Termination proof of /tmp/tmpCK5rlJ/Ex3_2

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

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

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

The replacement map contains the following entries:

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

↳ CSR

  ↳ CSRInnermostProof

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

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

The replacement map contains the following entries:

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

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

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

The replacement map contains the following entries:

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

Innermost Strategy.

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

        ↳ QCSDP

          ↳ QCSDependencyGraphProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

The ordinary context-sensitive dependency pairs DP_o are:

SEL(s(X), cons(Y, Z)) → SEL(X, Z)

The collapsing dependency pairs are DP_c:

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

The hidden terms of R are:

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

Every hiding context is built from:

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

Hence, the new unhiding pairs DP_u are :

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

        ↳ QCSDP

          ↳ QCSDependencyGraphProof

            ↳ QCSDP

              ↳ QCSDPForwardInstantiationProcessor

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

The TRS P consists of the following rules:

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

Using the Context-Sensitive Forward Instantiation Processor
the pair SEL(0, cons(X, Y)) → U(X)
was transformed to the following new pairs:

SEL(0, cons(dbl(z0), y1)) → U(dbl(z0))
SEL(0, cons(dbls(z0), y1)) → U(dbls(z0))
SEL(0, cons(sel(z0, z1), y1)) → U(sel(z0, z1))
SEL(0, cons(indx(z0, z1), y1)) → U(indx(z0, z1))

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

        ↳ QCSDP

          ↳ QCSDependencyGraphProof

            ↳ QCSDP

              ↳ QCSDPForwardInstantiationProcessor

                ↳ QCSDP

                  ↳ QCSDPForwardInstantiationProcessor

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

U(dbl(x_0)) → U(x_0)
SEL(0, cons(sel(z0, z1), y1)) → U(sel(z0, z1))
SEL(s(X), cons(Y, Z)) → SEL(X, Z)
SEL(0, cons(dbls(z0), y1)) → U(dbls(z0))
U(dbls(x_0)) → U(x_0)
SEL(s(X), cons(Y, Z)) → U(Z)
SEL(0, cons(indx(z0, z1), y1)) → U(indx(z0, z1))
SEL(s(X), cons(Y, Z)) → U(X)
U(sel(X, Z)) → SEL(X, Z)
SEL(0, cons(dbl(z0), y1)) → U(dbl(z0))
U(indx(x_0, x_1)) → U(x_0)
U(sel(x_0, x_1)) → U(x_0)
U(sel(x_0, x_1)) → U(x_1)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

Using the Context-Sensitive Forward Instantiation Processor
the pair U(dbl(x_0)) → U(x_0)
was transformed to the following new pairs:

U(dbl(dbl(z0))) → U(dbl(z0))
U(dbl(dbls(z0))) → U(dbls(z0))
U(dbl(sel(z0, z1))) → U(sel(z0, z1))
U(dbl(indx(z0, z1))) → U(indx(z0, z1))

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

        ↳ QCSDP

          ↳ QCSDependencyGraphProof

            ↳ QCSDP

              ↳ QCSDPForwardInstantiationProcessor

                ↳ QCSDP

                  ↳ QCSDPForwardInstantiationProcessor

                    ↳ QCSDP

                      ↳ QCSDependencyGraphProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

U(dbl(dbls(z0))) → U(dbls(z0))
SEL(0, cons(sel(z0, z1), y1)) → U(sel(z0, z1))
SEL(0, cons(dbls(z0), y1)) → U(dbls(z0))
SEL(s(X), cons(Y, Z)) → SEL(X, Z)
U(dbls(x_0)) → U(x_0)
U(dbl(dbl(z0))) → U(dbl(z0))
SEL(s(X), cons(Y, Z)) → U(Z)
SEL(0, cons(indx(z0, z1), y1)) → U(indx(z0, z1))
U(dbl(indx(z0, z1))) → U(indx(z0, z1))
SEL(0, cons(dbl(z0), y1)) → U(dbl(z0))
U(sel(X, Z)) → SEL(X, Z)
SEL(s(X), cons(Y, Z)) → U(X)
U(dbl(sel(z0, z1))) → U(sel(z0, z1))
U(indx(x_0, x_1)) → U(x_0)
U(sel(x_0, x_1)) → U(x_0)
U(sel(x_0, x_1)) → U(x_1)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

The approximation of the Context-Sensitive Dependency Graph contains 2 SCCs.
The rules U(dbl(dbl(z0))) → U(dbl(z0)) and U(dbls(x0)) → U(x0) form no chain, because ECap^µ(U(dbl(z0))) = U(dbl(z0)) does not unify with U(dbls(x0)). The rules U(dbl(dbl(z0))) → U(dbl(z0)) and U(sel(x0, x1)) → U(x0) form no chain, because ECap^µ(U(dbl(z0))) = U(dbl(z0)) does not unify with U(sel(x0, x1)). The rules U(dbl(dbl(z0))) → U(dbl(z0)) and U(sel(x0, x1)) → U(x1) form no chain, because ECap^µ(U(dbl(z0))) = U(dbl(z0)) does not unify with U(sel(x0, x1)). The rules U(dbl(dbl(z0))) → U(dbl(z0)) and U(indx(x0, x1)) → U(x0) form no chain, because ECap^µ(U(dbl(z0))) = U(dbl(z0)) does not unify with U(indx(x0, x1)). The rules U(dbl(dbl(z0))) → U(dbl(z0)) and U(sel(x0, x1)) → SEL(x0, x1) form no chain, because ECap^µ(U(dbl(z0))) = U(dbl(z0)) does not unify with U(sel(x0, x1)). The rules U(dbl(dbls(z0))) → U(dbls(z0)) and U(sel(x0, x1)) → U(x0) form no chain, because ECap^µ(U(dbls(z0))) = U(dbls(z0)) does not unify with U(sel(x0, x1)). The rules U(dbl(dbls(z0))) → U(dbls(z0)) and U(sel(x0, x1)) → U(x1) form no chain, because ECap^µ(U(dbls(z0))) = U(dbls(z0)) does not unify with U(sel(x0, x1)). The rules U(dbl(dbls(z0))) → U(dbls(z0)) and U(indx(x0, x1)) → U(x0) form no chain, because ECap^µ(U(dbls(z0))) = U(dbls(z0)) does not unify with U(indx(x0, x1)). The rules U(dbl(dbls(z0))) → U(dbls(z0)) and U(sel(x0, x1)) → SEL(x0, x1) form no chain, because ECap^µ(U(dbls(z0))) = U(dbls(z0)) does not unify with U(sel(x0, x1)). The rules U(dbl(dbls(z0))) → U(dbls(z0)) and U(dbl(dbls(x0))) → U(dbls(x0)) form no chain, because ECap^µ(U(dbls(z0))) = U(dbls(z0)) does not unify with U(dbl(dbls(x0))). The rules U(dbl(sel(z0, z1))) → U(sel(z0, z1)) and U(dbls(x0)) → U(x0) form no chain, because ECap^µ(U(sel(z0, z1))) = U(sel(z0, z1)) does not unify with U(dbls(x0)). The rules U(dbl(sel(z0, z1))) → U(sel(z0, z1)) and U(indx(x0, x1)) → U(x0) form no chain, because ECap^µ(U(sel(z0, z1))) = U(sel(z0, z1)) does not unify with U(indx(x0, x1)). The rules U(dbl(sel(z0, z1))) → U(sel(z0, z1)) and U(dbl(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(sel(z0, z1))) = U(sel(z0, z1)) does not unify with U(dbl(sel(x0, x1))). The rules U(dbl(indx(z0, z1))) → U(indx(z0, z1)) and U(dbls(x0)) → U(x0) form no chain, because ECap^µ(U(indx(z0, z1))) = U(indx(z0, z1)) does not unify with U(dbls(x0)). The rules U(dbl(indx(z0, z1))) → U(indx(z0, z1)) and U(sel(x0, x1)) → U(x0) form no chain, because ECap^µ(U(indx(z0, z1))) = U(indx(z0, z1)) does not unify with U(sel(x0, x1)). The rules U(dbl(indx(z0, z1))) → U(indx(z0, z1)) and U(sel(x0, x1)) → U(x1) form no chain, because ECap^µ(U(indx(z0, z1))) = U(indx(z0, z1)) does not unify with U(sel(x0, x1)). The rules U(dbl(indx(z0, z1))) → U(indx(z0, z1)) and U(sel(x0, x1)) → SEL(x0, x1) form no chain, because ECap^µ(U(indx(z0, z1))) = U(indx(z0, z1)) does not unify with U(sel(x0, x1)). The rules U(dbl(indx(z0, z1))) → U(indx(z0, z1)) and U(dbl(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(indx(z0, z1))) = U(indx(z0, z1)) does not unify with U(dbl(indx(x0, x1))).

