Termination proof of /tmp/tmpTEg8iO/Ex3_2

Termination of the following Term Rewriting System could not be shown:

Context-sensitive rewrite system:
The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(s(dbl(X)))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
from(X) → cons(X, from(s(X)))

The replacement map contains the following entries:

dbl: {1}
0: empty set
s: empty set
dbls: {1}
nil: empty set
cons: empty set
sel: {1, 2}
indx: {1}
from: empty set

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

Context-sensitive rewrite system:
The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(s(dbl(X)))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
from(X) → cons(X, from(s(X)))

The replacement map contains the following entries:

dbl: {1}
0: empty set
s: empty set
dbls: {1}
nil: empty set
cons: empty set
sel: {1, 2}
indx: {1}
from: empty set

The CSR is orthogonal. By [10] we can switch to innermost.

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

Context-sensitive rewrite system:
The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(s(dbl(X)))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
from(X) → cons(X, from(s(X)))

The replacement map contains the following entries:

dbl: {1}
0: empty set
s: empty set
dbls: {1}
nil: empty set
cons: empty set
sel: {1, 2}
indx: {1}
from: empty set

Innermost Strategy.

Using Improved CS-DPs we result in the following initial Q-CSDP problem.

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

        ↳ QCSDP

          ↳ QCSDependencyGraphProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

Q-restricted context-sensitive dependency pair problem:
The symbols in {dbl, dbls, sel, SEL, DBL, DBLS} are replacing on all positions.
For all symbols f in {indx, INDX} we have µ(f) = {1}.
The symbols in {s, cons, from, U, FROM} are not replacing on any position.

The ordinary context-sensitive dependency pairs DP_o are:

SEL(s(X), cons(Y, Z)) → SEL(X, Z)

The collapsing dependency pairs are DP_c:

SEL(0, cons(X, Y)) → X
SEL(s(X), cons(Y, Z)) → X
SEL(s(X), cons(Y, Z)) → Z

The hidden terms of R are:

dbl(X)
dbls(Y)
sel(X, Z)
indx(Y, Z)
from(s(X))

Every hiding context is built from:

dbl on positions {1}
dbls on positions {1}
sel on positions {1, 2}
indx on positions {1}

Hence, the new unhiding pairs DP_u are :

SEL(0, cons(X, Y)) → U(X)
SEL(s(X), cons(Y, Z)) → U(X)
SEL(s(X), cons(Y, Z)) → U(Z)
U(dbl(x_0)) → U(x_0)
U(dbls(x_0)) → U(x_0)
U(sel(x_0, x_1)) → U(x_0)
U(sel(x_0, x_1)) → U(x_1)
U(indx(x_0, x_1)) → U(x_0)
U(dbl(X)) → DBL(X)
U(dbls(Y)) → DBLS(Y)
U(sel(X, Z)) → SEL(X, Z)
U(indx(Y, Z)) → INDX(Y, Z)
U(from(s(X))) → FROM(s(X))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(s(dbl(X)))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
from(X) → cons(X, from(s(X)))

The set Q consists of the following terms:

dbl(0)
dbl(s(x0))
dbls(nil)
dbls(cons(x0, x1))
sel(0, cons(x0, x1))
sel(s(x0), cons(x1, x2))
indx(nil, x0)
indx(cons(x0, x1), x2)
from(x0)

The approximation of the Context-Sensitive Dependency Graph contains 1 SCC with 4 less nodes.

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

        ↳ QCSDP

          ↳ QCSDependencyGraphProof

            ↳ QCSDP

              ↳ ConvertedToQDPProblemProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

Q-restricted context-sensitive dependency pair problem:
The symbols in {dbl, dbls, sel, SEL} are replacing on all positions.
For all symbols f in {indx} we have µ(f) = {1}.
The symbols in {s, cons, from, U} are not replacing on any position.

The TRS P consists of the following rules:

SEL(s(X), cons(Y, Z)) → SEL(X, Z)
SEL(0, cons(X, Y)) → U(X)
U(dbl(x_0)) → U(x_0)
U(dbls(x_0)) → U(x_0)
U(sel(x_0, x_1)) → U(x_0)
U(sel(x_0, x_1)) → U(x_1)
U(indx(x_0, x_1)) → U(x_0)
U(sel(X, Z)) → SEL(X, Z)
SEL(s(X), cons(Y, Z)) → U(X)
SEL(s(X), cons(Y, Z)) → U(Z)

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(s(dbl(X)))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
from(X) → cons(X, from(s(X)))

The set Q consists of the following terms:

dbl(0)
dbl(s(x0))
dbls(nil)
dbls(cons(x0, x1))
sel(0, cons(x0, x1))
sel(s(x0), cons(x1, x2))
indx(nil, x0)
indx(cons(x0, x1), x2)
from(x0)

Converted QDP Problem, but could not keep Q or minimality.

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

        ↳ QCSDP

          ↳ QCSDependencyGraphProof

            ↳ QCSDP

              ↳ ConvertedToQDPProblemProof

                ↳ QDP

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

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

U(dbl(x_0)) → U(x_0)
SEL(0, cons(X, Y)) → U(X)
SEL(s(X), cons(Y, Z)) → U(X)
U(sel(X, Z)) → SEL(X, Z)
SEL(s(X), cons(Y, Z)) → SEL(X, Z)
U(indx(x_0, x_1)) → U(x_0)
U(dbls(x_0)) → U(x_0)
U(sel(x_0, x_1)) → U(x_0)
U(sel(x_0, x_1)) → U(x_1)
SEL(s(X), cons(Y, Z)) → U(Z)

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(s(dbl(X)))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
from(X) → cons(X, from(s(X)))

Q is empty.
We have to consider all (P,Q,R)-chains.
We applied the Incomplete Giesl Middeldorp transformation [11] to transform the context-sensitive TRS to a usual TRS.

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

  ↳ Trivial-Transformation

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

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(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:

SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(x1, x2)) → MARK(x1)
MARK(indx(x1, x2)) → INDXACTIVE(mark(x1), x2)
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(indx(x1, x2)) → MARK(x1)
MARK(dbl(x1)) → MARK(x1)
MARK(dbl(x1)) → DBLACTIVE(mark(x1))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(dbls(x1)) → MARK(x1)
MARK(dbls(x1)) → DBLSACTIVE(mark(x1))
MARK(from(x1)) → FROMACTIVE(x1)
MARK(sel(x1, x2)) → SELACTIVE(mark(x1), mark(x2))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(x1, x2)) → MARK(x2)

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(X)))

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

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

  ↳ Trivial-Transformation

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

SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(x1, x2)) → MARK(x1)
MARK(indx(x1, x2)) → INDXACTIVE(mark(x1), x2)
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(indx(x1, x2)) → MARK(x1)
MARK(dbl(x1)) → MARK(x1)
MARK(dbl(x1)) → DBLACTIVE(mark(x1))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(dbls(x1)) → MARK(x1)
MARK(dbls(x1)) → DBLSACTIVE(mark(x1))
MARK(from(x1)) → FROMACTIVE(x1)
MARK(sel(x1, x2)) → SELACTIVE(mark(x1), mark(x2))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(x1, x2)) → MARK(x2)

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(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.

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

  ↳ Trivial-Transformation

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

SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(x1, x2)) → MARK(x1)
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel(x1, x2)) → SELACTIVE(mark(x1), mark(x2))
MARK(dbl(x1)) → MARK(x1)
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(dbls(x1)) → MARK(x1)
MARK(sel(x1, x2)) → MARK(x2)

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(X)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(sel(x1, x2)) → SELACTIVE(mark(x1), mark(x2)) at position [0] we obtained the following new rules:

MARK(sel(indx(x0, x1), y1)) → SELACTIVE(indxActive(mark(x0), x1), mark(y1))
MARK(sel(from(x0), y1)) → SELACTIVE(fromActive(x0), mark(y1))
MARK(sel(s(x0), y1)) → SELACTIVE(s(x0), mark(y1))
MARK(sel(nil, y1)) → SELACTIVE(nil, mark(y1))
MARK(sel(dbl(x0), y1)) → SELACTIVE(dblActive(mark(x0)), mark(y1))
MARK(sel(sel(x0, x1), y1)) → SELACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(sel(0, y1)) → SELACTIVE(0, mark(y1))
MARK(sel(cons(x0, x1), y1)) → SELACTIVE(cons(x0, x1), mark(y1))
MARK(sel(dbls(x0), y1)) → SELACTIVE(dblsActive(mark(x0)), mark(y1))

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

  ↳ Trivial-Transformation

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

SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(indx(x0, x1), y1)) → SELACTIVE(indxActive(mark(x0), x1), mark(y1))
MARK(sel(x1, x2)) → MARK(x1)
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(indx(x1, x2)) → MARK(x1)
MARK(dbl(x1)) → MARK(x1)
MARK(sel(dbl(x0), y1)) → SELACTIVE(dblActive(mark(x0)), mark(y1))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(dbls(x1)) → MARK(x1)
MARK(sel(from(x0), y1)) → SELACTIVE(fromActive(x0), mark(y1))
MARK(sel(s(x0), y1)) → SELACTIVE(s(x0), mark(y1))
MARK(sel(nil, y1)) → SELACTIVE(nil, mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(sel(x0, x1), y1)) → SELACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(sel(0, y1)) → SELACTIVE(0, mark(y1))
MARK(sel(dbls(x0), y1)) → SELACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(sel(cons(x0, x1), y1)) → SELACTIVE(cons(x0, x1), mark(y1))
MARK(sel(x1, x2)) → MARK(x2)

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(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.

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

  ↳ Trivial-Transformation

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

SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(indx(x0, x1), y1)) → SELACTIVE(indxActive(mark(x0), x1), mark(y1))
MARK(sel(x1, x2)) → MARK(x1)
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(indx(x1, x2)) → MARK(x1)
MARK(dbl(x1)) → MARK(x1)
MARK(sel(dbl(x0), y1)) → SELACTIVE(dblActive(mark(x0)), mark(y1))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(dbls(x1)) → MARK(x1)
MARK(sel(from(x0), y1)) → SELACTIVE(fromActive(x0), mark(y1))
MARK(sel(s(x0), y1)) → SELACTIVE(s(x0), mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(sel(x0, x1), y1)) → SELACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(sel(0, y1)) → SELACTIVE(0, mark(y1))
MARK(sel(dbls(x0), y1)) → SELACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(sel(x1, x2)) → MARK(x2)

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(X)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(sel(indx(x0, x1), y1)) → SELACTIVE(indxActive(mark(x0), x1), mark(y1)) at position [1] we obtained the following new rules:

MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), s(x0))) → SELACTIVE(indxActive(mark(y0), y1), s(x0))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(indx(y0, y1), 0)) → SELACTIVE(indxActive(mark(y0), y1), 0)
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), nil)) → SELACTIVE(indxActive(mark(y0), y1), nil)
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

  ↳ Trivial-Transformation

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

MARK(indx(x1, x2)) → MARK(x1)
MARK(sel(indx(y0, y1), s(x0))) → SELACTIVE(indxActive(mark(y0), y1), s(x0))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(dbls(x1)) → MARK(x1)
MARK(sel(from(x0), y1)) → SELACTIVE(fromActive(x0), mark(y1))
MARK(sel(s(x0), y1)) → SELACTIVE(s(x0), mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(0, y1)) → SELACTIVE(0, mark(y1))
MARK(sel(indx(y0, y1), nil)) → SELACTIVE(indxActive(mark(y0), y1), nil)
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(dbl(x0), y1)) → SELACTIVE(dblActive(mark(x0)), mark(y1))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(indx(y0, y1), 0)) → SELACTIVE(indxActive(mark(y0), y1), 0)
MARK(sel(sel(x0, x1), y1)) → SELACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(dbls(x0), y1)) → SELACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(sel(x1, x2)) → MARK(x2)

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(X)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 3 less nodes.

