Termination w.r.t. Q proof of /tmp/tmpD0IqVb/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(activate(X))
activate(n__dbls(X)) → dbls(activate(X))
activate(n__sel(X1, X2)) → sel(activate(X1), activate(X2))
activate(n__indx(X1, X2)) → indx(activate(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(activate(X))
activate(n__dbls(X)) → dbls(activate(X))
activate(n__sel(X1, X2)) → sel(activate(X1), activate(X2))
activate(n__indx(X1, X2)) → indx(activate(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:

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))
ACTIVATE(n__dbls(X)) → ACTIVATE(X)
SEL1(s(X), cons(Y, Z)) → ACTIVATE(X)
ACTIVATE(n__dbl(X)) → 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)
ACTIVATE(n__sel(X1, X2)) → SEL(activate(X1), activate(X2))
QUOTE(dbl(X)) → DBL1(X)
QUOTE(sel(X, Y)) → SEL1(X, Y)
ACTIVATE(n__sel(X1, X2)) → ACTIVATE(X2)
INDX(cons(X, Y), Z) → ACTIVATE(X)
ACTIVATE(n__dbl(X)) → DBL(activate(X))
ACTIVATE(n__dbls(X)) → DBLS(activate(X))
ACTIVATE(n__sel(X1, X2)) → ACTIVATE(X1)
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))
ACTIVATE(n__indx(X1, X2)) → INDX(activate(X1), X2)
ACTIVATE(n__indx(X1, X2)) → ACTIVATE(X1)
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(activate(X))
activate(n__dbls(X)) → dbls(activate(X))
activate(n__sel(X1, X2)) → sel(activate(X1), activate(X2))
activate(n__indx(X1, X2)) → indx(activate(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:

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))
ACTIVATE(n__dbls(X)) → ACTIVATE(X)
SEL1(s(X), cons(Y, Z)) → ACTIVATE(X)
ACTIVATE(n__dbl(X)) → 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)
ACTIVATE(n__sel(X1, X2)) → SEL(activate(X1), activate(X2))
QUOTE(dbl(X)) → DBL1(X)
QUOTE(sel(X, Y)) → SEL1(X, Y)
ACTIVATE(n__sel(X1, X2)) → ACTIVATE(X2)
INDX(cons(X, Y), Z) → ACTIVATE(X)
ACTIVATE(n__dbl(X)) → DBL(activate(X))
ACTIVATE(n__dbls(X)) → DBLS(activate(X))
ACTIVATE(n__sel(X1, X2)) → ACTIVATE(X1)
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))
ACTIVATE(n__indx(X1, X2)) → INDX(activate(X1), X2)
ACTIVATE(n__indx(X1, X2)) → ACTIVATE(X1)
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(activate(X))
activate(n__dbls(X)) → dbls(activate(X))
activate(n__sel(X1, X2)) → sel(activate(X1), activate(X2))
activate(n__indx(X1, X2)) → indx(activate(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

            ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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(activate(X))
activate(n__dbls(X)) → dbls(activate(X))
activate(n__sel(X1, X2)) → sel(activate(X1), activate(X2))
activate(n__indx(X1, X2)) → indx(activate(X1), X2)
activate(n__from(X)) → from(X)
activate(X) → X

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

The following pairs can be oriented strictly and are deleted.

ACTIVATE(n__dbls(X)) → ACTIVATE(X)
DBLS(cons(X, Y)) → ACTIVATE(Y)
DBLS(cons(X, Y)) → ACTIVATE(X)

The remaining pairs can at least be oriented weakly.