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

        ↳ QCSDP

          ↳ QCSDependencyGraphProof

            ↳ QCSDP

              ↳ QCSDPForwardInstantiationProcessor

                ↳ QCSDP

                  ↳ QCSDPForwardInstantiationProcessor

                    ↳ QCSDP

                      ↳ QCSDependencyGraphProof

                        ↳ AND

                          ↳ QCSDP

                            ↳ QCSDPSubtermProof

                          ↳ QCSDP

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

Q-restricted context-sensitive dependency pair problem:
The symbols in {dbl, dbls, 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:

U(dbl(dbl(z0))) → U(dbl(z0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We use the subterm processor [20].

The following pairs can be oriented strictly and are deleted.

U(dbl(dbl(z0))) → U(dbl(z0))

The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
U(x1) = x1

Subterm Order

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

        ↳ QCSDP

          ↳ QCSDependencyGraphProof

            ↳ QCSDP

              ↳ QCSDPForwardInstantiationProcessor

                ↳ QCSDP

                  ↳ QCSDPForwardInstantiationProcessor

                    ↳ QCSDP

                      ↳ QCSDependencyGraphProof

                        ↳ AND

                          ↳ QCSDP

                            ↳ QCSDPSubtermProof

                              ↳ QCSDP

                                ↳ PIsEmptyProof

                          ↳ QCSDP

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

Q-restricted context-sensitive dependency pair problem:
The symbols in {dbl, dbls, sel} 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:
none

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

The TRS P is empty. Hence, there is no (P,Q,R,µ)-chain.

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

        ↳ QCSDP

          ↳ QCSDependencyGraphProof

            ↳ QCSDP

              ↳ QCSDPForwardInstantiationProcessor

                ↳ QCSDP

                  ↳ QCSDPForwardInstantiationProcessor

                    ↳ QCSDP

                      ↳ QCSDependencyGraphProof

                        ↳ AND

                          ↳ QCSDP

                          ↳ QCSDP

                            ↳ QCSDPForwardInstantiationProcessor

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

SEL(s(X), cons(Y, Z)) → SEL(X, Z)
SEL(s(X), cons(Y, Z)) → U(X)
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(Z)
U(dbl(dbls(z0))) → U(dbls(z0))
U(dbl(sel(z0, z1))) → U(sel(z0, z1))
U(dbl(indx(z0, z1))) → U(indx(z0, z1))
SEL(0, cons(dbl(z0), y1)) → U(dbl(z0))
SEL(0, cons(dbls(z0), y1)) → U(dbls(z0))
SEL(0, cons(sel(z0, z1), y1)) → U(sel(z0, z1))
SEL(0, cons(indx(z0, z1), y1)) → U(indx(z0, z1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

Using the Context-Sensitive Forward Instantiation Processor
the pair SEL(s(X), cons(Y, Z)) → U(X)
was transformed to the following new pairs:

SEL(s(dbls(z0)), cons(y1, y2)) → U(dbls(z0))
SEL(s(sel(z0, z1)), cons(y1, y2)) → U(sel(z0, z1))
SEL(s(indx(z0, z1)), cons(y1, y2)) → U(indx(z0, z1))
SEL(s(dbl(dbls(z0))), cons(y1, y2)) → U(dbl(dbls(z0)))
SEL(s(dbl(sel(z0, z1))), cons(y1, y2)) → U(dbl(sel(z0, z1)))
SEL(s(dbl(indx(z0, z1))), cons(y1, y2)) → U(dbl(indx(z0, z1)))

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

        ↳ QCSDP

          ↳ QCSDependencyGraphProof

            ↳ QCSDP

              ↳ QCSDPForwardInstantiationProcessor

                ↳ QCSDP

                  ↳ QCSDPForwardInstantiationProcessor

                    ↳ QCSDP

                      ↳ QCSDependencyGraphProof

                        ↳ AND

                          ↳ QCSDP

                          ↳ QCSDP

                            ↳ QCSDPForwardInstantiationProcessor

                              ↳ QCSDP

                                ↳ QCSDPForwardInstantiationProcessor

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

U(dbl(dbls(z0))) → U(dbls(z0))
SEL(s(dbl(dbls(z0))), cons(y1, y2)) → U(dbl(dbls(z0)))
SEL(0, cons(sel(z0, z1), y1)) → U(sel(z0, z1))
SEL(0, cons(dbls(z0), y1)) → U(dbls(z0))
SEL(s(X), cons(Y, Z)) → SEL(X, Z)
SEL(s(sel(z0, z1)), cons(y1, y2)) → U(sel(z0, z1))
SEL(s(dbls(z0)), cons(y1, y2)) → U(dbls(z0))
U(dbls(x_0)) → U(x_0)
SEL(s(indx(z0, z1)), cons(y1, y2)) → U(indx(z0, z1))
SEL(s(X), cons(Y, Z)) → U(Z)
SEL(s(dbl(sel(z0, z1))), cons(y1, y2)) → U(dbl(sel(z0, z1)))
SEL(s(dbl(indx(z0, z1))), cons(y1, y2)) → U(dbl(indx(z0, z1)))
SEL(0, cons(indx(z0, z1), y1)) → U(indx(z0, z1))
U(dbl(indx(z0, z1))) → U(indx(z0, z1))
SEL(0, cons(dbl(z0), y1)) → U(dbl(z0))
U(sel(X, Z)) → SEL(X, Z)
U(dbl(sel(z0, z1))) → U(sel(z0, z1))
U(indx(x_0, x_1)) → U(x_0)
U(sel(x_0, x_1)) → U(x_0)
U(sel(x_0, x_1)) → U(x_1)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

