Termination proof of /tmp/tmpRwyZdw/Ex5_7

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)))
dbl1(0) → 01
dbl1(s(X)) → s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) → X
sel1(s(X), cons(Y, Z)) → sel1(X, Z)
quote(0) → 01
quote(s(X)) → s1(quote(X))
quote(dbl(X)) → dbl1(X)
quote(sel(X, Y)) → sel1(X, Y)

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
dbl1: {1}
01: empty set
s1: {1}
sel1: {1, 2}
quote: {1}

↳ 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)))
dbl1(0) → 01
dbl1(s(X)) → s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) → X
sel1(s(X), cons(Y, Z)) → sel1(X, Z)
quote(0) → 01
quote(s(X)) → s1(quote(X))
quote(dbl(X)) → dbl1(X)
quote(sel(X, Y)) → sel1(X, Y)

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
dbl1: {1}
01: empty set
s1: {1}
sel1: {1, 2}
quote: {1}

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

↳ CSR

  ↳ CSDependencyPairsProof

    ↳ QCSDP

      ↳ QCSDependencyGraphProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

Q-restricted context-sensitive dependency pair problem:
The symbols in {dbl, dbls, sel, dbl1, s1, sel1, quote, SEL, DBL1, SEL1, QUOTE, 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)
DBL1(s(X)) → DBL1(X)
SEL1(s(X), cons(Y, Z)) → SEL1(X, Z)
QUOTE(s(X)) → QUOTE(X)
QUOTE(dbl(X)) → DBL1(X)
QUOTE(sel(X, Y)) → SEL1(X, Y)

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
DBL1(s(X)) → X
SEL1(0, cons(X, Y)) → X
SEL1(s(X), cons(Y, Z)) → X
SEL1(s(X), cons(Y, Z)) → Z
QUOTE(s(X)) → X

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)
DBL1(s(X)) → U(X)
SEL1(0, cons(X, Y)) → U(X)
SEL1(s(X), cons(Y, Z)) → U(X)
SEL1(s(X), cons(Y, Z)) → U(Z)
QUOTE(s(X)) → 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(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)))
dbl1(0) → 01
dbl1(s(X)) → s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) → X
sel1(s(X), cons(Y, Z)) → sel1(X, Z)
quote(0) → 01
quote(s(X)) → s1(quote(X))
quote(dbl(X)) → dbl1(X)
quote(sel(X, Y)) → sel1(X, Y)

Q is empty.

The approximation of the Context-Sensitive Dependency Graph contains 4 SCCs with 11 less nodes.

↳ CSR

  ↳ CSDependencyPairsProof

    ↳ QCSDP

      ↳ QCSDependencyGraphProof

        ↳ AND

          ↳ QCSDP

            ↳ ConvertedToQDPProblemProof

          ↳ QCSDP

          ↳ QCSDP

          ↳ QCSDP

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

Q-restricted context-sensitive dependency pair problem:
The symbols in {dbl, dbls, sel, dbl1, s1, sel1, quote, 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)))
dbl1(0) → 01
dbl1(s(X)) → s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) → X
sel1(s(X), cons(Y, Z)) → sel1(X, Z)
quote(0) → 01
quote(s(X)) → s1(quote(X))
quote(dbl(X)) → dbl1(X)
quote(sel(X, Y)) → sel1(X, Y)

Q is empty.

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

↳ CSR

  ↳ CSDependencyPairsProof

    ↳ QCSDP

      ↳ QCSDependencyGraphProof

        ↳ AND

          ↳ QCSDP

            ↳ ConvertedToQDPProblemProof

              ↳ QDP

          ↳ QCSDP

          ↳ QCSDP

          ↳ QCSDP

  ↳ 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)))
dbl1(0) → 01
dbl1(s(X)) → s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) → X
sel1(s(X), cons(Y, Z)) → sel1(X, Z)
quote(0) → 01
quote(s(X)) → s1(quote(X))
quote(dbl(X)) → dbl1(X)
quote(sel(X, Y)) → sel1(X, Y)

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

↳ CSR

  ↳ CSDependencyPairsProof

    ↳ QCSDP

      ↳ QCSDependencyGraphProof

        ↳ AND

          ↳ QCSDP

          ↳ QCSDP

          ↳ QCSDP

          ↳ QCSDP

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

Q-restricted context-sensitive dependency pair problem:
The symbols in {dbl, dbls, sel, dbl1, s1, sel1, quote, SEL1} are replacing on all positions.
For all symbols f in {indx} we have µ(f) = {1}.
The symbols in {s, cons, from} are not replacing on any position.

The TRS P consists of the following rules:

SEL1(s(X), cons(Y, Z)) → SEL1(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)))
dbl1(0) → 01
dbl1(s(X)) → s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) → X
sel1(s(X), cons(Y, Z)) → sel1(X, Z)
quote(0) → 01
quote(s(X)) → s1(quote(X))
quote(dbl(X)) → dbl1(X)
quote(sel(X, Y)) → sel1(X, Y)

Q is empty.

↳ CSR

  ↳ CSDependencyPairsProof

    ↳ QCSDP

      ↳ QCSDependencyGraphProof

        ↳ AND

          ↳ QCSDP

          ↳ QCSDP

          ↳ QCSDP

          ↳ QCSDP

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

Q-restricted context-sensitive dependency pair problem:
The symbols in {dbl, dbls, sel, dbl1, s1, sel1, quote, DBL1} are replacing on all positions.
For all symbols f in {indx} we have µ(f) = {1}.
The symbols in {s, cons, from} are not replacing on any position.

The TRS P consists of the following rules:

DBL1(s(X)) → DBL1(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)))
dbl1(0) → 01
dbl1(s(X)) → s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) → X
sel1(s(X), cons(Y, Z)) → sel1(X, Z)
quote(0) → 01
quote(s(X)) → s1(quote(X))
quote(dbl(X)) → dbl1(X)
quote(sel(X, Y)) → sel1(X, Y)

Q is empty.

↳ CSR

  ↳ CSDependencyPairsProof

    ↳ QCSDP

      ↳ QCSDependencyGraphProof

        ↳ AND

          ↳ QCSDP

          ↳ QCSDP

          ↳ QCSDP

          ↳ QCSDP

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

Q-restricted context-sensitive dependency pair problem:
The symbols in {dbl, dbls, sel, dbl1, s1, sel1, quote, QUOTE} are replacing on all positions.
For all symbols f in {indx} we have µ(f) = {1}.
The symbols in {s, cons, from} are not replacing on any position.

The TRS P consists of the following rules:

QUOTE(s(X)) → QUOTE(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)))
dbl1(0) → 01
dbl1(s(X)) → s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) → X
sel1(s(X), cons(Y, Z)) → sel1(X, Z)
quote(0) → 01
quote(s(X)) → s1(quote(X))
quote(dbl(X)) → dbl1(X)
quote(sel(X, Y)) → sel1(X, Y)

Q is empty.

We applied the Incomplete Giesl Middeldorp transformation [11] to transform the context-sensitive TRS to a usual TRS.

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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:

SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(indx(x1, x2)) → INDXACTIVE(mark(x1), x2)
MARK(indx(x1, x2)) → MARK(x1)
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(quote(x1)) → QUOTEACTIVE(mark(x1))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(dbl(x1)) → DBLACTIVE(mark(x1))
SELACTIVE(0, cons(X, Y)) → MARK(X)
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(dbls(x1)) → MARK(x1)
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
MARK(from(x1)) → FROMACTIVE(x1)
MARK(sel(x1, x2)) → SELACTIVE(mark(x1), mark(x2))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(quote(x1)) → MARK(x1)
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(x1, x2)) → MARK(x1)
DBL1ACTIVE(s(X)) → DBL1ACTIVE(mark(X))
QUOTEACTIVE(s(X)) → MARK(X)
MARK(dbl1(x1)) → DBL1ACTIVE(mark(x1))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(dbl(x1)) → MARK(x1)
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(x1, x2)) → SEL1ACTIVE(mark(x1), mark(x2))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(dbls(x1)) → DBLSACTIVE(mark(x1))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(sel1(x1, x2)) → MARK(x2)
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(x1, x2)) → MARK(x2)
QUOTEACTIVE(sel(X, Y)) → MARK(X)

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

  ↳ Trivial-Transformation

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

SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(indx(x1, x2)) → INDXACTIVE(mark(x1), x2)
MARK(indx(x1, x2)) → MARK(x1)
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(quote(x1)) → QUOTEACTIVE(mark(x1))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(dbl(x1)) → DBLACTIVE(mark(x1))
SELACTIVE(0, cons(X, Y)) → MARK(X)
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(dbls(x1)) → MARK(x1)
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
MARK(from(x1)) → FROMACTIVE(x1)
MARK(sel(x1, x2)) → SELACTIVE(mark(x1), mark(x2))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(quote(x1)) → MARK(x1)
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(x1, x2)) → MARK(x1)
DBL1ACTIVE(s(X)) → DBL1ACTIVE(mark(X))
QUOTEACTIVE(s(X)) → MARK(X)
MARK(dbl1(x1)) → DBL1ACTIVE(mark(x1))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(dbl(x1)) → MARK(x1)
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(x1, x2)) → SEL1ACTIVE(mark(x1), mark(x2))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(dbls(x1)) → DBLSACTIVE(mark(x1))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(sel1(x1, x2)) → MARK(x2)
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(x1, x2)) → MARK(x2)
QUOTEACTIVE(sel(X, Y)) → MARK(X)

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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

  ↳ CSDependencyPairsProof

  ↳ Incomplete Giesl Middeldorp-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ Narrowing

  ↳ Trivial-Transformation

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

SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(indx(x1, x2)) → MARK(x1)
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(quote(x1)) → QUOTEACTIVE(mark(x1))
DBL1ACTIVE(s(X)) → MARK(X)
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(dbls(x1)) → MARK(x1)
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
MARK(sel(x1, x2)) → SELACTIVE(mark(x1), mark(x2))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(quote(x1)) → MARK(x1)
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
DBL1ACTIVE(s(X)) → DBL1ACTIVE(mark(X))
MARK(sel(x1, x2)) → MARK(x1)
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(dbl1(x1)) → DBL1ACTIVE(mark(x1))
QUOTEACTIVE(s(X)) → MARK(X)
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(dbl(x1)) → MARK(x1)
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(x1, x2)) → SEL1ACTIVE(mark(x1), mark(x2))
MARK(sel1(x1, x2)) → MARK(x1)
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
MARK(sel1(x1, x2)) → MARK(x2)
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(x1, x2)) → MARK(x2)
QUOTEACTIVE(sel(X, Y)) → MARK(X)

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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(dbl1(x0), y1)) → SELACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel(indx(x0, x1), y1)) → SELACTIVE(indxActive(mark(x0), x1), mark(y1))
MARK(sel(quote(x0), y1)) → SELACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel(dbl(x0), y1)) → SELACTIVE(dblActive(mark(x0)), mark(y1))
MARK(sel(01, y1)) → SELACTIVE(01, mark(y1))
MARK(sel(sel1(x0, x1), y1)) → SELACTIVE(sel1Active(mark(x0), mark(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(sel(x0, x1), y1)) → SELACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(sel(0, y1)) → SELACTIVE(0, mark(y1))
MARK(sel(s1(x0), y1)) → SELACTIVE(s1(mark(x0)), 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))

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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:

SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(indx(x1, x2)) → MARK(x1)
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(quote(x0), y1)) → SELACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(quote(x1)) → QUOTEACTIVE(mark(x1))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel(01, y1)) → SELACTIVE(01, mark(y1))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel1(x0, x1), y1)) → SELACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
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))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
MARK(sel(nil, y1)) → SELACTIVE(nil, mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(0, y1)) → SELACTIVE(0, mark(y1))
MARK(quote(x1)) → MARK(x1)
MARK(sel(s1(x0), y1)) → SELACTIVE(s1(mark(x0)), mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(indx(x0, x1), y1)) → SELACTIVE(indxActive(mark(x0), x1), mark(y1))
MARK(sel(dbl1(x0), y1)) → SELACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel(x1, x2)) → MARK(x1)
DBL1ACTIVE(s(X)) → DBL1ACTIVE(mark(X))
QUOTEACTIVE(s(X)) → MARK(X)
MARK(dbl1(x1)) → DBL1ACTIVE(mark(x1))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(dbl(x1)) → MARK(x1)
MARK(sel(dbl(x0), y1)) → SELACTIVE(dblActive(mark(x0)), mark(y1))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(dbl(X)) → MARK(X)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel1(x1, x2)) → SEL1ACTIVE(mark(x1), mark(x2))
MARK(sel1(x1, x2)) → MARK(x1)
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(sel1(x1, x2)) → MARK(x2)
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(sel(x0, x1), y1)) → SELACTIVE(selActive(mark(x0), mark(x1)), 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))
MARK(sel(x1, x2)) → MARK(x2)
QUOTEACTIVE(sel(X, Y)) → MARK(X)

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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

  ↳ CSDependencyPairsProof

  ↳ 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:

SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(indx(x1, x2)) → MARK(x1)
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(quote(x1)) → QUOTEACTIVE(mark(x1))
MARK(sel(quote(x0), y1)) → SELACTIVE(quoteActive(mark(x0)), mark(y1))
DBL1ACTIVE(s(X)) → MARK(X)
SELACTIVE(0, cons(X, Y)) → MARK(X)
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel(sel1(x0, x1), y1)) → SELACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
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))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(0, y1)) → SELACTIVE(0, mark(y1))
MARK(quote(x1)) → MARK(x1)
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(dbl1(x0), y1)) → SELACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel(indx(x0, x1), y1)) → SELACTIVE(indxActive(mark(x0), x1), mark(y1))
DBL1ACTIVE(s(X)) → DBL1ACTIVE(mark(X))
MARK(sel(x1, x2)) → MARK(x1)
MARK(dbl1(x1)) → DBL1ACTIVE(mark(x1))
QUOTEACTIVE(s(X)) → MARK(X)
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(dbl(x1)) → MARK(x1)
MARK(sel(dbl(x0), y1)) → SELACTIVE(dblActive(mark(x0)), mark(y1))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(x1, x2)) → SEL1ACTIVE(mark(x1), mark(x2))
MARK(sel1(x1, x2)) → MARK(x1)
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(sel1(x1, x2)) → MARK(x2)
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(sel(x0, x1), y1)) → SELACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(sel(dbls(x0), y1)) → SELACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(sel(x1, x2)) → MARK(x2)
QUOTEACTIVE(sel(X, Y)) → MARK(X)

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(dbl1(x1)) → DBL1ACTIVE(mark(x1)) at position [0] we obtained the following new rules:

MARK(dbl1(0)) → DBL1ACTIVE(0)
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(dbl1(cons(x0, x1))) → DBL1ACTIVE(cons(x0, x1))
MARK(dbl1(from(x0))) → DBL1ACTIVE(fromActive(x0))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(dbl1(01)) → DBL1ACTIVE(01)
MARK(dbl1(nil)) → DBL1ACTIVE(nil)
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(dbl1(s1(x0))) → DBL1ACTIVE(s1(mark(x0)))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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:

SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(indx(x1, x2)) → MARK(x1)
MARK(dbl1(0)) → DBL1ACTIVE(0)
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(quote(x0), y1)) → SELACTIVE(quoteActive(mark(x0)), mark(y1))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
MARK(dbl1(01)) → DBL1ACTIVE(01)
MARK(dbl1(nil)) → DBL1ACTIVE(nil)
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(0, y1)) → SELACTIVE(0, mark(y1))
MARK(quote(x1)) → MARK(x1)
MARK(sel(dbl1(x0), y1)) → SELACTIVE(dbl1Active(mark(x0)), mark(y1))
DBL1ACTIVE(s(X)) → DBL1ACTIVE(mark(X))
MARK(sel(x1, x2)) → MARK(x1)
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(dbl(x1)) → MARK(x1)
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(x1, x2)) → SEL1ACTIVE(mark(x1), mark(x2))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel(dbls(x0), y1)) → SELACTIVE(dblsActive(mark(x0)), mark(y1))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(quote(x1)) → QUOTEACTIVE(mark(x1))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel(sel1(x0, x1), y1)) → SELACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(dbls(x1)) → MARK(x1)
MARK(dbl1(cons(x0, x1))) → DBL1ACTIVE(cons(x0, x1))
MARK(sel(from(x0), y1)) → SELACTIVE(fromActive(x0), mark(y1))
MARK(sel(s(x0), y1)) → SELACTIVE(s(x0), mark(y1))
MARK(dbl1(s1(x0))) → DBL1ACTIVE(s1(mark(x0)))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(indx(x0, x1), y1)) → SELACTIVE(indxActive(mark(x0), x1), mark(y1))
QUOTEACTIVE(s(X)) → MARK(X)
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(dbl(x0), y1)) → SELACTIVE(dblActive(mark(x0)), mark(y1))
MARK(dbl1(from(x0))) → DBL1ACTIVE(fromActive(x0))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel1(x1, x2)) → MARK(x2)
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(sel(x0, x1), y1)) → SELACTIVE(selActive(mark(x0), mark(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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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 5 less nodes.

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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:

SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(indx(x1, x2)) → MARK(x1)
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(quote(x1)) → QUOTEACTIVE(mark(x1))
MARK(sel(quote(x0), y1)) → SELACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
DBL1ACTIVE(s(X)) → MARK(X)
SELACTIVE(0, cons(X, Y)) → MARK(X)
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(sel1(x0, x1), y1)) → SELACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(from(x0), y1)) → SELACTIVE(fromActive(x0), mark(y1))
MARK(sel(s(x0), y1)) → SELACTIVE(s(x0), mark(y1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(0, y1)) → SELACTIVE(0, mark(y1))
MARK(quote(x1)) → MARK(x1)
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(dbl1(x0), y1)) → SELACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel(indx(x0, x1), y1)) → SELACTIVE(indxActive(mark(x0), x1), mark(y1))
DBL1ACTIVE(s(X)) → DBL1ACTIVE(mark(X))
MARK(sel(x1, x2)) → MARK(x1)
QUOTEACTIVE(s(X)) → MARK(X)
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(dbl(x0), y1)) → SELACTIVE(dblActive(mark(x0)), mark(y1))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(x1, x2)) → SEL1ACTIVE(mark(x1), mark(x2))
MARK(dbl1(from(x0))) → DBL1ACTIVE(fromActive(x0))
MARK(sel1(x1, x2)) → MARK(x1)
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(sel1(x1, x2)) → MARK(x2)
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(sel(x0, x1), y1)) → SELACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel(dbls(x0), y1)) → SELACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(sel(x1, x2)) → MARK(x2)
QUOTEACTIVE(sel(X, Y)) → MARK(X)

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(sel1(x1, x2)) → SEL1ACTIVE(mark(x1), mark(x2)) at position [0] we obtained the following new rules:

MARK(sel1(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
MARK(sel1(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(sel1(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
MARK(sel1(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel1(s1(x0), y1)) → SEL1ACTIVE(s1(mark(x0)), mark(y1))
MARK(sel1(0, y1)) → SEL1ACTIVE(0, mark(y1))
MARK(sel1(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel1(nil, y1)) → SEL1ACTIVE(nil, mark(y1))
MARK(sel1(cons(x0, x1), y1)) → SEL1ACTIVE(cons(x0, x1), mark(y1))
MARK(sel1(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel1(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(sel1(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel1(01, y1)) → SEL1ACTIVE(01, mark(y1))

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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:

SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(quote(x0), y1)) → SELACTIVE(quoteActive(mark(x0)), mark(y1))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(quote(x1)) → MARK(x1)
MARK(sel(0, y1)) → SELACTIVE(0, mark(y1))
MARK(sel(dbl1(x0), y1)) → SELACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel(x1, x2)) → MARK(x1)
DBL1ACTIVE(s(X)) → DBL1ACTIVE(mark(X))
MARK(sel1(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(dbl(x1)) → MARK(x1)
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(dbl(X)) → MARK(X)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel1(s1(x0), y1)) → SEL1ACTIVE(s1(mark(x0)), mark(y1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel1(cons(x0, x1), y1)) → SEL1ACTIVE(cons(x0, x1), mark(y1))
MARK(sel1(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel1(01, y1)) → SEL1ACTIVE(01, mark(y1))
MARK(sel(dbls(x0), y1)) → SELACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(sel1(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
MARK(quote(x1)) → QUOTEACTIVE(mark(x1))
MARK(sel1(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel1(0, y1)) → SEL1ACTIVE(0, mark(y1))
MARK(sel(sel1(x0, x1), y1)) → SELACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(dbls(x1)) → MARK(x1)
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel1(nil, y1)) → SEL1ACTIVE(nil, mark(y1))
MARK(sel(from(x0), y1)) → SELACTIVE(fromActive(x0), mark(y1))
MARK(sel(s(x0), y1)) → SELACTIVE(s(x0), mark(y1))
MARK(sel1(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(indx(x0, x1), y1)) → SELACTIVE(indxActive(mark(x0), x1), mark(y1))
QUOTEACTIVE(s(X)) → MARK(X)
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(dbl(x0), y1)) → SELACTIVE(dblActive(mark(x0)), mark(y1))
MARK(dbl1(from(x0))) → DBL1ACTIVE(fromActive(x0))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
MARK(sel1(x1, x2)) → MARK(x2)
MARK(dbl1(x1)) → MARK(x1)
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(sel(x0, x1), y1)) → SELACTIVE(selActive(mark(x0), mark(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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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

