEL(s(y0), cons(y1, n__sel(x0, x1))) → SEL(y0, sel(x0, x1))
SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(n__s(y0)), cons(y1, n__sel(x0, x1))) → SEL(s(y0), sel(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, n__sel(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))
SEL(s(y0), cons(y1, n__indx(x0, x1))) → SEL(y0, indx(x0, x1))
SEL(s(n__s(y0)), cons(y1, n__dbl(x0))) → SEL(s(y0), dbl(x0))
SEL(0, cons(X, Y)) → ACTIVATE(X)
SEL(s(n__dbl(y0)), cons(y1, n__s(x0))) → SEL(dbl(y0), s(x0))
SEL(s(y0), cons(y1, n__from(x0))) → SEL(y0, from(x0))
SEL(s(n__dbl(y0)), cons(y1, n__indx(x0, x1))) → SEL(dbl(y0), indx(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, n__dbls(x0))) → SEL(sel(y0, y1), dbls(x0))
SEL(s(y0), cons(y1, n__dbl(x0))) → SEL(y0, dbl(x0))
DBLS(cons(X, Y)) → ACTIVATE(Y)
SEL(s(n__sel(y0, y1)), cons(y2, n__from(x0))) → SEL(sel(y0, y1), from(x0))
SEL(s(n__dbl(y0)), cons(y1, n__sel(x0, x1))) → SEL(dbl(y0), sel(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, n__indx(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
ACTIVATE(n__from(X)) → FROM(X)
SEL(s(n__sel(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
ACTIVATE(n__indx(X1, X2)) → INDX(X1, X2)
ACTIVATE(n__dbls(X)) → DBLS(X)
SEL(s(n__dbl(y0)), cons(y1, n__dbl(x0))) → SEL(dbl(y0), dbl(x0))
SEL(s(n__dbl(y0)), cons(y1, n__from(x0))) → SEL(dbl(y0), from(x0))
SEL(s(y0), cons(y1, n__s(x0))) → SEL(y0, s(x0))
SEL(s(n__s(y0)), cons(y1, n__from(x0))) → SEL(s(y0), from(x0))
SEL(s(n__dbl(y0)), cons(y1, n__dbls(x0))) → SEL(dbl(y0), dbls(x0))
INDX(cons(X, Y), Z) → ACTIVATE(X)
SEL(s(n__dbl(y0)), cons(y1, x0)) → SEL(dbl(y0), x0)
ACTIVATE(n__dbl(X)) → DBL(X)
SEL(s(n__sel(y0, y1)), cons(y2, n__dbl(x0))) → SEL(sel(y0, y1), dbl(x0))
SEL(s(y0), cons(y1, n__dbls(x0))) → SEL(y0, dbls(x0))
FROM(X) → ACTIVATE(X)
SEL(s(n__s(y0)), cons(y1, n__dbls(x0))) → SEL(s(y0), dbls(x0))
DBLS(cons(X, Y)) → ACTIVATE(X)
SEL(s(X), cons(Y, Z)) → ACTIVATE(X)
DBL(s(X)) → ACTIVATE(X)
SEL(s(n__s(y0)), cons(y1, x0)) → SEL(s(y0), x0)
INDX(cons(X, Y), Z) → ACTIVATE(Y)
SEL(s(n__s(y0)), cons(y1, n__s(x0))) → SEL(s(y0), n__s(x0))
SEL(s(n__s(y0)), cons(y1, n__indx(x0, x1))) → SEL(s(y0), indx(x0, x1))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(n__dbl(activate(X)), n__dbls(activate(Y)))
sel(0, cons(X, Y)) → activate(X)
sel(s(X), cons(Y, Z)) → sel(activate(X), activate(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(n__sel(activate(X), activate(Z)), n__indx(activate(Y), activate(Z)))
from(X) → cons(activate(X), n__from(n__s(activate(X))))
dbl1(0) → 01
dbl1(s(X)) → s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) → activate(X)
sel1(s(X), cons(Y, Z)) → sel1(activate(X), activate(Z))
quote(0) → 01
quote(s(X)) → s1(quote(activate(X)))
quote(dbl(X)) → dbl1(X)
quote(sel(X, Y)) → sel1(X, Y)
s(X) → n__s(X)
dbl(X) → n__dbl(X)
dbls(X) → n__dbls(X)
sel(X1, X2) → n__sel(X1, X2)
indx(X1, X2) → n__indx(X1, X2)
from(X) → n__from(X)
activate(n__s(X)) → s(X)
activate(n__dbl(X)) → dbl(X)
activate(n__dbls(X)) → dbls(X)
activate(n__sel(X1, X2)) → sel(X1, X2)
activate(n__indx(X1, X2)) → indx(X1, X2)
activate(n__from(X)) → from(X)
activate(X) → X