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

Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0
POL(ACTIVATE(x₁)) = x₁
POL(DBL(x₁)) = x₁
POL(DBLS(x₁)) = 1 + x₁
POL(FROM(x₁)) = x₁
POL(INDX(x₁, x₂)) = x₁ + x₂
POL(SEL(x₁, x₂)) = x₁ + x₂
POL(activate(x₁)) = x₁
POL(cons(x₁, x₂)) = max(x₁, x₂)
POL(dbl(x₁)) = x₁
POL(dbls(x₁)) = 1 + x₁
POL(from(x₁)) = x₁
POL(indx(x₁, x₂)) = x₁ + x₂
POL(n__dbl(x₁)) = x₁
POL(n__dbls(x₁)) = 1 + x₁
POL(n__from(x₁)) = x₁
POL(n__indx(x₁, x₂)) = x₁ + x₂
POL(n__s(x₁)) = x₁
POL(n__sel(x₁, x₂)) = x₁ + x₂
POL(nil) = 0
POL(s(x₁)) = x₁
POL(sel(x₁, x₂)) = x₁ + x₂

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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(activate(X))
activate(n__dbls(X)) → dbls(activate(X))
activate(n__sel(X1, X2)) → sel(activate(X1), activate(X2))
activate(n__indx(X1, X2)) → indx(activate(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

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ QDP

                    ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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

The following pairs can be oriented strictly and are deleted.

SEL(0, cons(X, Y)) → ACTIVATE(X)

The remaining pairs can at least be oriented weakly.

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

Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 1
POL(ACTIVATE(x₁)) = x₁
POL(DBL(x₁)) = x₁
POL(FROM(x₁)) = x₁
POL(INDX(x₁, x₂)) = x₁ + x₂
POL(SEL(x₁, x₂)) = x₁ + x₂
POL(activate(x₁)) = x₁
POL(cons(x₁, x₂)) = max(x₁, x₂)
POL(dbl(x₁)) = x₁
POL(dbls(x₁)) = x₁
POL(from(x₁)) = x₁
POL(indx(x₁, x₂)) = x₁ + x₂
POL(n__dbl(x₁)) = x₁
POL(n__dbls(x₁)) = x₁
POL(n__from(x₁)) = x₁
POL(n__indx(x₁, x₂)) = x₁ + x₂
POL(n__s(x₁)) = x₁
POL(n__sel(x₁, x₂)) = x₁ + x₂
POL(nil) = 0
POL(s(x₁)) = x₁
POL(sel(x₁, x₂)) = x₁ + x₂

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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(activate(X))
activate(n__dbls(X)) → dbls(activate(X))
activate(n__sel(X1, X2)) → sel(activate(X1), activate(X2))
activate(n__indx(X1, X2)) → indx(activate(X1), X2)
activate(n__from(X)) → from(X)
activate(X) → X

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

The following pairs can be oriented strictly and are deleted.

FROM(X) → ACTIVATE(X)
ACTIVATE(n__indx(X1, X2)) → INDX(activate(X1), X2)
ACTIVATE(n__indx(X1, X2)) → ACTIVATE(X1)

The remaining pairs can at least be oriented weakly.

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

Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0
POL(ACTIVATE(x₁)) = x₁
POL(DBL(x₁)) = x₁
POL(FROM(x₁)) = 1 + x₁
POL(INDX(x₁, x₂)) = x₁ + x₂
POL(SEL(x₁, x₂)) = x₁ + x₂
POL(activate(x₁)) = x₁
POL(cons(x₁, x₂)) = max(x₁, x₂)
POL(dbl(x₁)) = x₁
POL(dbls(x₁)) = x₁
POL(from(x₁)) = 1 + x₁
POL(indx(x₁, x₂)) = 1 + x₁ + x₂
POL(n__dbl(x₁)) = x₁
POL(n__dbls(x₁)) = x₁
POL(n__from(x₁)) = 1 + x₁
POL(n__indx(x₁, x₂)) = 1 + x₁ + x₂
POL(n__s(x₁)) = x₁
POL(n__sel(x₁, x₂)) = x₁ + x₂
POL(nil) = 0
POL(s(x₁)) = x₁
POL(sel(x₁, x₂)) = x₁ + x₂