  ↳ CSDependencyPairsProof

  ↳ 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:

SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(indx(x1, x2)) → MARK(x1)
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
MARK(sel(quote(x0), y1)) → SELACTIVE(quoteActive(mark(x0)), mark(y1))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(quote(x1)) → MARK(x1)
MARK(sel(0, y1)) → SELACTIVE(0, mark(y1))
MARK(sel(dbl1(x0), y1)) → SELACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel(x1, x2)) → MARK(x1)
DBL1ACTIVE(s(X)) → DBL1ACTIVE(mark(X))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel1(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(dbl(x1)) → MARK(x1)
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel1(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel(dbls(x0), y1)) → SELACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(sel1(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
MARK(quote(x1)) → QUOTEACTIVE(mark(x1))
MARK(sel1(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel(sel1(x0, x1), y1)) → SELACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(dbls(x1)) → MARK(x1)
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel1(0, y1)) → SEL1ACTIVE(0, mark(y1))
MARK(sel(from(x0), y1)) → SELACTIVE(fromActive(x0), mark(y1))
MARK(sel(s(x0), y1)) → SELACTIVE(s(x0), mark(y1))
MARK(sel1(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(indx(x0, x1), y1)) → SELACTIVE(indxActive(mark(x0), x1), mark(y1))
QUOTEACTIVE(s(X)) → MARK(X)
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(dbl(x0), y1)) → SELACTIVE(dblActive(mark(x0)), mark(y1))
MARK(dbl1(from(x0))) → DBL1ACTIVE(fromActive(x0))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel1(x1, x2)) → MARK(x2)
MARK(dbl1(x1)) → MARK(x1)
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(sel(x0, x1), y1)) → SELACTIVE(selActive(mark(x0), mark(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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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

MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
MARK(quote(nil)) → QUOTEACTIVE(nil)
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(quote(s1(x0))) → QUOTEACTIVE(s1(mark(x0)))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(quote(from(x0))) → QUOTEACTIVE(fromActive(x0))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(quote(0)) → QUOTEACTIVE(0)
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
MARK(quote(01)) → QUOTEACTIVE(01)
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(quote(cons(x0, x1))) → QUOTEACTIVE(cons(x0, x1))
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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:

SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(quote(x0), y1)) → SELACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
DBL1ACTIVE(s(X)) → MARK(X)
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(quote(cons(x0, x1))) → QUOTEACTIVE(cons(x0, x1))
MARK(sel(0, y1)) → SELACTIVE(0, mark(y1))
MARK(quote(x1)) → MARK(x1)
MARK(sel1(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(quote(nil)) → QUOTEACTIVE(nil)
MARK(sel(dbl1(x0), y1)) → SELACTIVE(dbl1Active(mark(x0)), mark(y1))
DBL1ACTIVE(s(X)) → DBL1ACTIVE(mark(X))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(quote(s1(x0))) → QUOTEACTIVE(s1(mark(x0)))
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(dbl(x1)) → MARK(x1)
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(dbl(X)) → MARK(X)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel1(x1, x2)) → MARK(x1)
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel(dbls(x0), y1)) → SELACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(sel1(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
MARK(sel1(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel1(0, y1)) → SEL1ACTIVE(0, mark(y1))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(sel1(x0, x1), y1)) → SELACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(quote(0)) → QUOTEACTIVE(0)
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(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(quote(01)) → QUOTEACTIVE(01)
MARK(sel1(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(indx(x0, x1), y1)) → SELACTIVE(indxActive(mark(x0), x1), mark(y1))
QUOTEACTIVE(s(X)) → MARK(X)
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(dbl(x0), y1)) → SELACTIVE(dblActive(mark(x0)), mark(y1))
MARK(quote(from(x0))) → QUOTEACTIVE(fromActive(x0))
MARK(dbl1(from(x0))) → DBL1ACTIVE(fromActive(x0))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
MARK(sel1(x1, x2)) → MARK(x2)
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(sel(x0, x1), y1)) → SELACTIVE(selActive(mark(x0), mark(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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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 5 less nodes.

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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:

SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(indx(x1, x2)) → MARK(x1)
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
MARK(sel(quote(x0), y1)) → SELACTIVE(quoteActive(mark(x0)), mark(y1))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(0, y1)) → SELACTIVE(0, mark(y1))
MARK(quote(x1)) → MARK(x1)
MARK(sel1(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(sel(dbl1(x0), y1)) → SELACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel(x1, x2)) → MARK(x1)
DBL1ACTIVE(s(X)) → DBL1ACTIVE(mark(X))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel1(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(dbl(x1)) → MARK(x1)
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(x1, x2)) → MARK(x1)
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel(dbls(x0), y1)) → SELACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(sel1(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
MARK(sel1(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel(sel1(x0, x1), y1)) → SELACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(dbls(x1)) → MARK(x1)
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel1(0, y1)) → SEL1ACTIVE(0, mark(y1))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(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(sel1(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(indx(x0, x1), y1)) → SELACTIVE(indxActive(mark(x0), x1), mark(y1))
QUOTEACTIVE(s(X)) → MARK(X)
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(dbl(x0), y1)) → SELACTIVE(dblActive(mark(x0)), mark(y1))
MARK(quote(from(x0))) → QUOTEACTIVE(fromActive(x0))
MARK(dbl1(from(x0))) → DBL1ACTIVE(fromActive(x0))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel1(x1, x2)) → MARK(x2)
MARK(dbl1(x1)) → MARK(x1)
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(sel(x0, x1), y1)) → SELACTIVE(selActive(mark(x0), mark(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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(sel(dbl1(x0), y1)) → SELACTIVE(dbl1Active(mark(x0)), mark(y1)) at position [1] we obtained the following new rules:

MARK(sel(dbl1(y0), 01)) → SELACTIVE(dbl1Active(mark(y0)), 01)
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(sel(dbl1(y0), s(x0))) → SELACTIVE(dbl1Active(mark(y0)), s(x0))
MARK(sel(dbl1(y0), nil)) → SELACTIVE(dbl1Active(mark(y0)), nil)
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbl1(y0), 0)) → SELACTIVE(dbl1Active(mark(y0)), 0)
MARK(sel(dbl1(y0), s1(x0))) → SELACTIVE(dbl1Active(mark(y0)), s1(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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:

SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(quote(x0), y1)) → SELACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel(dbl1(y0), nil)) → SELACTIVE(dbl1Active(mark(y0)), nil)
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
DBL1ACTIVE(s(X)) → MARK(X)
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(quote(x1)) → MARK(x1)
MARK(sel(0, y1)) → SELACTIVE(0, mark(y1))
DBL1ACTIVE(s(X)) → DBL1ACTIVE(mark(X))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel(dbl1(y0), 01)) → SELACTIVE(dbl1Active(mark(y0)), 01)
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(dbl(x1)) → MARK(x1)
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(dbl(X)) → MARK(X)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel1(x1, x2)) → MARK(x1)
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel(dbls(x0), y1)) → SELACTIVE(dblsActive(mark(x0)), mark(y1))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel1(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
MARK(sel(dbl1(y0), s(x0))) → SELACTIVE(dbl1Active(mark(y0)), s(x0))
MARK(sel(dbl1(y0), 0)) → SELACTIVE(dbl1Active(mark(y0)), 0)
MARK(sel1(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel1(0, y1)) → SEL1ACTIVE(0, mark(y1))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(sel1(x0, x1), y1)) → SELACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel(from(x0), y1)) → SELACTIVE(fromActive(x0), mark(y1))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(s(x0), y1)) → SELACTIVE(s(x0), mark(y1))
MARK(sel1(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
MARK(sel(indx(x0, x1), y1)) → SELACTIVE(indxActive(mark(x0), x1), mark(y1))
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(dbl(x0), y1)) → SELACTIVE(dblActive(mark(x0)), mark(y1))
MARK(quote(from(x0))) → QUOTEACTIVE(fromActive(x0))
MARK(sel(dbl1(y0), s1(x0))) → SELACTIVE(dbl1Active(mark(y0)), s1(mark(x0)))
MARK(dbl1(from(x0))) → DBL1ACTIVE(fromActive(x0))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
MARK(sel1(x1, x2)) → MARK(x2)
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(x0, x1), y1)) → SELACTIVE(selActive(mark(x0), mark(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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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 5 less nodes.

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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:

SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
MARK(sel(quote(x0), y1)) → SELACTIVE(quoteActive(mark(x0)), mark(y1))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(0, y1)) → SELACTIVE(0, mark(y1))
MARK(quote(x1)) → MARK(x1)
MARK(sel1(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
DBL1ACTIVE(s(X)) → DBL1ACTIVE(mark(X))
MARK(sel(x1, x2)) → MARK(x1)
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel1(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(x1, x2)) → MARK(x1)
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbls(x0), y1)) → SELACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(sel1(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
MARK(sel1(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel(sel1(x0, x1), y1)) → SELACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(dbls(x1)) → MARK(x1)
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel1(0, y1)) → SEL1ACTIVE(0, mark(y1))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel(from(x0), y1)) → SELACTIVE(fromActive(x0), mark(y1))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(s(x0), y1)) → SELACTIVE(s(x0), mark(y1))
MARK(sel1(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
MARK(sel(indx(x0, x1), y1)) → SELACTIVE(indxActive(mark(x0), x1), mark(y1))
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(dbl(x0), y1)) → SELACTIVE(dblActive(mark(x0)), mark(y1))
MARK(quote(from(x0))) → QUOTEACTIVE(fromActive(x0))
MARK(dbl1(from(x0))) → DBL1ACTIVE(fromActive(x0))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel1(x1, x2)) → MARK(x2)
MARK(dbl1(x1)) → MARK(x1)
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(x0, x1), y1)) → SELACTIVE(selActive(mark(x0), mark(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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), 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)))
MARK(sel(indx(y0, y1), 01)) → SELACTIVE(indxActive(mark(y0), y1), 01)
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), s1(x0))) → SELACTIVE(indxActive(mark(y0), y1), s1(mark(x0)))
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(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(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)))

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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:

SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(x0), y1)) → SELACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), s1(x0))) → SELACTIVE(indxActive(mark(y0), y1), s1(mark(x0)))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(quote(x1)) → MARK(x1)
MARK(sel(0, y1)) → SELACTIVE(0, mark(y1))
MARK(sel(indx(y0, y1), nil)) → SELACTIVE(indxActive(mark(y0), y1), nil)
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
DBL1ACTIVE(s(X)) → DBL1ACTIVE(mark(X))
MARK(sel1(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(dbl(x1)) → MARK(x1)
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(dbl(X)) → MARK(X)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(indx(y0, y1), 01)) → SELACTIVE(indxActive(mark(y0), y1), 01)
MARK(sel1(x1, x2)) → MARK(x1)
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel(indx(y0, y1), 0)) → SELACTIVE(indxActive(mark(y0), y1), 0)
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel(dbls(x0), y1)) → SELACTIVE(dblsActive(mark(x0)), mark(y1))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel1(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
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(sel1(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel1(0, y1)) → SEL1ACTIVE(0, mark(y1))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(sel1(x0, x1), y1)) → SELACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(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(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(dbl(x0), y1)) → SELACTIVE(dblActive(mark(x0)), mark(y1))
MARK(quote(from(x0))) → QUOTEACTIVE(fromActive(x0))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(dbl1(from(x0))) → DBL1ACTIVE(fromActive(x0))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
MARK(sel1(x1, x2)) → MARK(x2)
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(x0, x1), y1)) → SELACTIVE(selActive(mark(x0), mark(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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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 5 less nodes.

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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:

SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
MARK(sel(quote(x0), y1)) → SELACTIVE(quoteActive(mark(x0)), mark(y1))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(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(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel(0, y1)) → SELACTIVE(0, mark(y1))
MARK(quote(x1)) → MARK(x1)
MARK(sel1(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(X)) → DBL1ACTIVE(mark(X))
MARK(sel(x1, x2)) → MARK(x1)
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel1(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(x1, x2)) → MARK(x1)
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbls(x0), y1)) → SELACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(sel1(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel1(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel(sel1(x0, x1), y1)) → SELACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(dbls(x1)) → MARK(x1)
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel1(0, y1)) → SEL1ACTIVE(0, mark(y1))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel(from(x0), y1)) → SELACTIVE(fromActive(x0), mark(y1))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(s(x0), y1)) → SELACTIVE(s(x0), mark(y1))
MARK(sel1(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(dbl(x0), y1)) → SELACTIVE(dblActive(mark(x0)), mark(y1))
MARK(quote(from(x0))) → QUOTEACTIVE(fromActive(x0))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(dbl1(from(x0))) → DBL1ACTIVE(fromActive(x0))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(sel1(x1, x2)) → MARK(x2)
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(dbl1(x1)) → MARK(x1)
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(x0, x1), y1)) → SELACTIVE(selActive(mark(x0), mark(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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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

MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel(quote(y0), 0)) → SELACTIVE(quoteActive(mark(y0)), 0)
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(sel(quote(y0), 01)) → SELACTIVE(quoteActive(mark(y0)), 01)
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(quote(y0), s1(x0))) → SELACTIVE(quoteActive(mark(y0)), s1(mark(x0)))
MARK(sel(quote(y0), s(x0))) → SELACTIVE(quoteActive(mark(y0)), s(x0))
MARK(sel(quote(y0), nil)) → SELACTIVE(quoteActive(mark(y0)), nil)

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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:

SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(quote(x1)) → MARK(x1)
MARK(sel(0, y1)) → SELACTIVE(0, mark(y1))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(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)
DBL1ACTIVE(s(X)) → DBL1ACTIVE(mark(X))
MARK(sel1(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(dbl(x1)) → MARK(x1)
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(dbl(X)) → MARK(X)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(quote(y0), 0)) → SELACTIVE(quoteActive(mark(y0)), 0)
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel(dbls(x0), y1)) → SELACTIVE(dblsActive(mark(x0)), mark(y1))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel1(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel1(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel1(0, y1)) → SEL1ACTIVE(0, mark(y1))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(sel1(x0, x1), y1)) → SELACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(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(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(sel(quote(y0), s(x0))) → SELACTIVE(quoteActive(mark(y0)), s(x0))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(dbl(x0), y1)) → SELACTIVE(dblActive(mark(x0)), mark(y1))
MARK(quote(from(x0))) → QUOTEACTIVE(fromActive(x0))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(dbl1(from(x0))) → DBL1ACTIVE(fromActive(x0))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
MARK(sel(quote(y0), 01)) → SELACTIVE(quoteActive(mark(y0)), 01)
MARK(sel1(x1, x2)) → MARK(x2)
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(x0, x1), y1)) → SELACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(sel(quote(y0), s1(x0))) → SELACTIVE(quoteActive(mark(y0)), s1(mark(x0)))
MARK(sel(x1, x2)) → MARK(x2)
MARK(sel(quote(y0), nil)) → SELACTIVE(quoteActive(mark(y0)), nil)

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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 5 less nodes.

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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:

SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel(0, y1)) → SELACTIVE(0, mark(y1))
MARK(quote(x1)) → MARK(x1)
MARK(sel1(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(X)) → DBL1ACTIVE(mark(X))
MARK(sel(x1, x2)) → MARK(x1)
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel1(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbls(x0), y1)) → SELACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(sel1(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel1(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(sel1(x0, x1), y1)) → SELACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel1(0, y1)) → SEL1ACTIVE(0, mark(y1))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel(from(x0), y1)) → SELACTIVE(fromActive(x0), mark(y1))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(s(x0), y1)) → SELACTIVE(s(x0), mark(y1))
MARK(sel1(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(dbl(x0), y1)) → SELACTIVE(dblActive(mark(x0)), mark(y1))
MARK(quote(from(x0))) → QUOTEACTIVE(fromActive(x0))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(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(dbl1(from(x0))) → DBL1ACTIVE(fromActive(x0))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(sel1(x1, x2)) → MARK(x2)
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(dbl1(x1)) → MARK(x1)
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(x0, x1), y1)) → SELACTIVE(selActive(mark(x0), mark(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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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), 01)) → SELACTIVE(dblActive(mark(y0)), 01)
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbl(y0), s1(x0))) → SELACTIVE(dblActive(mark(y0)), s1(mark(x0)))
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)))
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
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), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(dbl(y0), 0)) → SELACTIVE(dblActive(mark(y0)), 0)
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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)
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel(dbl(y0), nil)) → SELACTIVE(dblActive(mark(y0)), nil)
MARK(sel1(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel(dbl(y0), s(x0))) → SELACTIVE(dblActive(mark(y0)), s(x0))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel(0, y1)) → SELACTIVE(0, mark(y1))
MARK(quote(x1)) → MARK(x1)
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel1(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(dbl(x1)) → MARK(x1)
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel1(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel(s(x0), y1)) → SELACTIVE(s(x0), mark(y1))
MARK(sel(dbl(y0), 0)) → SELACTIVE(dblActive(mark(y0)), 0)
MARK(sel1(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(quote(from(x0))) → QUOTEACTIVE(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(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(sel(x0, x1), y1)) → SELACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
DBL1ACTIVE(s(X)) → DBL1ACTIVE(mark(X))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel(dbl(y0), 01)) → SELACTIVE(dblActive(mark(y0)), 01)
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), 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(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(sel(dbl(y0), s1(x0))) → SELACTIVE(dblActive(mark(y0)), s1(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(sel1(x0, x1), y1)) → SELACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(sel1(0, y1)) → SEL1ACTIVE(0, mark(y1))
MARK(sel(from(x0), y1)) → SELACTIVE(fromActive(x0), mark(y1))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(dbl1(from(x0))) → DBL1ACTIVE(fromActive(x0))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel1(x1, x2)) → MARK(x2)
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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 5 less nodes.

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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(indx(x1, x2)) → MARK(x1)
MARK(sel1(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(quote(x1)) → MARK(x1)
MARK(sel(0, y1)) → SELACTIVE(0, mark(y1))
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(dbl(x1)) → MARK(x1)
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel1(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel(s(x0), y1)) → SELACTIVE(s(x0), mark(y1))
MARK(sel1(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(quote(from(x0))) → QUOTEACTIVE(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(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(sel(x0, x1), y1)) → SELACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x1)
DBL1ACTIVE(s(X)) → DBL1ACTIVE(mark(X))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), 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))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, y1)) → SEL1ACTIVE(0, mark(y1))
MARK(sel(sel1(x0, x1), y1)) → SELACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(from(x0), y1)) → SELACTIVE(fromActive(x0), mark(y1))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(dbl1(from(x0))) → DBL1ACTIVE(fromActive(x0))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(x1, x2)) → MARK(x2)
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(sel(sel1(x0, x1), y1)) → SELACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1)) at position [1] we obtained the following new rules:

MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), 01)) → SELACTIVE(sel1Active(mark(y0), mark(y1)), 01)
MARK(sel(sel1(y0, y1), s1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), s1(mark(x0)))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(sel1(y0, y1), nil)) → SELACTIVE(sel1Active(mark(y0), mark(y1)), nil)
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(sel1(y0, y1), s(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), s(x0))
MARK(sel(sel1(y0, y1), 0)) → SELACTIVE(sel1Active(mark(y0), mark(y1)), 0)
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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(indx(x1, x2)) → MARK(x1)
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
MARK(sel(sel1(y0, y1), 01)) → SELACTIVE(sel1Active(mark(y0), mark(y1)), 01)
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), nil)) → SELACTIVE(sel1Active(mark(y0), mark(y1)), nil)
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel(0, y1)) → SELACTIVE(0, mark(y1))
MARK(quote(x1)) → MARK(x1)
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel1(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(dbl(x1)) → MARK(x1)
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
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(sel1(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel(s(x0), y1)) → SELACTIVE(s(x0), mark(y1))
MARK(sel1(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(quote(from(x0))) → QUOTEACTIVE(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(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(sel(x0, x1), y1)) → SELACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(sel1(y0, y1), s1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), s1(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
DBL1ACTIVE(s(X)) → DBL1ACTIVE(mark(X))
MARK(sel(x1, x2)) → MARK(x1)
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), 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(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(dbls(x1)) → MARK(x1)
MARK(sel1(0, y1)) → SEL1ACTIVE(0, mark(y1))
MARK(sel(from(x0), y1)) → SELACTIVE(fromActive(x0), mark(y1))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel(sel1(y0, y1), s(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), s(x0))
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(sel1(y0, y1), 0)) → SELACTIVE(sel1Active(mark(y0), mark(y1)), 0)
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(dbl1(from(x0))) → DBL1ACTIVE(fromActive(x0))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel1(x1, x2)) → MARK(x2)
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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 5 less nodes.