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

  ↳ Trivial-Transformation

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

MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(x1, x2)) → MARK(x1)
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(indx(x1, x2)) → MARK(x1)
MARK(dbl(x1)) → MARK(x1)
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(dbl(x0), y1)) → SELACTIVE(dblActive(mark(x0)), mark(y1))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(dbls(x1)) → MARK(x1)
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(from(x0), y1)) → SELACTIVE(fromActive(x0), mark(y1))
MARK(sel(s(x0), y1)) → SELACTIVE(s(x0), mark(y1))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(sel(x0, x1), y1)) → SELACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(sel(0, y1)) → SELACTIVE(0, mark(y1))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(dbls(x0), y1)) → SELACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(sel(x1, x2)) → MARK(x2)

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(X)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(sel(from(x0), y1)) → SELACTIVE(fromActive(x0), mark(y1)) at position [0] we obtained the following new rules:

MARK(sel(from(x0), y1)) → SELACTIVE(cons(x0, from(s(x0))), mark(y1))
MARK(sel(from(x0), y1)) → SELACTIVE(from(x0), mark(y1))

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

  ↳ Trivial-Transformation

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

SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(indx(x1, x2)) → MARK(x1)
MARK(dbl(x1)) → MARK(x1)
MARK(sel(dbl(x0), y1)) → SELACTIVE(dblActive(mark(x0)), mark(y1))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(dbls(x1)) → MARK(x1)
MARK(sel(from(x0), y1)) → SELACTIVE(cons(x0, from(s(x0))), mark(y1))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(s(x0), y1)) → SELACTIVE(s(x0), mark(y1))
MARK(sel(from(x0), y1)) → SELACTIVE(from(x0), mark(y1))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(sel(x0, x1), y1)) → SELACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(sel(0, y1)) → SELACTIVE(0, mark(y1))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x2)
MARK(sel(dbls(x0), y1)) → SELACTIVE(dblsActive(mark(x0)), mark(y1))

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(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.

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

  ↳ Trivial-Transformation

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

MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(x1, x2)) → MARK(x1)
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(indx(x1, x2)) → MARK(x1)
MARK(dbl(x1)) → MARK(x1)
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(dbl(x0), y1)) → SELACTIVE(dblActive(mark(x0)), mark(y1))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(dbls(x1)) → MARK(x1)
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(s(x0), y1)) → SELACTIVE(s(x0), mark(y1))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(sel(x0, x1), y1)) → SELACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(sel(0, y1)) → SELACTIVE(0, mark(y1))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(dbls(x0), y1)) → SELACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(sel(x1, x2)) → MARK(x2)

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(X)))

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

MARK(sel(s(y0), sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(s(y0), indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(sel(s(y0), 0)) → SELACTIVE(s(y0), 0)
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(s(y0), s(x0))) → SELACTIVE(s(y0), s(x0))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(sel(s(y0), nil)) → SELACTIVE(s(y0), nil)

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

  ↳ Trivial-Transformation

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

MARK(sel(s(y0), sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel(s(y0), 0)) → SELACTIVE(s(y0), 0)
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(dbls(x1)) → MARK(x1)
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(0, y1)) → SELACTIVE(0, mark(y1))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(x1, x2)) → MARK(x1)
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(dbl(x0), y1)) → SELACTIVE(dblActive(mark(x0)), mark(y1))
MARK(sel(s(y0), s(x0))) → SELACTIVE(s(y0), s(x0))
MARK(sel(s(y0), nil)) → SELACTIVE(s(y0), nil)
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(sel(x0, x1), y1)) → SELACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(dbls(x0), y1)) → SELACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(sel(x1, x2)) → MARK(x2)

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(X)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 3 less nodes.

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

  ↳ Trivial-Transformation

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

MARK(sel(s(y0), sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(dbls(x1)) → MARK(x1)
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(0, y1)) → SELACTIVE(0, mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(dbl(x0), y1)) → SELACTIVE(dblActive(mark(x0)), mark(y1))
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(sel(x0, x1), y1)) → SELACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(dbls(x0), y1)) → SELACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(sel(x1, x2)) → MARK(x2)

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(X)))

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

MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), s(x0))) → SELACTIVE(dblActive(mark(y0)), s(x0))
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(dbl(y0), 0)) → SELACTIVE(dblActive(mark(y0)), 0)
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), nil)) → SELACTIVE(dblActive(mark(y0)), nil)
MARK(sel(dbl(y0), indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

  ↳ Trivial-Transformation

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

MARK(sel(s(y0), sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), nil)) → SELACTIVE(dblActive(mark(y0)), nil)
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(dbl(y0), s(x0))) → SELACTIVE(dblActive(mark(y0)), s(x0))
MARK(sel(dbl(y0), 0)) → SELACTIVE(dblActive(mark(y0)), 0)
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(0, y1)) → SELACTIVE(0, mark(y1))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(dbl(y0), sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(dbl(y0), indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(sel(x0, x1), y1)) → SELACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x2)
MARK(sel(dbls(x0), y1)) → SELACTIVE(dblsActive(mark(x0)), mark(y1))

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(X)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 3 less nodes.

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

  ↳ Trivial-Transformation

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

MARK(sel(s(y0), sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(dbls(x1)) → MARK(x1)
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(0, y1)) → SELACTIVE(0, mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(dbl(y0), sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(dbl(y0), indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(sel(x0, x1), y1)) → SELACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(dbls(x0), y1)) → SELACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(sel(x1, x2)) → MARK(x2)

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(X)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(sel(sel(x0, x1), y1)) → SELACTIVE(selActive(mark(x0), mark(x1)), mark(y1)) at position [1] we obtained the following new rules:

MARK(sel(sel(y0, y1), indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(sel(y0, y1), 0)) → SELACTIVE(selActive(mark(y0), mark(y1)), 0)
MARK(sel(sel(y0, y1), dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(sel(y0, y1), s(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), s(x0))
MARK(sel(sel(y0, y1), dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), nil)) → SELACTIVE(selActive(mark(y0), mark(y1)), nil)

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

  ↳ Trivial-Transformation

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

MARK(sel(s(y0), sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(sel(y0, y1), s(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), s(x0))
MARK(sel(sel(y0, y1), dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(0, y1)) → SELACTIVE(0, mark(y1))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(sel(y0, y1), indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(x1, x2)) → MARK(x1)
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), 0)) → SELACTIVE(selActive(mark(y0), mark(y1)), 0)
MARK(sel(sel(y0, y1), dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(sel(y0, y1), nil)) → SELACTIVE(selActive(mark(y0), mark(y1)), nil)
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x2)
MARK(sel(dbls(x0), y1)) → SELACTIVE(dblsActive(mark(x0)), mark(y1))

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(X)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 3 less nodes.

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

  ↳ Trivial-Transformation

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

MARK(sel(s(y0), sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(dbls(x1)) → MARK(x1)
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(sel(y0, y1), dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(0, y1)) → SELACTIVE(0, mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(x1, x2)) → MARK(x1)
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(dbl(y0), indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(dbls(x0), y1)) → SELACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(sel(x1, x2)) → MARK(x2)

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(X)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(sel(0, y1)) → SELACTIVE(0, mark(y1)) at position [1] we obtained the following new rules:

MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(0, s(x0))) → SELACTIVE(0, s(x0))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(0, from(x0))) → SELACTIVE(0, fromActive(x0))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(0, 0)) → SELACTIVE(0, 0)
MARK(sel(0, nil)) → SELACTIVE(0, nil)

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

  ↳ Trivial-Transformation

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

MARK(sel(s(y0), sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel(0, s(x0))) → SELACTIVE(0, s(x0))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(sel(y0, y1), dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(sel(y0, y1), indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(0, from(x0))) → SELACTIVE(0, fromActive(x0))
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(dbl(y0), indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(0, 0)) → SELACTIVE(0, 0)
MARK(sel(0, nil)) → SELACTIVE(0, nil)
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x2)
MARK(sel(dbls(x0), y1)) → SELACTIVE(dblsActive(mark(x0)), mark(y1))

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(X)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 3 less nodes.

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

  ↳ Trivial-Transformation

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

MARK(sel(s(y0), sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(dbls(x1)) → MARK(x1)
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(sel(y0, y1), dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(x1, x2)) → MARK(x1)
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(0, from(x0))) → SELACTIVE(0, fromActive(x0))
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(dbl(y0), indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(dbls(x0), y1)) → SELACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(sel(x1, x2)) → MARK(x2)

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(X)))

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

MARK(sel(dbls(y0), nil)) → SELACTIVE(dblsActive(mark(y0)), nil)
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(dbls(y0), dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbls(y0), 0)) → SELACTIVE(dblsActive(mark(y0)), 0)
MARK(sel(dbls(y0), sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(dbls(y0), s(x0))) → SELACTIVE(dblsActive(mark(y0)), s(x0))
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(dbls(y0), indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

  ↳ Trivial-Transformation

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

MARK(indx(x1, x2)) → MARK(x1)
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(sel(y0, y1), dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(s(y0), sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(dbls(y0), 0)) → SELACTIVE(dblsActive(mark(y0)), 0)
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel(dbls(y0), s(x0))) → SELACTIVE(dblsActive(mark(y0)), s(x0))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(sel(y0, y1), indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(0, from(x0))) → SELACTIVE(0, fromActive(x0))
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(dbl(y0), indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), nil)) → SELACTIVE(dblsActive(mark(y0)), nil)
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(x1, x2)) → MARK(x2)

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(X)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 3 less nodes.

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

  ↳ Trivial-Transformation

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

MARK(sel(s(y0), sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(dbls(x1)) → MARK(x1)
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(dbls(y0), sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel(dbls(y0), dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(0, from(x0))) → SELACTIVE(0, fromActive(x0))
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(dbl(y0), indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x2)

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(X)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0)) at position [1] we obtained the following new rules:

MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), from(x0))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

  ↳ Trivial-Transformation

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

MARK(sel(s(y0), sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(sel(y0, y1), dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(sel(y0, y1), indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel(dbls(y0), dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(0, from(x0))) → SELACTIVE(0, fromActive(x0))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(dbl(y0), indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), from(x0))
MARK(sel(dbls(y0), indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x2)

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(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.

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

  ↳ Trivial-Transformation

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

MARK(sel(s(y0), sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(dbls(x1)) → MARK(x1)
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(dbls(y0), sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel(dbls(y0), dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(0, from(x0))) → SELACTIVE(0, fromActive(x0))
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(dbl(y0), indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x2)

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(X)))

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

SELACTIVE(s(nil), cons(y1, y2)) → SELACTIVE(nil, mark(y2))
SELACTIVE(s(0), cons(y1, y2)) → SELACTIVE(0, mark(y2))
SELACTIVE(s(indx(x0, x1)), cons(y1, y2)) → SELACTIVE(indxActive(mark(x0), x1), mark(y2))
SELACTIVE(s(dbls(x0)), cons(y1, y2)) → SELACTIVE(dblsActive(mark(x0)), mark(y2))
SELACTIVE(s(dbl(x0)), cons(y1, y2)) → SELACTIVE(dblActive(mark(x0)), mark(y2))
SELACTIVE(s(cons(x0, x1)), cons(y1, y2)) → SELACTIVE(cons(x0, x1), mark(y2))
SELACTIVE(s(from(x0)), cons(y1, y2)) → SELACTIVE(fromActive(x0), mark(y2))
SELACTIVE(s(s(x0)), cons(y1, y2)) → SELACTIVE(s(x0), mark(y2))
SELACTIVE(s(sel(x0, x1)), cons(y1, y2)) → SELACTIVE(selActive(mark(x0), mark(x1)), mark(y2))

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

  ↳ Trivial-Transformation

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

MARK(indx(x1, x2)) → MARK(x1)
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(dbls(x0)), cons(y1, y2)) → SELACTIVE(dblsActive(mark(x0)), mark(y2))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(dbls(y0), sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
SELACTIVE(s(s(x0)), cons(y1, y2)) → SELACTIVE(s(x0), mark(y2))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
SELACTIVE(s(nil), cons(y1, y2)) → SELACTIVE(nil, mark(y2))
MARK(sel(x1, x2)) → MARK(x1)
SELACTIVE(s(indx(x0, x1)), cons(y1, y2)) → SELACTIVE(indxActive(mark(x0), x1), mark(y2))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(s(y0), sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
SELACTIVE(s(dbl(x0)), cons(y1, y2)) → SELACTIVE(dblActive(mark(x0)), mark(y2))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SELACTIVE(s(cons(x0, x1)), cons(y1, y2)) → SELACTIVE(cons(x0, x1), mark(y2))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
SELACTIVE(s(from(x0)), cons(y1, y2)) → SELACTIVE(fromActive(x0), mark(y2))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(sel(y0, y1), indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(0, from(x0))) → SELACTIVE(0, fromActive(x0))
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(dbl(y0), indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
SELACTIVE(s(0), cons(y1, y2)) → SELACTIVE(0, mark(y2))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(x1, x2)) → MARK(x2)
SELACTIVE(s(sel(x0, x1)), cons(y1, y2)) → SELACTIVE(selActive(mark(x0), mark(x1)), mark(y2))

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(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.