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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(activate(X))
activate(n__dbls(X)) → dbls(activate(X))
activate(n__sel(X1, X2)) → sel(activate(X1), activate(X2))
activate(n__indx(X1, X2)) → indx(activate(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 4 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ACTIVATE(n__sel(X1, X2)) → ACTIVATE(X2)
SEL(s(X), cons(Y, Z)) → ACTIVATE(X)
ACTIVATE(n__dbl(X)) → DBL(activate(X))
DBL(s(X)) → ACTIVATE(X)
SEL(s(X), cons(Y, Z)) → ACTIVATE(Z)
ACTIVATE(n__sel(X1, X2)) → SEL(activate(X1), activate(X2))
ACTIVATE(n__sel(X1, X2)) → ACTIVATE(X1)
ACTIVATE(n__dbl(X)) → ACTIVATE(X)
SEL(s(X), cons(Y, Z)) → SEL(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(activate(X))
activate(n__dbls(X)) → dbls(activate(X))
activate(n__sel(X1, X2)) → sel(activate(X1), activate(X2))
activate(n__indx(X1, X2)) → indx(activate(X1), X2)
activate(n__from(X)) → from(X)
activate(X) → X

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

The following pairs can be oriented strictly and are deleted.

ACTIVATE(n__sel(X1, X2)) → ACTIVATE(X2)
SEL(s(X), cons(Y, Z)) → ACTIVATE(X)
SEL(s(X), cons(Y, Z)) → ACTIVATE(Z)
ACTIVATE(n__sel(X1, X2)) → ACTIVATE(X1)

The remaining pairs can at least be oriented weakly.

ACTIVATE(n__dbl(X)) → DBL(activate(X))
DBL(s(X)) → ACTIVATE(X)
ACTIVATE(n__sel(X1, X2)) → SEL(activate(X1), activate(X2))
ACTIVATE(n__dbl(X)) → ACTIVATE(X)
SEL(s(X), cons(Y, Z)) → SEL(activate(X), activate(Z))

Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 1
POL(ACTIVATE(x₁)) = x₁
POL(DBL(x₁)) = x₁
POL(SEL(x₁, x₂)) = 1 + x₁ + x₂
POL(activate(x₁)) = x₁
POL(cons(x₁, x₂)) = max(x₁, x₂)
POL(dbl(x₁)) = x₁
POL(dbls(x₁)) = x₁
POL(from(x₁)) = x₁
POL(indx(x₁, x₂)) = 1 + x₁ + x₂
POL(n__dbl(x₁)) = x₁
POL(n__dbls(x₁)) = x₁
POL(n__from(x₁)) = x₁
POL(n__indx(x₁, x₂)) = 1 + x₁ + x₂
POL(n__s(x₁)) = x₁
POL(n__sel(x₁, x₂)) = 1 + x₁ + x₂
POL(nil) = 1
POL(s(x₁)) = x₁
POL(sel(x₁, x₂)) = 1 + x₁ + x₂

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ACTIVATE(n__dbl(X)) → DBL(activate(X))
DBL(s(X)) → ACTIVATE(X)
ACTIVATE(n__sel(X1, X2)) → SEL(activate(X1), activate(X2))
ACTIVATE(n__dbl(X)) → ACTIVATE(X)
SEL(s(X), cons(Y, Z)) → SEL(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(activate(X))
activate(n__dbls(X)) → dbls(activate(X))
activate(n__sel(X1, X2)) → sel(activate(X1), activate(X2))
activate(n__indx(X1, X2)) → indx(activate(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 2 SCCs with 1 less node.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                          ↳ Narrowing

                                        ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

SEL(s(X), cons(Y, Z)) → SEL(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(activate(X))
activate(n__dbls(X)) → dbls(activate(X))
activate(n__sel(X1, X2)) → sel(activate(X1), activate(X2))
activate(n__indx(X1, X2)) → indx(activate(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__dbl(x0)), cons(y1, y2)) → SEL(dbl(activate(x0)), activate(y2))
SEL(s(n__s(x0)), cons(y1, y2)) → SEL(s(x0), activate(y2))
SEL(s(n__indx(x0, x1)), cons(y1, y2)) → SEL(indx(activate(x0), x1), activate(y2))
SEL(s(n__sel(x0, x1)), cons(y1, y2)) → SEL(sel(activate(x0), activate(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))
SEL(s(n__dbls(x0)), cons(y1, y2)) → SEL(dbls(activate(x0)), activate(y2))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                        ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