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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(indx(x1, x2)) → MARK(x1)
MARK(sel1(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(quote(x1)) → MARK(x1)
MARK(sel(0, y1)) → SELACTIVE(0, mark(y1))
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(dbl(x1)) → MARK(x1)
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel1(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel(s(x0), y1)) → SELACTIVE(s(x0), mark(y1))
MARK(sel1(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(quote(from(x0))) → QUOTEACTIVE(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(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(sel(x0, x1), y1)) → SELACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x1)
DBL1ACTIVE(s(X)) → DBL1ACTIVE(mark(X))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), 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))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, y1)) → SEL1ACTIVE(0, mark(y1))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(from(x0), y1)) → SELACTIVE(fromActive(x0), mark(y1))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(dbl1(from(x0))) → DBL1ACTIVE(fromActive(x0))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(x1, x2)) → MARK(x2)
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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

  ↳ CSDependencyPairsProof

  ↳ 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)
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel(0, y1)) → SELACTIVE(0, mark(y1))
MARK(quote(x1)) → MARK(x1)
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel1(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(dbl(x1)) → MARK(x1)
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
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(sel1(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel(s(x0), y1)) → SELACTIVE(s(x0), mark(y1))
MARK(sel1(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(quote(from(x0))) → QUOTEACTIVE(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(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(sel(x0, x1), y1)) → SELACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel(from(x0), y1)) → SELACTIVE(from(x0), mark(y1))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
DBL1ACTIVE(s(X)) → DBL1ACTIVE(mark(X))
MARK(sel(x1, x2)) → MARK(x1)
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), 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(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(dbls(x1)) → MARK(x1)
MARK(sel1(0, y1)) → SEL1ACTIVE(0, mark(y1))
MARK(sel(from(x0), y1)) → SELACTIVE(cons(x0, from(s(x0))), mark(y1))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(dbl1(from(x0))) → DBL1ACTIVE(fromActive(x0))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
MARK(sel1(x1, x2)) → MARK(x2)
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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

  ↳ CSDependencyPairsProof

  ↳ 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(sel1(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(quote(x1)) → MARK(x1)
MARK(sel(0, y1)) → SELACTIVE(0, mark(y1))
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(dbl(x1)) → MARK(x1)
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel1(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel(s(x0), y1)) → SELACTIVE(s(x0), mark(y1))
MARK(sel1(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(quote(from(x0))) → QUOTEACTIVE(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(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(sel(x0, x1), y1)) → SELACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x1)
DBL1ACTIVE(s(X)) → DBL1ACTIVE(mark(X))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), 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))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, y1)) → SEL1ACTIVE(0, mark(y1))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(dbl1(from(x0))) → DBL1ACTIVE(fromActive(x0))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(x1, x2)) → MARK(x2)
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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), indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(sel(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
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), s1(x0))) → SELACTIVE(s(y0), s1(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)
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(s(y0), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel(s(y0), 01)) → SELACTIVE(s(y0), 01)

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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:

MARK(indx(x1, x2)) → MARK(x1)
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel(0, y1)) → SELACTIVE(0, mark(y1))
MARK(quote(x1)) → MARK(x1)
MARK(sel(s(y0), 01)) → SELACTIVE(s(y0), 01)
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel1(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(s(y0), s1(x0))) → SELACTIVE(s(y0), s1(mark(x0)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
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(sel1(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(quote(from(x0))) → QUOTEACTIVE(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(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(sel(x0, x1), y1)) → SELACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(s(y0), 0)) → SELACTIVE(s(y0), 0)
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
DBL1ACTIVE(s(X)) → DBL1ACTIVE(mark(X))
MARK(sel(x1, x2)) → MARK(x1)
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(s(y0), nil)) → SELACTIVE(s(y0), nil)
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel(s(y0), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(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(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
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(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
MARK(sel1(0, y1)) → SEL1ACTIVE(0, mark(y1))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(s(y0), s(x0))) → SELACTIVE(s(y0), s(x0))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(dbl1(from(x0))) → DBL1ACTIVE(fromActive(x0))
MARK(quote(s(x0))) → QUOTEACTIVE(s(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(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
MARK(sel1(x1, x2)) → MARK(x2)
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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 5 less nodes.

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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)
MARK(sel1(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(0, y1)) → SELACTIVE(0, mark(y1))
MARK(quote(x1)) → MARK(x1)
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(dbl(x1)) → MARK(x1)
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
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(sel1(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(quote(from(x0))) → QUOTEACTIVE(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(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(sel(x0, x1), y1)) → SELACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x1)
DBL1ACTIVE(s(X)) → DBL1ACTIVE(mark(X))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel(s(y0), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(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))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
MARK(sel1(0, y1)) → SEL1ACTIVE(0, mark(y1))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(dbl1(from(x0))) → DBL1ACTIVE(fromActive(x0))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(x1, x2)) → MARK(x2)
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(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), 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), 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), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(sel(y0, y1), s1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), s1(mark(x0)))
MARK(sel(sel(y0, y1), 01)) → SELACTIVE(selActive(mark(y0), mark(y1)), 01)
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), nil)) → SELACTIVE(selActive(mark(y0), mark(y1)), nil)
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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:

MARK(indx(x1, x2)) → MARK(x1)
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
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(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(quote(x1)) → MARK(x1)
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)) → SELACTIVE(mark(X), mark(Z))
MARK(sel1(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(sel(y0, y1), 0)) → SELACTIVE(selActive(mark(y0), mark(y1)), 0)
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel(sel(y0, y1), 01)) → SELACTIVE(selActive(mark(y0), mark(y1)), 01)
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
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(sel1(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(quote(from(x0))) → QUOTEACTIVE(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(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(sel(y0, y1), nil)) → SELACTIVE(selActive(mark(y0), mark(y1)), nil)
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), s(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), s(x0))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
DBL1ACTIVE(s(X)) → DBL1ACTIVE(mark(X))
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(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(s(y0), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel(dbls(x0), y1)) → SELACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
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(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
MARK(sel1(0, y1)) → SEL1ACTIVE(0, mark(y1))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(dbl1(s(x0))) → DBL1ACTIVE(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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), s1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), s1(mark(x0)))
MARK(dbl1(from(x0))) → DBL1ACTIVE(fromActive(x0))
MARK(quote(s(x0))) → QUOTEACTIVE(s(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(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
MARK(sel1(x1, x2)) → MARK(x2)
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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 5 less nodes.

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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:

MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
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(quote(x1)) → MARK(x1)
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(dbl(x1)) → MARK(x1)
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
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(sel1(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(quote(from(x0))) → QUOTEACTIVE(fromActive(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(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x1)
DBL1ACTIVE(s(X)) → DBL1ACTIVE(mark(X))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel(s(y0), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(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))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
MARK(sel1(0, y1)) → SEL1ACTIVE(0, mark(y1))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(dbl1(s(x0))) → DBL1ACTIVE(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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(dbl1(from(x0))) → DBL1ACTIVE(fromActive(x0))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(x1, x2)) → MARK(x2)
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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, from(x0))) → SELACTIVE(0, fromActive(x0))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
MARK(sel(0, 01)) → SELACTIVE(0, 01)
MARK(sel(0, s1(x0))) → SELACTIVE(0, s1(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel(0, 0)) → SELACTIVE(0, 0)
MARK(sel(0, nil)) → SELACTIVE(0, nil)

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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:

MARK(indx(x1, x2)) → MARK(x1)
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
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(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(quote(x1)) → MARK(x1)
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel1(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
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))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel1(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(quote(from(x0))) → QUOTEACTIVE(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(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(0, 0)) → SELACTIVE(0, 0)
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(0, s(x0))) → SELACTIVE(0, s(x0))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel(0, s1(x0))) → SELACTIVE(0, s1(mark(x0)))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
DBL1ACTIVE(s(X)) → DBL1ACTIVE(mark(X))
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(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(0, 01)) → SELACTIVE(0, 01)
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
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), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel(dbls(x0), y1)) → SELACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
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(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
MARK(sel1(0, y1)) → SEL1ACTIVE(0, mark(y1))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(dbl1(s(x0))) → DBL1ACTIVE(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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(dbl1(from(x0))) → DBL1ACTIVE(fromActive(x0))
MARK(quote(s(x0))) → QUOTEACTIVE(s(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(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
MARK(sel1(x1, x2)) → MARK(x2)
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(0, nil)) → SELACTIVE(0, nil)

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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 5 less nodes.

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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:

MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
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(quote(x1)) → MARK(x1)
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
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))
MARK(sel1(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel1(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(quote(from(x0))) → QUOTEACTIVE(fromActive(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(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x1)
DBL1ACTIVE(s(X)) → DBL1ACTIVE(mark(X))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(s(y0), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(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))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
MARK(sel1(0, y1)) → SEL1ACTIVE(0, mark(y1))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(dbl1(s(x0))) → DBL1ACTIVE(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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(dbl1(from(x0))) → DBL1ACTIVE(fromActive(x0))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(x1, x2)) → MARK(x2)
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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), dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbls(y0), 0)) → SELACTIVE(dblsActive(mark(y0)), 0)
MARK(sel(dbls(y0), s1(x0))) → SELACTIVE(dblsActive(mark(y0)), s1(mark(x0)))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(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(dbls(y0), nil)) → SELACTIVE(dblsActive(mark(y0)), nil)
MARK(sel(dbls(y0), 01)) → SELACTIVE(dblsActive(mark(y0)), 01)
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(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(dbls(y0), s(x0))) → SELACTIVE(dblsActive(mark(y0)), s(x0))
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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:

MARK(indx(x1, x2)) → MARK(x1)
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(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), 01)) → SELACTIVE(dblsActive(mark(y0)), 01)
MARK(sel1(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
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(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(quote(x1)) → MARK(x1)
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)) → SELACTIVE(mark(X), mark(Z))
MARK(sel1(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
MARK(sel(dbls(y0), 0)) → SELACTIVE(dblsActive(mark(y0)), 0)
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))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel1(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(quote(from(x0))) → QUOTEACTIVE(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), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), nil)) → SELACTIVE(dblsActive(mark(y0)), nil)
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
DBL1ACTIVE(s(X)) → DBL1ACTIVE(mark(X))
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(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel(dbls(y0), s1(x0))) → SELACTIVE(dblsActive(mark(y0)), s1(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
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), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
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(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
MARK(sel1(0, y1)) → SEL1ACTIVE(0, mark(y1))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbls(y0), s(x0))) → SELACTIVE(dblsActive(mark(y0)), s(x0))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(dbl1(from(x0))) → DBL1ACTIVE(fromActive(x0))
MARK(quote(s(x0))) → QUOTEACTIVE(s(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(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
MARK(sel1(x1, x2)) → MARK(x2)
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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 5 less nodes.

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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:

MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(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(sel1(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
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(quote(x1)) → MARK(x1)
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(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(sel1(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
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))
MARK(sel1(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel1(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(quote(from(x0))) → QUOTEACTIVE(fromActive(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), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel1(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
DBL1ACTIVE(s(X)) → DBL1ACTIVE(mark(X))
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(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(s(y0), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
MARK(sel1(0, y1)) → SEL1ACTIVE(0, mark(y1))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(dbl1(from(x0))) → DBL1ACTIVE(fromActive(x0))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(x1, x2)) → MARK(x2)
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule DBL1ACTIVE(s(X)) → DBL1ACTIVE(mark(X)) at position [0] we obtained the following new rules:

DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
DBL1ACTIVE(s(cons(x0, x1))) → DBL1ACTIVE(cons(x0, x1))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
DBL1ACTIVE(s(nil)) → DBL1ACTIVE(nil)
DBL1ACTIVE(s(s1(x0))) → DBL1ACTIVE(s1(mark(x0)))
DBL1ACTIVE(s(0)) → DBL1ACTIVE(0)
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(01)) → DBL1ACTIVE(01)
DBL1ACTIVE(s(from(x0))) → DBL1ACTIVE(fromActive(x0))

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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:

MARK(indx(x1, x2)) → MARK(x1)
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
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(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(quote(x1)) → MARK(x1)
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(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)) → SELACTIVE(mark(X), mark(Z))
MARK(sel1(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
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))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel1(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(quote(from(x0))) → QUOTEACTIVE(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), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
DBL1ACTIVE(s(nil)) → DBL1ACTIVE(nil)
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
DBL1ACTIVE(s(01)) → DBL1ACTIVE(01)
MARK(sel(x1, x2)) → MARK(x2)
DBL1ACTIVE(s(cons(x0, x1))) → DBL1ACTIVE(cons(x0, x1))
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
DBL1ACTIVE(s(0)) → DBL1ACTIVE(0)
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(sel(x1, x2)) → MARK(x1)
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
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), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
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(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(dbls(x1)) → MARK(x1)
MARK(sel1(0, y1)) → SEL1ACTIVE(0, mark(y1))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(dbl1(s(x0))) → DBL1ACTIVE(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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(dbl1(from(x0))) → DBL1ACTIVE(fromActive(x0))
MARK(quote(s(x0))) → QUOTEACTIVE(s(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(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
DBL1ACTIVE(s(s1(x0))) → DBL1ACTIVE(s1(mark(x0)))
MARK(sel1(x1, x2)) → MARK(x2)
MARK(dbl1(x1)) → MARK(x1)
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
DBL1ACTIVE(s(from(x0))) → DBL1ACTIVE(fromActive(x0))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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 5 less nodes.

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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:

MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(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(sel1(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
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(quote(x1)) → MARK(x1)
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
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))
MARK(sel1(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel1(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(quote(from(x0))) → QUOTEACTIVE(fromActive(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), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
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)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(s(y0), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(0, y1)) → SEL1ACTIVE(0, mark(y1))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(dbl1(from(x0))) → DBL1ACTIVE(fromActive(x0))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(x1, x2)) → MARK(x2)
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(from(x0))) → DBL1ACTIVE(fromActive(x0))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule DBL1ACTIVE(s(from(x0))) → DBL1ACTIVE(fromActive(x0)) at position [0] we obtained the following new rules:

DBL1ACTIVE(s(from(x0))) → DBL1ACTIVE(cons(x0, from(s(x0))))
DBL1ACTIVE(s(from(x0))) → DBL1ACTIVE(from(x0))

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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:

MARK(indx(x1, x2)) → MARK(x1)
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
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(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(quote(x1)) → MARK(x1)
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)))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel1(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(from(x0))) → DBL1ACTIVE(cons(x0, from(s(x0))))
QUOTEACTIVE(dbl(X)) → MARK(X)
DBL1ACTIVE(s(from(x0))) → DBL1ACTIVE(from(x0))
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
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))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel1(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(quote(from(x0))) → QUOTEACTIVE(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), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(sel(x1, x2)) → MARK(x1)
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
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), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
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(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(dbls(x1)) → MARK(x1)
MARK(sel1(0, y1)) → SEL1ACTIVE(0, mark(y1))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(dbl1(s(x0))) → DBL1ACTIVE(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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(dbl1(from(x0))) → DBL1ACTIVE(fromActive(x0))
MARK(quote(s(x0))) → QUOTEACTIVE(s(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(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
MARK(sel1(x1, x2)) → MARK(x2)
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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

  ↳ CSDependencyPairsProof

  ↳ 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:

MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(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(sel1(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
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(quote(x1)) → MARK(x1)
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
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))
MARK(sel1(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel1(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(quote(from(x0))) → QUOTEACTIVE(fromActive(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), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
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)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(s(y0), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(0, y1)) → SEL1ACTIVE(0, mark(y1))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(dbl1(from(x0))) → DBL1ACTIVE(fromActive(x0))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(x1, x2)) → MARK(x2)
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(dbl1(from(x0))) → DBL1ACTIVE(fromActive(x0)) at position [0] we obtained the following new rules:

MARK(dbl1(from(x0))) → DBL1ACTIVE(cons(x0, from(s(x0))))
MARK(dbl1(from(x0))) → DBL1ACTIVE(from(x0))

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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:

MARK(indx(x1, x2)) → MARK(x1)
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
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(dbl1(from(x0))) → DBL1ACTIVE(from(x0))
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(quote(x1)) → MARK(x1)
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)))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel1(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
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))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel1(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(quote(from(x0))) → QUOTEACTIVE(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), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(sel(x1, x2)) → MARK(x1)
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
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), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
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(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(dbls(x1)) → MARK(x1)
MARK(sel1(0, y1)) → SEL1ACTIVE(0, mark(y1))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(dbl1(s(x0))) → DBL1ACTIVE(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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(quote(s(x0))) → QUOTEACTIVE(s(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(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
MARK(sel1(x1, x2)) → MARK(x2)
MARK(dbl1(from(x0))) → DBL1ACTIVE(cons(x0, from(s(x0))))
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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

  ↳ CSDependencyPairsProof

  ↳ 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:

MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(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(sel1(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
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(quote(x1)) → MARK(x1)
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
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))
MARK(sel1(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel1(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(quote(from(x0))) → QUOTEACTIVE(fromActive(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), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
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)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(s(y0), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(0, y1)) → SEL1ACTIVE(0, mark(y1))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(x1, x2)) → MARK(x2)
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(sel1(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1)) at position [1] we obtained the following new rules:

MARK(sel1(indx(y0, y1), s(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), s(x0))
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(indx(y0, y1), nil)) → SEL1ACTIVE(indxActive(mark(y0), y1), nil)
MARK(sel1(indx(y0, y1), s1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), s1(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel1(indx(y0, y1), 01)) → SEL1ACTIVE(indxActive(mark(y0), y1), 01)
MARK(sel1(indx(y0, y1), 0)) → SEL1ACTIVE(indxActive(mark(y0), y1), 0)
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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:

MARK(indx(x1, x2)) → MARK(x1)
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
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(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(quote(x1)) → MARK(x1)
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel1(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), nil)) → SEL1ACTIVE(indxActive(mark(y0), y1), nil)
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(indx(y0, y1), s(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), s(x0))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(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(sel1(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel1(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), s1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), s1(mark(x0)))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(quote(from(x0))) → QUOTEACTIVE(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), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(indx(y0, y1), 0)) → SEL1ACTIVE(indxActive(mark(y0), y1), 0)
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
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), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
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(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(dbls(x1)) → MARK(x1)
MARK(sel1(0, y1)) → SEL1ACTIVE(0, mark(y1))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(s(x0))) → DBL1ACTIVE(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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(indx(y0, y1), 01)) → SEL1ACTIVE(indxActive(mark(y0), y1), 01)
MARK(quote(s(x0))) → QUOTEACTIVE(s(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(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
MARK(sel1(x1, x2)) → MARK(x2)
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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 5 less nodes.

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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:

MARK(indx(x1, x2)) → MARK(x1)
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(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(sel1(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
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(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(quote(x1)) → MARK(x1)
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel1(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel1(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(quote(from(x0))) → QUOTEACTIVE(fromActive(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), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(s(y0), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(0, y1)) → SEL1ACTIVE(0, mark(y1))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(x1, x2)) → MARK(x2)
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(sel1(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1)) at position [1] we obtained the following new rules:

MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbls(y0), 01)) → SEL1ACTIVE(dblsActive(mark(y0)), 01)
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), 0)) → SEL1ACTIVE(dblsActive(mark(y0)), 0)
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), nil)) → SEL1ACTIVE(dblsActive(mark(y0)), nil)
MARK(sel1(dbls(y0), s(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), s(x0))
MARK(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(dbls(y0), s1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), s1(mark(x0)))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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:

MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
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(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(quote(x1)) → MARK(x1)
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
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))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel1(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(quote(from(x0))) → QUOTEACTIVE(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), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), s1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), s1(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x2)
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel1(dbls(y0), nil)) → SEL1ACTIVE(dblsActive(mark(y0)), nil)
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
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), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
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(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(dbls(x1)) → MARK(x1)
MARK(sel1(0, y1)) → SEL1ACTIVE(0, mark(y1))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel(sel(y0, y1), indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel1(dbls(y0), 01)) → SEL1ACTIVE(dblsActive(mark(y0)), 01)
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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel1(dbls(y0), 0)) → SEL1ACTIVE(dblsActive(mark(y0)), 0)
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(dbls(y0), s(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), s(x0))
MARK(quote(s(x0))) → QUOTEACTIVE(s(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(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
MARK(sel1(x1, x2)) → MARK(x2)
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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 5 less nodes.