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ 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

          ↳ QDP

          ↳ QDP

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

ACTIVATE(n__sel(X1, X2)) → SEL(X1, X2)
INDX(cons(X, Y), Z) → ACTIVATE(Z)
SEL(s(X), cons(Y, Z)) → ACTIVATE(Z)
SEL(s(y0), cons(y1, n__sel(x0, x1))) → SEL(y0, sel(x0, x1))
SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(n__s(y0)), cons(y1, n__sel(x0, x1))) → SEL(s(y0), sel(x0, x1))
SEL(s(y0), cons(y1, n__indx(x0, x1))) → SEL(y0, indx(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, n__sel(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))
SEL(s(n__dbl(y0)), cons(y1, n__s(x0))) → SEL(dbl(y0), s(x0))
SEL(0, cons(X, Y)) → ACTIVATE(X)
SEL(s(n__s(y0)), cons(y1, n__dbl(x0))) → SEL(s(y0), dbl(x0))
SEL(s(y0), cons(y1, n__from(x0))) → SEL(y0, from(x0))
SEL(s(n__dbl(y0)), cons(y1, n__indx(x0, x1))) → SEL(dbl(y0), indx(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, n__dbls(x0))) → SEL(sel(y0, y1), dbls(x0))
DBLS(cons(X, Y)) → ACTIVATE(Y)
SEL(s(y0), cons(y1, n__dbl(x0))) → SEL(y0, dbl(x0))
SEL(s(n__dbl(y0)), cons(y1, n__sel(x0, x1))) → SEL(dbl(y0), sel(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, n__from(x0))) → SEL(sel(y0, y1), from(x0))
ACTIVATE(n__from(X)) → FROM(X)
SEL(s(n__sel(y0, y1)), cons(y2, n__indx(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
ACTIVATE(n__indx(X1, X2)) → INDX(X1, X2)
SEL(s(n__dbl(y0)), cons(y1, n__dbl(x0))) → SEL(dbl(y0), dbl(x0))
ACTIVATE(n__dbls(X)) → DBLS(X)
SEL(s(n__dbl(y0)), cons(y1, n__from(x0))) → SEL(dbl(y0), from(x0))
SEL(s(y0), cons(y1, n__s(x0))) → SEL(y0, s(x0))
SEL(s(n__dbl(y0)), cons(y1, n__dbls(x0))) → SEL(dbl(y0), dbls(x0))
SEL(s(n__s(y0)), cons(y1, n__from(x0))) → SEL(s(y0), from(x0))
SEL(s(n__dbl(y0)), cons(y1, x0)) → SEL(dbl(y0), x0)
INDX(cons(X, Y), Z) → ACTIVATE(X)
ACTIVATE(n__dbl(X)) → DBL(X)
SEL(s(y0), cons(y1, n__dbls(x0))) → SEL(y0, dbls(x0))
SEL(s(n__sel(y0, y1)), cons(y2, n__dbl(x0))) → SEL(sel(y0, y1), dbl(x0))
FROM(X) → ACTIVATE(X)
SEL(s(n__s(y0)), cons(y1, n__dbls(x0))) → SEL(s(y0), dbls(x0))
SEL(s(X), cons(Y, Z)) → ACTIVATE(X)
DBLS(cons(X, Y)) → ACTIVATE(X)
DBL(s(X)) → ACTIVATE(X)
SEL(s(n__s(y0)), cons(y1, x0)) → SEL(s(y0), x0)
INDX(cons(X, Y), Z) → ACTIVATE(Y)
SEL(s(n__s(y0)), cons(y1, n__indx(x0, x1))) → SEL(s(y0), indx(x0, x1))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(n__dbl(activate(X)), n__dbls(activate(Y)))
sel(0, cons(X, Y)) → activate(X)
sel(s(X), cons(Y, Z)) → sel(activate(X), activate(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(n__sel(activate(X), activate(Z)), n__indx(activate(Y), activate(Z)))
from(X) → cons(activate(X), n__from(n__s(activate(X))))
dbl1(0) → 01
dbl1(s(X)) → s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) → activate(X)
sel1(s(X), cons(Y, Z)) → sel1(activate(X), activate(Z))
quote(0) → 01
quote(s(X)) → s1(quote(activate(X)))
quote(dbl(X)) → dbl1(X)
quote(sel(X, Y)) → sel1(X, Y)
s(X) → n__s(X)
dbl(X) → n__dbl(X)
dbls(X) → n__dbls(X)
sel(X1, X2) → n__sel(X1, X2)
indx(X1, X2) → n__indx(X1, X2)
from(X) → n__from(X)
activate(n__s(X)) → s(X)
activate(n__dbl(X)) → dbl(X)
activate(n__dbls(X)) → dbls(X)
activate(n__sel(X1, X2)) → sel(X1, X2)
activate(n__indx(X1, X2)) → indx(X1, X2)
activate(n__from(X)) → from(X)
activate(X) → X