SEL(s(n__dbl(x0)), cons(y1, y2)) → SEL(dbl(activate(x0)), activate(y2))
SEL(s(n__s(x0)), cons(y1, y2)) → SEL(s(x0), activate(y2))
SEL(s(n__indx(x0, x1)), cons(y1, y2)) → SEL(indx(activate(x0), x1), activate(y2))
SEL(s(x0), cons(y1, y2)) → SEL(x0, activate(y2))
SEL(s(n__from(x0)), cons(y1, y2)) → SEL(from(x0), activate(y2))
SEL(s(n__sel(x0, x1)), cons(y1, y2)) → SEL(sel(activate(x0), activate(x1)), activate(y2))
SEL(s(n__dbls(x0)), cons(y1, y2)) → SEL(dbls(activate(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(activate(X))
activate(n__dbls(X)) → dbls(activate(X))
activate(n__sel(X1, X2)) → sel(activate(X1), activate(X2))
activate(n__indx(X1, X2)) → indx(activate(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(activate(x0)), activate(y2)) at position [1] we obtained the following new rules:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                        ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

SEL(s(n__dbl(y0)), cons(y1, n__indx(x0, x1))) → SEL(dbl(activate(y0)), indx(activate(x0), x1))
SEL(s(n__indx(x0, x1)), cons(y1, y2)) → SEL(indx(activate(x0), x1), activate(y2))
SEL(s(n__dbl(y0)), cons(y1, n__sel(x0, x1))) → SEL(dbl(activate(y0)), sel(activate(x0), activate(x1)))
SEL(s(n__sel(x0, x1)), cons(y1, y2)) → SEL(sel(activate(x0), activate(x1)), activate(y2))
SEL(s(n__dbl(y0)), cons(y1, n__dbls(x0))) → SEL(dbl(activate(y0)), dbls(activate(x0)))
SEL(s(n__s(x0)), cons(y1, y2)) → SEL(s(x0), activate(y2))
SEL(s(n__dbl(y0)), cons(y1, n__dbl(x0))) → SEL(dbl(activate(y0)), dbl(activate(x0)))
SEL(s(n__dbl(y0)), cons(y1, x0)) → SEL(dbl(activate(y0)), x0)
SEL(s(n__dbl(y0)), cons(y1, n__from(x0))) → SEL(dbl(activate(y0)), from(x0))
SEL(s(x0), cons(y1, y2)) → SEL(x0, activate(y2))
SEL(s(n__from(x0)), cons(y1, y2)) → SEL(from(x0), activate(y2))
SEL(s(n__dbl(y0)), cons(y1, n__s(x0))) → SEL(dbl(activate(y0)), s(x0))
SEL(s(n__dbls(x0)), cons(y1, y2)) → SEL(dbls(activate(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(activate(X))
activate(n__dbls(X)) → dbls(activate(X))
activate(n__sel(X1, X2)) → sel(activate(X1), activate(X2))
activate(n__indx(X1, X2)) → indx(activate(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(activate(x0), activate(x1)))
SEL(s(n__s(y0)), cons(y1, n__dbl(x0))) → SEL(s(y0), dbl(activate(x0)))
SEL(s(n__s(y0)), cons(y1, n__dbls(x0))) → SEL(s(y0), dbls(activate(x0)))
SEL(s(n__s(y0)), cons(y1, n__indx(x0, x1))) → SEL(s(y0), indx(activate(x0), x1))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ Narrowing