Using the Context-Sensitive Forward Instantiation Processor
the pair U(dbls(x_0)) → U(x_0)
was transformed to the following new pairs:

U(dbls(dbls(z0))) → U(dbls(z0))
U(dbls(sel(z0, z1))) → U(sel(z0, z1))
U(dbls(indx(z0, z1))) → U(indx(z0, z1))
U(dbls(dbl(dbls(z0)))) → U(dbl(dbls(z0)))
U(dbls(dbl(sel(z0, z1)))) → U(dbl(sel(z0, z1)))
U(dbls(dbl(indx(z0, z1)))) → U(dbl(indx(z0, z1)))

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

        ↳ QCSDP

          ↳ QCSDependencyGraphProof

            ↳ QCSDP

              ↳ QCSDPForwardInstantiationProcessor

                ↳ QCSDP

                  ↳ QCSDPForwardInstantiationProcessor

                    ↳ QCSDP

                      ↳ QCSDependencyGraphProof

                        ↳ AND

                          ↳ QCSDP

                          ↳ QCSDP

                            ↳ QCSDPForwardInstantiationProcessor

                              ↳ QCSDP

                                ↳ QCSDPForwardInstantiationProcessor

                                  ↳ QCSDP

                                    ↳ QCSDependencyGraphProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

U(dbl(dbls(z0))) → U(dbls(z0))
SEL(0, cons(sel(z0, z1), y1)) → U(sel(z0, z1))
U(dbls(sel(z0, z1))) → U(sel(z0, z1))
SEL(s(X), cons(Y, Z)) → SEL(X, Z)
U(dbls(dbl(indx(z0, z1)))) → U(dbl(indx(z0, z1)))
U(dbls(dbl(dbls(z0)))) → U(dbl(dbls(z0)))
SEL(s(indx(z0, z1)), cons(y1, y2)) → U(indx(z0, z1))
SEL(s(X), cons(Y, Z)) → U(Z)
U(indx(x_0, x_1)) → U(x_0)
SEL(s(dbl(dbls(z0))), cons(y1, y2)) → U(dbl(dbls(z0)))
U(dbls(dbls(z0))) → U(dbls(z0))
SEL(0, cons(dbls(z0), y1)) → U(dbls(z0))
SEL(s(sel(z0, z1)), cons(y1, y2)) → U(sel(z0, z1))
SEL(s(dbls(z0)), cons(y1, y2)) → U(dbls(z0))
SEL(s(dbl(sel(z0, z1))), cons(y1, y2)) → U(dbl(sel(z0, z1)))
SEL(s(dbl(indx(z0, z1))), cons(y1, y2)) → U(dbl(indx(z0, z1)))
SEL(0, cons(indx(z0, z1), y1)) → U(indx(z0, z1))
U(dbl(indx(z0, z1))) → U(indx(z0, z1))
SEL(0, cons(dbl(z0), y1)) → U(dbl(z0))
U(sel(X, Z)) → SEL(X, Z)
U(dbl(sel(z0, z1))) → U(sel(z0, z1))
U(dbls(dbl(sel(z0, z1)))) → U(dbl(sel(z0, z1)))
U(sel(x_0, x_1)) → U(x_0)
U(dbls(indx(z0, z1))) → U(indx(z0, z1))
U(sel(x_0, x_1)) → U(x_1)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