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

  ↳ Trivial-Transformation

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

MARK(indx(x1, x2)) → MARK(x1)
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(dbls(x0)), cons(y1, y2)) → SELACTIVE(dblsActive(mark(x0)), mark(y2))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(sel(y0, y1), dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
SELACTIVE(s(s(x0)), cons(y1, y2)) → SELACTIVE(s(x0), mark(y2))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
SELACTIVE(s(indx(x0, x1)), cons(y1, y2)) → SELACTIVE(indxActive(mark(x0), x1), mark(y2))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(s(y0), sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
SELACTIVE(s(dbl(x0)), cons(y1, y2)) → SELACTIVE(dblActive(mark(x0)), mark(y2))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
SELACTIVE(s(from(x0)), cons(y1, y2)) → SELACTIVE(fromActive(x0), mark(y2))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(sel(y0, y1), indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(0, from(x0))) → SELACTIVE(0, fromActive(x0))
MARK(sel(dbl(y0), indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbls(y0), indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
SELACTIVE(s(0), cons(y1, y2)) → SELACTIVE(0, mark(y2))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(x1, x2)) → MARK(x2)
SELACTIVE(s(sel(x0, x1)), cons(y1, y2)) → SELACTIVE(selActive(mark(x0), mark(x1)), mark(y2))

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(X)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule SELACTIVE(s(0), cons(y1, y2)) → SELACTIVE(0, mark(y2)) at position [1] we obtained the following new rules:

SELACTIVE(s(0), cons(y0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
SELACTIVE(s(0), cons(y0, s(x0))) → SELACTIVE(0, s(x0))
SELACTIVE(s(0), cons(y0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
SELACTIVE(s(0), cons(y0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(0), cons(y0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
SELACTIVE(s(0), cons(y0, 0)) → SELACTIVE(0, 0)
SELACTIVE(s(0), cons(y0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
SELACTIVE(s(0), cons(y0, from(x0))) → SELACTIVE(0, fromActive(x0))
SELACTIVE(s(0), cons(y0, nil)) → SELACTIVE(0, nil)

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

  ↳ Trivial-Transformation

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

SELACTIVE(s(0), cons(y0, s(x0))) → SELACTIVE(0, s(x0))
MARK(indx(x1, x2)) → MARK(x1)
SELACTIVE(s(0), cons(y0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(dbls(x0)), cons(y1, y2)) → SELACTIVE(dblsActive(mark(x0)), mark(y2))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(0), cons(y0, from(x0))) → SELACTIVE(0, fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(dbls(y0), sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
SELACTIVE(s(0), cons(y0, 0)) → SELACTIVE(0, 0)
SELACTIVE(s(0), cons(y0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
SELACTIVE(s(0), cons(y0, nil)) → SELACTIVE(0, nil)
SELACTIVE(s(s(x0)), cons(y1, y2)) → SELACTIVE(s(x0), mark(y2))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
SELACTIVE(s(indx(x0, x1)), cons(y1, y2)) → SELACTIVE(indxActive(mark(x0), x1), mark(y2))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(s(y0), sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(0), cons(y0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
SELACTIVE(s(dbl(x0)), cons(y1, y2)) → SELACTIVE(dblActive(mark(x0)), mark(y2))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
SELACTIVE(s(0), cons(y0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
SELACTIVE(s(from(x0)), cons(y1, y2)) → SELACTIVE(fromActive(x0), mark(y2))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(sel(y0, y1), indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(0, from(x0))) → SELACTIVE(0, fromActive(x0))
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(dbl(y0), indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
SELACTIVE(s(0), cons(y0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(x1, x2)) → MARK(x2)
SELACTIVE(s(sel(x0, x1)), cons(y1, y2)) → SELACTIVE(selActive(mark(x0), mark(x1)), mark(y2))

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(X)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 3 less nodes.

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

  ↳ Trivial-Transformation

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

MARK(indx(x1, x2)) → MARK(x1)
SELACTIVE(s(0), cons(y0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(dbls(x0)), cons(y1, y2)) → SELACTIVE(dblsActive(mark(x0)), mark(y2))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(0), cons(y0, from(x0))) → SELACTIVE(0, fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(dbls(y0), sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
SELACTIVE(s(0), cons(y0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
SELACTIVE(s(s(x0)), cons(y1, y2)) → SELACTIVE(s(x0), mark(y2))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
SELACTIVE(s(indx(x0, x1)), cons(y1, y2)) → SELACTIVE(indxActive(mark(x0), x1), mark(y2))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(s(y0), sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(0), cons(y0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
SELACTIVE(s(dbl(x0)), cons(y1, y2)) → SELACTIVE(dblActive(mark(x0)), mark(y2))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
SELACTIVE(s(0), cons(y0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
SELACTIVE(s(from(x0)), cons(y1, y2)) → SELACTIVE(fromActive(x0), mark(y2))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(sel(y0, y1), indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(0, from(x0))) → SELACTIVE(0, fromActive(x0))
MARK(sel(dbl(y0), indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbls(y0), indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
SELACTIVE(s(0), cons(y0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(x1, x2)) → MARK(x2)
SELACTIVE(s(sel(x0, x1)), cons(y1, y2)) → SELACTIVE(selActive(mark(x0), mark(x1)), mark(y2))

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(X)))

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

SELACTIVE(s(indx(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(indx(y0, y1)), cons(y2, s(x0))) → SELACTIVE(indxActive(mark(y0), y1), s(x0))
SELACTIVE(s(indx(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
SELACTIVE(s(indx(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, nil)) → SELACTIVE(indxActive(mark(y0), y1), nil)
SELACTIVE(s(indx(y0, y1)), cons(y2, 0)) → SELACTIVE(indxActive(mark(y0), y1), 0)
SELACTIVE(s(indx(y0, y1)), cons(y2, dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
SELACTIVE(s(indx(y0, y1)), cons(y2, from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

  ↳ Trivial-Transformation

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

SELACTIVE(s(indx(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(indx(x1, x2)) → MARK(x1)
SELACTIVE(s(indx(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(0), cons(y0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
SELACTIVE(s(dbls(x0)), cons(y1, y2)) → SELACTIVE(dblsActive(mark(x0)), mark(y2))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(0), cons(y0, from(x0))) → SELACTIVE(0, fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(sel(y0, y1), dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(indx(y0, y1)), cons(y2, nil)) → SELACTIVE(indxActive(mark(y0), y1), nil)
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
SELACTIVE(s(0), cons(y0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
SELACTIVE(s(s(x0)), cons(y1, y2)) → SELACTIVE(s(x0), mark(y2))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(indx(y0, y1)), cons(y2, dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
SELACTIVE(s(indx(y0, y1)), cons(y2, 0)) → SELACTIVE(indxActive(mark(y0), y1), 0)
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(s(y0), sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(0), cons(y0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
SELACTIVE(s(dbl(x0)), cons(y1, y2)) → SELACTIVE(dblActive(mark(x0)), mark(y2))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(s(indx(y0, y1)), cons(y2, from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
SELACTIVE(s(0), cons(y0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
SELACTIVE(s(from(x0)), cons(y1, y2)) → SELACTIVE(fromActive(x0), mark(y2))
SELACTIVE(s(indx(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
SELACTIVE(s(indx(y0, y1)), cons(y2, s(x0))) → SELACTIVE(indxActive(mark(y0), y1), s(x0))
MARK(sel(sel(y0, y1), indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(0, from(x0))) → SELACTIVE(0, fromActive(x0))
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(dbl(y0), indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
SELACTIVE(s(0), cons(y0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(x1, x2)) → MARK(x2)
SELACTIVE(s(sel(x0, x1)), cons(y1, y2)) → SELACTIVE(selActive(mark(x0), mark(x1)), mark(y2))

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(X)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 3 less nodes.

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

  ↳ Trivial-Transformation

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

SELACTIVE(s(indx(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(indx(x1, x2)) → MARK(x1)
SELACTIVE(s(indx(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
SELACTIVE(s(0), cons(y0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(dbls(x0)), cons(y1, y2)) → SELACTIVE(dblsActive(mark(x0)), mark(y2))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(0), cons(y0, from(x0))) → SELACTIVE(0, fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(dbls(y0), sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
SELACTIVE(s(0), cons(y0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
SELACTIVE(s(s(x0)), cons(y1, y2)) → SELACTIVE(s(x0), mark(y2))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(s(y0), sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(0), cons(y0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
SELACTIVE(s(dbl(x0)), cons(y1, y2)) → SELACTIVE(dblActive(mark(x0)), mark(y2))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
SELACTIVE(s(indx(y0, y1)), cons(y2, from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
SELACTIVE(s(0), cons(y0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
SELACTIVE(s(from(x0)), cons(y1, y2)) → SELACTIVE(fromActive(x0), mark(y2))
SELACTIVE(s(indx(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(sel(y0, y1), indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(0, from(x0))) → SELACTIVE(0, fromActive(x0))
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(dbl(y0), indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbls(y0), indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
SELACTIVE(s(0), cons(y0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(x1, x2)) → MARK(x2)
SELACTIVE(s(sel(x0, x1)), cons(y1, y2)) → SELACTIVE(selActive(mark(x0), mark(x1)), mark(y2))

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(X)))

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

SELACTIVE(s(dbls(y0)), cons(y1, from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
SELACTIVE(s(dbls(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(dbls(y0)), cons(y1, dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, nil)) → SELACTIVE(dblsActive(mark(y0)), nil)
SELACTIVE(s(dbls(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
SELACTIVE(s(dbls(y0)), cons(y1, 0)) → SELACTIVE(dblsActive(mark(y0)), 0)
SELACTIVE(s(dbls(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, s(x0))) → SELACTIVE(dblsActive(mark(y0)), s(x0))
SELACTIVE(s(dbls(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

  ↳ Trivial-Transformation

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

SELACTIVE(s(dbls(y0)), cons(y1, dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(s(indx(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(indx(x1, x2)) → MARK(x1)
SELACTIVE(s(indx(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(0), cons(y0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, 0)) → SELACTIVE(dblsActive(mark(y0)), 0)
SELACTIVE(s(0), cons(y0, from(x0))) → SELACTIVE(0, fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(sel(y0, y1), dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
SELACTIVE(s(0), cons(y0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, s(x0))) → SELACTIVE(dblsActive(mark(y0)), s(x0))
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
SELACTIVE(s(s(x0)), cons(y1, y2)) → SELACTIVE(s(x0), mark(y2))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(indx(y0, y1)), cons(y2, dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(dbls(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(s(y0), sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(0), cons(y0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
SELACTIVE(s(dbl(x0)), cons(y1, y2)) → SELACTIVE(dblActive(mark(x0)), mark(y2))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SELACTIVE(s(indx(y0, y1)), cons(y2, from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
SELACTIVE(s(0), cons(y0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
SELACTIVE(s(from(x0)), cons(y1, y2)) → SELACTIVE(fromActive(x0), mark(y2))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
SELACTIVE(s(indx(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(0, from(x0))) → SELACTIVE(0, fromActive(x0))
MARK(sel(sel(y0, y1), dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(dbl(y0), indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
SELACTIVE(s(dbls(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(dbls(y0)), cons(y1, from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
SELACTIVE(s(dbls(y0)), cons(y1, nil)) → SELACTIVE(dblsActive(mark(y0)), nil)
SELACTIVE(s(0), cons(y0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(x1, x2)) → MARK(x2)
SELACTIVE(s(sel(x0, x1)), cons(y1, y2)) → SELACTIVE(selActive(mark(x0), mark(x1)), mark(y2))

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(X)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 3 less nodes.