                                        ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

Q DP problem:
The TRS P consists of the following 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, n__sel(x0, x1))) → SEL(s(y0), sel(activate(x0), activate(x1)))
SEL(s(n__dbl(y0)), cons(y1, n__indx(x0, x1))) → SEL(dbl(activate(y0)), indx(activate(x0), x1))
SEL(s(n__dbl(y0)), cons(y1, n__sel(x0, x1))) → SEL(dbl(activate(y0)), sel(activate(x0), activate(x1)))
SEL(s(n__indx(x0, x1)), cons(y1, y2)) → SEL(indx(activate(x0), x1), activate(y2))
SEL(s(n__s(y0)), cons(y1, n__dbl(x0))) → SEL(s(y0), dbl(activate(x0)))
SEL(s(n__sel(x0, x1)), cons(y1, y2)) → SEL(sel(activate(x0), activate(x1)), activate(y2))
SEL(s(n__s(y0)), cons(y1, n__dbls(x0))) → SEL(s(y0), dbls(activate(x0)))
SEL(s(n__s(y0)), cons(y1, n__indx(x0, x1))) → SEL(s(y0), indx(activate(x0), x1))
SEL(s(n__dbl(y0)), cons(y1, n__dbls(x0))) → SEL(dbl(activate(y0)), dbls(activate(x0)))
SEL(s(n__dbl(y0)), cons(y1, n__dbl(x0))) → SEL(dbl(activate(y0)), dbl(activate(x0)))
SEL(s(n__dbl(y0)), cons(y1, n__from(x0))) → SEL(dbl(activate(y0)), from(x0))
SEL(s(n__dbl(y0)), cons(y1, x0)) → SEL(dbl(activate(y0)), x0)
SEL(s(n__s(y0)), cons(y1, x0)) → SEL(s(y0), x0)
SEL(s(n__from(x0)), cons(y1, y2)) → SEL(from(x0), activate(y2))
SEL(s(x0), cons(y1, y2)) → SEL(x0, activate(y2))
SEL(s(n__dbl(y0)), cons(y1, n__s(x0))) → SEL(dbl(activate(y0)), s(x0))
SEL(s(n__dbls(x0)), cons(y1, y2)) → SEL(dbls(activate(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(activate(X))
activate(n__dbls(X)) → dbls(activate(X))
activate(n__sel(X1, X2)) → sel(activate(X1), activate(X2))
activate(n__indx(X1, X2)) → indx(activate(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(activate(x0), x1), activate(y2)) at position [1] we obtained the following new rules:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ Narrowing

                                        ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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(activate(X))
activate(n__dbls(X)) → dbls(activate(X))
activate(n__sel(X1, X2)) → sel(activate(X1), activate(X2))
activate(n__indx(X1, X2)) → indx(activate(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(activate(x0), activate(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(activate(y0), activate(y1)), dbls(activate(x0)))
SEL(s(n__sel(y0, y1)), cons(y2, n__s(x0))) → SEL(sel(activate(y0), activate(y1)), s(x0))
SEL(s(n__sel(y0, y1)), cons(y2, n__dbl(x0))) → SEL(sel(activate(y0), activate(y1)), dbl(activate(x0)))
SEL(s(n__sel(y0, y1)), cons(y2, n__from(x0))) → SEL(sel(activate(y0), activate(y1)), from(x0))
SEL(s(n__sel(y0, y1)), cons(y2, n__indx(x0, x1))) → SEL(sel(activate(y0), activate(y1)), indx(activate(x0), x1))
SEL(s(n__sel(y0, y1)), cons(y2, n__sel(x0, x1))) → SEL(sel(activate(y0), activate(y1)), sel(activate(x0), activate(x1)))
SEL(s(n__sel(y0, y1)), cons(y2, x0)) → SEL(sel(activate(y0), activate(y1)), x0)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ Narrowing

                                        ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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(activate(X))
activate(n__dbls(X)) → dbls(activate(X))
activate(n__sel(X1, X2)) → sel(activate(X1), activate(X2))
activate(n__indx(X1, X2)) → indx(activate(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

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                        ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