The approximation of the Context-Sensitive Dependency Graph contains 2 SCCs.
The rules U(dbls(dbls(z0))) → U(dbls(z0)) and U(sel(x0, x1)) → U(x0) form no chain, because ECap^µ(U(dbls(z0))) = U(dbls(z0)) does not unify with U(sel(x0, x1)). The rules U(dbls(dbls(z0))) → U(dbls(z0)) and U(sel(x0, x1)) → U(x1) form no chain, because ECap^µ(U(dbls(z0))) = U(dbls(z0)) does not unify with U(sel(x0, x1)). The rules U(dbls(dbls(z0))) → U(dbls(z0)) and U(indx(x0, x1)) → U(x0) form no chain, because ECap^µ(U(dbls(z0))) = U(dbls(z0)) does not unify with U(indx(x0, x1)). The rules U(dbls(dbls(z0))) → U(dbls(z0)) and U(sel(x0, x1)) → SEL(x0, x1) form no chain, because ECap^µ(U(dbls(z0))) = U(dbls(z0)) does not unify with U(sel(x0, x1)). The rules U(dbls(dbls(z0))) → U(dbls(z0)) and U(dbl(dbls(x0))) → U(dbls(x0)) form no chain, because ECap^µ(U(dbls(z0))) = U(dbls(z0)) does not unify with U(dbl(dbls(x0))). The rules U(dbls(dbls(z0))) → U(dbls(z0)) and U(dbl(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(dbls(z0))) = U(dbls(z0)) does not unify with U(dbl(sel(x0, x1))). The rules U(dbls(dbls(z0))) → U(dbls(z0)) and U(dbl(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(dbls(z0))) = U(dbls(z0)) does not unify with U(dbl(indx(x0, x1))). The rules U(dbls(sel(z0, z1))) → U(sel(z0, z1)) and U(indx(x0, x1)) → U(x0) form no chain, because ECap^µ(U(sel(z0, z1))) = U(sel(z0, z1)) does not unify with U(indx(x0, x1)). The rules U(dbls(sel(z0, z1))) → U(sel(z0, z1)) and U(dbl(dbls(x0))) → U(dbls(x0)) form no chain, because ECap^µ(U(sel(z0, z1))) = U(sel(z0, z1)) does not unify with U(dbl(dbls(x0))). The rules U(dbls(sel(z0, z1))) → U(sel(z0, z1)) and U(dbl(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(sel(z0, z1))) = U(sel(z0, z1)) does not unify with U(dbl(sel(x0, x1))). The rules U(dbls(sel(z0, z1))) → U(sel(z0, z1)) and U(dbl(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(sel(z0, z1))) = U(sel(z0, z1)) does not unify with U(dbl(indx(x0, x1))). The rules U(dbls(sel(z0, z1))) → U(sel(z0, z1)) and U(dbls(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(sel(z0, z1))) = U(sel(z0, z1)) does not unify with U(dbls(sel(x0, x1))). The rules U(dbls(indx(z0, z1))) → U(indx(z0, z1)) and U(sel(x0, x1)) → U(x0) form no chain, because ECap^µ(U(indx(z0, z1))) = U(indx(z0, z1)) does not unify with U(sel(x0, x1)). The rules U(dbls(indx(z0, z1))) → U(indx(z0, z1)) and U(sel(x0, x1)) → U(x1) form no chain, because ECap^µ(U(indx(z0, z1))) = U(indx(z0, z1)) does not unify with U(sel(x0, x1)). The rules U(dbls(indx(z0, z1))) → U(indx(z0, z1)) and U(sel(x0, x1)) → SEL(x0, x1) form no chain, because ECap^µ(U(indx(z0, z1))) = U(indx(z0, z1)) does not unify with U(sel(x0, x1)). The rules U(dbls(indx(z0, z1))) → U(indx(z0, z1)) and U(dbl(dbls(x0))) → U(dbls(x0)) form no chain, because ECap^µ(U(indx(z0, z1))) = U(indx(z0, z1)) does not unify with U(dbl(dbls(x0))). The rules U(dbls(indx(z0, z1))) → U(indx(z0, z1)) and U(dbl(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(indx(z0, z1))) = U(indx(z0, z1)) does not unify with U(dbl(sel(x0, x1))). The rules U(dbls(indx(z0, z1))) → U(indx(z0, z1)) and U(dbl(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(indx(z0, z1))) = U(indx(z0, z1)) does not unify with U(dbl(indx(x0, x1))). The rules U(dbls(indx(z0, z1))) → U(indx(z0, z1)) and U(dbls(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(indx(z0, z1))) = U(indx(z0, z1)) does not unify with U(dbls(indx(x0, x1))). The rules U(dbls(dbl(dbls(z0)))) → U(dbl(dbls(z0))) and U(sel(x0, x1)) → U(x0) form no chain, because ECap^µ(U(dbl(dbls(z0)))) = U(dbl(dbls(z0))) does not unify with U(sel(x0, x1)). The rules U(dbls(dbl(dbls(z0)))) → U(dbl(dbls(z0))) and U(sel(x0, x1)) → U(x1) form no chain, because ECap^µ(U(dbl(dbls(z0)))) = U(dbl(dbls(z0))) does not unify with U(sel(x0, x1)). The rules U(dbls(dbl(dbls(z0)))) → U(dbl(dbls(z0))) and U(indx(x0, x1)) → U(x0) form no chain, because ECap^µ(U(dbl(dbls(z0)))) = U(dbl(dbls(z0))) does not unify with U(indx(x0, x1)). The rules U(dbls(dbl(dbls(z0)))) → U(dbl(dbls(z0))) and U(sel(x0, x1)) → SEL(x0, x1) form no chain, because ECap^µ(U(dbl(dbls(z0)))) = U(dbl(dbls(z0))) does not unify with U(sel(x0, x1)). The rules U(dbls(dbl(dbls(z0)))) → U(dbl(dbls(z0))) and U(dbl(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(dbl(dbls(z0)))) = U(dbl(dbls(z0))) does not unify with U(dbl(sel(x0, x1))). The rules U(dbls(dbl(dbls(z0)))) → U(dbl(dbls(z0))) and U(dbl(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(dbl(dbls(z0)))) = U(dbl(dbls(z0))) does not unify with U(dbl(indx(x0, x1))). The rules U(dbls(dbl(dbls(z0)))) → U(dbl(dbls(z0))) and U(dbls(dbl(dbls(x0)))) → U(dbl(dbls(x0))) form no chain, because ECap^µ(U(dbl(dbls(z0)))) = U(dbl(dbls(z0))) does not unify with U(dbls(dbl(dbls(x0)))). The rules U(dbls(dbl(sel(z0, z1)))) → U(dbl(sel(z0, z1))) and U(sel(x0, x1)) → U(x0) form no chain, because ECap^µ(U(dbl(sel(z0, z1)))) = U(dbl(sel(z0, z1))) does not unify with U(sel(x0, x1)). The rules U(dbls(dbl(sel(z0, z1)))) → U(dbl(sel(z0, z1))) and U(sel(x0, x1)) → U(x1) form no chain, because ECap^µ(U(dbl(sel(z0, z1)))) = U(dbl(sel(z0, z1))) does not unify with U(sel(x0, x1)). The rules U(dbls(dbl(sel(z0, z1)))) → U(dbl(sel(z0, z1))) and U(indx(x0, x1)) → U(x0) form no chain, because ECap^µ(U(dbl(sel(z0, z1)))) = U(dbl(sel(z0, z1))) does not unify with U(indx(x0, x1)). The rules U(dbls(dbl(sel(z0, z1)))) → U(dbl(sel(z0, z1))) and U(sel(x0, x1)) → SEL(x0, x1) form no chain, because ECap^µ(U(dbl(sel(z0, z1)))) = U(dbl(sel(z0, z1))) does not unify with U(sel(x0, x1)). The rules U(dbls(dbl(sel(z0, z1)))) → U(dbl(sel(z0, z1))) and U(dbl(dbls(x0))) → U(dbls(x0)) form no chain, because ECap^µ(U(dbl(sel(z0, z1)))) = U(dbl(sel(z0, z1))) does not unify with U(dbl(dbls(x0))). The rules U(dbls(dbl(sel(z0, z1)))) → U(dbl(sel(z0, z1))) and U(dbl(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(dbl(sel(z0, z1)))) = U(dbl(sel(z0, z1))) does not unify with U(dbl(indx(x0, x1))). The rules U(dbls(dbl(sel(z0, z1)))) → U(dbl(sel(z0, z1))) and U(dbls(dbl(sel(x0, x1)))) → U(dbl(sel(x0, x1))) form no chain, because ECap^µ(U(dbl(sel(z0, z1)))) = U(dbl(sel(z0, z1))) does not unify with U(dbls(dbl(sel(x0, x1)))). The rules U(dbls(dbl(indx(z0, z1)))) → U(dbl(indx(z0, z1))) and U(sel(x0, x1)) → U(x0) form no chain, because ECap^µ(U(dbl(indx(z0, z1)))) = U(dbl(indx(z0, z1))) does not unify with U(sel(x0, x1)). The rules U(dbls(dbl(indx(z0, z1)))) → U(dbl(indx(z0, z1))) and U(sel(x0, x1)) → U(x1) form no chain, because ECap^µ(U(dbl(indx(z0, z1)))) = U(dbl(indx(z0, z1))) does not unify with U(sel(x0, x1)). The rules U(dbls(dbl(indx(z0, z1)))) → U(dbl(indx(z0, z1))) and U(indx(x0, x1)) → U(x0) form no chain, because ECap^µ(U(dbl(indx(z0, z1)))) = U(dbl(indx(z0, z1))) does not unify with U(indx(x0, x1)). The rules U(dbls(dbl(indx(z0, z1)))) → U(dbl(indx(z0, z1))) and U(sel(x0, x1)) → SEL(x0, x1) form no chain, because ECap^µ(U(dbl(indx(z0, z1)))) = U(dbl(indx(z0, z1))) does not unify with U(sel(x0, x1)). The rules U(dbls(dbl(indx(z0, z1)))) → U(dbl(indx(z0, z1))) and U(dbl(dbls(x0))) → U(dbls(x0)) form no chain, because ECap^µ(U(dbl(indx(z0, z1)))) = U(dbl(indx(z0, z1))) does not unify with U(dbl(dbls(x0))). The rules U(dbls(dbl(indx(z0, z1)))) → U(dbl(indx(z0, z1))) and U(dbl(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(dbl(indx(z0, z1)))) = U(dbl(indx(z0, z1))) does not unify with U(dbl(sel(x0, x1))). The rules U(dbls(dbl(indx(z0, z1)))) → U(dbl(indx(z0, z1))) and U(dbls(dbl(indx(x0, x1)))) → U(dbl(indx(x0, x1))) form no chain, because ECap^µ(U(dbl(indx(z0, z1)))) = U(dbl(indx(z0, z1))) does not unify with U(dbls(dbl(indx(x0, x1)))).