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

  ↳ Trivial-Transformation

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

SELACTIVE(s(dbls(y0)), cons(y1, dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(s(indx(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(indx(x1, x2)) → MARK(x1)
SELACTIVE(s(indx(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(0), cons(y0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(0), cons(y0, from(x0))) → SELACTIVE(0, fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(dbls(y0), sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
SELACTIVE(s(0), cons(y0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
SELACTIVE(s(s(x0)), cons(y1, y2)) → SELACTIVE(s(x0), mark(y2))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(s(y0), sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(0), cons(y0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
SELACTIVE(s(dbl(x0)), cons(y1, y2)) → SELACTIVE(dblActive(mark(x0)), mark(y2))
SELACTIVE(s(dbls(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
SELACTIVE(s(indx(y0, y1)), cons(y2, from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
SELACTIVE(s(dbls(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
SELACTIVE(s(0), cons(y0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
SELACTIVE(s(from(x0)), cons(y1, y2)) → SELACTIVE(fromActive(x0), mark(y2))
SELACTIVE(s(indx(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(sel(y0, y1), indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(0, from(x0))) → SELACTIVE(0, fromActive(x0))
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(dbls(y0), indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
SELACTIVE(s(dbls(y0)), cons(y1, from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
SELACTIVE(s(dbls(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
SELACTIVE(s(0), cons(y0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(x1, x2)) → MARK(x2)
SELACTIVE(s(sel(x0, x1)), cons(y1, y2)) → SELACTIVE(selActive(mark(x0), mark(x1)), mark(y2))

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(X)))

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

SELACTIVE(s(dbl(y0)), cons(y1, 0)) → SELACTIVE(dblActive(mark(y0)), 0)
SELACTIVE(s(dbl(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
SELACTIVE(s(dbl(y0)), cons(y1, dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, nil)) → SELACTIVE(dblActive(mark(y0)), nil)
SELACTIVE(s(dbl(y0)), cons(y1, s(x0))) → SELACTIVE(dblActive(mark(y0)), s(x0))
SELACTIVE(s(dbl(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
SELACTIVE(s(dbl(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(dbl(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

  ↳ Trivial-Transformation

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

SELACTIVE(s(dbl(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(s(indx(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(indx(x1, x2)) → MARK(x1)
SELACTIVE(s(indx(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
SELACTIVE(s(0), cons(y0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(0), cons(y0, from(x0))) → SELACTIVE(0, fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(sel(y0, y1), dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(dbl(y0)), cons(y1, s(x0))) → SELACTIVE(dblActive(mark(y0)), s(x0))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
SELACTIVE(s(0), cons(y0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
SELACTIVE(s(s(x0)), cons(y1, y2)) → SELACTIVE(s(x0), mark(y2))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
SELACTIVE(s(dbl(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(indx(y0, y1)), cons(y2, dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(dbls(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(s(y0), sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
SELACTIVE(s(dbl(y0)), cons(y1, from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(s(y0), indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(0), cons(y0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, nil)) → SELACTIVE(dblActive(mark(y0)), nil)
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SELACTIVE(s(dbls(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(0), cons(y0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
SELACTIVE(s(from(x0)), cons(y1, y2)) → SELACTIVE(fromActive(x0), mark(y2))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
SELACTIVE(s(indx(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(sel(y0, y1), indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(0, from(x0))) → SELACTIVE(0, fromActive(x0))
MARK(sel(dbl(y0), sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(dbl(y0), indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel(y0, y1), dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbls(y0), indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
SELACTIVE(s(dbl(y0)), cons(y1, 0)) → SELACTIVE(dblActive(mark(y0)), 0)
SELACTIVE(s(dbls(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(dbls(y0)), cons(y1, from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
SELACTIVE(s(0), cons(y0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(dbl(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(x1, x2)) → MARK(x2)
SELACTIVE(s(sel(x0, x1)), cons(y1, y2)) → SELACTIVE(selActive(mark(x0), mark(x1)), mark(y2))

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(X)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 3 less nodes.

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Narrowing

  ↳ Trivial-Transformation

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

SELACTIVE(s(dbls(y0)), cons(y1, dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(indx(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(indx(x1, x2)) → MARK(x1)
SELACTIVE(s(indx(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(0), cons(y0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(0), cons(y0, from(x0))) → SELACTIVE(0, fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(dbls(y0), sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
SELACTIVE(s(0), cons(y0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
SELACTIVE(s(s(x0)), cons(y1, y2)) → SELACTIVE(s(x0), mark(y2))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
SELACTIVE(s(dbl(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(s(y0), sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
SELACTIVE(s(dbl(y0)), cons(y1, from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(s(y0), indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(0), cons(y0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
SELACTIVE(s(dbls(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
SELACTIVE(s(indx(y0, y1)), cons(y2, from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
SELACTIVE(s(dbls(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(0), cons(y0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
SELACTIVE(s(from(x0)), cons(y1, y2)) → SELACTIVE(fromActive(x0), mark(y2))
SELACTIVE(s(indx(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(sel(y0, y1), indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
SELACTIVE(s(dbl(y0)), cons(y1, dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbls(y0), dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(0, from(x0))) → SELACTIVE(0, fromActive(x0))
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(dbl(y0), indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
SELACTIVE(s(dbls(y0)), cons(y1, from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
SELACTIVE(s(dbls(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
SELACTIVE(s(0), cons(y0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(dbl(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(x1, x2)) → MARK(x2)
SELACTIVE(s(sel(x0, x1)), cons(y1, y2)) → SELACTIVE(selActive(mark(x0), mark(x1)), mark(y2))

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(X)))

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

SELACTIVE(s(from(x0)), cons(y1, y2)) → SELACTIVE(cons(x0, from(s(x0))), mark(y2))
SELACTIVE(s(from(x0)), cons(y1, y2)) → SELACTIVE(from(x0), mark(y2))

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Narrowing

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ DependencyGraphProof

  ↳ Trivial-Transformation

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

SELACTIVE(s(dbl(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(s(indx(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(indx(x1, x2)) → MARK(x1)
SELACTIVE(s(indx(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
SELACTIVE(s(0), cons(y0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(0), cons(y0, from(x0))) → SELACTIVE(0, fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(sel(y0, y1), dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
SELACTIVE(s(0), cons(y0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
SELACTIVE(s(s(x0)), cons(y1, y2)) → SELACTIVE(s(x0), mark(y2))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
SELACTIVE(s(dbl(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(from(x0)), cons(y1, y2)) → SELACTIVE(from(x0), mark(y2))
SELACTIVE(s(indx(y0, y1)), cons(y2, dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(dbls(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(s(y0), sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(dbl(y0)), cons(y1, from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
SELACTIVE(s(0), cons(y0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(s(from(x0)), cons(y1, y2)) → SELACTIVE(cons(x0, from(s(x0))), mark(y2))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SELACTIVE(s(dbls(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(0), cons(y0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
SELACTIVE(s(indx(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(sel(y0, y1), indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(0, from(x0))) → SELACTIVE(0, fromActive(x0))
MARK(sel(dbl(y0), indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbls(y0), indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
SELACTIVE(s(dbls(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(dbls(y0)), cons(y1, from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
SELACTIVE(s(0), cons(y0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(dbl(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(x1, x2)) → MARK(x2)
SELACTIVE(s(sel(x0, x1)), cons(y1, y2)) → SELACTIVE(selActive(mark(x0), mark(x1)), mark(y2))

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(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.

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Narrowing

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Narrowing

  ↳ Trivial-Transformation

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

SELACTIVE(s(dbls(y0)), cons(y1, dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(indx(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(indx(x1, x2)) → MARK(x1)
SELACTIVE(s(indx(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(0), cons(y0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(0), cons(y0, from(x0))) → SELACTIVE(0, fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(dbls(y0), sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
SELACTIVE(s(0), cons(y0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
SELACTIVE(s(s(x0)), cons(y1, y2)) → SELACTIVE(s(x0), mark(y2))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
SELACTIVE(s(dbl(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(s(y0), sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
SELACTIVE(s(dbl(y0)), cons(y1, from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(s(y0), indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(0), cons(y0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
SELACTIVE(s(dbls(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
SELACTIVE(s(indx(y0, y1)), cons(y2, from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
SELACTIVE(s(dbls(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(0), cons(y0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
SELACTIVE(s(indx(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(sel(y0, y1), indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
SELACTIVE(s(dbl(y0)), cons(y1, dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbls(y0), dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(0, from(x0))) → SELACTIVE(0, fromActive(x0))
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(dbl(y0), indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
SELACTIVE(s(dbls(y0)), cons(y1, from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
SELACTIVE(s(dbls(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
SELACTIVE(s(0), cons(y0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(dbl(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(x1, x2)) → MARK(x2)
SELACTIVE(s(sel(x0, x1)), cons(y1, y2)) → SELACTIVE(selActive(mark(x0), mark(x1)), mark(y2))

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(X)))

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

SELACTIVE(s(s(y0)), cons(y1, nil)) → SELACTIVE(s(y0), nil)
SELACTIVE(s(s(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
SELACTIVE(s(s(y0)), cons(y1, 0)) → SELACTIVE(s(y0), 0)
SELACTIVE(s(s(y0)), cons(y1, from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SELACTIVE(s(s(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(s(y0)), cons(y1, dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
SELACTIVE(s(s(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
SELACTIVE(s(s(y0)), cons(y1, s(x0))) → SELACTIVE(s(y0), s(x0))
SELACTIVE(s(s(y0)), cons(y1, dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Narrowing

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Narrowing

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ DependencyGraphProof

  ↳ Trivial-Transformation

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

SELACTIVE(s(dbl(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(s(indx(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(indx(x1, x2)) → MARK(x1)
SELACTIVE(s(indx(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
SELACTIVE(s(0), cons(y0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(0), cons(y0, from(x0))) → SELACTIVE(0, fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(s(y0)), cons(y1, 0)) → SELACTIVE(s(y0), 0)
SELACTIVE(s(s(y0)), cons(y1, from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(sel(sel(y0, y1), dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
SELACTIVE(s(0), cons(y0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
SELACTIVE(s(dbl(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(s(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(s(y0)), cons(y1, dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
SELACTIVE(s(indx(y0, y1)), cons(y2, dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(dbls(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(s(y0), sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(dbl(y0)), cons(y1, from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
SELACTIVE(s(0), cons(y0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SELACTIVE(s(dbls(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(dbls(x1)) → MARK(x1)
SELACTIVE(s(s(y0)), cons(y1, s(x0))) → SELACTIVE(s(y0), s(x0))
SELACTIVE(s(s(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(0), cons(y0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
SELACTIVE(s(indx(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(sel(y0, y1), indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(0, from(x0))) → SELACTIVE(0, fromActive(x0))
MARK(sel(dbl(y0), indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbls(y0), indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))
SELACTIVE(s(s(y0)), cons(y1, dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
SELACTIVE(s(s(y0)), cons(y1, nil)) → SELACTIVE(s(y0), nil)
SELACTIVE(s(dbls(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(dbls(y0)), cons(y1, from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
SELACTIVE(s(0), cons(y0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(dbl(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(s(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(x1, x2)) → MARK(x2)
SELACTIVE(s(sel(x0, x1)), cons(y1, y2)) → SELACTIVE(selActive(mark(x0), mark(x1)), mark(y2))

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(X)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 3 less nodes.