SEL(s(n__sel(y0, y1)), cons(y2, n__s(x0))) → SEL(sel(activate(y0), activate(y1)), s(x0))
SEL(s(n__s(y0)), cons(y1, n__s(x0))) → SEL(s(y0), s(x0))
SEL(s(n__dbl(y0)), cons(y1, n__indx(x0, x1))) → SEL(dbl(activate(y0)), indx(activate(x0), x1))
SEL(s(n__sel(y0, y1)), cons(y2, n__dbl(x0))) → SEL(sel(activate(y0), activate(y1)), dbl(activate(x0)))
SEL(s(n__indx(y0, y1)), cons(y2, n__s(x0))) → SEL(indx(activate(y0), y1), s(x0))
SEL(s(n__indx(y0, y1)), cons(y2, n__sel(x0, x1))) → SEL(indx(activate(y0), y1), sel(activate(x0), activate(x1)))
SEL(s(n__dbl(y0)), cons(y1, n__dbls(x0))) → SEL(dbl(activate(y0)), dbls(activate(x0)))
SEL(s(n__dbl(y0)), cons(y1, n__dbl(x0))) → SEL(dbl(activate(y0)), dbl(activate(x0)))
SEL(s(n__dbl(y0)), cons(y1, n__from(x0))) → SEL(dbl(activate(y0)), from(x0))
SEL(s(n__sel(y0, y1)), cons(y2, n__indx(x0, x1))) → SEL(sel(activate(y0), activate(y1)), indx(activate(x0), x1))
SEL(s(n__indx(y0, y1)), cons(y2, n__dbls(x0))) → SEL(indx(activate(y0), y1), dbls(activate(x0)))
SEL(s(n__sel(y0, y1)), cons(y2, n__sel(x0, x1))) → SEL(sel(activate(y0), activate(y1)), sel(activate(x0), activate(x1)))
SEL(s(n__from(x0)), cons(y1, y2)) → SEL(cons(activate(x0), n__from(n__s(activate(x0)))), activate(y2))
SEL(s(n__indx(y0, y1)), cons(y2, n__dbl(x0))) → SEL(indx(activate(y0), y1), dbl(activate(x0)))
SEL(s(n__dbls(x0)), cons(y1, y2)) → SEL(dbls(activate(x0)), activate(y2))
SEL(s(n__sel(y0, y1)), cons(y2, x0)) → SEL(sel(activate(y0), activate(y1)), x0)
SEL(s(n__sel(y0, y1)), cons(y2, n__dbls(x0))) → SEL(sel(activate(y0), activate(y1)), dbls(activate(x0)))
SEL(s(n__s(y0)), cons(y1, n__from(x0))) → SEL(s(y0), from(x0))
SEL(s(n__s(y0)), cons(y1, n__sel(x0, x1))) → SEL(s(y0), sel(activate(x0), activate(x1)))
SEL(s(n__dbl(y0)), cons(y1, n__sel(x0, x1))) → SEL(dbl(activate(y0)), sel(activate(x0), activate(x1)))
SEL(s(n__s(y0)), cons(y1, n__dbl(x0))) → SEL(s(y0), dbl(activate(x0)))
SEL(s(n__indx(y0, y1)), cons(y2, n__indx(x0, x1))) → SEL(indx(activate(y0), y1), indx(activate(x0), x1))
SEL(s(n__indx(y0, y1)), cons(y2, n__from(x0))) → SEL(indx(activate(y0), y1), from(x0))
SEL(s(n__s(y0)), cons(y1, n__dbls(x0))) → SEL(s(y0), dbls(activate(x0)))
SEL(s(n__s(y0)), cons(y1, n__indx(x0, x1))) → SEL(s(y0), indx(activate(x0), x1))
SEL(s(n__sel(y0, y1)), cons(y2, n__from(x0))) → SEL(sel(activate(y0), activate(y1)), from(x0))
SEL(s(n__indx(y0, y1)), cons(y2, x0)) → SEL(indx(activate(y0), y1), x0)
SEL(s(n__dbl(y0)), cons(y1, x0)) → SEL(dbl(activate(y0)), x0)
SEL(s(n__s(y0)), cons(y1, x0)) → SEL(s(y0), x0)
SEL(s(x0), cons(y1, y2)) → SEL(x0, activate(y2))
SEL(s(n__dbl(y0)), cons(y1, n__s(x0))) → SEL(dbl(activate(y0)), s(x0))
SEL(s(n__from(x0)), cons(y1, y2)) → SEL(n__from(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(activate(X))
activate(n__dbls(X)) → dbls(activate(X))
activate(n__sel(X1, X2)) → sel(activate(X1), activate(X2))
activate(n__indx(X1, X2)) → indx(activate(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