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

        ↳ QCSDP

          ↳ QCSDependencyGraphProof

            ↳ QCSDP

              ↳ QCSDPForwardInstantiationProcessor

                ↳ QCSDP

                  ↳ QCSDPForwardInstantiationProcessor

                    ↳ QCSDP

                      ↳ QCSDependencyGraphProof

                        ↳ AND

                          ↳ QCSDP

                          ↳ QCSDP

                            ↳ QCSDPForwardInstantiationProcessor

                              ↳ QCSDP

                                ↳ QCSDPForwardInstantiationProcessor

                                  ↳ QCSDP

                                    ↳ QCSDependencyGraphProof

                                      ↳ AND

                                        ↳ QCSDP

                                          ↳ QCSDPSubtermProof

                                        ↳ QCSDP

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

U(dbls(dbls(z0))) → U(dbls(z0))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We use the subterm processor [20].

The following pairs can be oriented strictly and are deleted.

U(dbls(dbls(z0))) → U(dbls(z0))

The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
U(x1) = x1

Subterm Order

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

        ↳ QCSDP

          ↳ QCSDependencyGraphProof

            ↳ QCSDP

              ↳ QCSDPForwardInstantiationProcessor

                ↳ QCSDP

                  ↳ QCSDPForwardInstantiationProcessor

                    ↳ QCSDP

                      ↳ QCSDependencyGraphProof

                        ↳ AND

                          ↳ QCSDP

                          ↳ QCSDP

                            ↳ QCSDPForwardInstantiationProcessor

                              ↳ QCSDP

                                ↳ QCSDPForwardInstantiationProcessor

                                  ↳ QCSDP

                                    ↳ QCSDependencyGraphProof

                                      ↳ AND

                                        ↳ QCSDP

                                          ↳ QCSDPSubtermProof

                                            ↳ QCSDP

                                              ↳ PIsEmptyProof

                                        ↳ QCSDP

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

Q-restricted context-sensitive dependency pair problem:
The symbols in {dbl, dbls, sel} 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:
none

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

The TRS P is empty. Hence, there is no (P,Q,R,µ)-chain.

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

        ↳ QCSDP

          ↳ QCSDependencyGraphProof

            ↳ QCSDP

              ↳ QCSDPForwardInstantiationProcessor

                ↳ QCSDP

                  ↳ QCSDPForwardInstantiationProcessor

                    ↳ QCSDP

                      ↳ QCSDependencyGraphProof

                        ↳ AND

                          ↳ QCSDP

                          ↳ QCSDP

                            ↳ QCSDPForwardInstantiationProcessor

                              ↳ QCSDP

                                ↳ QCSDPForwardInstantiationProcessor

                                  ↳ QCSDP

                                    ↳ QCSDependencyGraphProof

                                      ↳ AND

                                        ↳ QCSDP

                                        ↳ QCSDP

                                          ↳ QCSDPForwardInstantiationProcessor

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

SEL(s(X), cons(Y, Z)) → SEL(X, Z)
SEL(s(X), cons(Y, Z)) → U(Z)
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(0, cons(dbl(z0), y1)) → U(dbl(z0))
U(dbl(dbls(z0))) → U(dbls(z0))
U(dbls(sel(z0, z1))) → U(sel(z0, z1))
U(dbls(indx(z0, z1))) → U(indx(z0, z1))
U(dbls(dbl(dbls(z0)))) → U(dbl(dbls(z0)))
U(dbls(dbl(sel(z0, z1)))) → U(dbl(sel(z0, z1)))
U(dbl(sel(z0, z1))) → U(sel(z0, z1))
U(dbls(dbl(indx(z0, z1)))) → U(dbl(indx(z0, z1)))
U(dbl(indx(z0, z1))) → U(indx(z0, z1))
SEL(0, cons(dbls(z0), y1)) → U(dbls(z0))
SEL(0, cons(sel(z0, z1), y1)) → U(sel(z0, z1))
SEL(0, cons(indx(z0, z1), y1)) → U(indx(z0, z1))
SEL(s(dbls(z0)), cons(y1, y2)) → U(dbls(z0))
SEL(s(sel(z0, z1)), cons(y1, y2)) → U(sel(z0, z1))
SEL(s(indx(z0, z1)), cons(y1, y2)) → U(indx(z0, z1))
SEL(s(dbl(dbls(z0))), cons(y1, y2)) → U(dbl(dbls(z0)))
SEL(s(dbl(sel(z0, z1))), cons(y1, y2)) → U(dbl(sel(z0, z1)))
SEL(s(dbl(indx(z0, z1))), cons(y1, y2)) → U(dbl(indx(z0, z1)))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

Using the Context-Sensitive Forward Instantiation Processor
the pair SEL(s(X), cons(Y, Z)) → U(Z)
was transformed to the following new pairs:

SEL(s(y0), cons(y1, sel(z0, z1))) → U(sel(z0, z1))
SEL(s(y0), cons(y1, indx(z0, z1))) → U(indx(z0, z1))
SEL(s(y0), cons(y1, dbl(dbls(z0)))) → U(dbl(dbls(z0)))
SEL(s(y0), cons(y1, dbl(sel(z0, z1)))) → U(dbl(sel(z0, z1)))
SEL(s(y0), cons(y1, dbl(indx(z0, z1)))) → U(dbl(indx(z0, z1)))
SEL(s(y0), cons(y1, dbls(sel(z0, z1)))) → U(dbls(sel(z0, z1)))
SEL(s(y0), cons(y1, dbls(indx(z0, z1)))) → U(dbls(indx(z0, z1)))
SEL(s(y0), cons(y1, dbls(dbl(dbls(z0))))) → U(dbls(dbl(dbls(z0))))
SEL(s(y0), cons(y1, dbls(dbl(sel(z0, z1))))) → U(dbls(dbl(sel(z0, z1))))
SEL(s(y0), cons(y1, dbls(dbl(indx(z0, z1))))) → U(dbls(dbl(indx(z0, z1))))

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

        ↳ QCSDP

          ↳ QCSDependencyGraphProof

            ↳ QCSDP

              ↳ QCSDPForwardInstantiationProcessor

                ↳ QCSDP

                  ↳ QCSDPForwardInstantiationProcessor

                    ↳ QCSDP

                      ↳ QCSDependencyGraphProof

                        ↳ AND

                          ↳ QCSDP

                          ↳ QCSDP

                            ↳ QCSDPForwardInstantiationProcessor

                              ↳ QCSDP

                                ↳ QCSDPForwardInstantiationProcessor

                                  ↳ QCSDP

                                    ↳ QCSDependencyGraphProof

                                      ↳ AND

                                        ↳ QCSDP

                                        ↳ QCSDP

                                          ↳ QCSDPForwardInstantiationProcessor

                                            ↳ QCSDP

                                              ↳ QCSDPForwardInstantiationProcessor

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

U(dbl(dbls(z0))) → U(dbls(z0))
U(dbls(sel(z0, z1))) → U(sel(z0, z1))
SEL(0, cons(sel(z0, z1), y1)) → U(sel(z0, z1))
SEL(s(y0), cons(y1, dbls(sel(z0, z1)))) → U(dbls(sel(z0, z1)))
SEL(s(X), cons(Y, Z)) → SEL(X, Z)
U(dbls(dbl(indx(z0, z1)))) → U(dbl(indx(z0, z1)))
U(dbls(dbl(dbls(z0)))) → U(dbl(dbls(z0)))
SEL(s(indx(z0, z1)), cons(y1, y2)) → U(indx(z0, z1))
SEL(s(y0), cons(y1, dbl(dbls(z0)))) → U(dbl(dbls(z0)))
SEL(s(y0), cons(y1, dbls(indx(z0, z1)))) → U(dbls(indx(z0, z1)))
SEL(s(y0), cons(y1, dbls(dbl(sel(z0, z1))))) → U(dbls(dbl(sel(z0, z1))))
U(indx(x_0, x_1)) → U(x_0)
SEL(s(y0), cons(y1, dbls(dbl(indx(z0, z1))))) → U(dbls(dbl(indx(z0, z1))))
SEL(s(dbl(dbls(z0))), cons(y1, y2)) → U(dbl(dbls(z0)))
SEL(s(y0), cons(y1, dbl(indx(z0, z1)))) → U(dbl(indx(z0, z1)))
SEL(0, cons(dbls(z0), y1)) → U(dbls(z0))
SEL(s(dbls(z0)), cons(y1, y2)) → U(dbls(z0))
SEL(s(sel(z0, z1)), cons(y1, y2)) → U(sel(z0, z1))
SEL(s(y0), cons(y1, dbls(dbl(dbls(z0))))) → U(dbls(dbl(dbls(z0))))
SEL(s(y0), cons(y1, sel(z0, z1))) → U(sel(z0, z1))
SEL(s(y0), cons(y1, indx(z0, z1))) → U(indx(z0, z1))
SEL(s(dbl(sel(z0, z1))), cons(y1, y2)) → U(dbl(sel(z0, z1)))
SEL(s(y0), cons(y1, dbl(sel(z0, z1)))) → U(dbl(sel(z0, z1)))
SEL(s(dbl(indx(z0, z1))), cons(y1, y2)) → U(dbl(indx(z0, z1)))
SEL(0, cons(indx(z0, z1), y1)) → U(indx(z0, z1))
U(dbl(indx(z0, z1))) → U(indx(z0, z1))
SEL(0, cons(dbl(z0), y1)) → U(dbl(z0))
U(sel(X, Z)) → SEL(X, Z)
U(dbl(sel(z0, z1))) → U(sel(z0, z1))
U(dbls(dbl(sel(z0, z1)))) → U(dbl(sel(z0, z1)))
U(sel(x_0, x_1)) → U(x_0)
U(dbls(indx(z0, z1))) → U(indx(z0, z1))
U(sel(x_0, x_1)) → U(x_1)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

Using the Context-Sensitive Forward Instantiation Processor
the pair U(dbl(dbls(z0))) → U(dbls(z0))
was transformed to the following new pairs:

U(dbl(dbls(sel(z0, z1)))) → U(dbls(sel(z0, z1)))
U(dbl(dbls(indx(z0, z1)))) → U(dbls(indx(z0, z1)))
U(dbl(dbls(dbl(dbls(z0))))) → U(dbls(dbl(dbls(z0))))
U(dbl(dbls(dbl(sel(z0, z1))))) → U(dbls(dbl(sel(z0, z1))))
U(dbl(dbls(dbl(indx(z0, z1))))) → U(dbls(dbl(indx(z0, z1))))