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Narrowing

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Narrowing

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ Narrowing

  ↳ Trivial-Transformation

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

SELACTIVE(s(dbls(y0)), cons(y1, dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(indx(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(indx(x1, x2)) → MARK(x1)
SELACTIVE(s(indx(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(0), cons(y0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(0), cons(y0, from(x0))) → SELACTIVE(0, fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(s(y0)), cons(y1, from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(sel(dbls(y0), sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
SELACTIVE(s(0), cons(y0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
SELACTIVE(s(dbl(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(s(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(s(y0)), cons(y1, dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(s(y0), sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
SELACTIVE(s(dbl(y0)), cons(y1, from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(s(y0), indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(0), cons(y0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
SELACTIVE(s(dbls(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SELACTIVE(s(indx(y0, y1)), cons(y2, from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(dbls(x1)) → MARK(x1)
SELACTIVE(s(dbls(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(s(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(0), cons(y0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
SELACTIVE(s(indx(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(sel(y0, y1), indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
SELACTIVE(s(dbl(y0)), cons(y1, dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbls(y0), dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(0, from(x0))) → SELACTIVE(0, fromActive(x0))
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(dbl(y0), indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))
SELACTIVE(s(s(y0)), cons(y1, dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
SELACTIVE(s(dbls(y0)), cons(y1, from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
SELACTIVE(s(dbls(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
SELACTIVE(s(0), cons(y0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(dbl(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(s(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(x1, x2)) → MARK(x2)
SELACTIVE(s(sel(x0, x1)), cons(y1, y2)) → SELACTIVE(selActive(mark(x0), mark(x1)), mark(y2))

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(X)))

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

SELACTIVE(s(sel(y0, y1)), cons(y2, from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(sel(y0, y1)), cons(y2, s(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), s(x0))
SELACTIVE(s(sel(y0, y1)), cons(y2, dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
SELACTIVE(s(sel(y0, y1)), cons(y2, 0)) → SELACTIVE(selActive(mark(y0), mark(y1)), 0)
SELACTIVE(s(sel(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
SELACTIVE(s(sel(y0, y1)), cons(y2, nil)) → SELACTIVE(selActive(mark(y0), mark(y1)), nil)

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Narrowing

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Narrowing

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ Narrowing

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ DependencyGraphProof

  ↳ Trivial-Transformation

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

SELACTIVE(s(dbl(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(s(indx(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(indx(x1, x2)) → MARK(x1)
SELACTIVE(s(indx(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
SELACTIVE(s(0), cons(y0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(sel(y0, y1)), cons(y2, s(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), s(x0))
SELACTIVE(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(0), cons(y0, from(x0))) → SELACTIVE(0, fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(s(y0)), cons(y1, from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(sel(sel(y0, y1), dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
SELACTIVE(s(0), cons(y0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
SELACTIVE(s(sel(y0, y1)), cons(y2, dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(dbl(x1)) → MARK(x1)
SELACTIVE(s(dbl(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
SELACTIVE(s(s(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(s(y0)), cons(y1, dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
SELACTIVE(s(indx(y0, y1)), cons(y2, dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(dbls(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(s(y0), sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(dbl(y0)), cons(y1, from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
SELACTIVE(s(0), cons(y0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(dbls(x1)) → MARK(x1)
SELACTIVE(s(indx(y0, y1)), cons(y2, from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SELACTIVE(s(s(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
SELACTIVE(s(sel(y0, y1)), cons(y2, 0)) → SELACTIVE(selActive(mark(y0), mark(y1)), 0)
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(sel(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
SELACTIVE(s(0), cons(y0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
SELACTIVE(s(indx(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(sel(y0, y1), indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
SELACTIVE(s(sel(y0, y1)), cons(y2, from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(0, from(x0))) → SELACTIVE(0, fromActive(x0))
MARK(sel(dbl(y0), indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbls(y0), indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
SELACTIVE(s(s(y0)), cons(y1, dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(dbls(y0)), cons(y1, from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
SELACTIVE(s(0), cons(y0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(dbl(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(sel(y0, y1)), cons(y2, nil)) → SELACTIVE(selActive(mark(y0), mark(y1)), nil)
SELACTIVE(s(s(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(x1, x2)) → MARK(x2)

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(X)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 3 less nodes.

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Narrowing

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Narrowing

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ Narrowing

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Narrowing

  ↳ Trivial-Transformation

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

SELACTIVE(s(dbls(y0)), cons(y1, dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(indx(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(indx(x1, x2)) → MARK(x1)
SELACTIVE(s(indx(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(0), cons(y0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
SELACTIVE(s(0), cons(y0, from(x0))) → SELACTIVE(0, fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(s(y0)), cons(y1, from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(sel(dbls(y0), sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
SELACTIVE(s(0), cons(y0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
SELACTIVE(s(sel(y0, y1)), cons(y2, dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
SELACTIVE(s(s(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(s(y0)), cons(y1, dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(s(y0), sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(dbl(y0)), cons(y1, from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
SELACTIVE(s(0), cons(y0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
SELACTIVE(s(dbls(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SELACTIVE(s(indx(y0, y1)), cons(y2, from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(dbls(x1)) → MARK(x1)
SELACTIVE(s(dbls(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(s(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(0), cons(y0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
SELACTIVE(s(sel(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(sel(y0, y1), indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
SELACTIVE(s(dbl(y0)), cons(y1, dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbls(y0), dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(0, from(x0))) → SELACTIVE(0, fromActive(x0))
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(dbl(y0), indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))
SELACTIVE(s(s(y0)), cons(y1, dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
SELACTIVE(s(dbls(y0)), cons(y1, from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
SELACTIVE(s(dbls(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
SELACTIVE(s(0), cons(y0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(dbl(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(s(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(x1, x2)) → MARK(x2)

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(X)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule SELACTIVE(s(0), cons(y0, from(x0))) → SELACTIVE(0, fromActive(x0)) at position [1] we obtained the following new rules:

SELACTIVE(s(0), cons(y0, from(x0))) → SELACTIVE(0, from(x0))
SELACTIVE(s(0), cons(y0, from(x0))) → SELACTIVE(0, cons(x0, from(s(x0))))

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Narrowing

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Narrowing

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ Narrowing

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Narrowing

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ DependencyGraphProof

  ↳ Trivial-Transformation

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

SELACTIVE(s(dbl(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(s(indx(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(indx(x1, x2)) → MARK(x1)
SELACTIVE(s(indx(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
SELACTIVE(s(0), cons(y0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(s(y0)), cons(y1, from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(sel(sel(y0, y1), dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(0), cons(y0, from(x0))) → SELACTIVE(0, from(x0))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
SELACTIVE(s(0), cons(y0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
SELACTIVE(s(sel(y0, y1)), cons(y2, dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(dbl(x1)) → MARK(x1)
SELACTIVE(s(dbl(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
SELACTIVE(s(s(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(s(y0)), cons(y1, dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
SELACTIVE(s(indx(y0, y1)), cons(y2, dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(dbls(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(s(y0), sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
SELACTIVE(s(dbl(y0)), cons(y1, from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(s(y0), indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(0), cons(y0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(dbls(x1)) → MARK(x1)
SELACTIVE(s(indx(y0, y1)), cons(y2, from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SELACTIVE(s(s(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(sel(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
SELACTIVE(s(0), cons(y0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
SELACTIVE(s(indx(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(sel(y0, y1), indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
SELACTIVE(s(sel(y0, y1)), cons(y2, from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(0, from(x0))) → SELACTIVE(0, fromActive(x0))
MARK(sel(dbl(y0), indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbls(y0), indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))
SELACTIVE(s(0), cons(y0, from(x0))) → SELACTIVE(0, cons(x0, from(s(x0))))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
SELACTIVE(s(s(y0)), cons(y1, dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(dbls(y0)), cons(y1, from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
SELACTIVE(s(0), cons(y0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(dbl(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(s(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(x1, x2)) → MARK(x2)

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(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.

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Narrowing

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Narrowing

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ Narrowing

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Narrowing

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ Narrowing

  ↳ Trivial-Transformation

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

SELACTIVE(s(dbls(y0)), cons(y1, dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(indx(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(indx(x1, x2)) → MARK(x1)
SELACTIVE(s(indx(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(0), cons(y0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(s(y0)), cons(y1, from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(sel(dbls(y0), sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
SELACTIVE(s(0), cons(y0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
SELACTIVE(s(sel(y0, y1)), cons(y2, dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
SELACTIVE(s(s(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(s(y0)), cons(y1, dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(s(y0), sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(dbl(y0)), cons(y1, from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
SELACTIVE(s(0), cons(y0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
SELACTIVE(s(dbls(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SELACTIVE(s(indx(y0, y1)), cons(y2, from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(dbls(x1)) → MARK(x1)
SELACTIVE(s(dbls(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(s(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(0), cons(y0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
SELACTIVE(s(sel(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(sel(y0, y1), indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
SELACTIVE(s(dbl(y0)), cons(y1, dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbls(y0), dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(0, from(x0))) → SELACTIVE(0, fromActive(x0))
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(dbl(y0), indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))
SELACTIVE(s(s(y0)), cons(y1, dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
SELACTIVE(s(0), cons(y0, from(x0))) → SELACTIVE(0, cons(x0, from(s(x0))))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
SELACTIVE(s(dbls(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(dbls(y0)), cons(y1, from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
SELACTIVE(s(0), cons(y0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(dbl(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(s(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(x1, x2)) → MARK(x2)

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(X)))

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

SELACTIVE(s(indx(y0, y1)), cons(y2, from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))
SELACTIVE(s(indx(y0, y1)), cons(y2, from(x0))) → SELACTIVE(indxActive(mark(y0), y1), from(x0))

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Narrowing

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Narrowing

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ Narrowing

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Narrowing

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ DependencyGraphProof

  ↳ Trivial-Transformation

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

SELACTIVE(s(dbl(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(s(indx(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
SELACTIVE(s(indx(y0, y1)), cons(y2, from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))
MARK(indx(x1, x2)) → MARK(x1)
SELACTIVE(s(indx(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
SELACTIVE(s(0), cons(y0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(s(y0)), cons(y1, from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(sel(sel(y0, y1), dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
SELACTIVE(s(0), cons(y0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
SELACTIVE(s(sel(y0, y1)), cons(y2, dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(dbl(x1)) → MARK(x1)
SELACTIVE(s(dbl(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
SELACTIVE(s(s(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(s(y0)), cons(y1, dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
SELACTIVE(s(indx(y0, y1)), cons(y2, dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(dbls(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(s(y0), sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
SELACTIVE(s(dbl(y0)), cons(y1, from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(s(y0), indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(0), cons(y0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SELACTIVE(s(s(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(sel(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
SELACTIVE(s(0), cons(y0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
SELACTIVE(s(indx(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(sel(y0, y1), indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
SELACTIVE(s(sel(y0, y1)), cons(y2, from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(indx(y0, y1)), cons(y2, from(x0))) → SELACTIVE(indxActive(mark(y0), y1), from(x0))
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(0, from(x0))) → SELACTIVE(0, fromActive(x0))
MARK(sel(dbl(y0), indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbls(y0), indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
SELACTIVE(s(0), cons(y0, from(x0))) → SELACTIVE(0, cons(x0, from(s(x0))))
SELACTIVE(s(s(y0)), cons(y1, dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
SELACTIVE(s(dbls(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
SELACTIVE(s(0), cons(y0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(dbl(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(s(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(x1, x2)) → MARK(x2)

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(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.

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Narrowing

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Narrowing

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ Narrowing

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Narrowing

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ Narrowing

  ↳ Trivial-Transformation

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

SELACTIVE(s(dbls(y0)), cons(y1, dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(indx(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(indx(x1, x2)) → MARK(x1)
SELACTIVE(s(indx(y0, y1)), cons(y2, from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))
SELACTIVE(s(indx(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(0), cons(y0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(s(y0)), cons(y1, from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(sel(dbls(y0), sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
SELACTIVE(s(0), cons(y0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
SELACTIVE(s(sel(y0, y1)), cons(y2, dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
SELACTIVE(s(s(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(s(y0)), cons(y1, dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(s(y0), sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(dbl(y0)), cons(y1, from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
SELACTIVE(s(0), cons(y0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
SELACTIVE(s(dbls(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(dbls(x1)) → MARK(x1)
SELACTIVE(s(dbls(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(s(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(0), cons(y0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
SELACTIVE(s(sel(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(sel(y0, y1), indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
SELACTIVE(s(dbl(y0)), cons(y1, dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbls(y0), dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(0, from(x0))) → SELACTIVE(0, fromActive(x0))
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(dbl(y0), indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))
SELACTIVE(s(s(y0)), cons(y1, dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
SELACTIVE(s(0), cons(y0, from(x0))) → SELACTIVE(0, cons(x0, from(s(x0))))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
SELACTIVE(s(dbls(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(dbls(y0)), cons(y1, from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
SELACTIVE(s(0), cons(y0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(dbl(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(s(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(x1, x2)) → MARK(x2)

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(X)))

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

SELACTIVE(s(dbls(y0)), cons(y1, from(x0))) → SELACTIVE(dblsActive(mark(y0)), from(x0))
SELACTIVE(s(dbls(y0)), cons(y1, from(x0))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, from(s(x0))))

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Narrowing

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Narrowing

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ Narrowing

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Narrowing

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                        ↳ QDP

                                                                                                                                                                          ↳ DependencyGraphProof

  ↳ Trivial-Transformation

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

SELACTIVE(s(dbl(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(s(indx(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
SELACTIVE(s(dbls(y0)), cons(y1, from(x0))) → SELACTIVE(dblsActive(mark(y0)), from(x0))
SELACTIVE(s(indx(y0, y1)), cons(y2, from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))
MARK(indx(x1, x2)) → MARK(x1)
SELACTIVE(s(indx(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
SELACTIVE(s(0), cons(y0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(0, cons(X, Y)) → MARK(X)
SELACTIVE(s(dbls(y0)), cons(y1, from(x0))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, from(s(x0))))
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(s(y0)), cons(y1, from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(sel(sel(y0, y1), dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
SELACTIVE(s(0), cons(y0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
SELACTIVE(s(sel(y0, y1)), cons(y2, dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(dbl(x1)) → MARK(x1)
SELACTIVE(s(dbl(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
SELACTIVE(s(s(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(s(y0)), cons(y1, dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
SELACTIVE(s(indx(y0, y1)), cons(y2, dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(dbls(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(s(y0), sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
SELACTIVE(s(dbl(y0)), cons(y1, from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(s(y0), indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(0), cons(y0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SELACTIVE(s(s(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(sel(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
SELACTIVE(s(0), cons(y0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
SELACTIVE(s(indx(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(sel(y0, y1), indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
SELACTIVE(s(sel(y0, y1)), cons(y2, from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(0, from(x0))) → SELACTIVE(0, fromActive(x0))
MARK(sel(dbl(y0), indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbls(y0), indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
SELACTIVE(s(0), cons(y0, from(x0))) → SELACTIVE(0, cons(x0, from(s(x0))))
SELACTIVE(s(s(y0)), cons(y1, dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
SELACTIVE(s(0), cons(y0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(dbl(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(s(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(x1, x2)) → MARK(x2)

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(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.