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

SEL(s(n__dbl(y0)), cons(y1, n__s(x0))) → SEL(dbl(y0), n__s(x0))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ 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

                                                                            ↳ DependencyGraphProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ACTIVATE(n__sel(X1, X2)) → SEL(X1, X2)
INDX(cons(X, Y), Z) → ACTIVATE(Z)
SEL(s(X), cons(Y, Z)) → ACTIVATE(Z)
SEL(s(y0), cons(y1, n__sel(x0, x1))) → SEL(y0, sel(x0, x1))
SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(n__s(y0)), cons(y1, n__sel(x0, x1))) → SEL(s(y0), sel(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, n__sel(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))
SEL(s(y0), cons(y1, n__indx(x0, x1))) → SEL(y0, indx(x0, x1))
SEL(s(n__s(y0)), cons(y1, n__dbl(x0))) → SEL(s(y0), dbl(x0))
SEL(0, cons(X, Y)) → ACTIVATE(X)
SEL(s(y0), cons(y1, n__from(x0))) → SEL(y0, from(x0))
SEL(s(n__dbl(y0)), cons(y1, n__indx(x0, x1))) → SEL(dbl(y0), indx(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, n__dbls(x0))) → SEL(sel(y0, y1), dbls(x0))
SEL(s(y0), cons(y1, n__dbl(x0))) → SEL(y0, dbl(x0))
DBLS(cons(X, Y)) → ACTIVATE(Y)
SEL(s(n__sel(y0, y1)), cons(y2, n__from(x0))) → SEL(sel(y0, y1), from(x0))
SEL(s(n__dbl(y0)), cons(y1, n__sel(x0, x1))) → SEL(dbl(y0), sel(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, n__indx(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
ACTIVATE(n__from(X)) → FROM(X)
SEL(s(n__sel(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
ACTIVATE(n__indx(X1, X2)) → INDX(X1, X2)
SEL(s(n__dbl(y0)), cons(y1, n__s(x0))) → SEL(dbl(y0), n__s(x0))
ACTIVATE(n__dbls(X)) → DBLS(X)
SEL(s(n__dbl(y0)), cons(y1, n__dbl(x0))) → SEL(dbl(y0), dbl(x0))