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

        ↳ QCSDP

          ↳ QCSDependencyGraphProof

            ↳ QCSDP

              ↳ QCSDPForwardInstantiationProcessor

                ↳ QCSDP

                  ↳ QCSDPForwardInstantiationProcessor

                    ↳ QCSDP

                      ↳ QCSDependencyGraphProof

                        ↳ AND

                          ↳ QCSDP

                          ↳ QCSDP

                            ↳ QCSDPForwardInstantiationProcessor

                              ↳ QCSDP

                                ↳ QCSDPForwardInstantiationProcessor

                                  ↳ QCSDP

                                    ↳ QCSDependencyGraphProof

                                      ↳ AND

                                        ↳ QCSDP

                                        ↳ QCSDP

                                          ↳ QCSDPForwardInstantiationProcessor

                                            ↳ QCSDP

                                              ↳ QCSDPForwardInstantiationProcessor

                                                ↳ QCSDP

                                                  ↳ QCSDPForwardInstantiationProcessor

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

U(dbl(dbls(sel(z0, z1)))) → U(dbls(sel(z0, z1)))
SEL(0, cons(sel(z0, z1), y1)) → U(sel(z0, z1))
U(dbls(sel(z0, z1))) → U(sel(z0, z1))
SEL(s(y0), cons(y1, dbls(sel(z0, z1)))) → U(dbls(sel(z0, z1)))
U(dbl(dbls(dbl(dbls(z0))))) → U(dbls(dbl(dbls(z0))))
SEL(s(X), cons(Y, Z)) → SEL(X, Z)
U(dbls(dbl(indx(z0, z1)))) → U(dbl(indx(z0, z1)))
U(dbls(dbl(dbls(z0)))) → U(dbl(dbls(z0)))
SEL(s(indx(z0, z1)), cons(y1, y2)) → U(indx(z0, z1))
SEL(s(y0), cons(y1, dbl(dbls(z0)))) → U(dbl(dbls(z0)))
SEL(s(y0), cons(y1, dbls(indx(z0, z1)))) → U(dbls(indx(z0, z1)))
U(dbl(dbls(dbl(sel(z0, z1))))) → U(dbls(dbl(sel(z0, z1))))
SEL(s(y0), cons(y1, dbls(dbl(sel(z0, z1))))) → U(dbls(dbl(sel(z0, z1))))
U(indx(x_0, x_1)) → U(x_0)
SEL(s(y0), cons(y1, dbls(dbl(indx(z0, z1))))) → U(dbls(dbl(indx(z0, z1))))
SEL(s(dbl(dbls(z0))), cons(y1, y2)) → U(dbl(dbls(z0)))
SEL(s(y0), cons(y1, dbl(indx(z0, z1)))) → U(dbl(indx(z0, z1)))
SEL(0, cons(dbls(z0), y1)) → U(dbls(z0))
SEL(s(sel(z0, z1)), cons(y1, y2)) → U(sel(z0, z1))
SEL(s(dbls(z0)), cons(y1, y2)) → U(dbls(z0))
SEL(s(y0), cons(y1, dbls(dbl(dbls(z0))))) → U(dbls(dbl(dbls(z0))))
SEL(s(y0), cons(y1, sel(z0, z1))) → U(sel(z0, z1))
SEL(s(y0), cons(y1, indx(z0, z1))) → U(indx(z0, z1))
SEL(s(dbl(sel(z0, z1))), cons(y1, y2)) → U(dbl(sel(z0, z1)))
U(dbl(dbls(dbl(indx(z0, z1))))) → U(dbls(dbl(indx(z0, z1))))
SEL(s(y0), cons(y1, dbl(sel(z0, z1)))) → U(dbl(sel(z0, z1)))
SEL(s(dbl(indx(z0, z1))), cons(y1, y2)) → U(dbl(indx(z0, z1)))
SEL(0, cons(indx(z0, z1), y1)) → U(indx(z0, z1))
U(dbl(indx(z0, z1))) → U(indx(z0, z1))
U(sel(X, Z)) → SEL(X, Z)
SEL(0, cons(dbl(z0), y1)) → U(dbl(z0))
U(dbl(sel(z0, z1))) → U(sel(z0, z1))
U(dbls(dbl(sel(z0, z1)))) → U(dbl(sel(z0, z1)))
U(dbl(dbls(indx(z0, z1)))) → U(dbls(indx(z0, z1)))
U(sel(x_0, x_1)) → U(x_0)
U(dbls(indx(z0, z1))) → U(indx(z0, z1))
U(sel(x_0, x_1)) → U(x_1)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

Using the Context-Sensitive Forward Instantiation Processor
the pair U(dbls(dbl(dbls(z0)))) → U(dbl(dbls(z0)))
was transformed to the following new pairs:

U(dbls(dbl(dbls(sel(z0, z1))))) → U(dbl(dbls(sel(z0, z1))))
U(dbls(dbl(dbls(indx(z0, z1))))) → U(dbl(dbls(indx(z0, z1))))
U(dbls(dbl(dbls(dbl(dbls(z0)))))) → U(dbl(dbls(dbl(dbls(z0)))))
U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) → U(dbl(dbls(dbl(sel(z0, z1)))))
U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) → U(dbl(dbls(dbl(indx(z0, z1)))))