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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:

MARK(indx(x1, x2)) → MARK(x1)
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(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(sel1(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
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(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(quote(x1)) → MARK(x1)
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(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)) → SELACTIVE(mark(X), mark(Z))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel1(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel1(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(quote(from(x0))) → QUOTEACTIVE(fromActive(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), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x2)
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(s(y0), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(0, y1)) → SEL1ACTIVE(0, mark(y1))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(x1, x2)) → MARK(x2)
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(sel1(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1)) at position [1] we obtained the following new rules:

MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(dbl(y0), 0)) → SEL1ACTIVE(dblActive(mark(y0)), 0)
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbl(y0), s(x0))) → SEL1ACTIVE(dblActive(mark(y0)), s(x0))
MARK(sel1(dbl(y0), 01)) → SEL1ACTIVE(dblActive(mark(y0)), 01)
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(dbl(y0), nil)) → SEL1ACTIVE(dblActive(mark(y0)), nil)
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), s1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), s1(mark(x0)))

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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:

MARK(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(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)
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(quote(x1)) → MARK(x1)
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel1(dbl(y0), nil)) → SEL1ACTIVE(dblActive(mark(y0)), nil)
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(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))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel1(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(quote(from(x0))) → QUOTEACTIVE(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(sel1(dbl(y0), s(x0))) → SEL1ACTIVE(dblActive(mark(y0)), s(x0))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(dbl(y0), s1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), s1(mark(x0)))
MARK(sel(x1, x2)) → MARK(x2)
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel1(dbl(y0), 0)) → SEL1ACTIVE(dblActive(mark(y0)), 0)
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
MARK(sel1(dbl(y0), 01)) → SEL1ACTIVE(dblActive(mark(y0)), 01)
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
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), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
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(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(dbls(x1)) → MARK(x1)
MARK(sel1(0, y1)) → SEL1ACTIVE(0, mark(y1))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(s(x0))) → DBL1ACTIVE(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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(quote(s(x0))) → QUOTEACTIVE(s(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(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
MARK(sel1(x1, x2)) → MARK(x2)
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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 5 less nodes.

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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:

MARK(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(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(sel1(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(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)
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(quote(x1)) → MARK(x1)
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(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)) → SELACTIVE(mark(X), mark(Z))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel1(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel1(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(quote(from(x0))) → QUOTEACTIVE(fromActive(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), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x2)
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(s(y0), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(0, y1)) → SEL1ACTIVE(0, mark(y1))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(x1, x2)) → MARK(x2)
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(sel1(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1)) at position [1] we obtained the following new rules:

MARK(sel1(quote(y0), s1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), s1(mark(x0)))
MARK(sel1(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(sel1(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(quote(y0), 01)) → SEL1ACTIVE(quoteActive(mark(y0)), 01)
MARK(sel1(quote(y0), s(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), s(x0))
MARK(sel1(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(quote(y0), nil)) → SEL1ACTIVE(quoteActive(mark(y0)), nil)
MARK(sel1(quote(y0), 0)) → SEL1ACTIVE(quoteActive(mark(y0)), 0)
MARK(sel1(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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:

MARK(sel1(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel1(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
MARK(sel1(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(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)
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(quote(x1)) → MARK(x1)
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(quote(y0), s1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), s1(mark(x0)))
MARK(sel1(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(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))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(quote(y0), s(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), s(x0))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel1(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(quote(from(x0))) → QUOTEACTIVE(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), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(quote(y0), 0)) → SEL1ACTIVE(quoteActive(mark(y0)), 0)
MARK(sel(x1, x2)) → MARK(x2)
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel1(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
MARK(sel1(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
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), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
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(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(dbls(x1)) → MARK(x1)
MARK(sel1(0, y1)) → SEL1ACTIVE(0, mark(y1))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel1(quote(y0), nil)) → SEL1ACTIVE(quoteActive(mark(y0)), nil)
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(s(x0))) → DBL1ACTIVE(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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(quote(s(x0))) → QUOTEACTIVE(s(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(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
MARK(sel1(quote(y0), 01)) → SEL1ACTIVE(quoteActive(mark(y0)), 01)
MARK(sel1(x1, x2)) → MARK(x2)
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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 5 less nodes.

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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:

MARK(sel1(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(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(sel1(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel1(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
MARK(sel1(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(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)
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(quote(x1)) → MARK(x1)
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(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)) → SELACTIVE(mark(X), mark(Z))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(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(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel1(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(quote(from(x0))) → QUOTEACTIVE(fromActive(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), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x2)
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel1(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
MARK(sel1(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(s(y0), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(0, y1)) → SEL1ACTIVE(0, mark(y1))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(x1, x2)) → MARK(x2)
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(sel1(0, y1)) → SEL1ACTIVE(0, mark(y1)) at position [1] we obtained the following new rules:

MARK(sel1(0, s1(x0))) → SEL1ACTIVE(0, s1(mark(x0)))
MARK(sel1(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel1(0, s(x0))) → SEL1ACTIVE(0, s(x0))
MARK(sel1(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel1(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
MARK(sel1(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(sel1(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
MARK(sel1(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel1(0, 01)) → SEL1ACTIVE(0, 01)
MARK(sel1(0, nil)) → SEL1ACTIVE(0, nil)
MARK(sel1(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
MARK(sel1(0, 0)) → SEL1ACTIVE(0, 0)
MARK(sel1(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
MARK(sel1(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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:

MARK(sel1(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(0, s(x0))) → SEL1ACTIVE(0, s(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(0, 01)) → SEL1ACTIVE(0, 01)
MARK(sel1(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel1(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(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)
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(quote(x1)) → MARK(x1)
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), 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))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel1(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(quote(from(x0))) → QUOTEACTIVE(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), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x2)
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel1(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
MARK(sel1(0, 0)) → SEL1ACTIVE(0, 0)
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
MARK(sel1(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(0, nil)) → SEL1ACTIVE(0, nil)
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
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), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(sel1(0, s1(x0))) → SEL1ACTIVE(0, s1(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(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
MARK(sel1(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(s(x0))) → DBL1ACTIVE(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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel1(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(quote(s(x0))) → QUOTEACTIVE(s(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(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
MARK(sel1(x1, x2)) → MARK(x2)
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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 5 less nodes.

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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:

MARK(sel1(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel1(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(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(sel1(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel1(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
MARK(sel1(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(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)
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(quote(x1)) → MARK(x1)
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(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)) → SELACTIVE(mark(X), mark(Z))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(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(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel1(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(quote(from(x0))) → QUOTEACTIVE(fromActive(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), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x2)
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel1(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel1(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel1(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
MARK(sel1(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(s(y0), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
MARK(sel1(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel1(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(x1, x2)) → MARK(x2)
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(sel1(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1)) at position [0] we obtained the following new rules:

MARK(sel1(from(x0), y1)) → SEL1ACTIVE(cons(x0, from(s(x0))), mark(y1))
MARK(sel1(from(x0), y1)) → SEL1ACTIVE(from(x0), mark(y1))

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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:

MARK(sel1(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), 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(sel1(from(x0), y1)) → SEL1ACTIVE(from(x0), mark(y1))
MARK(sel1(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(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)
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(quote(x1)) → MARK(x1)
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(from(x0), y1)) → SEL1ACTIVE(cons(x0, from(s(x0))), mark(y1))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), 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))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel1(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(quote(from(x0))) → QUOTEACTIVE(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), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x2)
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel1(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
MARK(sel1(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
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), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
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(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
MARK(sel1(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(s(x0))) → DBL1ACTIVE(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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel1(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(quote(s(x0))) → QUOTEACTIVE(s(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(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
MARK(sel1(x1, x2)) → MARK(x2)
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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

  ↳ CSDependencyPairsProof

  ↳ 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

                                                                                                                                                                                                      ↳ Narrowing

  ↳ Trivial-Transformation

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

MARK(sel1(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel1(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(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(sel1(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
MARK(sel1(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(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)
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(quote(x1)) → MARK(x1)
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(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)) → SELACTIVE(mark(X), mark(Z))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(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(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel1(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(quote(from(x0))) → QUOTEACTIVE(fromActive(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), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x2)
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel1(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel1(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel1(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel1(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
MARK(sel1(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(s(y0), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
MARK(sel1(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel1(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(x1, x2)) → MARK(x2)
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(sel1(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1)) at position [1] we obtained the following new rules:

MARK(sel1(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(dbl1(y0), 0)) → SEL1ACTIVE(dbl1Active(mark(y0)), 0)
MARK(sel1(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(sel1(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbl1(y0), nil)) → SEL1ACTIVE(dbl1Active(mark(y0)), nil)
MARK(sel1(dbl1(y0), s(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), s(x0))
MARK(sel1(dbl1(y0), s1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), s1(mark(x0)))
MARK(sel1(dbl1(y0), 01)) → SEL1ACTIVE(dbl1Active(mark(y0)), 01)

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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

                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ DependencyGraphProof

  ↳ Trivial-Transformation

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

MARK(sel1(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(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)
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbl1(y0), nil)) → SEL1ACTIVE(dbl1Active(mark(y0)), nil)
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(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(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(quote(from(x0))) → QUOTEACTIVE(fromActive(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), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel1(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(s(y0), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(x1, x2)) → MARK(x2)
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel1(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(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(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
MARK(sel1(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbl1(y0), s(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), s(x0))
MARK(quote(x1)) → MARK(x1)
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel1(dbl1(y0), s1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), s1(mark(x0)))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel1(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel1(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
MARK(sel1(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
MARK(sel1(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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(sel1(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), 0)) → SEL1ACTIVE(dbl1Active(mark(y0)), 0)
MARK(dbls(x1)) → MARK(x1)
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
MARK(dbl1(x1)) → MARK(x1)
MARK(sel1(dbl1(y0), 01)) → SEL1ACTIVE(dbl1Active(mark(y0)), 01)
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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 5 less nodes.

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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

                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

  ↳ Trivial-Transformation

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

MARK(sel1(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel1(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(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(sel1(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
MARK(sel1(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(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)
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(quote(x1)) → MARK(x1)
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(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)) → SELACTIVE(mark(X), mark(Z))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(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(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel1(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(quote(from(x0))) → QUOTEACTIVE(fromActive(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), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x2)
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel1(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel1(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel1(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel1(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
MARK(sel1(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(s(y0), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel1(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(x1, x2)) → MARK(x2)
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(sel1(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1)) at position [1] we obtained the following new rules:

MARK(sel1(sel1(y0, y1), dbl1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(sel1(y0, y1), quote(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(sel1(y0, y1), s(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), s(x0))
MARK(sel1(sel1(y0, y1), cons(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(sel1(y0, y1), s1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), s1(mark(x0)))
MARK(sel1(sel1(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), from(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
MARK(sel1(sel1(y0, y1), nil)) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), nil)
MARK(sel1(sel1(y0, y1), 01)) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), 01)
MARK(sel1(sel1(y0, y1), indx(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel1(sel1(y0, y1), dbls(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(sel1(y0, y1), sel(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), dbl(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel1(sel1(y0, y1), 0)) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), 0)

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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

                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

  ↳ Trivial-Transformation

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

MARK(sel1(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), quote(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(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)))
MARK(sel1(sel1(y0, y1), from(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(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(sel1(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(sel1(y0, y1), s1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), s1(mark(x0)))
MARK(sel1(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(quote(from(x0))) → QUOTEACTIVE(fromActive(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), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(sel1(sel1(y0, y1), sel(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel1(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(s(y0), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(sel1(y0, y1), 0)) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), 0)
MARK(sel1(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel1(y0, y1), indx(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(x1, x2)) → MARK(x2)
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel1(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(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(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
MARK(sel1(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(quote(x1)) → MARK(x1)
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(sel1(sel1(y0, y1), dbls(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(sel1(y0, y1), dbl(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel1(sel1(y0, y1), s(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), s(x0))
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(sel1(y0, y1), cons(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
MARK(sel1(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
MARK(sel1(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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(sel1(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), dbl1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel1(sel1(y0, y1), nil)) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), nil)
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), 01)) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), 01)
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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 5 less nodes.

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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

                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Narrowing

  ↳ Trivial-Transformation

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

MARK(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), quote(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(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)
MARK(sel1(sel1(y0, y1), from(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(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(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(quote(from(x0))) → QUOTEACTIVE(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), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(sel1(sel1(y0, y1), sel(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(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), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel1(y0, y1), indx(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(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(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel1(x1, x2)) → MARK(x2)
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel1(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), 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(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
MARK(sel1(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(quote(x1)) → MARK(x1)
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(sel1(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(sel1(sel1(y0, y1), dbls(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(sel1(y0, y1), dbl(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel1(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(sel1(y0, y1), cons(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(s1(x1)) → MARK(x1)
MARK(sel1(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
MARK(sel1(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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(sel1(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), dbl1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
MARK(sel1(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(sel1(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1)) at position [1] we obtained the following new rules:

MARK(sel1(sel(y0, y1), 0)) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), 0)
MARK(sel1(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel1(sel(y0, y1), s(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), s(x0))
MARK(sel1(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(sel(y0, y1), s1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), s1(mark(x0)))
MARK(sel1(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel1(sel(y0, y1), 01)) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), 01)
MARK(sel1(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), nil)) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), nil)
MARK(sel1(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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

                                                                                                                                                                                                      ↳ 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(sel1(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(sel1(y0, y1), quote(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), s(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), s(x0))
MARK(sel1(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(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)))
MARK(sel1(sel1(y0, y1), from(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(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(sel1(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), s1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), s1(mark(x0)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(quote(from(x0))) → QUOTEACTIVE(fromActive(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), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(sel1(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), sel(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel1(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel1(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(s(y0), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), 01)) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), 01)
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel1(sel1(y0, y1), indx(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(x1, x2)) → MARK(x2)
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), 0)) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), 0)
MARK(sel1(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(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(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
MARK(sel1(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(quote(x1)) → MARK(x1)
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), dbls(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel1(y0, y1), dbl(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(sel1(y0, y1), cons(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
MARK(sel1(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
MARK(sel1(sel(y0, y1), nil)) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), nil)
MARK(sel1(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
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(sel1(sel1(y0, y1), dbl1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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 5 less nodes.

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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

                                                                                                                                                                                                      ↳ 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(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), quote(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(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)
MARK(sel1(sel1(y0, y1), from(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(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(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(quote(from(x0))) → QUOTEACTIVE(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), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(sel1(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), sel(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel1(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(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), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel1(sel1(y0, y1), indx(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel1(x1, x2)) → MARK(x2)
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel1(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), 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(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
MARK(sel1(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(quote(x1)) → MARK(x1)
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(sel1(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(sel1(sel1(y0, y1), dbls(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(sel1(y0, y1), dbl(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel1(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(sel1(y0, y1), cons(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(s1(x1)) → MARK(x1)
MARK(sel1(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
MARK(sel1(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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(sel1(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), dbl1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
MARK(sel1(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(sel1(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1)) at position [1] we obtained the following new rules:

MARK(sel1(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel1(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
MARK(sel1(s(y0), 0)) → SEL1ACTIVE(s(y0), 0)
MARK(sel1(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
MARK(sel1(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
MARK(sel1(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(sel1(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel1(s(y0), nil)) → SEL1ACTIVE(s(y0), nil)
MARK(sel1(s(y0), 01)) → SEL1ACTIVE(s(y0), 01)
MARK(sel1(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel1(s(y0), s1(x0))) → SEL1ACTIVE(s(y0), s1(mark(x0)))
MARK(sel1(s(y0), s(x0))) → SEL1ACTIVE(s(y0), s(x0))

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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

                                                                                                                                                                                                      ↳ 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(sel1(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(sel1(y0, y1), quote(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(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)))
MARK(sel1(sel1(y0, y1), from(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(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(sel1(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(sel1(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(quote(from(x0))) → QUOTEACTIVE(fromActive(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), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(sel1(sel1(y0, y1), sel(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel1(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel1(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(s(y0), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel1(y0, y1), indx(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(x1, x2)) → MARK(x2)
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel1(s(y0), s(x0))) → SEL1ACTIVE(s(y0), s(x0))
MARK(sel1(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel1(s(y0), 0)) → SEL1ACTIVE(s(y0), 0)
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(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(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
MARK(sel1(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(s(y0), 01)) → SEL1ACTIVE(s(y0), 01)
MARK(quote(x1)) → MARK(x1)
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(s(y0), s1(x0))) → SEL1ACTIVE(s(y0), s1(mark(x0)))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel1(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(sel1(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), dbls(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel1(y0, y1), dbl(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(sel1(y0, y1), cons(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
MARK(sel1(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
MARK(sel1(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(sel1(s(y0), nil)) → SEL1ACTIVE(s(y0), nil)
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
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(sel1(sel1(y0, y1), dbl1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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 5 less nodes.

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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

                                                                                                                                                                                                      ↳ 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(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), quote(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(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)
MARK(sel1(sel1(y0, y1), from(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel1(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(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(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(quote(from(x0))) → QUOTEACTIVE(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), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(s(X)) → QUOTEACTIVE(mark(X))
MARK(sel1(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), sel(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel1(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(sel1(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(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), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel1(sel1(y0, y1), indx(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel1(x1, x2)) → MARK(x2)
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel1(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), 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(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
MARK(sel1(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(quote(x1)) → MARK(x1)
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel1(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(sel1(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), dbls(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(sel1(y0, y1), dbl(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel1(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(sel1(y0, y1), cons(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel1(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(s1(x1)) → MARK(x1)
MARK(sel1(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
MARK(sel1(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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(sel1(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), dbl1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
MARK(sel1(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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

QUOTEACTIVE(s(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
QUOTEACTIVE(s(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(s(01)) → QUOTEACTIVE(01)
QUOTEACTIVE(s(s1(x0))) → QUOTEACTIVE(s1(mark(x0)))
QUOTEACTIVE(s(cons(x0, x1))) → QUOTEACTIVE(cons(x0, x1))
QUOTEACTIVE(s(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(s(from(x0))) → QUOTEACTIVE(fromActive(x0))
QUOTEACTIVE(s(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(s(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(s(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(s(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(s(0)) → QUOTEACTIVE(0)
QUOTEACTIVE(s(nil)) → QUOTEACTIVE(nil)
QUOTEACTIVE(s(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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

                                                                                                                                                                                                      ↳ 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(sel1(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(sel1(y0, y1), quote(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
QUOTEACTIVE(s(0)) → QUOTEACTIVE(0)
MARK(sel1(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(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)))
MARK(sel1(sel1(y0, y1), from(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
QUOTEACTIVE(s(cons(x0, x1))) → QUOTEACTIVE(cons(x0, x1))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(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)))
QUOTEACTIVE(s(from(x0))) → QUOTEACTIVE(fromActive(x0))
MARK(sel1(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(sel1(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
QUOTEACTIVE(s(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(quote(from(x0))) → QUOTEACTIVE(fromActive(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), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel1(sel1(y0, y1), sel(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel1(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel1(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(s(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(s(y0), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
QUOTEACTIVE(s(01)) → QUOTEACTIVE(01)
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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel1(y0, y1), indx(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(s(nil)) → QUOTEACTIVE(nil)
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(x1, x2)) → MARK(x2)
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel1(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
QUOTEACTIVE(s(s1(x0))) → QUOTEACTIVE(s1(mark(x0)))
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(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(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
MARK(sel1(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(quote(x1)) → MARK(x1)
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel1(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
QUOTEACTIVE(s(s(x0))) → QUOTEACTIVE(s(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(s(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(sel1(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
MARK(sel1(sel1(y0, y1), dbls(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel1(y0, y1), dbl(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(sel1(y0, y1), cons(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
MARK(sel1(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
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(sel1(sel1(y0, y1), dbl1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
QUOTEACTIVE(s(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(s(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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 5 less nodes.