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Narrowing

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Narrowing

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ Narrowing

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Narrowing

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                        ↳ QDP

                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                            ↳ QDP

                                                                                                                                                                              ↳ Narrowing

  ↳ Trivial-Transformation

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

SELACTIVE(s(dbls(y0)), cons(y1, dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(indx(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(indx(x1, x2)) → MARK(x1)
SELACTIVE(s(indx(y0, y1)), cons(y2, from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))
SELACTIVE(s(indx(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(0), cons(y0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
SELACTIVE(0, cons(X, Y)) → MARK(X)
SELACTIVE(s(dbls(y0)), cons(y1, from(x0))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, from(s(x0))))
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(s(y0)), cons(y1, from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(sel(dbls(y0), sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
SELACTIVE(s(0), cons(y0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
SELACTIVE(s(sel(y0, y1)), cons(y2, dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
SELACTIVE(s(s(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(s(y0)), cons(y1, dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(s(y0), sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(dbl(y0)), cons(y1, from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
SELACTIVE(s(0), cons(y0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(dbls(x1)) → MARK(x1)
SELACTIVE(s(dbls(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(s(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(0), cons(y0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
SELACTIVE(s(sel(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(sel(y0, y1), indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
SELACTIVE(s(dbl(y0)), cons(y1, dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbls(y0), dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(0, from(x0))) → SELACTIVE(0, fromActive(x0))
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(dbl(y0), indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))
SELACTIVE(s(s(y0)), cons(y1, dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
SELACTIVE(s(0), cons(y0, from(x0))) → SELACTIVE(0, cons(x0, from(s(x0))))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
SELACTIVE(s(dbls(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
SELACTIVE(s(0), cons(y0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(dbl(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(s(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(x1, x2)) → MARK(x2)

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(X)))

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

SELACTIVE(s(dbl(y0)), cons(y1, from(x0))) → SELACTIVE(dblActive(mark(y0)), cons(x0, from(s(x0))))
SELACTIVE(s(dbl(y0)), cons(y1, from(x0))) → SELACTIVE(dblActive(mark(y0)), from(x0))

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Narrowing

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Narrowing

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ Narrowing

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Narrowing

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                        ↳ QDP

                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                            ↳ QDP

                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ DependencyGraphProof

  ↳ Trivial-Transformation

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

SELACTIVE(s(dbl(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(s(indx(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
SELACTIVE(s(indx(y0, y1)), cons(y2, from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))
MARK(indx(x1, x2)) → MARK(x1)
SELACTIVE(s(indx(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
SELACTIVE(s(0), cons(y0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(0, cons(X, Y)) → MARK(X)
SELACTIVE(s(dbls(y0)), cons(y1, from(x0))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, from(s(x0))))
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(s(y0)), cons(y1, from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(sel(sel(y0, y1), dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
SELACTIVE(s(0), cons(y0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
SELACTIVE(s(sel(y0, y1)), cons(y2, dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(dbl(x1)) → MARK(x1)
SELACTIVE(s(dbl(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
SELACTIVE(s(s(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(s(y0)), cons(y1, dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
SELACTIVE(s(indx(y0, y1)), cons(y2, dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(dbls(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(s(y0), sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(0), cons(y0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
SELACTIVE(s(dbls(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SELACTIVE(s(s(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(sel(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
SELACTIVE(s(0), cons(y0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
SELACTIVE(s(dbl(y0)), cons(y1, from(x0))) → SELACTIVE(dblActive(mark(y0)), from(x0))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
SELACTIVE(s(indx(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(sel(y0, y1), indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
SELACTIVE(s(sel(y0, y1)), cons(y2, from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(0, from(x0))) → SELACTIVE(0, fromActive(x0))
MARK(sel(dbl(y0), indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbls(y0), indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
SELACTIVE(s(0), cons(y0, from(x0))) → SELACTIVE(0, cons(x0, from(s(x0))))
SELACTIVE(s(s(y0)), cons(y1, dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
SELACTIVE(s(0), cons(y0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(dbl(y0)), cons(y1, from(x0))) → SELACTIVE(dblActive(mark(y0)), cons(x0, from(s(x0))))
SELACTIVE(s(dbl(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(s(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(x1, x2)) → MARK(x2)

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(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.

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Narrowing

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Narrowing

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ Narrowing

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Narrowing

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                        ↳ QDP

                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                            ↳ QDP

                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ Narrowing

  ↳ Trivial-Transformation

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

SELACTIVE(s(dbls(y0)), cons(y1, dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(indx(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(indx(x1, x2)) → MARK(x1)
SELACTIVE(s(indx(y0, y1)), cons(y2, from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))
SELACTIVE(s(indx(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(0), cons(y0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
SELACTIVE(0, cons(X, Y)) → MARK(X)
SELACTIVE(s(dbls(y0)), cons(y1, from(x0))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, from(s(x0))))
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(s(y0)), cons(y1, from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(sel(dbls(y0), sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
SELACTIVE(s(0), cons(y0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
SELACTIVE(s(sel(y0, y1)), cons(y2, dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
SELACTIVE(s(s(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(s(y0)), cons(y1, dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(s(y0), sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(0), cons(y0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(dbls(x1)) → MARK(x1)
SELACTIVE(s(dbls(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(s(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(0), cons(y0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
SELACTIVE(s(sel(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(sel(y0, y1), indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
SELACTIVE(s(dbl(y0)), cons(y1, dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbls(y0), dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(0, from(x0))) → SELACTIVE(0, fromActive(x0))
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(dbl(y0), indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))
SELACTIVE(s(s(y0)), cons(y1, dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
SELACTIVE(s(0), cons(y0, from(x0))) → SELACTIVE(0, cons(x0, from(s(x0))))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
SELACTIVE(s(dbls(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
SELACTIVE(s(0), cons(y0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(dbl(y0)), cons(y1, from(x0))) → SELACTIVE(dblActive(mark(y0)), cons(x0, from(s(x0))))
SELACTIVE(s(dbl(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(s(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(x1, x2)) → MARK(x2)

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(X)))

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

SELACTIVE(s(s(y0)), cons(y1, from(x0))) → SELACTIVE(s(y0), cons(x0, from(s(x0))))
SELACTIVE(s(s(y0)), cons(y1, from(x0))) → SELACTIVE(s(y0), from(x0))

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Narrowing

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Narrowing

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ Narrowing

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Narrowing

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                        ↳ QDP

                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                            ↳ QDP

                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ DependencyGraphProof

  ↳ Trivial-Transformation

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

SELACTIVE(s(dbl(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(s(indx(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
SELACTIVE(s(indx(y0, y1)), cons(y2, from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))
MARK(indx(x1, x2)) → MARK(x1)
SELACTIVE(s(indx(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
SELACTIVE(s(0), cons(y0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(0, cons(X, Y)) → MARK(X)
SELACTIVE(s(dbls(y0)), cons(y1, from(x0))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, from(s(x0))))
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(sel(y0, y1), dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
SELACTIVE(s(0), cons(y0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
SELACTIVE(s(sel(y0, y1)), cons(y2, dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(dbl(x1)) → MARK(x1)
SELACTIVE(s(dbl(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
SELACTIVE(s(s(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(s(y0)), cons(y1, dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
SELACTIVE(s(indx(y0, y1)), cons(y2, dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(dbls(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(s(y0), sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(0), cons(y0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
SELACTIVE(s(dbls(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SELACTIVE(s(s(y0)), cons(y1, from(x0))) → SELACTIVE(s(y0), cons(x0, from(s(x0))))
SELACTIVE(s(s(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(sel(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
SELACTIVE(s(0), cons(y0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
SELACTIVE(s(indx(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(sel(y0, y1), indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
SELACTIVE(s(sel(y0, y1)), cons(y2, from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(s(y0)), cons(y1, from(x0))) → SELACTIVE(s(y0), from(x0))
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(0, from(x0))) → SELACTIVE(0, fromActive(x0))
MARK(sel(dbl(y0), indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbls(y0), indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
SELACTIVE(s(0), cons(y0, from(x0))) → SELACTIVE(0, cons(x0, from(s(x0))))
SELACTIVE(s(s(y0)), cons(y1, dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
SELACTIVE(s(0), cons(y0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(dbl(y0)), cons(y1, from(x0))) → SELACTIVE(dblActive(mark(y0)), cons(x0, from(s(x0))))
SELACTIVE(s(dbl(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(s(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(x1, x2)) → MARK(x2)

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(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.

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Narrowing

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Narrowing

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ Narrowing

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Narrowing

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                        ↳ QDP

                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                            ↳ QDP

                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Narrowing

  ↳ Trivial-Transformation

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

SELACTIVE(s(dbls(y0)), cons(y1, dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(indx(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(indx(x1, x2)) → MARK(x1)
SELACTIVE(s(indx(y0, y1)), cons(y2, from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))
SELACTIVE(s(indx(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(0), cons(y0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
SELACTIVE(0, cons(X, Y)) → MARK(X)
SELACTIVE(s(dbls(y0)), cons(y1, from(x0))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, from(s(x0))))
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(dbls(y0), sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
SELACTIVE(s(0), cons(y0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
SELACTIVE(s(sel(y0, y1)), cons(y2, dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
SELACTIVE(s(s(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(s(y0)), cons(y1, dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(s(y0), sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(0), cons(y0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(dbls(x1)) → MARK(x1)
SELACTIVE(s(dbls(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(s(y0)), cons(y1, from(x0))) → SELACTIVE(s(y0), cons(x0, from(s(x0))))
SELACTIVE(s(s(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(0), cons(y0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
SELACTIVE(s(sel(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(sel(y0, y1), indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
SELACTIVE(s(dbl(y0)), cons(y1, dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbls(y0), dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(0, from(x0))) → SELACTIVE(0, fromActive(x0))
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(dbl(y0), indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))
SELACTIVE(s(s(y0)), cons(y1, dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
SELACTIVE(s(0), cons(y0, from(x0))) → SELACTIVE(0, cons(x0, from(s(x0))))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
SELACTIVE(s(dbls(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
SELACTIVE(s(0), cons(y0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(dbl(y0)), cons(y1, from(x0))) → SELACTIVE(dblActive(mark(y0)), cons(x0, from(s(x0))))
SELACTIVE(s(dbl(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(s(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(x1, x2)) → MARK(x2)

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(X)))

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

SELACTIVE(s(sel(y0, y1)), cons(y2, from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, from(s(x0))))
SELACTIVE(s(sel(y0, y1)), cons(y2, from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), from(x0))

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Narrowing

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Narrowing

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ Narrowing

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Narrowing

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                        ↳ QDP

                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                            ↳ QDP

                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ DependencyGraphProof

  ↳ Trivial-Transformation

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

SELACTIVE(s(dbl(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(s(indx(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
SELACTIVE(s(indx(y0, y1)), cons(y2, from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))
MARK(indx(x1, x2)) → MARK(x1)
SELACTIVE(s(indx(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
SELACTIVE(s(0), cons(y0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(0, cons(X, Y)) → MARK(X)
SELACTIVE(s(dbls(y0)), cons(y1, from(x0))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, from(s(x0))))
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(sel(y0, y1), dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
SELACTIVE(s(0), cons(y0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
SELACTIVE(s(sel(y0, y1)), cons(y2, dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(dbl(x1)) → MARK(x1)
SELACTIVE(s(dbl(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
SELACTIVE(s(s(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(s(y0)), cons(y1, dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
SELACTIVE(s(indx(y0, y1)), cons(y2, dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(dbls(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(s(y0), sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(0), cons(y0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
SELACTIVE(s(dbls(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SELACTIVE(s(s(y0)), cons(y1, from(x0))) → SELACTIVE(s(y0), cons(x0, from(s(x0))))
SELACTIVE(s(s(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(sel(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
SELACTIVE(s(0), cons(y0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
SELACTIVE(s(sel(y0, y1)), cons(y2, from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), from(x0))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
SELACTIVE(s(indx(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(sel(y0, y1), indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(0, from(x0))) → SELACTIVE(0, fromActive(x0))
MARK(sel(dbl(y0), indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbls(y0), indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
SELACTIVE(s(0), cons(y0, from(x0))) → SELACTIVE(0, cons(x0, from(s(x0))))
SELACTIVE(s(s(y0)), cons(y1, dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
SELACTIVE(s(sel(y0, y1)), cons(y2, from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, from(s(x0))))
SELACTIVE(s(0), cons(y0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(dbl(y0)), cons(y1, from(x0))) → SELACTIVE(dblActive(mark(y0)), cons(x0, from(s(x0))))
SELACTIVE(s(dbl(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(s(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(x1, x2)) → MARK(x2)

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(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.