SEL(s(n__dbl(y0)), cons(y1, n__from(x0))) → SEL(dbl(y0), from(x0))
SEL(s(y0), cons(y1, n__s(x0))) → SEL(y0, s(x0))
SEL(s(n__s(y0)), cons(y1, n__from(x0))) → SEL(s(y0), from(x0))
SEL(s(n__dbl(y0)), cons(y1, n__dbls(x0))) → SEL(dbl(y0), dbls(x0))
INDX(cons(X, Y), Z) → ACTIVATE(X)
SEL(s(n__dbl(y0)), cons(y1, x0)) → SEL(dbl(y0), x0)
ACTIVATE(n__dbl(X)) → DBL(X)
SEL(s(n__sel(y0, y1)), cons(y2, n__dbl(x0))) → SEL(sel(y0, y1), dbl(x0))
SEL(s(y0), cons(y1, n__dbls(x0))) → SEL(y0, dbls(x0))
FROM(X) → ACTIVATE(X)
SEL(s(n__s(y0)), cons(y1, n__dbls(x0))) → SEL(s(y0), dbls(x0))
DBLS(cons(X, Y)) → ACTIVATE(X)
SEL(s(X), cons(Y, Z)) → ACTIVATE(X)
DBL(s(X)) → ACTIVATE(X)
SEL(s(n__s(y0)), cons(y1, x0)) → SEL(s(y0), x0)
INDX(cons(X, Y), Z) → ACTIVATE(Y)
SEL(s(n__s(y0)), cons(y1, n__indx(x0, x1))) → SEL(s(y0), indx(x0, x1))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(n__dbl(activate(X)), n__dbls(activate(Y)))
sel(0, cons(X, Y)) → activate(X)
sel(s(X), cons(Y, Z)) → sel(activate(X), activate(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(n__sel(activate(X), activate(Z)), n__indx(activate(Y), activate(Z)))
from(X) → cons(activate(X), n__from(n__s(activate(X))))
dbl1(0) → 01
dbl1(s(X)) → s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) → activate(X)
sel1(s(X), cons(Y, Z)) → sel1(activate(X), activate(Z))
quote(0) → 01
quote(s(X)) → s1(quote(activate(X)))
quote(dbl(X)) → dbl1(X)
quote(sel(X, Y)) → sel1(X, Y)
s(X) → n__s(X)
dbl(X) → n__dbl(X)
dbls(X) → n__dbls(X)
sel(X1, X2) → n__sel(X1, X2)
indx(X1, X2) → n__indx(X1, X2)
from(X) → n__from(X)
activate(n__s(X)) → s(X)
activate(n__dbl(X)) → dbl(X)
activate(n__dbls(X)) → dbls(X)
activate(n__sel(X1, X2)) → sel(X1, X2)
activate(n__indx(X1, X2)) → indx(X1, X2)
activate(n__from(X)) → from(X)
activate(X) → X