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                        ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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(activate(X))
activate(n__dbls(X)) → dbls(activate(X))
activate(n__sel(X1, X2)) → sel(activate(X1), activate(X2))
activate(n__indx(X1, X2)) → indx(activate(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(activate(x0)))
SEL(s(y0), cons(y1, n__sel(x0, x1))) → SEL(y0, sel(activate(x0), activate(x1)))
SEL(s(y0), cons(y1, n__dbls(x0))) → SEL(y0, dbls(activate(x0)))
SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(y0), cons(y1, n__from(x0))) → SEL(y0, from(x0))
SEL(s(y0), cons(y1, n__indx(x0, x1))) → SEL(y0, indx(activate(x0), x1))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                        ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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(activate(X))
activate(n__dbls(X)) → dbls(activate(X))
activate(n__sel(X1, X2)) → sel(activate(X1), activate(X2))
activate(n__indx(X1, X2)) → indx(activate(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(activate(x0)), activate(y2)) at position [1] we obtained the following new rules:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                        ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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(activate(X))
activate(n__dbls(X)) → dbls(activate(X))
activate(n__sel(X1, X2)) → sel(activate(X1), activate(X2))
activate(n__indx(X1, X2)) → indx(activate(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(activate(y0)), s(x0)) at position [1] we obtained the following new rules:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                        ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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(activate(X))
activate(n__dbls(X)) → dbls(activate(X))
activate(n__sel(X1, X2)) → sel(activate(X1), activate(X2))
activate(n__indx(X1, X2)) → indx(activate(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

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                        ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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(activate(X))
activate(n__dbls(X)) → dbls(activate(X))
activate(n__sel(X1, X2)) → sel(activate(X1), activate(X2))
activate(n__indx(X1, X2)) → indx(activate(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

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                        ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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(activate(X))
activate(n__dbls(X)) → dbls(activate(X))
activate(n__sel(X1, X2)) → sel(activate(X1), activate(X2))
activate(n__indx(X1, X2)) → indx(activate(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

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                        ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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(activate(X))
activate(n__dbls(X)) → dbls(activate(X))
activate(n__sel(X1, X2)) → sel(activate(X1), activate(X2))
activate(n__indx(X1, X2)) → indx(activate(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(y0, y1)), cons(y2, n__s(x0))) → SEL(indx(activate(y0), y1), s(x0)) at position [1] we obtained the following new rules:

SEL(s(n__indx(y0, y1)), cons(y2, n__s(x0))) → SEL(indx(activate(y0), y1), n__s(x0))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                        ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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(activate(X))
activate(n__dbls(X)) → dbls(activate(X))
activate(n__sel(X1, X2)) → sel(activate(X1), activate(X2))
activate(n__indx(X1, X2)) → indx(activate(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

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                        ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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(activate(X))
activate(n__dbls(X)) → dbls(activate(X))
activate(n__sel(X1, X2)) → sel(activate(X1), activate(X2))
activate(n__indx(X1, X2)) → indx(activate(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(activate(y0), activate(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(activate(y0), activate(y1)), n__s(x0))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                        ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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(activate(X))
activate(n__dbls(X)) → dbls(activate(X))
activate(n__sel(X1, X2)) → sel(activate(X1), activate(X2))
activate(n__indx(X1, X2)) → indx(activate(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

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                        ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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(activate(X))
activate(n__dbls(X)) → dbls(activate(X))
activate(n__sel(X1, X2)) → sel(activate(X1), activate(X2))
activate(n__indx(X1, X2)) → indx(activate(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