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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

                                                                                                                                                                                                      ↳ 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(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(sel1(y0, y1), quote(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(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)))
MARK(sel1(sel1(y0, y1), from(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel1(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(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)))
QUOTEACTIVE(s(from(x0))) → QUOTEACTIVE(fromActive(x0))
MARK(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
QUOTEACTIVE(s(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(quote(from(x0))) → QUOTEACTIVE(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), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), sel(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel1(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(sel1(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(s(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(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), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel1(y0, y1), indx(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(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(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel1(x1, x2)) → MARK(x2)
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel1(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), 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(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
MARK(sel1(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(quote(x1)) → MARK(x1)
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
QUOTEACTIVE(s(s(x0))) → QUOTEACTIVE(s(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(s(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
QUOTEACTIVE(s(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel1(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(sel1(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
MARK(sel1(sel1(y0, y1), dbls(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(sel1(y0, y1), dbl(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel1(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(sel1(y0, y1), cons(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel1(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(s1(x1)) → MARK(x1)
MARK(sel1(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
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(sel1(sel1(y0, y1), dbl1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
MARK(sel1(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
QUOTEACTIVE(s(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X))
QUOTEACTIVE(s(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule QUOTEACTIVE(dbl(X)) → DBL1ACTIVE(mark(X)) at position [0] we obtained the following new rules:

QUOTEACTIVE(dbl(cons(x0, x1))) → DBL1ACTIVE(cons(x0, x1))
QUOTEACTIVE(dbl(from(x0))) → DBL1ACTIVE(fromActive(x0))
QUOTEACTIVE(dbl(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(dbl(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(dbl(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(dbl(nil)) → DBL1ACTIVE(nil)
QUOTEACTIVE(dbl(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(dbl(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
QUOTEACTIVE(dbl(0)) → DBL1ACTIVE(0)
QUOTEACTIVE(dbl(s1(x0))) → DBL1ACTIVE(s1(mark(x0)))
QUOTEACTIVE(dbl(s(x0))) → DBL1ACTIVE(s(x0))
QUOTEACTIVE(dbl(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(01)) → DBL1ACTIVE(01)

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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

                                                                                                                                                                                                      ↳ 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:

QUOTEACTIVE(dbl(cons(x0, x1))) → DBL1ACTIVE(cons(x0, x1))
MARK(sel1(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), quote(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
QUOTEACTIVE(dbl(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(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)
MARK(sel1(sel1(y0, y1), from(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
QUOTEACTIVE(dbl(from(x0))) → DBL1ACTIVE(fromActive(x0))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(s(from(x0))) → QUOTEACTIVE(fromActive(x0))
MARK(sel1(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
QUOTEACTIVE(dbl(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(sel1(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
QUOTEACTIVE(s(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(quote(from(x0))) → QUOTEACTIVE(fromActive(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), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel1(sel1(y0, y1), sel(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel1(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel1(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(s(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(s(y0), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel1(sel1(y0, y1), indx(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(x1, x2)) → MARK(x2)
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel1(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(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(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
MARK(sel1(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(quote(x1)) → MARK(x1)
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel1(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
QUOTEACTIVE(s(s(x0))) → QUOTEACTIVE(s(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(s(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(dbl(s(x0))) → DBL1ACTIVE(s(x0))
QUOTEACTIVE(dbl(s1(x0))) → DBL1ACTIVE(s1(mark(x0)))
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(sel1(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
QUOTEACTIVE(dbl(nil)) → DBL1ACTIVE(nil)
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), dbls(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(dbl(0)) → DBL1ACTIVE(0)
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel1(y0, y1), dbl(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
QUOTEACTIVE(dbl(01)) → DBL1ACTIVE(01)
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(sel1(y0, y1), cons(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(dbl(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(dbl(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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(sel1(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), dbl1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
QUOTEACTIVE(dbl(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(s(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(dbl(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(s(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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 5 less nodes.

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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

                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                        ↳ 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(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(sel1(y0, y1), quote(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(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)))
MARK(sel1(sel1(y0, y1), from(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel1(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(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)))
QUOTEACTIVE(dbl(from(x0))) → DBL1ACTIVE(fromActive(x0))
QUOTEACTIVE(s(from(x0))) → QUOTEACTIVE(fromActive(x0))
MARK(sel1(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
QUOTEACTIVE(s(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(quote(from(x0))) → QUOTEACTIVE(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), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), sel(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel1(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(sel1(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(s(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(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), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel1(y0, y1), indx(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(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(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel1(x1, x2)) → MARK(x2)
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel1(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), 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(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y))
MARK(sel1(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(quote(x1)) → MARK(x1)
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
QUOTEACTIVE(s(s(x0))) → QUOTEACTIVE(s(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(s(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
QUOTEACTIVE(s(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(dbl(s(x0))) → DBL1ACTIVE(s(x0))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(sel1(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
MARK(sel1(sel1(y0, y1), dbls(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(sel1(y0, y1), dbl(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel1(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(sel1(y0, y1), cons(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(dbl(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(dbl(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
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(sel1(sel1(y0, y1), dbl1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
MARK(sel1(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
QUOTEACTIVE(dbl(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(s(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(dbl(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(s(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(dbl(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule QUOTEACTIVE(sel(X, Y)) → SEL1ACTIVE(mark(X), mark(Y)) at position [0] we obtained the following new rules:

QUOTEACTIVE(sel(0, y1)) → SEL1ACTIVE(0, mark(y1))
QUOTEACTIVE(sel(01, y1)) → SEL1ACTIVE(01, mark(y1))
QUOTEACTIVE(sel(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
QUOTEACTIVE(sel(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
QUOTEACTIVE(sel(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
QUOTEACTIVE(sel(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
QUOTEACTIVE(sel(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
QUOTEACTIVE(sel(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
QUOTEACTIVE(sel(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
QUOTEACTIVE(sel(cons(x0, x1), y1)) → SEL1ACTIVE(cons(x0, x1), mark(y1))
QUOTEACTIVE(sel(nil, y1)) → SEL1ACTIVE(nil, mark(y1))
QUOTEACTIVE(sel(s1(x0), y1)) → SEL1ACTIVE(s1(mark(x0)), mark(y1))
QUOTEACTIVE(sel(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
QUOTEACTIVE(sel(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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

                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                            ↳ 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(sel1(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), quote(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
QUOTEACTIVE(dbl(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(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)
MARK(sel1(sel1(y0, y1), from(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(sel(0, y1)) → SEL1ACTIVE(0, mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
QUOTEACTIVE(dbl(from(x0))) → DBL1ACTIVE(fromActive(x0))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(s(from(x0))) → QUOTEACTIVE(fromActive(x0))
MARK(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
QUOTEACTIVE(dbl(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(sel1(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
QUOTEACTIVE(s(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(quote(from(x0))) → QUOTEACTIVE(fromActive(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), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
MARK(sel1(sel1(y0, y1), sel(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel1(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
QUOTEACTIVE(sel(cons(x0, x1), y1)) → SEL1ACTIVE(cons(x0, x1), mark(y1))
QUOTEACTIVE(sel(s1(x0), y1)) → SEL1ACTIVE(s1(mark(x0)), mark(y1))
MARK(sel1(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel1(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(s(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(sel(nil, y1)) → SEL1ACTIVE(nil, mark(y1))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(s(y0), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel1(sel1(y0, y1), indx(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(x1, x2)) → MARK(x2)
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel1(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(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))
QUOTEACTIVE(sel(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(sel1(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(quote(x1)) → MARK(x1)
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel1(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
QUOTEACTIVE(s(s(x0))) → QUOTEACTIVE(s(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(s(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
QUOTEACTIVE(sel(01, y1)) → SEL1ACTIVE(01, mark(y1))
MARK(sel1(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(sel1(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(dbl(s(x0))) → DBL1ACTIVE(s(x0))
QUOTEACTIVE(sel(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(sel1(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), dbls(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel1(y0, y1), dbl(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
QUOTEACTIVE(sel(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(sel1(y0, y1), cons(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel1(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(dbl(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(dbl(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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(sel1(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), dbl1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
QUOTEACTIVE(s(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(dbl(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(dbl(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
QUOTEACTIVE(sel(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(s(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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

  ↳ CSDependencyPairsProof

  ↳ 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

                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ 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(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(sel1(y0, y1), quote(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(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)))
MARK(sel1(sel1(y0, y1), from(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel1(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
QUOTEACTIVE(sel(0, y1)) → SEL1ACTIVE(0, mark(y1))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
QUOTEACTIVE(dbl(from(x0))) → DBL1ACTIVE(fromActive(x0))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(s(from(x0))) → QUOTEACTIVE(fromActive(x0))
MARK(sel1(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
QUOTEACTIVE(s(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(quote(from(x0))) → QUOTEACTIVE(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), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
MARK(sel1(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), sel(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel1(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(sel1(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(s(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(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), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel1(y0, y1), indx(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(x1, x2)) → MARK(x2)
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel1(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(sel(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(sel1(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(quote(x1)) → MARK(x1)
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
QUOTEACTIVE(s(s(x0))) → QUOTEACTIVE(s(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(s(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
QUOTEACTIVE(s(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(dbl(s(x0))) → DBL1ACTIVE(s(x0))
QUOTEACTIVE(sel(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(sel1(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
MARK(sel1(sel1(y0, y1), dbls(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(sel1(y0, y1), dbl(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(sel1(y0, y1), cons(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(dbl(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(dbl(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
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(sel1(sel1(y0, y1), dbl1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
QUOTEACTIVE(dbl(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(s(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(dbl(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
QUOTEACTIVE(sel(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(s(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(dbl(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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

MARK(quote(from(x0))) → QUOTEACTIVE(cons(x0, from(s(x0))))
MARK(quote(from(x0))) → QUOTEACTIVE(from(x0))

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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

                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ 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(sel1(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), quote(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
QUOTEACTIVE(dbl(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(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)
MARK(sel1(sel1(y0, y1), from(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(sel(0, y1)) → SEL1ACTIVE(0, mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(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)))
QUOTEACTIVE(dbl(from(x0))) → DBL1ACTIVE(fromActive(x0))
QUOTEACTIVE(s(from(x0))) → QUOTEACTIVE(fromActive(x0))
MARK(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
QUOTEACTIVE(dbl(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(sel1(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
QUOTEACTIVE(s(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(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), 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))
QUOTEACTIVE(sel(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
MARK(sel1(sel1(y0, y1), sel(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
MARK(quote(from(x0))) → QUOTEACTIVE(from(x0))
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel1(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel1(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(s(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(s(y0), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(sel(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel1(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel1(sel1(y0, y1), indx(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel1(x1, x2)) → MARK(x2)
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel1(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(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))
QUOTEACTIVE(sel(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel1(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(quote(x1)) → MARK(x1)
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel1(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
QUOTEACTIVE(s(s(x0))) → QUOTEACTIVE(s(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(s(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel1(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(dbl(s(x0))) → DBL1ACTIVE(s(x0))
QUOTEACTIVE(sel(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(sel1(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
MARK(quote(from(x0))) → QUOTEACTIVE(cons(x0, from(s(x0))))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), dbls(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel1(sel1(y0, y1), dbl(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
QUOTEACTIVE(sel(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel1(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(sel1(y0, y1), cons(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel1(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(dbl(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(dbl(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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(sel1(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), dbl1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
MARK(sel1(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
QUOTEACTIVE(s(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(dbl(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(dbl(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel1(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(s(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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

  ↳ CSDependencyPairsProof

  ↳ 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

                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                        ↳ 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(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(sel1(y0, y1), quote(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(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)))
MARK(sel1(sel1(y0, y1), from(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel1(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
QUOTEACTIVE(sel(0, y1)) → SEL1ACTIVE(0, mark(y1))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(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)))
QUOTEACTIVE(dbl(from(x0))) → DBL1ACTIVE(fromActive(x0))
QUOTEACTIVE(s(from(x0))) → QUOTEACTIVE(fromActive(x0))
MARK(sel1(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
QUOTEACTIVE(s(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(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(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))
QUOTEACTIVE(sel(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
MARK(sel1(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), sel(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel1(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(sel1(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(s(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(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), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel1(y0, y1), indx(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(x1, x2)) → MARK(x2)
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel1(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(sel(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(quote(x1)) → MARK(x1)
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
QUOTEACTIVE(s(s(x0))) → QUOTEACTIVE(s(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(s(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
QUOTEACTIVE(s(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(dbl(s(x0))) → DBL1ACTIVE(s(x0))
QUOTEACTIVE(sel(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(sel1(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
MARK(sel1(sel1(y0, y1), dbls(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(sel1(y0, y1), dbl(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel1(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(sel1(y0, y1), cons(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(dbl(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(dbl(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
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(sel1(sel1(y0, y1), dbl1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
MARK(sel1(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
QUOTEACTIVE(dbl(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(s(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(dbl(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
QUOTEACTIVE(sel(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(s(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(dbl(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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

QUOTEACTIVE(s(from(x0))) → QUOTEACTIVE(from(x0))
QUOTEACTIVE(s(from(x0))) → QUOTEACTIVE(cons(x0, from(s(x0))))

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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

                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                            ↳ 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(sel1(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), quote(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
QUOTEACTIVE(dbl(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(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)
MARK(sel1(sel1(y0, y1), from(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(sel(0, y1)) → SEL1ACTIVE(0, mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(dbl(x1)) → MARK(x1)
QUOTEACTIVE(s(from(x0))) → QUOTEACTIVE(from(x0))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
QUOTEACTIVE(dbl(from(x0))) → DBL1ACTIVE(fromActive(x0))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
QUOTEACTIVE(dbl(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(sel1(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
QUOTEACTIVE(s(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(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), 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))
QUOTEACTIVE(sel(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
MARK(sel1(sel1(y0, y1), sel(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel1(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel1(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(s(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(s(y0), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(sel(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel1(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel1(sel1(y0, y1), indx(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel1(x1, x2)) → MARK(x2)
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel1(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
QUOTEACTIVE(s(from(x0))) → QUOTEACTIVE(cons(x0, from(s(x0))))
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(sel(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(sel1(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(quote(x1)) → MARK(x1)
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel1(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
QUOTEACTIVE(s(s(x0))) → QUOTEACTIVE(s(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(s(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel1(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(dbl(s(x0))) → DBL1ACTIVE(s(x0))
QUOTEACTIVE(sel(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(sel1(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), dbls(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel1(y0, y1), dbl(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
QUOTEACTIVE(sel(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(sel1(y0, y1), cons(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel1(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(dbl(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(dbl(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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(sel1(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), dbl1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
QUOTEACTIVE(s(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(dbl(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(dbl(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel1(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(s(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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

  ↳ CSDependencyPairsProof

  ↳ 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

                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                ↳ 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(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(sel1(y0, y1), quote(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(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)))
MARK(sel1(sel1(y0, y1), from(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel1(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
QUOTEACTIVE(sel(0, y1)) → SEL1ACTIVE(0, mark(y1))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(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)))
QUOTEACTIVE(dbl(from(x0))) → DBL1ACTIVE(fromActive(x0))
MARK(sel1(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
QUOTEACTIVE(s(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(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(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))
QUOTEACTIVE(sel(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
MARK(sel1(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), sel(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel1(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(sel1(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(s(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(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), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel1(y0, y1), indx(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(x1, x2)) → MARK(x2)
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel1(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(sel(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(quote(x1)) → MARK(x1)
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
QUOTEACTIVE(s(s(x0))) → QUOTEACTIVE(s(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(s(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
QUOTEACTIVE(s(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(dbl(s(x0))) → DBL1ACTIVE(s(x0))
QUOTEACTIVE(sel(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(sel1(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
MARK(sel1(sel1(y0, y1), dbls(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(sel1(y0, y1), dbl(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel1(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(sel1(y0, y1), cons(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(dbl(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(dbl(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
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(sel1(sel1(y0, y1), dbl1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
MARK(sel1(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
QUOTEACTIVE(s(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(dbl(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(dbl(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
QUOTEACTIVE(sel(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(s(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(dbl(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule QUOTEACTIVE(dbl(from(x0))) → DBL1ACTIVE(fromActive(x0)) at position [0] we obtained the following new rules:

QUOTEACTIVE(dbl(from(x0))) → DBL1ACTIVE(cons(x0, from(s(x0))))
QUOTEACTIVE(dbl(from(x0))) → DBL1ACTIVE(from(x0))

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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

                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                    ↳ 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(sel1(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), quote(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
QUOTEACTIVE(dbl(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(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)
MARK(sel1(sel1(y0, y1), from(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(sel(0, y1)) → SEL1ACTIVE(0, mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(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(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
QUOTEACTIVE(dbl(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(sel1(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
QUOTEACTIVE(s(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(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), 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))
QUOTEACTIVE(sel(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
MARK(sel1(sel1(y0, y1), sel(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel1(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel1(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(s(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(dbl(from(x0))) → DBL1ACTIVE(cons(x0, from(s(x0))))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(s(y0), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(sel(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel1(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel1(sel1(y0, y1), indx(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel1(x1, x2)) → MARK(x2)
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel1(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(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))
QUOTEACTIVE(sel(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(sel1(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(quote(x1)) → MARK(x1)
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel1(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
QUOTEACTIVE(dbl(from(x0))) → DBL1ACTIVE(from(x0))
QUOTEACTIVE(s(s(x0))) → QUOTEACTIVE(s(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(s(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel1(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(dbl(s(x0))) → DBL1ACTIVE(s(x0))
QUOTEACTIVE(sel(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(sel1(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), dbls(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel1(y0, y1), dbl(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
QUOTEACTIVE(sel(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(sel1(y0, y1), cons(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel1(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(dbl(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(dbl(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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(sel1(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), dbl1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
QUOTEACTIVE(dbl(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(s(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(dbl(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel1(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(s(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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

  ↳ CSDependencyPairsProof

  ↳ 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

                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                        ↳ 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(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(sel1(y0, y1), quote(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(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)))
MARK(sel1(sel1(y0, y1), from(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel1(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
QUOTEACTIVE(sel(0, y1)) → SEL1ACTIVE(0, mark(y1))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(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(sel1(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
QUOTEACTIVE(s(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
MARK(sel1(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), sel(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel1(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(sel1(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(s(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(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), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel1(y0, y1), indx(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(x1, x2)) → MARK(x2)
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel1(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(sel(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(quote(x1)) → MARK(x1)
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
QUOTEACTIVE(s(s(x0))) → QUOTEACTIVE(s(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(s(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
QUOTEACTIVE(s(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(dbl(s(x0))) → DBL1ACTIVE(s(x0))
QUOTEACTIVE(sel(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(sel1(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
MARK(sel1(sel1(y0, y1), dbls(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(sel1(y0, y1), dbl(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel1(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(sel1(y0, y1), cons(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel1(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(dbl(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(dbl(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
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(sel1(sel1(y0, y1), dbl1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
MARK(sel1(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
QUOTEACTIVE(s(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(dbl(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(dbl(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
QUOTEACTIVE(sel(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(s(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(dbl(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule QUOTEACTIVE(sel(0, y1)) → SEL1ACTIVE(0, mark(y1)) at position [1] we obtained the following new rules:

QUOTEACTIVE(sel(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
QUOTEACTIVE(sel(0, s(x0))) → SEL1ACTIVE(0, s(x0))
QUOTEACTIVE(sel(0, s1(x0))) → SEL1ACTIVE(0, s1(mark(x0)))
QUOTEACTIVE(sel(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
QUOTEACTIVE(sel(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
QUOTEACTIVE(sel(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
QUOTEACTIVE(sel(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(sel(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
QUOTEACTIVE(sel(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
QUOTEACTIVE(sel(0, 0)) → SEL1ACTIVE(0, 0)
QUOTEACTIVE(sel(0, nil)) → SEL1ACTIVE(0, nil)
QUOTEACTIVE(sel(0, 01)) → SEL1ACTIVE(0, 01)

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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

                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                            ↳ 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(sel1(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), quote(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(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)))
QUOTEACTIVE(sel(0, nil)) → SEL1ACTIVE(0, nil)
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(sel1(y0, y1), from(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
QUOTEACTIVE(sel(0, 0)) → SEL1ACTIVE(0, 0)
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(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(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
QUOTEACTIVE(dbl(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(sel1(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
QUOTEACTIVE(s(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
MARK(sel1(sel1(y0, y1), sel(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
QUOTEACTIVE(sel(0, 01)) → SEL1ACTIVE(0, 01)
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(sel(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel1(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel1(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(s(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(s(y0), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(sel(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
MARK(sel1(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
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)))
QUOTEACTIVE(sel(0, s1(x0))) → SEL1ACTIVE(0, s1(mark(x0)))
MARK(sel(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel1(sel1(y0, y1), indx(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel1(x1, x2)) → MARK(x2)
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel1(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(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))
QUOTEACTIVE(sel(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(sel1(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(quote(x1)) → MARK(x1)
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
QUOTEACTIVE(sel(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel1(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
QUOTEACTIVE(sel(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
QUOTEACTIVE(s(s(x0))) → QUOTEACTIVE(s(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(s(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(0, s(x0))) → SEL1ACTIVE(0, s(x0))
MARK(sel1(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel1(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(dbl(s(x0))) → DBL1ACTIVE(s(x0))
QUOTEACTIVE(sel(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(sel1(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), dbls(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel1(y0, y1), dbl(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
QUOTEACTIVE(sel(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(sel1(y0, y1), cons(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
QUOTEACTIVE(sel(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(dbl(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(dbl(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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(sel1(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), dbl1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
QUOTEACTIVE(sel(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(s(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(dbl(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel1(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(s(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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 5 less nodes.