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ 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

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ACTIVATE(n__sel(X1, X2)) → SEL(X1, X2)
INDX(cons(X, Y), Z) → ACTIVATE(Z)
SEL(s(X), cons(Y, Z)) → ACTIVATE(Z)
SEL(s(y0), cons(y1, n__sel(x0, x1))) → SEL(y0, sel(x0, x1))
SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(n__s(y0)), cons(y1, n__sel(x0, x1))) → SEL(s(y0), sel(x0, x1))
SEL(s(y0), cons(y1, n__indx(x0, x1))) → SEL(y0, indx(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, n__sel(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))
SEL(0, cons(X, Y)) → ACTIVATE(X)
SEL(s(n__s(y0)), cons(y1, n__dbl(x0))) → SEL(s(y0), dbl(x0))
SEL(s(y0), cons(y1, n__from(x0))) → SEL(y0, from(x0))
SEL(s(n__dbl(y0)), cons(y1, n__indx(x0, x1))) → SEL(dbl(y0), indx(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, n__dbls(x0))) → SEL(sel(y0, y1), dbls(x0))
DBLS(cons(X, Y)) → ACTIVATE(Y)
SEL(s(y0), cons(y1, n__dbl(x0))) → SEL(y0, dbl(x0))
SEL(s(n__dbl(y0)), cons(y1, n__sel(x0, x1))) → SEL(dbl(y0), sel(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, n__from(x0))) → SEL(sel(y0, y1), from(x0))
ACTIVATE(n__from(X)) → FROM(X)
SEL(s(n__sel(y0, y1)), cons(y2, n__indx(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
ACTIVATE(n__indx(X1, X2)) → INDX(X1, X2)
SEL(s(n__dbl(y0)), cons(y1, n__dbl(x0))) → SEL(dbl(y0), dbl(x0))
ACTIVATE(n__dbls(X)) → DBLS(X)
SEL(s(n__dbl(y0)), cons(y1, n__from(x0))) → SEL(dbl(y0), from(x0))
SEL(s(y0), cons(y1, n__s(x0))) → SEL(y0, s(x0))
SEL(s(n__dbl(y0)), cons(y1, n__dbls(x0))) → SEL(dbl(y0), dbls(x0))
SEL(s(n__s(y0)), cons(y1, n__from(x0))) → SEL(s(y0), from(x0))
SEL(s(n__dbl(y0)), cons(y1, x0)) → SEL(dbl(y0), x0)
INDX(cons(X, Y), Z) → ACTIVATE(X)
ACTIVATE(n__dbl(X)) → DBL(X)
SEL(s(y0), cons(y1, n__dbls(x0))) → SEL(y0, dbls(x0))
SEL(s(n__sel(y0, y1)), cons(y2, n__dbl(x0))) → SEL(sel(y0, y1), dbl(x0))
FROM(X) → ACTIVATE(X)
SEL(s(n__s(y0)), cons(y1, n__dbls(x0))) → SEL(s(y0), dbls(x0))
SEL(s(X), cons(Y, Z)) → ACTIVATE(X)
DBLS(cons(X, Y)) → ACTIVATE(X)
DBL(s(X)) → ACTIVATE(X)
SEL(s(n__s(y0)), cons(y1, x0)) → SEL(s(y0), x0)
INDX(cons(X, Y), Z) → ACTIVATE(Y)
SEL(s(n__s(y0)), cons(y1, n__indx(x0, x1))) → SEL(s(y0), indx(x0, x1))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(n__dbl(activate(X)), n__dbls(activate(Y)))
sel(0, cons(X, Y)) → activate(X)
sel(s(X), cons(Y, Z)) → sel(activate(X), activate(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(n__sel(activate(X), activate(Z)), n__indx(activate(Y), activate(Z)))
from(X) → cons(activate(X), n__from(n__s(activate(X))))
dbl1(0) → 01
dbl1(s(X)) → s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) → activate(X)
sel1(s(X), cons(Y, Z)) → sel1(activate(X), activate(Z))
quote(0) → 01
quote(s(X)) → s1(quote(activate(X)))
quote(dbl(X)) → dbl1(X)
quote(sel(X, Y)) → sel1(X, Y)
s(X) → n__s(X)
dbl(X) → n__dbl(X)
dbls(X) → n__dbls(X)
sel(X1, X2) → n__sel(X1, X2)
indx(X1, X2) → n__indx(X1, X2)
from(X) → n__from(X)
activate(n__s(X)) → s(X)
activate(n__dbl(X)) → dbl(X)
activate(n__dbls(X)) → dbls(X)
activate(n__sel(X1, X2)) → sel(X1, X2)
activate(n__indx(X1, X2)) → indx(X1, X2)
activate(n__from(X)) → from(X)
activate(X) → X

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

SEL(s(y0), cons(y1, n__s(x0))) → SEL(y0, n__s(x0))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ 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