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Narrowing

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Narrowing

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ Narrowing

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Narrowing

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                        ↳ QDP

                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                            ↳ QDP

                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ QDPOrderProof

  ↳ Trivial-Transformation

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

SELACTIVE(s(dbls(y0)), cons(y1, dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(indx(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(indx(x1, x2)) → MARK(x1)
SELACTIVE(s(indx(y0, y1)), cons(y2, from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))
SELACTIVE(s(indx(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(0), cons(y0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
SELACTIVE(0, cons(X, Y)) → MARK(X)
SELACTIVE(s(dbls(y0)), cons(y1, from(x0))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, from(s(x0))))
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(dbls(y0), sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
SELACTIVE(s(0), cons(y0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
SELACTIVE(s(sel(y0, y1)), cons(y2, dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
SELACTIVE(s(s(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(s(y0)), cons(y1, dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(s(y0), sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(0), cons(y0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(dbls(x1)) → MARK(x1)
SELACTIVE(s(dbls(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(s(y0)), cons(y1, from(x0))) → SELACTIVE(s(y0), cons(x0, from(s(x0))))
SELACTIVE(s(s(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(0), cons(y0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
SELACTIVE(s(sel(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(sel(y0, y1), indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
SELACTIVE(s(dbl(y0)), cons(y1, dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbls(y0), dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(0, from(x0))) → SELACTIVE(0, fromActive(x0))
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(dbl(y0), indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))
SELACTIVE(s(s(y0)), cons(y1, dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
SELACTIVE(s(0), cons(y0, from(x0))) → SELACTIVE(0, cons(x0, from(s(x0))))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
SELACTIVE(s(dbls(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
SELACTIVE(s(sel(y0, y1)), cons(y2, from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, from(s(x0))))
SELACTIVE(s(0), cons(y0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(dbl(y0)), cons(y1, from(x0))) → SELACTIVE(dblActive(mark(y0)), cons(x0, from(s(x0))))
SELACTIVE(s(dbl(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(s(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(x1, x2)) → MARK(x2)

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(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.

SELACTIVE(s(dbls(y0)), cons(y1, dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(s(0), cons(y0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
SELACTIVE(s(indx(y0, y1)), cons(y2, dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbls(y0), dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(sel(y0, y1), dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
SELACTIVE(s(s(y0)), cons(y1, dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))

The remaining pairs can at least be oriented weakly.

SELACTIVE(s(dbl(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(indx(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(indx(x1, x2)) → MARK(x1)
SELACTIVE(s(indx(y0, y1)), cons(y2, from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))
SELACTIVE(s(indx(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
SELACTIVE(0, cons(X, Y)) → MARK(X)
SELACTIVE(s(dbls(y0)), cons(y1, from(x0))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, from(s(x0))))
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(dbls(y0), sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
SELACTIVE(s(0), cons(y0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
SELACTIVE(s(dbl(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
SELACTIVE(s(s(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(s(y0)), cons(y1, dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(dbls(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(s(y0), sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(0), cons(y0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(dbls(x1)) → MARK(x1)
SELACTIVE(s(dbls(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(s(y0)), cons(y1, from(x0))) → SELACTIVE(s(y0), cons(x0, from(s(x0))))
SELACTIVE(s(s(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
SELACTIVE(s(dbl(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(0), cons(y0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
SELACTIVE(s(sel(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(sel(y0, y1), indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(0, from(x0))) → SELACTIVE(0, fromActive(x0))
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(dbl(y0), sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(dbl(y0), indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))
SELACTIVE(s(0), cons(y0, from(x0))) → SELACTIVE(0, cons(x0, from(s(x0))))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
SELACTIVE(s(dbls(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
SELACTIVE(s(sel(y0, y1)), cons(y2, from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, from(s(x0))))
SELACTIVE(s(0), cons(y0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(dbl(y0)), cons(y1, from(x0))) → SELACTIVE(dblActive(mark(y0)), cons(x0, from(s(x0))))
SELACTIVE(s(dbl(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(s(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(x1, x2)) → MARK(x2)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(MARK(x₁)) = 1
POL(SELACTIVE(x₁, x₂)) = x₂
POL(cons(x₁, x₂)) = 1
POL(dbl(x₁)) = 0
POL(dblActive(x₁)) = 0
POL(dbls(x₁)) = 0
POL(dblsActive(x₁)) = 1
POL(from(x₁)) = 0
POL(fromActive(x₁)) = 1
POL(indx(x₁, x₂)) = 0
POL(indxActive(x₁, x₂)) = x₁
POL(mark(x₁)) = 1
POL(nil) = 0
POL(s(x₁)) = 0
POL(sel(x₁, x₂)) = 1
POL(selActive(x₁, x₂)) = 1

The following usable rules [17] were oriented:

mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(0, cons(X, Y)) → mark(X)
selActive(x1, x2) → sel(x1, x2)
fromActive(X) → cons(X, from(s(X)))
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
dblsActive(nil) → nil
dblActive(s(X)) → s(s(dbl(X)))
dblActive(0) → 0
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
indxActive(nil, X) → nil

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Narrowing

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Narrowing

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ Narrowing

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Narrowing

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                        ↳ QDP

                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                            ↳ QDP

                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Narrowing

  ↳ Trivial-Transformation

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

SELACTIVE(s(dbl(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(indx(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(indx(x1, x2)) → MARK(x1)
SELACTIVE(s(indx(y0, y1)), cons(y2, from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))
SELACTIVE(s(indx(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
SELACTIVE(0, cons(X, Y)) → MARK(X)
SELACTIVE(s(dbls(y0)), cons(y1, from(x0))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, from(s(x0))))
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(sel(y0, y1), dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
SELACTIVE(s(0), cons(y0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
SELACTIVE(s(dbl(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
SELACTIVE(s(s(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(s(y0)), cons(y1, dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(dbls(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(s(y0), sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(0), cons(y0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(dbls(x1)) → MARK(x1)
SELACTIVE(s(dbls(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(s(y0)), cons(y1, from(x0))) → SELACTIVE(s(y0), cons(x0, from(s(x0))))
SELACTIVE(s(s(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
SELACTIVE(s(dbl(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(0), cons(y0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
SELACTIVE(s(sel(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(sel(y0, y1), indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(0, from(x0))) → SELACTIVE(0, fromActive(x0))
MARK(sel(dbl(y0), indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(dbls(y0), indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
SELACTIVE(s(0), cons(y0, from(x0))) → SELACTIVE(0, cons(x0, from(s(x0))))
SELACTIVE(s(dbls(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
SELACTIVE(s(sel(y0, y1)), cons(y2, from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, from(s(x0))))
SELACTIVE(s(0), cons(y0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(dbl(y0)), cons(y1, from(x0))) → SELACTIVE(dblActive(mark(y0)), cons(x0, from(s(x0))))
SELACTIVE(s(dbl(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(s(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(x1, x2)) → MARK(x2)

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(X)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0)) at position [1] we obtained the following new rules:

MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), from(x0))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, from(s(x0))))

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Narrowing

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Narrowing

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ Narrowing

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Narrowing

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                        ↳ QDP

                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                            ↳ QDP

                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ DependencyGraphProof

  ↳ Trivial-Transformation

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

SELACTIVE(s(dbl(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(indx(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
SELACTIVE(s(indx(y0, y1)), cons(y2, from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))
MARK(indx(x1, x2)) → MARK(x1)
SELACTIVE(s(indx(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
SELACTIVE(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(0, cons(X, Y)) → MARK(X)
SELACTIVE(s(dbls(y0)), cons(y1, from(x0))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, from(s(x0))))
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(dbls(y0), sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
SELACTIVE(s(0), cons(y0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
SELACTIVE(s(dbl(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
SELACTIVE(s(s(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(s(y0)), cons(y1, dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(dbls(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, from(s(x0))))
MARK(sel(s(y0), sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(0), cons(y0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
SELACTIVE(s(dbls(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
SELACTIVE(s(dbls(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SELACTIVE(s(s(y0)), cons(y1, from(x0))) → SELACTIVE(s(y0), cons(x0, from(s(x0))))
SELACTIVE(s(s(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
SELACTIVE(s(dbl(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(sel(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
SELACTIVE(s(0), cons(y0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
SELACTIVE(s(indx(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(0, from(x0))) → SELACTIVE(0, fromActive(x0))
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), from(x0))
MARK(sel(dbl(y0), sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(dbl(y0), indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))
SELACTIVE(s(0), cons(y0, from(x0))) → SELACTIVE(0, cons(x0, from(s(x0))))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
SELACTIVE(s(dbls(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
SELACTIVE(s(sel(y0, y1)), cons(y2, from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, from(s(x0))))
SELACTIVE(s(0), cons(y0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(dbl(y0)), cons(y1, from(x0))) → SELACTIVE(dblActive(mark(y0)), cons(x0, from(s(x0))))
SELACTIVE(s(dbl(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(s(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(x1, x2)) → MARK(x2)

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(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.

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ Narrowing

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ Narrowing

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ Narrowing

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ QDP

                                              ↳ Narrowing

                                                ↳ QDP

                                                  ↳ DependencyGraphProof

                                                    ↳ QDP

                                                      ↳ Narrowing

                                                        ↳ QDP

                                                          ↳ DependencyGraphProof

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ DependencyGraphProof

                                                                    ↳ QDP

                                                                      ↳ Narrowing

                                                                        ↳ QDP

                                                                          ↳ DependencyGraphProof

                                                                            ↳ QDP

                                                                              ↳ Narrowing

                                                                                ↳ QDP

                                                                                  ↳ DependencyGraphProof

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ DependencyGraphProof

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Narrowing

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                    ↳ QDP

                                                                                                                                      ↳ Narrowing

                                                                                                                                        ↳ QDP

                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                            ↳ QDP

                                                                                                                                              ↳ Narrowing

                                                                                                                                                ↳ QDP

                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                    ↳ QDP

                                                                                                                                                      ↳ Narrowing

                                                                                                                                                        ↳ QDP

                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                            ↳ QDP

                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                ↳ QDP

                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                    ↳ QDP

                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                        ↳ QDP

                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                            ↳ QDP

                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                      ↳ QDPOrderProof

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ Narrowing

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                                                                                                ↳ QDP