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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

                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                ↳ 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(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(sel1(y0, y1), quote(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(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)))
MARK(sel1(sel1(y0, y1), from(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel1(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(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(sel1(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
QUOTEACTIVE(s(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
MARK(sel1(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), sel(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(sel(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel1(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(sel1(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(s(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(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), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel1(y0, y1), indx(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(x1, x2)) → MARK(x2)
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel1(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(sel(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(quote(x1)) → MARK(x1)
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
MARK(sel1(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(sel(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
QUOTEACTIVE(s(s(x0))) → QUOTEACTIVE(s(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(s(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
QUOTEACTIVE(s(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(dbl(s(x0))) → DBL1ACTIVE(s(x0))
QUOTEACTIVE(sel(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(sel1(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
MARK(sel1(sel1(y0, y1), dbls(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(sel1(y0, y1), dbl(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel1(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(sel1(y0, y1), cons(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
QUOTEACTIVE(sel(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel1(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(dbl(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(dbl(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
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(sel1(sel1(y0, y1), dbl1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
MARK(sel1(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
QUOTEACTIVE(sel(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(s(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(dbl(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(dbl(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
QUOTEACTIVE(sel(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(s(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(dbl(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule QUOTEACTIVE(sel(s(x0), y1)) → SEL1ACTIVE(s(x0), mark(y1)) at position [1] we obtained the following new rules:

QUOTEACTIVE(sel(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), 0)) → SEL1ACTIVE(s(y0), 0)
QUOTEACTIVE(sel(s(y0), nil)) → SEL1ACTIVE(s(y0), nil)
QUOTEACTIVE(sel(s(y0), 01)) → SEL1ACTIVE(s(y0), 01)
QUOTEACTIVE(sel(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
QUOTEACTIVE(sel(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(s(y0), s(x0))) → SEL1ACTIVE(s(y0), s(x0))
QUOTEACTIVE(sel(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(sel(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), s1(x0))) → SEL1ACTIVE(s(y0), s1(mark(x0)))
QUOTEACTIVE(sel(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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

                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                    ↳ 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(sel1(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), quote(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
QUOTEACTIVE(dbl(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(sel(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(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)
MARK(sel1(sel1(y0, y1), from(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(s(y0), s1(x0))) → SEL1ACTIVE(s(y0), s1(mark(x0)))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
QUOTEACTIVE(sel(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
QUOTEACTIVE(dbl(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(sel1(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
QUOTEACTIVE(s(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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))
QUOTEACTIVE(sel(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
MARK(sel1(sel1(y0, y1), sel(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel1(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel1(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(s(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(s(y0), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(s(y0), 01)) → SEL1ACTIVE(s(y0), 01)
QUOTEACTIVE(sel(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(mark(x0)))
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)))
QUOTEACTIVE(sel(s(y0), nil)) → SEL1ACTIVE(s(y0), nil)
MARK(sel(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel1(sel1(y0, y1), indx(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel1(x1, x2)) → MARK(x2)
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel1(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
QUOTEACTIVE(sel(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(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))
QUOTEACTIVE(sel(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(sel1(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(quote(x1)) → MARK(x1)
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel1(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
QUOTEACTIVE(sel(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
QUOTEACTIVE(sel(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
QUOTEACTIVE(s(s(x0))) → QUOTEACTIVE(s(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(s(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(s(y0), 0)) → SEL1ACTIVE(s(y0), 0)
MARK(sel1(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel1(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(dbl(s(x0))) → DBL1ACTIVE(s(x0))
QUOTEACTIVE(sel(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(sel1(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), dbls(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel1(y0, y1), dbl(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
QUOTEACTIVE(sel(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
QUOTEACTIVE(sel(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(sel1(y0, y1), cons(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
QUOTEACTIVE(sel(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(dbl(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(dbl(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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(sel1(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), dbl1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
QUOTEACTIVE(sel(s(y0), s(x0))) → SEL1ACTIVE(s(y0), s(x0))
MARK(dbls(x1)) → MARK(x1)
QUOTEACTIVE(sel(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(s(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(dbl(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel1(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(s(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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 5 less nodes.

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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

                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                        ↳ 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(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(sel1(y0, y1), quote(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(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)))
MARK(sel1(sel1(y0, y1), from(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel1(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
QUOTEACTIVE(sel(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
QUOTEACTIVE(s(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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))
QUOTEACTIVE(sel(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
MARK(sel1(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), sel(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
QUOTEACTIVE(sel(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel1(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(sel1(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(s(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(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), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel1(y0, y1), indx(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(x1, x2)) → MARK(x2)
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel1(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(sel(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(quote(x1)) → MARK(x1)
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
MARK(sel1(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(sel(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
QUOTEACTIVE(s(s(x0))) → QUOTEACTIVE(s(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(s(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
QUOTEACTIVE(s(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(dbl(s(x0))) → DBL1ACTIVE(s(x0))
QUOTEACTIVE(sel(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(sel1(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
MARK(sel1(sel1(y0, y1), dbls(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(sel1(y0, y1), dbl(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel1(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(sel1(y0, y1), cons(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
QUOTEACTIVE(sel(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel1(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(dbl(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(dbl(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
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(sel1(sel1(y0, y1), dbl1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
MARK(sel1(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
QUOTEACTIVE(sel(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(s(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(dbl(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(dbl(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
QUOTEACTIVE(sel(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1))
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(s(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(dbl(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule QUOTEACTIVE(sel(quote(x0), y1)) → SEL1ACTIVE(quoteActive(mark(x0)), mark(y1)) at position [1] we obtained the following new rules:

QUOTEACTIVE(sel(quote(y0), 01)) → SEL1ACTIVE(quoteActive(mark(y0)), 01)
QUOTEACTIVE(sel(quote(y0), s(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), s(x0))
QUOTEACTIVE(sel(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(quote(y0), s1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), s1(mark(x0)))
QUOTEACTIVE(sel(quote(y0), 0)) → SEL1ACTIVE(quoteActive(mark(y0)), 0)
QUOTEACTIVE(sel(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), nil)) → SEL1ACTIVE(quoteActive(mark(y0)), nil)
QUOTEACTIVE(sel(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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

                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                            ↳ 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(sel1(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), quote(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
QUOTEACTIVE(dbl(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(sel(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(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)
MARK(sel1(sel1(y0, y1), from(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(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)))
QUOTEACTIVE(sel(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
QUOTEACTIVE(dbl(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(sel1(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
QUOTEACTIVE(s(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(sel(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
MARK(sel1(sel1(y0, y1), sel(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel1(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
QUOTEACTIVE(sel(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel1(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(s(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(s(y0), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), s(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), s(x0))
MARK(sel(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(mark(x0)))
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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel1(sel1(y0, y1), indx(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel1(x1, x2)) → MARK(x2)
QUOTEACTIVE(sel(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel1(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
QUOTEACTIVE(sel(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(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))
QUOTEACTIVE(sel(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(sel1(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(quote(x1)) → MARK(x1)
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel1(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
QUOTEACTIVE(sel(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
QUOTEACTIVE(sel(quote(y0), 0)) → SEL1ACTIVE(quoteActive(mark(y0)), 0)
QUOTEACTIVE(sel(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
QUOTEACTIVE(s(s(x0))) → QUOTEACTIVE(s(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(s(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel1(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(dbl(s(x0))) → DBL1ACTIVE(s(x0))
QUOTEACTIVE(sel(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
QUOTEACTIVE(sel(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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)))
QUOTEACTIVE(sel(quote(y0), nil)) → SEL1ACTIVE(quoteActive(mark(y0)), nil)
MARK(sel1(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), dbls(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel1(y0, y1), dbl(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
QUOTEACTIVE(sel(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(y0), s1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), s1(mark(x0)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
QUOTEACTIVE(sel(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
MARK(sel1(sel1(y0, y1), cons(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
QUOTEACTIVE(sel(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
QUOTEACTIVE(sel(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(dbl(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(dbl(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(y0), 01)) → SEL1ACTIVE(quoteActive(mark(y0)), 01)
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(sel1(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), dbl1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
QUOTEACTIVE(sel(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(s(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(dbl(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(s(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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 5 less nodes.

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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

                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                                ↳ 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(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(sel1(y0, y1), quote(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(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)))
MARK(sel1(sel1(y0, y1), from(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel1(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
QUOTEACTIVE(sel(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
QUOTEACTIVE(s(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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))
QUOTEACTIVE(sel(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
MARK(sel1(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), sel(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
QUOTEACTIVE(sel(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel1(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
QUOTEACTIVE(sel(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(sel1(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(s(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(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), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel1(y0, y1), indx(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(x1, x2)) → MARK(x2)
QUOTEACTIVE(sel(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel1(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(sel(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(quote(x1)) → MARK(x1)
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
MARK(sel1(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(sel(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
QUOTEACTIVE(s(s(x0))) → QUOTEACTIVE(s(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(s(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
QUOTEACTIVE(s(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(dbl(s(x0))) → DBL1ACTIVE(s(x0))
QUOTEACTIVE(sel(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(sel1(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
MARK(sel1(sel1(y0, y1), dbls(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(sel1(y0, y1), dbl(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel1(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(sel1(y0, y1), cons(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
QUOTEACTIVE(sel(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel1(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(dbl(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(dbl(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
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(sel1(sel1(y0, y1), dbl1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
MARK(sel1(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
QUOTEACTIVE(sel(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(s(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(dbl(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(dbl(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(s(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(dbl(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule QUOTEACTIVE(sel(sel(x0, x1), y1)) → SEL1ACTIVE(selActive(mark(x0), mark(x1)), mark(y1)) at position [1] we obtained the following new rules:

QUOTEACTIVE(sel(sel(y0, y1), nil)) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), nil)
QUOTEACTIVE(sel(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), s1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), s1(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), 0)) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), 0)
QUOTEACTIVE(sel(sel(y0, y1), 01)) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), 01)
QUOTEACTIVE(sel(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
QUOTEACTIVE(sel(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(sel(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(sel(y0, y1), s(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), s(x0))
QUOTEACTIVE(sel(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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

                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                    ↳ 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(sel1(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), quote(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(sel(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), s1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), s1(mark(x0)))
QUOTEACTIVE(dbl(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(sel(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel(y0, y1), 01)) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), 01)
MARK(sel1(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(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)
MARK(sel1(sel1(y0, y1), from(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
QUOTEACTIVE(sel(sel(y0, y1), s(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), s(x0))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), nil)) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), nil)
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)))
QUOTEACTIVE(sel(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(sel1(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
QUOTEACTIVE(s(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(sel(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
MARK(sel1(sel1(y0, y1), sel(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel1(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(s(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(s(y0), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(mark(x0)))
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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel1(sel1(y0, y1), indx(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel1(x1, x2)) → MARK(x2)
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
QUOTEACTIVE(sel(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(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(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(sel1(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(quote(x1)) → MARK(x1)
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel1(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
QUOTEACTIVE(sel(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
QUOTEACTIVE(sel(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
QUOTEACTIVE(s(s(x0))) → QUOTEACTIVE(s(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(s(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel1(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(dbl(s(x0))) → DBL1ACTIVE(s(x0))
QUOTEACTIVE(sel(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
QUOTEACTIVE(sel(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(sel1(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), dbls(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel1(y0, y1), dbl(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
QUOTEACTIVE(sel(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
MARK(sel1(sel1(y0, y1), cons(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
QUOTEACTIVE(sel(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
QUOTEACTIVE(sel(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(dbl(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(dbl(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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(sel1(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), dbl1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
QUOTEACTIVE(sel(sel(y0, y1), 0)) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), 0)
QUOTEACTIVE(sel(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(s(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(dbl(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
QUOTEACTIVE(sel(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(s(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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 5 less nodes.

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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

                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                                        ↳ 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(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(sel1(y0, y1), quote(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
QUOTEACTIVE(sel(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(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)))
MARK(sel1(sel1(y0, y1), from(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
QUOTEACTIVE(sel(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
QUOTEACTIVE(sel(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
QUOTEACTIVE(s(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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))
QUOTEACTIVE(sel(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1))
MARK(sel1(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), sel(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
QUOTEACTIVE(sel(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel1(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
QUOTEACTIVE(sel(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(sel1(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(s(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(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), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel1(y0, y1), indx(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(x1, x2)) → MARK(x2)
QUOTEACTIVE(sel(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel1(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), 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(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(quote(x1)) → MARK(x1)
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
MARK(sel1(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(sel(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
QUOTEACTIVE(s(s(x0))) → QUOTEACTIVE(s(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(s(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
QUOTEACTIVE(s(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(dbl(s(x0))) → DBL1ACTIVE(s(x0))
QUOTEACTIVE(sel(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(sel1(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
MARK(sel1(sel1(y0, y1), dbls(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(sel1(y0, y1), dbl(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel1(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(sel1(y0, y1), cons(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
QUOTEACTIVE(sel(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel1(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(dbl(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(dbl(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
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(sel1(sel1(y0, y1), dbl1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
MARK(sel1(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
QUOTEACTIVE(sel(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(s(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(dbl(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(dbl(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
QUOTEACTIVE(sel(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(s(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(dbl(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule QUOTEACTIVE(sel(dbl(x0), y1)) → SEL1ACTIVE(dblActive(mark(x0)), mark(y1)) at position [1] we obtained the following new rules:

QUOTEACTIVE(sel(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl(y0), nil)) → SEL1ACTIVE(dblActive(mark(y0)), nil)
QUOTEACTIVE(sel(dbl(y0), s1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), s1(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(dbl(y0), 01)) → SEL1ACTIVE(dblActive(mark(y0)), 01)
QUOTEACTIVE(sel(dbl(y0), 0)) → SEL1ACTIVE(dblActive(mark(y0)), 0)
QUOTEACTIVE(sel(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(dbl(y0), s(x0))) → SEL1ACTIVE(dblActive(mark(y0)), s(x0))
QUOTEACTIVE(sel(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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

                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                            ↳ 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:

QUOTEACTIVE(sel(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), quote(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(sel(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
QUOTEACTIVE(dbl(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(sel(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(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)
MARK(sel1(sel1(y0, y1), from(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl(y0), s1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), s1(mark(x0)))
QUOTEACTIVE(sel(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), s(x0))) → SEL1ACTIVE(dblActive(mark(y0)), s(x0))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(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)))
QUOTEACTIVE(sel(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(sel1(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
QUOTEACTIVE(s(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
MARK(sel1(sel1(y0, y1), sel(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel1(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(s(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(s(y0), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(mark(x0)))
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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), nil)) → SEL1ACTIVE(dblActive(mark(y0)), nil)
MARK(sel1(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel1(sel1(y0, y1), indx(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel1(x1, x2)) → MARK(x2)
QUOTEACTIVE(sel(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
QUOTEACTIVE(sel(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(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(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(sel1(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(quote(x1)) → MARK(x1)
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel1(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
QUOTEACTIVE(sel(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
QUOTEACTIVE(sel(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
QUOTEACTIVE(s(s(x0))) → QUOTEACTIVE(s(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(s(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel1(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(dbl(s(x0))) → DBL1ACTIVE(s(x0))
QUOTEACTIVE(sel(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
QUOTEACTIVE(sel(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(sel1(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), dbls(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel1(y0, y1), dbl(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
QUOTEACTIVE(sel(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
MARK(sel1(sel1(y0, y1), cons(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
QUOTEACTIVE(sel(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(dbl(y0), 0)) → SEL1ACTIVE(dblActive(mark(y0)), 0)
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
QUOTEACTIVE(sel(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(dbl(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(dbl(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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(sel1(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), dbl1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
QUOTEACTIVE(sel(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(s(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(dbl(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
QUOTEACTIVE(sel(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(s(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), 01)) → SEL1ACTIVE(dblActive(mark(y0)), 01)
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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 5 less nodes.

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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

                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                                                ↳ 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:

QUOTEACTIVE(sel(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(sel1(y0, y1), quote(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
QUOTEACTIVE(sel(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(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)))
MARK(sel1(sel1(y0, y1), from(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
QUOTEACTIVE(sel(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
QUOTEACTIVE(sel(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
QUOTEACTIVE(s(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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))
QUOTEACTIVE(sel(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), sel(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
QUOTEACTIVE(sel(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel1(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
QUOTEACTIVE(sel(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(sel1(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(s(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(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), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel1(y0, y1), indx(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(x1, x2)) → MARK(x2)
QUOTEACTIVE(sel(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel1(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), 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(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(quote(x1)) → MARK(x1)
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
MARK(sel1(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(sel(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
QUOTEACTIVE(s(s(x0))) → QUOTEACTIVE(s(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(s(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
QUOTEACTIVE(s(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(dbl(s(x0))) → DBL1ACTIVE(s(x0))
QUOTEACTIVE(sel(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(sel1(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
MARK(sel1(sel1(y0, y1), dbls(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(sel1(y0, y1), dbl(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel1(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(sel1(y0, y1), cons(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
QUOTEACTIVE(sel(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel1(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(dbl(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(dbl(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
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(sel1(sel1(y0, y1), dbl1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
MARK(sel1(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
QUOTEACTIVE(sel(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(s(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(dbl(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(dbl(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
QUOTEACTIVE(sel(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(s(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(dbl(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule QUOTEACTIVE(sel(dbl1(x0), y1)) → SEL1ACTIVE(dbl1Active(mark(x0)), mark(y1)) at position [1] we obtained the following new rules:

QUOTEACTIVE(sel(dbl1(y0), 01)) → SEL1ACTIVE(dbl1Active(mark(y0)), 01)
QUOTEACTIVE(sel(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), 0)) → SEL1ACTIVE(dbl1Active(mark(y0)), 0)
QUOTEACTIVE(sel(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), nil)) → SEL1ACTIVE(dbl1Active(mark(y0)), nil)
QUOTEACTIVE(sel(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(dbl1(y0), s1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), s1(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), s(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), s(x0))
QUOTEACTIVE(sel(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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

                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                    ↳ 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(sel1(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), quote(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(sel(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(sel(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(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)
MARK(sel1(sel1(y0, y1), from(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(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)))
QUOTEACTIVE(sel(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(sel1(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
QUOTEACTIVE(s(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(sel(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(sel1(sel1(y0, y1), sel(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), s1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), s1(mark(x0)))
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel1(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
QUOTEACTIVE(sel(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(sel(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(s(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(s(y0), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(mark(x0)))
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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel1(sel1(y0, y1), indx(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel1(x1, x2)) → MARK(x2)
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl1(y0), 0)) → SEL1ACTIVE(dbl1Active(mark(y0)), 0)
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
QUOTEACTIVE(sel(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(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(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(sel1(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(quote(x1)) → MARK(x1)
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel1(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
QUOTEACTIVE(sel(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
QUOTEACTIVE(sel(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
QUOTEACTIVE(s(s(x0))) → QUOTEACTIVE(s(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(s(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel1(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(dbl(s(x0))) → DBL1ACTIVE(s(x0))
QUOTEACTIVE(sel(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
QUOTEACTIVE(sel(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(dbl1(y0), 01)) → SEL1ACTIVE(dbl1Active(mark(y0)), 01)
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(sel1(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), dbls(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel1(y0, y1), dbl(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel(x1, x2)) → MARK(x2)
QUOTEACTIVE(sel(dbl1(y0), s(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), s(x0))
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
QUOTEACTIVE(sel(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
MARK(sel1(sel1(y0, y1), cons(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
QUOTEACTIVE(sel(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
QUOTEACTIVE(sel(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
QUOTEACTIVE(sel(dbl1(y0), nil)) → SEL1ACTIVE(dbl1Active(mark(y0)), nil)
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(dbl(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(dbl(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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(sel1(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), dbl1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
QUOTEACTIVE(sel(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(s(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(dbl(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
QUOTEACTIVE(sel(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(s(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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 5 less nodes.