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ACTIVATE(n__sel(X1, X2)) → SEL(X1, X2)
INDX(cons(X, Y), Z) → ACTIVATE(Z)
SEL(s(X), cons(Y, Z)) → ACTIVATE(Z)
SEL(s(y0), cons(y1, n__sel(x0, x1))) → SEL(y0, sel(x0, x1))
SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(n__s(y0)), cons(y1, n__sel(x0, x1))) → SEL(s(y0), sel(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, n__sel(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))
SEL(s(y0), cons(y1, n__indx(x0, x1))) → SEL(y0, indx(x0, x1))
SEL(s(n__s(y0)), cons(y1, n__dbl(x0))) → SEL(s(y0), dbl(x0))
SEL(0, cons(X, Y)) → ACTIVATE(X)
SEL(s(y0), cons(y1, n__from(x0))) → SEL(y0, from(x0))
SEL(s(n__dbl(y0)), cons(y1, n__indx(x0, x1))) → SEL(dbl(y0), indx(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, n__dbls(x0))) → SEL(sel(y0, y1), dbls(x0))
SEL(s(y0), cons(y1, n__dbl(x0))) → SEL(y0, dbl(x0))
DBLS(cons(X, Y)) → ACTIVATE(Y)
SEL(s(n__sel(y0, y1)), cons(y2, n__from(x0))) → SEL(sel(y0, y1), from(x0))
SEL(s(n__dbl(y0)), cons(y1, n__sel(x0, x1))) → SEL(dbl(y0), sel(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, n__indx(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
ACTIVATE(n__from(X)) → FROM(X)
SEL(s(n__sel(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
ACTIVATE(n__indx(X1, X2)) → INDX(X1, X2)
ACTIVATE(n__dbls(X)) → DBLS(X)
SEL(s(n__dbl(y0)), cons(y1, n__dbl(x0))) → SEL(dbl(y0), dbl(x0))
SEL(s(n__dbl(y0)), cons(y1, n__from(x0))) → SEL(dbl(y0), from(x0))
SEL(s(n__s(y0)), cons(y1, n__from(x0))) → SEL(s(y0), from(x0))
SEL(s(n__dbl(y0)), cons(y1, n__dbls(x0))) → SEL(dbl(y0), dbls(x0))
INDX(cons(X, Y), Z) → ACTIVATE(X)
SEL(s(n__dbl(y0)), cons(y1, x0)) → SEL(dbl(y0), x0)
ACTIVATE(n__dbl(X)) → DBL(X)
SEL(s(n__sel(y0, y1)), cons(y2, n__dbl(x0))) → SEL(sel(y0, y1), dbl(x0))
SEL(s(y0), cons(y1, n__dbls(x0))) → SEL(y0, dbls(x0))
FROM(X) → ACTIVATE(X)
SEL(s(n__s(y0)), cons(y1, n__dbls(x0))) → SEL(s(y0), dbls(x0))
DBLS(cons(X, Y)) → ACTIVATE(X)
SEL(s(X), cons(Y, Z)) → ACTIVATE(X)
DBL(s(X)) → ACTIVATE(X)
SEL(s(n__s(y0)), cons(y1, x0)) → SEL(s(y0), x0)
INDX(cons(X, Y), Z) → ACTIVATE(Y)
SEL(s(y0), cons(y1, n__s(x0))) → SEL(y0, n__s(x0))
SEL(s(n__s(y0)), cons(y1, n__indx(x0, x1))) → SEL(s(y0), indx(x0, x1))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(n__dbl(activate(X)), n__dbls(activate(Y)))
sel(0, cons(X, Y)) → activate(X)
sel(s(X), cons(Y, Z)) → sel(activate(X), activate(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(n__sel(activate(X), activate(Z)), n__indx(activate(Y), activate(Z)))
from(X) → cons(activate(X), n__from(n__s(activate(X))))
dbl1(0) → 01
dbl1(s(X)) → s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) → activate(X)
sel1(s(X), cons(Y, Z)) → sel1(activate(X), activate(Z))
quote(0) → 01
quote(s(X)) → s1(quote(activate(X)))
quote(dbl(X)) → dbl1(X)
quote(sel(X, Y)) → sel1(X, Y)
s(X) → n__s(X)
dbl(X) → n__dbl(X)
dbls(X) → n__dbls(X)
sel(X1, X2) → n__sel(X1, X2)
indx(X1, X2) → n__indx(X1, X2)
from(X) → n__from(X)
activate(n__s(X)) → s(X)
activate(n__dbl(X)) → dbl(X)
activate(n__dbls(X)) → dbls(X)
activate(n__sel(X1, X2)) → sel(X1, X2)
activate(n__indx(X1, X2)) → indx(X1, X2)
activate(n__from(X)) → from(X)
activate(X) → X

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ Narrowing

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ 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