  ↳ Trivial-Transformation

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

SELACTIVE(s(dbl(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(indx(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(indx(x1, x2)) → MARK(x1)
SELACTIVE(s(indx(y0, y1)), cons(y2, from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))
SELACTIVE(s(indx(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
SELACTIVE(0, cons(X, Y)) → MARK(X)
SELACTIVE(s(dbls(y0)), cons(y1, from(x0))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, from(s(x0))))
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(dbls(y0), sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
SELACTIVE(s(0), cons(y0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
SELACTIVE(s(dbl(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
SELACTIVE(s(s(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(s(y0)), cons(y1, dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(dbls(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, from(s(x0))))
MARK(sel(s(y0), sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
SELACTIVE(s(0), cons(y0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
SELACTIVE(s(dbls(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SELACTIVE(s(s(y0)), cons(y1, from(x0))) → SELACTIVE(s(y0), cons(x0, from(s(x0))))
SELACTIVE(s(s(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
SELACTIVE(s(dbl(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(0), cons(y0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
SELACTIVE(s(sel(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(sel(y0, y1), indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(0, from(x0))) → SELACTIVE(0, fromActive(x0))
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(dbl(y0), sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(dbl(y0), indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, from(s(x0))))
SELACTIVE(s(0), cons(y0, from(x0))) → SELACTIVE(0, cons(x0, from(s(x0))))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
SELACTIVE(s(dbls(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
SELACTIVE(s(sel(y0, y1)), cons(y2, from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, from(s(x0))))
SELACTIVE(s(0), cons(y0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(dbl(y0)), cons(y1, from(x0))) → SELACTIVE(dblActive(mark(y0)), cons(x0, from(s(x0))))
SELACTIVE(s(dbl(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(s(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(x1, x2)) → MARK(x2)

The TRS R consists of the following rules:

mark(dbl(x1)) → dblActive(mark(x1))
dblActive(x1) → dbl(x1)
mark(dbls(x1)) → dblsActive(mark(x1))
dblsActive(x1) → dbls(x1)
mark(sel(x1, x2)) → selActive(mark(x1), mark(x2))
selActive(x1, x2) → sel(x1, x2)
mark(indx(x1, x2)) → indxActive(mark(x1), x2)
indxActive(x1, x2) → indx(x1, x2)
mark(from(x1)) → fromActive(x1)
fromActive(x1) → from(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
dblActive(0) → 0
dblActive(s(X)) → s(s(dbl(X)))
dblsActive(nil) → nil
dblsActive(cons(X, Y)) → cons(dbl(X), dbls(Y))
selActive(0, cons(X, Y)) → mark(X)
selActive(s(X), cons(Y, Z)) → selActive(mark(X), mark(Z))
indxActive(nil, X) → nil
indxActive(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
fromActive(X) → cons(X, from(s(X)))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We applied the Trivial transformation to transform the context-sensitive TRS to a usual TRS.

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

    ↳ QTRS

      ↳ Overlay + Local Confluence

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

dbl(0) → 0
dbl(s(X)) → s(s(dbl(X)))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
from(X) → cons(X, from(s(X)))

Q is empty.

The TRS is overlay and locally confluent. By [19] we can switch to innermost.

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

    ↳ QTRS

      ↳ Overlay + Local Confluence

        ↳ QTRS

          ↳ DependencyPairsProof

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

dbl(0) → 0
dbl(s(X)) → s(s(dbl(X)))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
from(X) → cons(X, from(s(X)))

The set Q consists of the following terms:

dbl(0)
dbl(s(x0))
dbls(nil)
dbls(cons(x0, x1))
sel(0, cons(x0, x1))
sel(s(x0), cons(x1, x2))
indx(nil, x0)
indx(cons(x0, x1), x2)
from(x0)

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

DBLS(cons(X, Y)) → DBLS(Y)
SEL(s(X), cons(Y, Z)) → SEL(X, Z)
DBLS(cons(X, Y)) → DBL(X)
DBL(s(X)) → DBL(X)
FROM(X) → FROM(s(X))
INDX(cons(X, Y), Z) → INDX(Y, Z)
INDX(cons(X, Y), Z) → SEL(X, Z)

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(s(dbl(X)))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
from(X) → cons(X, from(s(X)))

The set Q consists of the following terms:

dbl(0)
dbl(s(x0))
dbls(nil)
dbls(cons(x0, x1))
sel(0, cons(x0, x1))
sel(s(x0), cons(x1, x2))
indx(nil, x0)
indx(cons(x0, x1), x2)
from(x0)

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

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

    ↳ QTRS

      ↳ Overlay + Local Confluence

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

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

DBLS(cons(X, Y)) → DBLS(Y)
SEL(s(X), cons(Y, Z)) → SEL(X, Z)
DBLS(cons(X, Y)) → DBL(X)
DBL(s(X)) → DBL(X)
FROM(X) → FROM(s(X))
INDX(cons(X, Y), Z) → INDX(Y, Z)
INDX(cons(X, Y), Z) → SEL(X, Z)

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(s(dbl(X)))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
from(X) → cons(X, from(s(X)))

The set Q consists of the following terms:

dbl(0)
dbl(s(x0))
dbls(nil)
dbls(cons(x0, x1))
sel(0, cons(x0, x1))
sel(s(x0), cons(x1, x2))
indx(nil, x0)
indx(cons(x0, x1), x2)
from(x0)

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 5 SCCs with 2 less nodes.

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

    ↳ QTRS

      ↳ Overlay + Local Confluence

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ UsableRulesProof

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

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

FROM(X) → FROM(s(X))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(s(dbl(X)))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
from(X) → cons(X, from(s(X)))

The set Q consists of the following terms:

dbl(0)
dbl(s(x0))
dbls(nil)
dbls(cons(x0, x1))
sel(0, cons(x0, x1))
sel(s(x0), cons(x1, x2))
indx(nil, x0)
indx(cons(x0, x1), x2)
from(x0)

We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

    ↳ QTRS

      ↳ Overlay + Local Confluence

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

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

FROM(X) → FROM(s(X))

R is empty.
The set Q consists of the following terms:

dbl(0)
dbl(s(x0))
dbls(nil)
dbls(cons(x0, x1))
sel(0, cons(x0, x1))
sel(s(x0), cons(x1, x2))
indx(nil, x0)
indx(cons(x0, x1), x2)
from(x0)

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

dbl(0)
dbl(s(x0))
dbls(nil)
dbls(cons(x0, x1))
sel(0, cons(x0, x1))
sel(s(x0), cons(x1, x2))
indx(nil, x0)
indx(cons(x0, x1), x2)
from(x0)

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

    ↳ QTRS

      ↳ Overlay + Local Confluence

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

                          ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

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

FROM(X) → FROM(s(X))

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

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

    ↳ QTRS

      ↳ Overlay + Local Confluence

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

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

SEL(s(X), cons(Y, Z)) → SEL(X, Z)

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(s(dbl(X)))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
from(X) → cons(X, from(s(X)))

The set Q consists of the following terms:

dbl(0)
dbl(s(x0))
dbls(nil)
dbls(cons(x0, x1))
sel(0, cons(x0, x1))
sel(s(x0), cons(x1, x2))
indx(nil, x0)
indx(cons(x0, x1), x2)
from(x0)

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

    ↳ QTRS

      ↳ Overlay + Local Confluence

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

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

SEL(s(X), cons(Y, Z)) → SEL(X, Z)

R is empty.
The set Q consists of the following terms:

dbl(0)
dbl(s(x0))
dbls(nil)
dbls(cons(x0, x1))
sel(0, cons(x0, x1))
sel(s(x0), cons(x1, x2))
indx(nil, x0)
indx(cons(x0, x1), x2)
from(x0)

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

dbl(0)
dbl(s(x0))
dbls(nil)
dbls(cons(x0, x1))
sel(0, cons(x0, x1))
sel(s(x0), cons(x1, x2))
indx(nil, x0)
indx(cons(x0, x1), x2)
from(x0)

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

    ↳ QTRS

      ↳ Overlay + Local Confluence

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

                          ↳ QDP

                            ↳ QDPSizeChangeProof

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

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

SEL(s(X), cons(Y, Z)) → SEL(X, Z)

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

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

SEL(s(X), cons(Y, Z)) → SEL(X, Z)
The graph contains the following edges 1 > 1, 2 > 2

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

    ↳ QTRS

      ↳ Overlay + Local Confluence

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                  ↳ QDP

                  ↳ QDP

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

INDX(cons(X, Y), Z) → INDX(Y, Z)

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(s(dbl(X)))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
from(X) → cons(X, from(s(X)))

The set Q consists of the following terms:

dbl(0)
dbl(s(x0))
dbls(nil)
dbls(cons(x0, x1))
sel(0, cons(x0, x1))
sel(s(x0), cons(x1, x2))
indx(nil, x0)
indx(cons(x0, x1), x2)
from(x0)

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

    ↳ QTRS

      ↳ Overlay + Local Confluence

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

                  ↳ QDP

                  ↳ QDP

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

INDX(cons(X, Y), Z) → INDX(Y, Z)

R is empty.
The set Q consists of the following terms:

dbl(0)
dbl(s(x0))
dbls(nil)
dbls(cons(x0, x1))
sel(0, cons(x0, x1))
sel(s(x0), cons(x1, x2))
indx(nil, x0)
indx(cons(x0, x1), x2)
from(x0)

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

dbl(0)
dbl(s(x0))
dbls(nil)
dbls(cons(x0, x1))
sel(0, cons(x0, x1))
sel(s(x0), cons(x1, x2))
indx(nil, x0)
indx(cons(x0, x1), x2)
from(x0)

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

    ↳ QTRS

      ↳ Overlay + Local Confluence

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

                          ↳ QDP

                            ↳ QDPSizeChangeProof

                  ↳ QDP

                  ↳ QDP

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

INDX(cons(X, Y), Z) → INDX(Y, Z)

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

INDX(cons(X, Y), Z) → INDX(Y, Z)
The graph contains the following edges 1 > 1, 2 >= 2

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

    ↳ QTRS

      ↳ Overlay + Local Confluence

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                  ↳ QDP

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

DBL(s(X)) → DBL(X)

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(s(dbl(X)))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
from(X) → cons(X, from(s(X)))

The set Q consists of the following terms:

dbl(0)
dbl(s(x0))
dbls(nil)
dbls(cons(x0, x1))
sel(0, cons(x0, x1))
sel(s(x0), cons(x1, x2))
indx(nil, x0)
indx(cons(x0, x1), x2)
from(x0)

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

    ↳ QTRS

      ↳ Overlay + Local Confluence

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

                  ↳ QDP

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

DBL(s(X)) → DBL(X)

R is empty.
The set Q consists of the following terms:

dbl(0)
dbl(s(x0))
dbls(nil)
dbls(cons(x0, x1))
sel(0, cons(x0, x1))
sel(s(x0), cons(x1, x2))
indx(nil, x0)
indx(cons(x0, x1), x2)
from(x0)

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

dbl(0)
dbl(s(x0))
dbls(nil)
dbls(cons(x0, x1))
sel(0, cons(x0, x1))
sel(s(x0), cons(x1, x2))
indx(nil, x0)
indx(cons(x0, x1), x2)
from(x0)

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

    ↳ QTRS

      ↳ Overlay + Local Confluence

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

                          ↳ QDP

                            ↳ QDPSizeChangeProof

                  ↳ QDP

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

DBL(s(X)) → DBL(X)

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

DBL(s(X)) → DBL(X)
The graph contains the following edges 1 > 1

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

    ↳ QTRS

      ↳ Overlay + Local Confluence

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

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

DBLS(cons(X, Y)) → DBLS(Y)

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(s(dbl(X)))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
sel(0, cons(X, Y)) → X
sel(s(X), cons(Y, Z)) → sel(X, Z)
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
from(X) → cons(X, from(s(X)))

The set Q consists of the following terms:

dbl(0)
dbl(s(x0))
dbls(nil)
dbls(cons(x0, x1))
sel(0, cons(x0, x1))
sel(s(x0), cons(x1, x2))
indx(nil, x0)
indx(cons(x0, x1), x2)
from(x0)

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

    ↳ QTRS

      ↳ Overlay + Local Confluence

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

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

DBLS(cons(X, Y)) → DBLS(Y)

R is empty.
The set Q consists of the following terms:

dbl(0)
dbl(s(x0))
dbls(nil)
dbls(cons(x0, x1))
sel(0, cons(x0, x1))
sel(s(x0), cons(x1, x2))
indx(nil, x0)
indx(cons(x0, x1), x2)
from(x0)

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.

dbl(0)
dbl(s(x0))
dbls(nil)
dbls(cons(x0, x1))
sel(0, cons(x0, x1))
sel(s(x0), cons(x1, x2))
indx(nil, x0)
indx(cons(x0, x1), x2)
from(x0)

↳ CSR

  ↳ CSRInnermostProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

    ↳ QTRS

      ↳ Overlay + Local Confluence

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

                          ↳ QDP

                            ↳ QDPSizeChangeProof

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

DBLS(cons(X, Y)) → DBLS(Y)

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

DBLS(cons(X, Y)) → DBLS(Y)
The graph contains the following edges 1 > 1