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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

                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                                                        ↳ 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:

QUOTEACTIVE(sel(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(sel1(y0, y1), quote(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
QUOTEACTIVE(sel(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(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)))
MARK(sel1(sel1(y0, y1), from(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
QUOTEACTIVE(sel(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
QUOTEACTIVE(sel(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
QUOTEACTIVE(s(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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))
QUOTEACTIVE(sel(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(sel1(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), sel(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
QUOTEACTIVE(sel(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel1(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
QUOTEACTIVE(sel(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(sel1(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(sel(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(s(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(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), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel1(y0, y1), indx(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(x1, x2)) → MARK(x2)
QUOTEACTIVE(sel(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel1(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), 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(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(quote(x1)) → MARK(x1)
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
MARK(sel1(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(sel(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
QUOTEACTIVE(s(s(x0))) → QUOTEACTIVE(s(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(s(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
QUOTEACTIVE(s(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(dbl(s(x0))) → DBL1ACTIVE(s(x0))
QUOTEACTIVE(sel(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(sel1(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
MARK(sel1(sel1(y0, y1), dbls(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(sel1(y0, y1), dbl(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel1(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(sel1(y0, y1), cons(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
QUOTEACTIVE(sel(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel1(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(dbl(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(dbl(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
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(sel1(sel1(y0, y1), dbl1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
MARK(sel1(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
QUOTEACTIVE(sel(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(s(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(dbl(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(dbl(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
QUOTEACTIVE(sel(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(s(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(dbl(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule QUOTEACTIVE(sel(indx(x0, x1), y1)) → SEL1ACTIVE(indxActive(mark(x0), x1), mark(y1)) at position [1] we obtained the following new rules:

QUOTEACTIVE(sel(indx(y0, y1), s1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), s1(mark(x0)))
QUOTEACTIVE(sel(indx(y0, y1), 0)) → SEL1ACTIVE(indxActive(mark(y0), y1), 0)
QUOTEACTIVE(sel(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
QUOTEACTIVE(sel(indx(y0, y1), 01)) → SEL1ACTIVE(indxActive(mark(y0), y1), 01)
QUOTEACTIVE(sel(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
QUOTEACTIVE(sel(indx(y0, y1), s(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), s(x0))
QUOTEACTIVE(sel(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(indx(y0, y1), nil)) → SEL1ACTIVE(indxActive(mark(y0), y1), nil)
QUOTEACTIVE(sel(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
QUOTEACTIVE(sel(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
QUOTEACTIVE(sel(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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

                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                            ↳ 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(sel1(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), quote(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(sel(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(indx(y0, y1), s(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), s(x0))
QUOTEACTIVE(dbl(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(sel(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(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)
MARK(sel1(sel1(y0, y1), from(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), 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)))
QUOTEACTIVE(sel(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(sel1(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
QUOTEACTIVE(s(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(sel(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(sel1(sel1(y0, y1), sel(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel1(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(s(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(s(y0), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(sel(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(mark(x0)))
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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
QUOTEACTIVE(sel(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel1(sel1(y0, y1), indx(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel1(x1, x2)) → MARK(x2)
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
QUOTEACTIVE(sel(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(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(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(sel1(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(quote(x1)) → MARK(x1)
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel1(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
QUOTEACTIVE(sel(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
QUOTEACTIVE(sel(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
QUOTEACTIVE(sel(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(s(s(x0))) → QUOTEACTIVE(s(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(s(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel1(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(dbl(s(x0))) → DBL1ACTIVE(s(x0))
QUOTEACTIVE(sel(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
QUOTEACTIVE(sel(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(s(X)) → MARK(X)
QUOTEACTIVE(sel(indx(y0, y1), s1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), s1(mark(x0)))
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(sel1(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), dbls(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel1(y0, y1), dbl(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
QUOTEACTIVE(sel(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
QUOTEACTIVE(sel(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
QUOTEACTIVE(sel(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
QUOTEACTIVE(sel(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
MARK(sel1(sel1(y0, y1), cons(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
QUOTEACTIVE(sel(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(indx(y0, y1), 0)) → SEL1ACTIVE(indxActive(mark(y0), y1), 0)
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
QUOTEACTIVE(sel(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(dbl(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(dbl(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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(sel1(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), dbl1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(indx(y0, y1), 01)) → SEL1ACTIVE(indxActive(mark(y0), y1), 01)
MARK(sel1(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
QUOTEACTIVE(sel(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(s(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(dbl(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
QUOTEACTIVE(sel(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(indx(y0, y1), nil)) → SEL1ACTIVE(indxActive(mark(y0), y1), nil)
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(s(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
QUOTEACTIVE(dbl(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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 5 less nodes.

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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

                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                                                                ↳ 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:

QUOTEACTIVE(sel(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(sel1(y0, y1), quote(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
QUOTEACTIVE(sel(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(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)))
MARK(sel1(sel1(y0, y1), from(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
QUOTEACTIVE(sel(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
QUOTEACTIVE(sel(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
QUOTEACTIVE(s(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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))
QUOTEACTIVE(sel(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(sel1(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), sel(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
QUOTEACTIVE(sel(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel1(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
QUOTEACTIVE(sel(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(sel1(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(sel(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(s(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(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), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(sel(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
QUOTEACTIVE(sel(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel1(y0, y1), indx(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(x1, x2)) → MARK(x2)
QUOTEACTIVE(sel(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel1(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), 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(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(quote(x1)) → MARK(x1)
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
MARK(sel1(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(sel(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
QUOTEACTIVE(sel(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(s(s(x0))) → QUOTEACTIVE(s(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(s(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
QUOTEACTIVE(s(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(dbl(s(x0))) → DBL1ACTIVE(s(x0))
QUOTEACTIVE(sel(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(sel1(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
MARK(sel1(sel1(y0, y1), dbls(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(sel1(y0, y1), dbl(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
QUOTEACTIVE(sel(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel1(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
QUOTEACTIVE(sel(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(sel1(y0, y1), cons(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
QUOTEACTIVE(sel(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel1(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(dbl(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(dbl(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
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(sel1(sel1(y0, y1), dbl1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
MARK(sel1(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
QUOTEACTIVE(sel(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(s(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(dbl(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(dbl(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
QUOTEACTIVE(sel(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(s(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
QUOTEACTIVE(dbl(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule QUOTEACTIVE(sel(dbls(x0), y1)) → SEL1ACTIVE(dblsActive(mark(x0)), mark(y1)) at position [1] we obtained the following new rules:

QUOTEACTIVE(sel(dbls(y0), 0)) → SEL1ACTIVE(dblsActive(mark(y0)), 0)
QUOTEACTIVE(sel(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(sel(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(dbls(y0), s(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), s(x0))
QUOTEACTIVE(sel(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(dbls(y0), nil)) → SEL1ACTIVE(dblsActive(mark(y0)), nil)
QUOTEACTIVE(sel(dbls(y0), s1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), s1(mark(x0)))
QUOTEACTIVE(sel(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbls(y0), 01)) → SEL1ACTIVE(dblsActive(mark(y0)), 01)

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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

                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                    ↳ 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:

QUOTEACTIVE(sel(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(sel(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), quote(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(sel(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(sel(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(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)
MARK(sel1(sel1(y0, y1), from(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
QUOTEACTIVE(sel(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(sel1(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
QUOTEACTIVE(s(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(sel(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(sel1(sel1(y0, y1), sel(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
QUOTEACTIVE(sel(dbls(y0), 0)) → SEL1ACTIVE(dblsActive(mark(y0)), 0)
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel1(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(s(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
QUOTEACTIVE(sel(dbls(y0), s1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), s1(mark(x0)))
MARK(sel(s(y0), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(sel(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(mark(x0)))
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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
QUOTEACTIVE(sel(dbls(y0), s(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), s(x0))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel1(sel1(y0, y1), indx(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel1(x1, x2)) → MARK(x2)
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
QUOTEACTIVE(sel(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(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(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(sel1(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(quote(x1)) → MARK(x1)
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel1(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
QUOTEACTIVE(sel(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
QUOTEACTIVE(sel(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
QUOTEACTIVE(s(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(sel(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(s(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel1(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
QUOTEACTIVE(sel(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(dbl(s(x0))) → DBL1ACTIVE(s(x0))
QUOTEACTIVE(sel(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
QUOTEACTIVE(sel(dbls(y0), 01)) → SEL1ACTIVE(dblsActive(mark(y0)), 01)
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
QUOTEACTIVE(sel(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(sel1(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), dbls(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbls(y0), nil)) → SEL1ACTIVE(dblsActive(mark(y0)), nil)
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel1(y0, y1), dbl(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
QUOTEACTIVE(sel(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
QUOTEACTIVE(sel(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
QUOTEACTIVE(sel(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
MARK(sel1(sel1(y0, y1), cons(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
QUOTEACTIVE(sel(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
QUOTEACTIVE(sel(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(dbl(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(dbl(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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(sel1(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), dbl1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
QUOTEACTIVE(sel(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(s(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(dbl(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
QUOTEACTIVE(sel(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(s(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
QUOTEACTIVE(dbl(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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 5 less nodes.

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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

                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                                                                        ↳ 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:

QUOTEACTIVE(sel(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(sel(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(sel1(y0, y1), quote(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
QUOTEACTIVE(sel(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(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)))
MARK(sel1(sel1(y0, y1), from(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
QUOTEACTIVE(sel(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
QUOTEACTIVE(sel(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
QUOTEACTIVE(s(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(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(dbls(y0), indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(sel1(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), sel(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
QUOTEACTIVE(sel(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel1(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
QUOTEACTIVE(sel(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(sel1(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(sel(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(s(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(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), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(sel(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
QUOTEACTIVE(sel(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel1(y0, y1), indx(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(x1, x2)) → MARK(x2)
QUOTEACTIVE(sel(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel1(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), 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(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(sel1(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(quote(x1)) → MARK(x1)
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
QUOTEACTIVE(sel(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(sel(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
QUOTEACTIVE(sel(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(s(s(x0))) → QUOTEACTIVE(s(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(s(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
QUOTEACTIVE(s(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
QUOTEACTIVE(sel(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(dbl(s(x0))) → DBL1ACTIVE(s(x0))
QUOTEACTIVE(sel(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(s(X)) → MARK(X)
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(sel1(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
MARK(sel1(sel1(y0, y1), dbls(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(sel1(y0, y1), dbl(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
QUOTEACTIVE(sel(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
QUOTEACTIVE(sel(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(sel1(y0, y1), cons(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
QUOTEACTIVE(sel(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(dbl(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(dbl(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
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(sel1(sel1(y0, y1), dbl1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
QUOTEACTIVE(sel(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(s(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(dbl(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(dbl(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
QUOTEACTIVE(sel(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(s(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
QUOTEACTIVE(dbl(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule QUOTEACTIVE(sel(sel1(x0, x1), y1)) → SEL1ACTIVE(sel1Active(mark(x0), mark(x1)), mark(y1)) at position [1] we obtained the following new rules:

QUOTEACTIVE(sel(sel1(y0, y1), from(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
QUOTEACTIVE(sel(sel1(y0, y1), 0)) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), 0)
QUOTEACTIVE(sel(sel1(y0, y1), indx(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(sel1(y0, y1), nil)) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), nil)
QUOTEACTIVE(sel(sel1(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel1(y0, y1), sel(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel1(y0, y1), quote(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(sel1(y0, y1), cons(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(sel(sel1(y0, y1), dbls(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(sel1(y0, y1), s(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), s(x0))
QUOTEACTIVE(sel(sel1(y0, y1), 01)) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), 01)
QUOTEACTIVE(sel(sel1(y0, y1), s1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), s1(mark(x0)))
QUOTEACTIVE(sel(sel1(y0, y1), dbl1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(sel1(y0, y1), dbl(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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

                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                            ↳ 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:

QUOTEACTIVE(sel(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), quote(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(sel(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel1(y0, y1), 0)) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), 0)
QUOTEACTIVE(dbl(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(sel(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(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)
MARK(sel1(sel1(y0, y1), from(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(sel(sel1(y0, y1), dbl(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
QUOTEACTIVE(sel(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(sel1(y0, y1), indx(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(sel1(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
QUOTEACTIVE(s(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(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), 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))
QUOTEACTIVE(sel(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(sel(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(sel1(y0, y1), s(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), s(x0))
MARK(sel1(sel1(y0, y1), sel(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel1(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(s(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(s(y0), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(sel(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(mark(x0)))
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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(sel1(y0, y1), cons(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel1(sel1(y0, y1), indx(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel1(x1, x2)) → MARK(x2)
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(sel1(y0, y1), from(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
MARK(sel1(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
QUOTEACTIVE(sel(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(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(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel1(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(quote(x1)) → MARK(x1)
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(sel(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
MARK(sel1(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
QUOTEACTIVE(sel(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
QUOTEACTIVE(s(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(sel(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(s(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel1(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
QUOTEACTIVE(sel(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(dbl(s(x0))) → DBL1ACTIVE(s(x0))
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
QUOTEACTIVE(sel(sel1(y0, y1), nil)) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), nil)
QUOTEACTIVE(sel(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(s(X)) → MARK(X)
QUOTEACTIVE(sel(sel1(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(sel1(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), dbls(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel1(sel1(y0, y1), dbl(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
QUOTEACTIVE(sel(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel1(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
QUOTEACTIVE(sel(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
QUOTEACTIVE(sel(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
MARK(sel1(sel1(y0, y1), cons(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
QUOTEACTIVE(sel(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel1(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
QUOTEACTIVE(sel(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(dbl(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(dbl(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(sel1(y0, y1), 01)) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), 01)
QUOTEACTIVE(sel(sel1(y0, y1), s1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), s1(mark(x0)))
QUOTEACTIVE(sel(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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(sel1(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), dbl1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
MARK(sel1(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
QUOTEACTIVE(sel(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(s(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(dbl(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
QUOTEACTIVE(sel(sel1(y0, y1), sel(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel1(y0, y1), quote(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(sel1(y0, y1), dbls(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(sel(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(s(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
QUOTEACTIVE(dbl(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel1(y0, y1), dbl1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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 5 less nodes.

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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

                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                                                                                ↳ 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:

QUOTEACTIVE(sel(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(sel(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(sel1(y0, y1), quote(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
QUOTEACTIVE(sel(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(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)))
MARK(sel1(sel1(y0, y1), from(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
QUOTEACTIVE(sel(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(sel(sel1(y0, y1), dbl(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
QUOTEACTIVE(sel(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel1(y0, y1), indx(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel1(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
QUOTEACTIVE(s(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(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(dbls(y0), indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(sel1(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), sel(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
QUOTEACTIVE(sel(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel1(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
QUOTEACTIVE(sel(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(sel1(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(sel(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(s(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(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), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(sel(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
QUOTEACTIVE(sel(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(sel1(y0, y1), cons(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel1(y0, y1), indx(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(x1, x2)) → MARK(x2)
QUOTEACTIVE(sel(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel1(y0, y1), from(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
MARK(sel1(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
QUOTEACTIVE(sel(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(sel1(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), 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(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(sel1(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(quote(x1)) → MARK(x1)
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
QUOTEACTIVE(sel(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(sel(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
QUOTEACTIVE(sel(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(s(s(x0))) → QUOTEACTIVE(s(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(s(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
QUOTEACTIVE(s(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
QUOTEACTIVE(sel(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(dbl(s(x0))) → DBL1ACTIVE(s(x0))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(s(X)) → MARK(X)
QUOTEACTIVE(sel(sel1(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(sel1(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
MARK(sel1(sel1(y0, y1), dbls(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(sel1(y0, y1), dbl(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
QUOTEACTIVE(sel(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
QUOTEACTIVE(sel(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(sel1(y0, y1), cons(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
QUOTEACTIVE(sel(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(dbl(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(dbl(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
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(sel1(sel1(y0, y1), dbl1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
QUOTEACTIVE(sel(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(s(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(dbl(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(dbl(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
QUOTEACTIVE(sel(sel1(y0, y1), sel(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel1(y0, y1), quote(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(sel1(y0, y1), dbls(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(s(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
QUOTEACTIVE(dbl(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel1(y0, y1), dbl1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule QUOTEACTIVE(sel(from(x0), y1)) → SEL1ACTIVE(fromActive(x0), mark(y1)) at position [0] we obtained the following new rules:

QUOTEACTIVE(sel(from(x0), y1)) → SEL1ACTIVE(from(x0), mark(y1))
QUOTEACTIVE(sel(from(x0), y1)) → SEL1ACTIVE(cons(x0, from(s(x0))), mark(y1))

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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

                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                                    ↳ 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:

QUOTEACTIVE(sel(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), quote(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(sel(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(sel(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(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)
MARK(sel1(sel1(y0, y1), from(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel1(y0, y1), dbl(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
QUOTEACTIVE(sel(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
QUOTEACTIVE(sel(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(sel1(y0, y1), indx(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel1(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(sel1(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
QUOTEACTIVE(s(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(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), 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))
QUOTEACTIVE(sel(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(sel(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(from(x0), y1)) → SEL1ACTIVE(from(x0), mark(y1))
MARK(sel1(sel1(y0, y1), sel(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel1(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(s(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(s(y0), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(sel(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(mark(x0)))
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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(sel1(y0, y1), cons(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel1(sel1(y0, y1), indx(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel1(x1, x2)) → MARK(x2)
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(sel1(y0, y1), from(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
MARK(sel1(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(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(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel1(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(quote(x1)) → MARK(x1)
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(sel(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
MARK(sel1(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
QUOTEACTIVE(sel(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
QUOTEACTIVE(s(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(sel(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(s(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel1(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
QUOTEACTIVE(sel(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(from(x0), y1)) → SEL1ACTIVE(cons(x0, from(s(x0))), mark(y1))
QUOTEACTIVE(sel(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(dbl(s(x0))) → DBL1ACTIVE(s(x0))
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
QUOTEACTIVE(sel(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(s(X)) → MARK(X)
QUOTEACTIVE(sel(sel1(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(sel1(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), dbls(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel1(sel1(y0, y1), dbl(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel1(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
QUOTEACTIVE(sel(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
QUOTEACTIVE(sel(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
MARK(sel1(sel1(y0, y1), cons(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
QUOTEACTIVE(sel(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel1(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
QUOTEACTIVE(sel(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(dbl(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(dbl(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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(sel1(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), dbl1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
MARK(sel1(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
QUOTEACTIVE(sel(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(s(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(dbl(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
QUOTEACTIVE(sel(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel1(y0, y1), sel(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(sel1(y0, y1), quote(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(sel(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel1(y0, y1), dbls(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
QUOTEACTIVE(s(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
QUOTEACTIVE(dbl(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel1(y0, y1), dbl1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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

  ↳ CSDependencyPairsProof

  ↳ 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

                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                                                                                        ↳ 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:

QUOTEACTIVE(sel(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(sel(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(sel1(y0, y1), quote(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
QUOTEACTIVE(sel(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(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)))
MARK(sel1(sel1(y0, y1), from(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
QUOTEACTIVE(sel(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(sel(sel1(y0, y1), dbl(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
QUOTEACTIVE(sel(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel1(y0, y1), indx(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel1(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
QUOTEACTIVE(s(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(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(dbls(y0), indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(sel1(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), sel(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
QUOTEACTIVE(sel(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel1(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
QUOTEACTIVE(sel(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(sel1(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(sel(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(s(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(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), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(sel(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
QUOTEACTIVE(sel(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(sel1(y0, y1), cons(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel1(y0, y1), indx(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(x1, x2)) → MARK(x2)
QUOTEACTIVE(sel(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel1(y0, y1), from(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
MARK(sel1(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
QUOTEACTIVE(sel(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(sel1(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), 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(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(sel1(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(quote(x1)) → MARK(x1)
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
QUOTEACTIVE(sel(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(sel(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
QUOTEACTIVE(sel(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(s(s(x0))) → QUOTEACTIVE(s(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(s(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
QUOTEACTIVE(s(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
QUOTEACTIVE(sel(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(dbl(s(x0))) → DBL1ACTIVE(s(x0))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(s(X)) → MARK(X)
QUOTEACTIVE(sel(sel1(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(sel1(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
MARK(sel1(sel1(y0, y1), dbls(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(sel1(y0, y1), dbl(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
QUOTEACTIVE(sel(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
QUOTEACTIVE(sel(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(sel1(y0, y1), cons(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
QUOTEACTIVE(sel(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(dbl(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(dbl(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
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(sel1(sel1(y0, y1), dbl1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
QUOTEACTIVE(sel(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(s(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(dbl(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(dbl(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
QUOTEACTIVE(sel(sel1(y0, y1), sel(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel1(y0, y1), quote(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(sel1(y0, y1), dbls(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(s(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
QUOTEACTIVE(dbl(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel1(y0, y1), dbl1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule QUOTEACTIVE(sel(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0)) at position [1] we obtained the following new rules:

QUOTEACTIVE(sel(0, from(x0))) → SEL1ACTIVE(0, cons(x0, from(s(x0))))
QUOTEACTIVE(sel(0, from(x0))) → SEL1ACTIVE(0, from(x0))

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ 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

                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                                            ↳ 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:

QUOTEACTIVE(sel(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), quote(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(sel(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(sel(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(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)
MARK(sel1(sel1(y0, y1), from(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel1(y0, y1), dbl(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
QUOTEACTIVE(sel(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
QUOTEACTIVE(sel(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(sel1(y0, y1), indx(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel1(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(sel1(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
QUOTEACTIVE(s(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(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), 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))
QUOTEACTIVE(sel(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(sel(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(sel1(sel1(y0, y1), sel(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel1(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(0, from(x0))) → SEL1ACTIVE(0, from(x0))
QUOTEACTIVE(sel(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(s(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(s(y0), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel1(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(sel(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(mark(x0)))
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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(sel1(y0, y1), cons(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel1(sel1(y0, y1), indx(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel1(x1, x2)) → MARK(x2)
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(sel1(y0, y1), from(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
MARK(sel1(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(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(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel1(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(quote(x1)) → MARK(x1)
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(sel(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
MARK(sel1(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
QUOTEACTIVE(sel(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(sel(0, from(x0))) → SEL1ACTIVE(0, cons(x0, from(s(x0))))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
QUOTEACTIVE(s(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(sel(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(s(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel1(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
QUOTEACTIVE(sel(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(dbl(s(x0))) → DBL1ACTIVE(s(x0))
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
QUOTEACTIVE(sel(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(s(X)) → MARK(X)
QUOTEACTIVE(sel(sel1(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(sel1(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), dbls(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel1(sel1(y0, y1), dbl(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel1(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
QUOTEACTIVE(sel(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
QUOTEACTIVE(sel(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
MARK(sel1(sel1(y0, y1), cons(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
QUOTEACTIVE(sel(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel1(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
QUOTEACTIVE(sel(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(dbl(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(dbl(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
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(sel1(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), dbl1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
MARK(sel1(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
QUOTEACTIVE(sel(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(s(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(dbl(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
QUOTEACTIVE(sel(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel1(y0, y1), sel(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(sel1(y0, y1), quote(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(sel(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel1(y0, y1), dbls(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
QUOTEACTIVE(s(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
QUOTEACTIVE(dbl(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel1(y0, y1), dbl1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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