                                                                            ↳ DependencyGraphProof

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ DependencyGraphProof

                                                                                      ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ACTIVATE(n__sel(X1, X2)) → SEL(X1, X2)
INDX(cons(X, Y), Z) → ACTIVATE(Z)
SEL(s(X), cons(Y, Z)) → ACTIVATE(Z)
SEL(s(y0), cons(y1, n__sel(x0, x1))) → SEL(y0, sel(x0, x1))
SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(n__s(y0)), cons(y1, n__sel(x0, x1))) → SEL(s(y0), sel(x0, x1))
SEL(s(y0), cons(y1, n__indx(x0, x1))) → SEL(y0, indx(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, n__sel(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))
SEL(0, cons(X, Y)) → ACTIVATE(X)
SEL(s(n__s(y0)), cons(y1, n__dbl(x0))) → SEL(s(y0), dbl(x0))
SEL(s(y0), cons(y1, n__from(x0))) → SEL(y0, from(x0))
SEL(s(n__dbl(y0)), cons(y1, n__indx(x0, x1))) → SEL(dbl(y0), indx(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, n__dbls(x0))) → SEL(sel(y0, y1), dbls(x0))
DBLS(cons(X, Y)) → ACTIVATE(Y)
SEL(s(y0), cons(y1, n__dbl(x0))) → SEL(y0, dbl(x0))
SEL(s(n__dbl(y0)), cons(y1, n__sel(x0, x1))) → SEL(dbl(y0), sel(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, n__from(x0))) → SEL(sel(y0, y1), from(x0))
ACTIVATE(n__from(X)) → FROM(X)
SEL(s(n__sel(y0, y1)), cons(y2, n__indx(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
SEL(s(n__sel(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
ACTIVATE(n__indx(X1, X2)) → INDX(X1, X2)
SEL(s(n__dbl(y0)), cons(y1, n__dbl(x0))) → SEL(dbl(y0), dbl(x0))
ACTIVATE(n__dbls(X)) → DBLS(X)
SEL(s(n__dbl(y0)), cons(y1, n__from(x0))) → SEL(dbl(y0), from(x0))
SEL(s(n__dbl(y0)), cons(y1, n__dbls(x0))) → SEL(dbl(y0), dbls(x0))
SEL(s(n__s(y0)), cons(y1, n__from(x0))) → SEL(s(y0), from(x0))
SEL(s(n__dbl(y0)), cons(y1, x0)) → SEL(dbl(y0), x0)
INDX(cons(X, Y), Z) → ACTIVATE(X)
ACTIVATE(n__dbl(X)) → DBL(X)
SEL(s(y0), cons(y1, n__dbls(x0))) → SEL(y0, dbls(x0))
SEL(s(n__sel(y0, y1)), cons(y2, n__dbl(x0))) → SEL(sel(y0, y1), dbl(x0))
FROM(X) → ACTIVATE(X)
SEL(s(n__s(y0)), cons(y1, n__dbls(x0))) → SEL(s(y0), dbls(x0))
SEL(s(X), cons(Y, Z)) → ACTIVATE(X)
DBLS(cons(X, Y)) → ACTIVATE(X)
DBL(s(X)) → ACTIVATE(X)
SEL(s(n__s(y0)), cons(y1, x0)) → SEL(s(y0), x0)
INDX(cons(X, Y), Z) → ACTIVATE(Y)
SEL(s(n__s(y0)), cons(y1, n__indx(x0, x1))) → SEL(s(y0), indx(x0, x1))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(n__dbl(activate(X)), n__dbls(activate(Y)))
sel(0, cons(X, Y)) → activate(X)
sel(s(X), cons(Y, Z)) → sel(activate(X), activate(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(n__sel(activate(X), activate(Z)), n__indx(activate(Y), activate(Z)))
from(X) → cons(activate(X), n__from(n__s(activate(X))))
dbl1(0) → 01
dbl1(s(X)) → s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) → activate(X)
sel1(s(X), cons(Y, Z)) → sel1(activate(X), activate(Z))
quote(0) → 01
quote(s(X)) → s1(quote(activate(X)))
quote(dbl(X)) → dbl1(X)
quote(sel(X, Y)) → sel1(X, Y)
s(X) → n__s(X)
dbl(X) → n__dbl(X)
dbls(X) → n__dbls(X)
sel(X1, X2) → n__sel(X1, X2)
indx(X1, X2) → n__indx(X1, X2)
from(X) → n__from(X)
activate(n__s(X)) → s(X)
activate(n__dbl(X)) → dbl(X)
activate(n__dbls(X)) → dbls(X)
activate(n__sel(X1, X2)) → sel(X1, X2)
activate(n__indx(X1, X2)) → indx(X1, X2)
activate(n__from(X)) → from(X)
activate(X) → X