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                        ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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(activate(X))
activate(n__dbls(X)) → dbls(activate(X))
activate(n__sel(X1, X2)) → sel(activate(X1), activate(X2))
activate(n__indx(X1, X2)) → indx(activate(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

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                        ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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(activate(X))
activate(n__dbls(X)) → dbls(activate(X))
activate(n__sel(X1, X2)) → sel(activate(X1), activate(X2))
activate(n__indx(X1, X2)) → indx(activate(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(y0)), cons(y1, n__s(x0))) → SEL(dbls(activate(y0)), s(x0)) at position [1] we obtained the following new rules:

SEL(s(n__dbls(y0)), cons(y1, n__s(x0))) → SEL(dbls(activate(y0)), n__s(x0))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                        ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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(activate(X))
activate(n__dbls(X)) → dbls(activate(X))
activate(n__sel(X1, X2)) → sel(activate(X1), activate(X2))
activate(n__indx(X1, X2)) → indx(activate(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

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ QDP

                                        ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

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(activate(X))
activate(n__dbls(X)) → dbls(activate(X))
activate(n__sel(X1, X2)) → sel(activate(X1), activate(X2))
activate(n__indx(X1, X2)) → indx(activate(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

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ QDP

                                          ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ACTIVATE(n__dbl(X)) → DBL(activate(X))
DBL(s(X)) → ACTIVATE(X)
ACTIVATE(n__dbl(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(activate(X))
activate(n__dbls(X)) → dbls(activate(X))
activate(n__sel(X1, X2)) → sel(activate(X1), activate(X2))
activate(n__indx(X1, X2)) → indx(activate(X1), X2)
activate(n__from(X)) → from(X)
activate(X) → X

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

The following pairs can be oriented strictly and are deleted.

ACTIVATE(n__dbl(X)) → DBL(activate(X))
ACTIVATE(n__dbl(X)) → ACTIVATE(X)

The remaining pairs can at least be oriented weakly.

DBL(s(X)) → ACTIVATE(X)

Used ordering: Polynomial interpretation with max and min functions [25]:

POL(0) = 0
POL(ACTIVATE(x₁)) = x₁
POL(DBL(x₁)) = x₁
POL(activate(x₁)) = x₁
POL(cons(x₁, x₂)) = max(x₁, x₂)
POL(dbl(x₁)) = 1 + x₁
POL(dbls(x₁)) = 1 + x₁
POL(from(x₁)) = 1 + x₁
POL(indx(x₁, x₂)) = x₂
POL(n__dbl(x₁)) = 1 + x₁
POL(n__dbls(x₁)) = 1 + x₁
POL(n__from(x₁)) = 1 + x₁
POL(n__indx(x₁, x₂)) = x₂
POL(n__s(x₁)) = x₁
POL(n__sel(x₁, x₂)) = x₂
POL(nil) = 0
POL(s(x₁)) = x₁
POL(sel(x₁, x₂)) = x₂

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ DependencyGraphProof

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ QDP

                                          ↳ QDPOrderProof

                                            ↳ QDP

                                              ↳ DependencyGraphProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

DBL(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(activate(X))
activate(n__dbls(X)) → dbls(activate(X))
activate(n__sel(X1, X2)) → sel(activate(X1), activate(X2))
activate(n__indx(X1, X2)) → indx(activate(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 0 SCCs with 1 less node.

↳ 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(activate(X))
activate(n__dbls(X)) → dbls(activate(X))
activate(n__sel(X1, X2)) → sel(activate(X1), activate(X2))
activate(n__indx(X1, X2)) → indx(activate(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(activate(X))
activate(n__dbls(X)) → dbls(activate(X))
activate(n__sel(X1, X2)) → sel(activate(X1), activate(X2))
activate(n__indx(X1, X2)) → indx(activate(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(activate(X))
activate(n__dbls(X)) → dbls(activate(X))
activate(n__sel(X1, X2)) → sel(activate(X1), activate(X2))
activate(n__indx(X1, X2)) → indx(activate(X1), X2)
activate(n__from(X)) → from(X)
activate(X) → X

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