  ↳ CSDependencyPairsProof

  ↳ 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

                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                                                                            ↳ QDP

                                                                                                                                                                                                                                                                                                                                                              ↳ Narrowing

                                                                                                                                                                                                                                                                                                                                                                ↳ QDP

                                                                                                                                                                                                                                                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                                                                                    ↳ QDP

                                                                                                                                                                                                                                                                                                                                                                      ↳ Narrowing

                                                                                                                                                                                                                                                                                                                                                                        ↳ QDP

                                                                                                                                                                                                                                                                                                                                                                          ↳ DependencyGraphProof

                                                                                                                                                                                                                                                                                                                                                                            ↳ QDP

  ↳ Trivial-Transformation

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

QUOTEACTIVE(sel(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(sel(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel1(sel1(y0, y1), quote(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(dbl1(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), dbls(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), sel1(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
QUOTEACTIVE(sel(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
MARK(sel1(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(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)))
MARK(sel1(sel1(y0, y1), from(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(indx(y0, y1), dbl1(x0))) → SELACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
QUOTEACTIVE(sel(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
DBL1ACTIVE(s(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(X), cons(Y, Z)) → SELACTIVE(mark(X), mark(Z))
MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
MARK(dbl(x1)) → MARK(x1)
MARK(dbl1(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), from(x0))) → SELACTIVE(quoteActive(mark(y0)), fromActive(x0))
MARK(sel1(dbls(y0), dbls(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), dbl(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbl1(x0))) → SELACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), indx(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(dbl1(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
QUOTEACTIVE(sel(sel1(y0, y1), dbl(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
QUOTEACTIVE(sel(s(y0), sel1(x0, x1))) → SEL1ACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel1(y0, y1), indx(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel1(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(sel(y0, y1), quote(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), sel(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel(y0, y1), sel(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(dbl(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
QUOTEACTIVE(s(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(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(dbls(y0), indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(sel1(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(sel1(y0, y1), sel(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(dbl1(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
QUOTEACTIVE(sel(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
MARK(sel1(dbls(y0), cons(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), cons(x0, x1))
DBL1ACTIVE(s(s(x0))) → DBL1ACTIVE(s(x0))
MARK(sel1(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), sel1(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
QUOTEACTIVE(sel(dbl1(y0), quote(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), dbl1(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(quote(y0), indx(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel1(y0, y1), from(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), dbl(x0))) → SELACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), quote(x0))) → SELACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), from(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), dbls(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel1(dbls(y0), sel(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel1(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(quote(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(sel1(s(y0), cons(x0, x1))) → SEL1ACTIVE(s(y0), cons(x0, x1))
QUOTEACTIVE(sel(X, Y)) → MARK(Y)
MARK(sel(sel1(y0, y1), indx(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(sel(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel(y0, y1), cons(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
QUOTEACTIVE(s(sel(x0, x1))) → QUOTEACTIVE(selActive(mark(x0), mark(x1)))
MARK(sel(quote(y0), quote(x0))) → SELACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(quote(quote(x0))) → QUOTEACTIVE(quoteActive(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), quote(x0))) → SELACTIVE(s(y0), quoteActive(mark(x0)))
MARK(sel1(0, sel1(x0, x1))) → SEL1ACTIVE(0, sel1Active(mark(x0), mark(x1)))
MARK(sel1(quote(y0), quote(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel(s(y0), dbl1(x0))) → SELACTIVE(s(y0), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
DBL1ACTIVE(s(indx(x0, x1))) → DBL1ACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(sel(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel1(indx(y0, y1), dbl1(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dbl1Active(mark(x0)))
MARK(sel1(dbl(y0), indx(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(s(y0), quote(x0))) → SEL1ACTIVE(s(y0), quoteActive(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(indx(y0, y1), sel1(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
MARK(sel1(0, dbls(x0))) → SEL1ACTIVE(0, dblsActive(mark(x0)))
QUOTEACTIVE(sel(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(quote(y0), sel(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), quote(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(sel1(y0, y1), cons(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(dbl1(y0), indx(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(sel1(y0, y1), indx(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(dbl1(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
MARK(sel1(x1, x2)) → MARK(x2)
QUOTEACTIVE(sel(dbl(y0), quote(x0))) → SEL1ACTIVE(dblActive(mark(y0)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), dbl(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblActive(mark(x0)))
DBL1ACTIVE(s(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel1(y0, y1), from(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), fromActive(x0))
MARK(sel1(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(X)
QUOTEACTIVE(sel(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(sel1(sel(y0, y1), dbl(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), from(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel1(dbl(y0), sel(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), selActive(mark(x0), 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(dbl1(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(quote(y0), cons(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), cons(x0, x1))
MARK(sel1(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(quote(x1)) → MARK(x1)
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(indx(y0, y1), quote(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(s(y0), dbls(x0))) → SEL1ACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel1(sel(y0, y1), from(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
QUOTEACTIVE(sel(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
DBL1ACTIVE(s(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(sel(0, quote(x0))) → SEL1ACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(sel(0, from(x0))) → SEL1ACTIVE(0, cons(x0, from(s(x0))))
QUOTEACTIVE(dbl(X)) → MARK(X)
MARK(sel1(0, from(x0))) → SEL1ACTIVE(0, fromActive(x0))
QUOTEACTIVE(sel(indx(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(s(s(x0))) → QUOTEACTIVE(s(x0))
MARK(sel(s(y0), sel1(x0, x1))) → SELACTIVE(s(y0), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), sel1(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl1(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl(y0), dbl(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
QUOTEACTIVE(s(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
QUOTEACTIVE(s(quote(x0))) → QUOTEACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbls(y0), dbl(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbls(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(dbl1(y0), cons(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(sel1(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel1(y0, y1), cons(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel1(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
SEL1ACTIVE(s(X), cons(Y, Z)) → SEL1ACTIVE(mark(X), mark(Z))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(quote(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
MARK(sel1(indx(y0, y1), dbl(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
QUOTEACTIVE(sel(dbls(y0), indx(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel1(dbl(y0), cons(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
QUOTEACTIVE(dbl(s(x0))) → DBL1ACTIVE(s(x0))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(quote(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), indx(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), indxActive(mark(x0), x1))
QUOTEACTIVE(s(X)) → MARK(X)
QUOTEACTIVE(sel(sel1(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), from(x0))) → SELACTIVE(dbl1Active(mark(y0)), fromActive(x0))
MARK(quote(sel(x0, x1))) → QUOTEACTIVE(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(sel1(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
MARK(sel1(sel1(y0, y1), dbls(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel1(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel1(y0, y1), sel(x0, x1))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), sel1(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbl1(y0), dbl1(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
DBL1ACTIVE(s(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(sel1(y0, y1), dbl(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(quote(y0), dbls(x0))) → SELACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), quote(x0))) → SELACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
MARK(sel1(dbl(y0), sel1(x0, x1))) → SEL1ACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(indx(y0, y1), from(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), fromActive(x0))
QUOTEACTIVE(sel(dbl1(y0), sel(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel(y0, y1), dbl1(x0))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel(x1, x2)) → MARK(x2)
SEL1ACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(quote(y0), sel1(x0, x1))) → SELACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), quote(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
MARK(sel(quote(y0), dbl1(x0))) → SELACTIVE(quoteActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), quote(x0))) → SELACTIVE(dbl1Active(mark(y0)), quoteActive(mark(x0)))
DBL1ACTIVE(s(X)) → MARK(X)
MARK(sel1(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
QUOTEACTIVE(sel(indx(y0, y1), dbls(x0))) → SEL1ACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), from(x0))) → SEL1ACTIVE(s(y0), fromActive(x0))
DBL1ACTIVE(s(dbl1(x0))) → DBL1ACTIVE(dbl1Active(mark(x0)))
MARK(sel1(sel1(y0, y1), cons(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), cons(x0, x1))
MARK(dbl1(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
QUOTEACTIVE(sel(s(y0), sel(x0, x1))) → SEL1ACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), cons(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), cons(x0, x1))
MARK(sel1(0, cons(x0, x1))) → SEL1ACTIVE(0, cons(x0, x1))
QUOTEACTIVE(sel(quote(y0), cons(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), cons(x0, x1))
QUOTEACTIVE(sel(0, dbl(x0))) → SEL1ACTIVE(0, dblActive(mark(x0)))
MARK(sel1(indx(y0, y1), indx(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), sel1(x0, x1))) → SELACTIVE(dblActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel(sel1(y0, y1), dbl(x0))) → SELACTIVE(sel1Active(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), quote(x0))) → SELACTIVE(indxActive(mark(y0), y1), quoteActive(mark(x0)))
MARK(sel1(dbl1(y0), dbl(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblActive(mark(x0)))
MARK(sel1(dbl(y0), dbl1(x0))) → SEL1ACTIVE(dblActive(mark(y0)), dbl1Active(mark(x0)))
MARK(sel1(s(y0), dbl1(x0))) → SEL1ACTIVE(s(y0), dbl1Active(mark(x0)))
DBL1ACTIVE(s(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(quote(dbl1(x0))) → QUOTEACTIVE(dbl1Active(mark(x0)))
MARK(sel(dbl1(y0), dbls(x0))) → SELACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), dbl1(x0))) → SELACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
MARK(s1(x1)) → MARK(x1)
QUOTEACTIVE(dbl(dbl(x0))) → DBL1ACTIVE(dblActive(mark(x0)))
QUOTEACTIVE(dbl(dbls(x0))) → DBL1ACTIVE(dblsActive(mark(x0)))
MARK(sel1(0, indx(x0, x1))) → SEL1ACTIVE(0, indxActive(mark(x0), x1))
MARK(sel1(x1, x2)) → MARK(x1)
MARK(sel(0, quote(x0))) → SELACTIVE(0, quoteActive(mark(x0)))
QUOTEACTIVE(s(dbl(x0))) → QUOTEACTIVE(dblActive(mark(x0)))
MARK(sel1(dbl1(y0), from(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), fromActive(x0))
QUOTEACTIVE(sel(sel(y0, y1), indx(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(s(y0), dbl(x0))) → SEL1ACTIVE(s(y0), dblActive(mark(x0)))
MARK(quote(sel1(x0, x1))) → QUOTEACTIVE(sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(X, Y)) → MARK(X)
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(sel1(sel1(y0, y1), dbl1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
MARK(sel1(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(dbls(y0), dbl1(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), dbls(x0))) → SEL1ACTIVE(dbl1Active(mark(y0)), dblsActive(mark(x0)))
MARK(sel1(0, dbl1(x0))) → SEL1ACTIVE(0, dbl1Active(mark(x0)))
MARK(sel(0, dbl1(x0))) → SELACTIVE(0, dbl1Active(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
QUOTEACTIVE(sel(indx(y0, y1), sel(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(0, sel(x0, x1))) → SEL1ACTIVE(0, selActive(mark(x0), mark(x1)))
QUOTEACTIVE(s(indx(x0, x1))) → QUOTEACTIVE(indxActive(mark(x0), x1))
QUOTEACTIVE(dbl(sel1(x0, x1))) → DBL1ACTIVE(sel1Active(mark(x0), mark(x1)))
MARK(sel(dbl1(y0), dbl1(x0))) → SELACTIVE(dbl1Active(mark(y0)), dbl1Active(mark(x0)))
QUOTEACTIVE(dbl(quote(x0))) → DBL1ACTIVE(quoteActive(mark(x0)))
MARK(sel1(dbl(y0), from(x0))) → SEL1ACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(dbl1(s(x0))) → DBL1ACTIVE(s(x0))
QUOTEACTIVE(sel(sel1(y0, y1), sel(x0, x1))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel(y0, y1), sel1(x0, x1))) → SEL1ACTIVE(selActive(mark(y0), mark(y1)), sel1Active(mark(x0), mark(x1)))
SEL1ACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(0, sel1(x0, x1))) → SELACTIVE(0, sel1Active(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel1(y0, y1), quote(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(dbls(y0), quote(x0))) → SEL1ACTIVE(dblsActive(mark(y0)), quoteActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), dbls(x0))) → SEL1ACTIVE(quoteActive(mark(y0)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(quote(y0), sel1(x0, x1))) → SEL1ACTIVE(quoteActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(sel1(s(y0), indx(x0, x1))) → SEL1ACTIVE(s(y0), indxActive(mark(x0), x1))
QUOTEACTIVE(sel(sel1(y0, y1), dbls(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dblsActive(mark(x0)))
QUOTEACTIVE(sel(dbls(y0), sel1(x0, x1))) → SEL1ACTIVE(dblsActive(mark(y0)), sel1Active(mark(x0), mark(x1)))
MARK(quote(s(x0))) → QUOTEACTIVE(s(x0))
QUOTEACTIVE(s(dbls(x0))) → QUOTEACTIVE(dblsActive(mark(x0)))
MARK(dbl1(x1)) → MARK(x1)
MARK(sel(dbl1(y0), sel(x0, x1))) → SELACTIVE(dbl1Active(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), dbl1(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(indx(y0, y1), cons(x0, x1))) → SEL1ACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
QUOTEACTIVE(dbl(sel(x0, x1))) → DBL1ACTIVE(selActive(mark(x0), mark(x1)))
QUOTEACTIVE(sel(sel1(y0, y1), dbl1(x0))) → SEL1ACTIVE(sel1Active(mark(y0), mark(y1)), dbl1Active(mark(x0)))
QUOTEACTIVE(sel(dbl1(y0), sel1(x0, x1))) → SEL1ACTIVE(dbl1Active(mark(y0)), sel1Active(mark(x0), mark(x1)))

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(dbl1(x1)) → dbl1Active(mark(x1))
dbl1Active(x1) → dbl1(x1)
mark(sel1(x1, x2)) → sel1Active(mark(x1), mark(x2))
sel1Active(x1, x2) → sel1(x1, x2)
mark(quote(x1)) → quoteActive(mark(x1))
quoteActive(x1) → quote(x1)
mark(0) → 0
mark(s(x1)) → s(x1)
mark(nil) → nil
mark(cons(x1, x2)) → cons(x1, x2)
mark(01) → 01
mark(s1(x1)) → s1(mark(x1))
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)))
dbl1Active(0) → 01
dbl1Active(s(X)) → s1(s1(dbl1Active(mark(X))))
sel1Active(0, cons(X, Y)) → mark(X)
sel1Active(s(X), cons(Y, Z)) → sel1Active(mark(X), mark(Z))
quoteActive(0) → 01
quoteActive(s(X)) → s1(quoteActive(mark(X)))
quoteActive(dbl(X)) → dbl1Active(mark(X))
quoteActive(sel(X, Y)) → sel1Active(mark(X), mark(Y))

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

  ↳ CSDependencyPairsProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

    ↳ 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)))
dbl1(0) → 01
dbl1(s(X)) → s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) → X
sel1(s(X), cons(Y, Z)) → sel1(X, Z)
quote(0) → 01
quote(s(X)) → s1(quote(X))
quote(dbl(X)) → dbl1(X)
quote(sel(X, Y)) → sel1(X, Y)

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:

DBLS(cons(X, Y)) → DBLS(Y)
SEL(s(X), cons(Y, Z)) → SEL(X, Z)
DBLS(cons(X, Y)) → DBL(X)
QUOTE(dbl(X)) → DBL1(X)
DBL(s(X)) → DBL(X)
QUOTE(sel(X, Y)) → SEL1(X, Y)
DBL1(s(X)) → DBL1(X)
SEL1(s(X), cons(Y, Z)) → SEL1(X, Z)
FROM(X) → FROM(s(X))
QUOTE(s(X)) → QUOTE(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)))
dbl1(0) → 01
dbl1(s(X)) → s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) → X
sel1(s(X), cons(Y, Z)) → sel1(X, Z)
quote(0) → 01
quote(s(X)) → s1(quote(X))
quote(dbl(X)) → dbl1(X)
quote(sel(X, Y)) → sel1(X, Y)

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

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

    ↳ 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)
QUOTE(dbl(X)) → DBL1(X)
DBL(s(X)) → DBL(X)
QUOTE(sel(X, Y)) → SEL1(X, Y)
DBL1(s(X)) → DBL1(X)
SEL1(s(X), cons(Y, Z)) → SEL1(X, Z)
FROM(X) → FROM(s(X))
QUOTE(s(X)) → QUOTE(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)))
dbl1(0) → 01
dbl1(s(X)) → s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) → X
sel1(s(X), cons(Y, Z)) → sel1(X, Z)
quote(0) → 01
quote(s(X)) → s1(quote(X))
quote(dbl(X)) → dbl1(X)
quote(sel(X, Y)) → sel1(X, Y)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 8 SCCs with 4 less nodes.

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

SEL1(s(X), cons(Y, Z)) → SEL1(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)))
dbl1(0) → 01
dbl1(s(X)) → s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) → X
sel1(s(X), cons(Y, Z)) → sel1(X, Z)
quote(0) → 01
quote(s(X)) → s1(quote(X))
quote(dbl(X)) → dbl1(X)
quote(sel(X, Y)) → sel1(X, Y)

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

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

SEL1(s(X), cons(Y, Z)) → SEL1(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:

SEL1(s(X), cons(Y, Z)) → SEL1(X, Z)
The graph contains the following edges 1 > 1, 2 > 2

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

DBL1(s(X)) → DBL1(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)))
dbl1(0) → 01
dbl1(s(X)) → s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) → X
sel1(s(X), cons(Y, Z)) → sel1(X, Z)
quote(0) → 01
quote(s(X)) → s1(quote(X))
quote(dbl(X)) → dbl1(X)
quote(sel(X, Y)) → sel1(X, Y)

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

DBL1(s(X)) → DBL1(X)

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

DBL1(s(X)) → DBL1(X)
The graph contains the following edges 1 > 1

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

QUOTE(s(X)) → QUOTE(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)))
dbl1(0) → 01
dbl1(s(X)) → s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) → X
sel1(s(X), cons(Y, Z)) → sel1(X, Z)
quote(0) → 01
quote(s(X)) → s1(quote(X))
quote(dbl(X)) → dbl1(X)
quote(sel(X, Y)) → sel1(X, Y)

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

QUOTE(s(X)) → QUOTE(X)

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

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

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ 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)))
dbl1(0) → 01
dbl1(s(X)) → s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) → X
sel1(s(X), cons(Y, Z)) → sel1(X, Z)
quote(0) → 01
quote(s(X)) → s1(quote(X))
quote(dbl(X)) → dbl1(X)
quote(sel(X, Y)) → sel1(X, Y)

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ 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

  ↳ CSDependencyPairsProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ 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)))
dbl1(0) → 01
dbl1(s(X)) → s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) → X
sel1(s(X), cons(Y, Z)) → sel1(X, Z)
quote(0) → 01
quote(s(X)) → s1(quote(X))
quote(dbl(X)) → dbl1(X)
quote(sel(X, Y)) → sel1(X, Y)

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ 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)

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

  ↳ CSDependencyPairsProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ 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)))
dbl1(0) → 01
dbl1(s(X)) → s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) → X
sel1(s(X), cons(Y, Z)) → sel1(X, Z)
quote(0) → 01
quote(s(X)) → s1(quote(X))
quote(dbl(X)) → dbl1(X)
quote(sel(X, Y)) → sel1(X, Y)

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ 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

  ↳ CSDependencyPairsProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ 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)))
dbl1(0) → 01
dbl1(s(X)) → s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) → X
sel1(s(X), cons(Y, Z)) → sel1(X, Z)
quote(0) → 01
quote(s(X)) → s1(quote(X))
quote(dbl(X)) → dbl1(X)
quote(sel(X, Y)) → sel1(X, Y)

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ 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

  ↳ CSDependencyPairsProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ 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)))
dbl1(0) → 01
dbl1(s(X)) → s1(s1(dbl1(X)))
sel1(0, cons(X, Y)) → X
sel1(s(X), cons(Y, Z)) → sel1(X, Z)
quote(0) → 01
quote(s(X)) → s1(quote(X))
quote(dbl(X)) → dbl1(X)
quote(sel(X, Y)) → sel1(X, Y)

↳ CSR

  ↳ CSDependencyPairsProof

  ↳ Incomplete Giesl Middeldorp-Transformation

  ↳ Trivial-Transformation

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ 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