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

SEL1(s(X), cons(Y, Z)) → SEL1(activate(X), activate(Z))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(n__dbl(activate(X)), n__dbls(activate(Y)))
sel(0, cons(X, Y)) → activate(X)
sel(s(X), cons(Y, Z)) → sel(activate(X), activate(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(n__sel(activate(X), activate(Z)), n__indx(activate(Y), activate(Z)))
from(X) → cons(activate(X), n__from(n__s(activate(X))))
dbl1(0) → 01
dbl1(s(X)) → s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) → activate(X)
sel1(s(X), cons(Y, Z)) → sel1(activate(X), activate(Z))
quote(0) → 01
quote(s(X)) → s1(quote(activate(X)))
quote(dbl(X)) → dbl1(X)
quote(sel(X, Y)) → sel1(X, Y)
s(X) → n__s(X)
dbl(X) → n__dbl(X)
dbls(X) → n__dbls(X)
sel(X1, X2) → n__sel(X1, X2)
indx(X1, X2) → n__indx(X1, X2)
from(X) → n__from(X)
activate(n__s(X)) → s(X)
activate(n__dbl(X)) → dbl(X)
activate(n__dbls(X)) → dbls(X)
activate(n__sel(X1, X2)) → sel(X1, X2)
activate(n__indx(X1, X2)) → indx(X1, X2)
activate(n__from(X)) → from(X)
activate(X) → X

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

DBL1(s(X)) → DBL1(activate(X))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(n__dbl(activate(X)), n__dbls(activate(Y)))
sel(0, cons(X, Y)) → activate(X)
sel(s(X), cons(Y, Z)) → sel(activate(X), activate(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(n__sel(activate(X), activate(Z)), n__indx(activate(Y), activate(Z)))
from(X) → cons(activate(X), n__from(n__s(activate(X))))
dbl1(0) → 01
dbl1(s(X)) → s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) → activate(X)
sel1(s(X), cons(Y, Z)) → sel1(activate(X), activate(Z))
quote(0) → 01
quote(s(X)) → s1(quote(activate(X)))
quote(dbl(X)) → dbl1(X)
quote(sel(X, Y)) → sel1(X, Y)
s(X) → n__s(X)
dbl(X) → n__dbl(X)
dbls(X) → n__dbls(X)
sel(X1, X2) → n__sel(X1, X2)
indx(X1, X2) → n__indx(X1, X2)
from(X) → n__from(X)
activate(n__s(X)) → s(X)
activate(n__dbl(X)) → dbl(X)
activate(n__dbls(X)) → dbls(X)
activate(n__sel(X1, X2)) → sel(X1, X2)
activate(n__indx(X1, X2)) → indx(X1, X2)
activate(n__from(X)) → from(X)
activate(X) → X

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

QUOTE(s(X)) → QUOTE(activate(X))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(n__s(n__dbl(activate(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(n__dbl(activate(X)), n__dbls(activate(Y)))
sel(0, cons(X, Y)) → activate(X)
sel(s(X), cons(Y, Z)) → sel(activate(X), activate(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(n__sel(activate(X), activate(Z)), n__indx(activate(Y), activate(Z)))
from(X) → cons(activate(X), n__from(n__s(activate(X))))
dbl1(0) → 01
dbl1(s(X)) → s1(s1(dbl1(activate(X))))
sel1(0, cons(X, Y)) → activate(X)
sel1(s(X), cons(Y, Z)) → sel1(activate(X), activate(Z))
quote(0) → 01
quote(s(X)) → s1(quote(activate(X)))
quote(dbl(X)) → dbl1(X)
quote(sel(X, Y)) → sel1(X, Y)
s(X) → n__s(X)
dbl(X) → n__dbl(X)
dbls(X) → n__dbls(X)
sel(X1, X2) → n__sel(X1, X2)
indx(X1, X2) → n__indx(X1, X2)
from(X) → n__from(X)
activate(n__s(X)) → s(X)
activate(n__dbl(X)) → dbl(X)
activate(n__dbls(X)) → dbls(X)
activate(n__sel(X1, X2)) → sel(X1, X2)
activate(n__indx(X1, X2)) → indx(X1, X2)
activate(n__from(X)) → from(X)
activate(X) → X

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