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

        ↳ QCSDP

          ↳ QCSDependencyGraphProof

            ↳ QCSDP

              ↳ QCSDPForwardInstantiationProcessor

                ↳ QCSDP

                  ↳ QCSDPForwardInstantiationProcessor

                    ↳ QCSDP

                      ↳ QCSDependencyGraphProof

                        ↳ AND

                          ↳ QCSDP

                          ↳ QCSDP

                            ↳ QCSDPForwardInstantiationProcessor

                              ↳ QCSDP

                                ↳ QCSDPForwardInstantiationProcessor

                                  ↳ QCSDP

                                    ↳ QCSDependencyGraphProof

                                      ↳ AND

                                        ↳ QCSDP

                                        ↳ QCSDP

                                          ↳ QCSDPForwardInstantiationProcessor

                                            ↳ QCSDP

                                              ↳ QCSDPForwardInstantiationProcessor

                                                ↳ QCSDP

                                                  ↳ QCSDPForwardInstantiationProcessor

                                                    ↳ QCSDP

                                                      ↳ QCSDPForwardInstantiationProcessor

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

U(dbl(dbls(sel(z0, z1)))) → U(dbls(sel(z0, z1)))
U(dbls(sel(z0, z1))) → U(sel(z0, z1))
SEL(0, cons(sel(z0, z1), y1)) → U(sel(z0, z1))
SEL(s(y0), cons(y1, dbls(sel(z0, z1)))) → U(dbls(sel(z0, z1)))
U(dbl(dbls(dbl(dbls(z0))))) → U(dbls(dbl(dbls(z0))))
SEL(s(X), cons(Y, Z)) → SEL(X, Z)
U(dbls(dbl(dbls(dbl(dbls(z0)))))) → U(dbl(dbls(dbl(dbls(z0)))))
U(dbls(dbl(indx(z0, z1)))) → U(dbl(indx(z0, z1)))
SEL(s(indx(z0, z1)), cons(y1, y2)) → U(indx(z0, z1))
SEL(s(y0), cons(y1, dbl(dbls(z0)))) → U(dbl(dbls(z0)))
SEL(s(y0), cons(y1, dbls(indx(z0, z1)))) → U(dbls(indx(z0, z1)))
U(dbl(dbls(dbl(sel(z0, z1))))) → U(dbls(dbl(sel(z0, z1))))
SEL(s(y0), cons(y1, dbls(dbl(sel(z0, z1))))) → U(dbls(dbl(sel(z0, z1))))
U(indx(x_0, x_1)) → U(x_0)
U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) → U(dbl(dbls(dbl(sel(z0, z1)))))
U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) → U(dbl(dbls(dbl(indx(z0, z1)))))
SEL(s(y0), cons(y1, dbls(dbl(indx(z0, z1))))) → U(dbls(dbl(indx(z0, z1))))
SEL(s(dbl(dbls(z0))), cons(y1, y2)) → U(dbl(dbls(z0)))
SEL(s(y0), cons(y1, dbl(indx(z0, z1)))) → U(dbl(indx(z0, z1)))
SEL(0, cons(dbls(z0), y1)) → U(dbls(z0))
SEL(s(dbls(z0)), cons(y1, y2)) → U(dbls(z0))
SEL(s(sel(z0, z1)), cons(y1, y2)) → U(sel(z0, z1))
SEL(s(y0), cons(y1, dbls(dbl(dbls(z0))))) → U(dbls(dbl(dbls(z0))))
SEL(s(y0), cons(y1, sel(z0, z1))) → U(sel(z0, z1))
U(dbls(dbl(dbls(indx(z0, z1))))) → U(dbl(dbls(indx(z0, z1))))
SEL(s(y0), cons(y1, indx(z0, z1))) → U(indx(z0, z1))
SEL(s(dbl(sel(z0, z1))), cons(y1, y2)) → U(dbl(sel(z0, z1)))
U(dbl(dbls(dbl(indx(z0, z1))))) → U(dbls(dbl(indx(z0, z1))))
SEL(s(y0), cons(y1, dbl(sel(z0, z1)))) → U(dbl(sel(z0, z1)))
SEL(s(dbl(indx(z0, z1))), cons(y1, y2)) → U(dbl(indx(z0, z1)))
SEL(0, cons(indx(z0, z1), y1)) → U(indx(z0, z1))
U(dbl(indx(z0, z1))) → U(indx(z0, z1))
SEL(0, cons(dbl(z0), y1)) → U(dbl(z0))
U(sel(X, Z)) → SEL(X, Z)
U(dbls(dbl(dbls(sel(z0, z1))))) → U(dbl(dbls(sel(z0, z1))))
U(dbl(sel(z0, z1))) → U(sel(z0, z1))
U(dbls(dbl(sel(z0, z1)))) → U(dbl(sel(z0, z1)))
U(sel(x_0, x_1)) → U(x_0)
U(dbl(dbls(indx(z0, z1)))) → U(dbls(indx(z0, z1)))
U(dbls(indx(z0, z1))) → U(indx(z0, z1))
U(sel(x_0, x_1)) → U(x_1)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

Using the Context-Sensitive Forward Instantiation Processor
the pair U(dbl(dbls(dbl(dbls(z0))))) → U(dbls(dbl(dbls(z0))))
was transformed to the following new pairs:

U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) → U(dbls(dbl(dbls(sel(z0, z1)))))
U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) → U(dbls(dbl(dbls(indx(z0, z1)))))
U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) → U(dbls(dbl(dbls(dbl(dbls(z0))))))
U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) → U(dbls(dbl(dbls(dbl(sel(z0, z1))))))
U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) → U(dbls(dbl(dbls(dbl(indx(z0, z1))))))

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

        ↳ QCSDP

          ↳ QCSDependencyGraphProof

            ↳ QCSDP

              ↳ QCSDPForwardInstantiationProcessor

                ↳ QCSDP

                  ↳ QCSDPForwardInstantiationProcessor

                    ↳ QCSDP

                      ↳ QCSDependencyGraphProof

                        ↳ AND

                          ↳ QCSDP

                          ↳ QCSDP

                            ↳ QCSDPForwardInstantiationProcessor

                              ↳ QCSDP

                                ↳ QCSDPForwardInstantiationProcessor

                                  ↳ QCSDP

                                    ↳ QCSDependencyGraphProof

                                      ↳ AND

                                        ↳ QCSDP

                                        ↳ QCSDP

                                          ↳ QCSDPForwardInstantiationProcessor

                                            ↳ QCSDP

                                              ↳ QCSDPForwardInstantiationProcessor

                                                ↳ QCSDP

                                                  ↳ QCSDPForwardInstantiationProcessor

                                                    ↳ QCSDP

                                                      ↳ QCSDPForwardInstantiationProcessor

                                                        ↳ QCSDP

                                                          ↳ QCSDPForwardInstantiationProcessor

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

U(dbl(dbls(sel(z0, z1)))) → U(dbls(sel(z0, z1)))
U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) → U(dbls(dbl(dbls(indx(z0, z1)))))
SEL(0, cons(sel(z0, z1), y1)) → U(sel(z0, z1))
U(dbls(sel(z0, z1))) → U(sel(z0, z1))
SEL(s(y0), cons(y1, dbls(sel(z0, z1)))) → U(dbls(sel(z0, z1)))
SEL(s(X), cons(Y, Z)) → SEL(X, Z)
U(dbls(dbl(dbls(dbl(dbls(z0)))))) → U(dbl(dbls(dbl(dbls(z0)))))
U(dbls(dbl(indx(z0, z1)))) → U(dbl(indx(z0, z1)))
SEL(s(indx(z0, z1)), cons(y1, y2)) → U(indx(z0, z1))
SEL(s(y0), cons(y1, dbl(dbls(z0)))) → U(dbl(dbls(z0)))
U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) → U(dbls(dbl(dbls(dbl(sel(z0, z1))))))
SEL(s(y0), cons(y1, dbls(indx(z0, z1)))) → U(dbls(indx(z0, z1)))
U(dbl(dbls(dbl(sel(z0, z1))))) → U(dbls(dbl(sel(z0, z1))))
SEL(s(y0), cons(y1, dbls(dbl(sel(z0, z1))))) → U(dbls(dbl(sel(z0, z1))))
U(indx(x_0, x_1)) → U(x_0)
U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) → U(dbl(dbls(dbl(sel(z0, z1)))))
U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) → U(dbl(dbls(dbl(indx(z0, z1)))))
SEL(s(y0), cons(y1, dbls(dbl(indx(z0, z1))))) → U(dbls(dbl(indx(z0, z1))))
SEL(s(dbl(dbls(z0))), cons(y1, y2)) → U(dbl(dbls(z0)))
U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) → U(dbls(dbl(dbls(dbl(indx(z0, z1))))))
SEL(s(y0), cons(y1, dbl(indx(z0, z1)))) → U(dbl(indx(z0, z1)))
SEL(0, cons(dbls(z0), y1)) → U(dbls(z0))
SEL(s(sel(z0, z1)), cons(y1, y2)) → U(sel(z0, z1))
SEL(s(dbls(z0)), cons(y1, y2)) → U(dbls(z0))
SEL(s(y0), cons(y1, dbls(dbl(dbls(z0))))) → U(dbls(dbl(dbls(z0))))
SEL(s(y0), cons(y1, sel(z0, z1))) → U(sel(z0, z1))
U(dbls(dbl(dbls(indx(z0, z1))))) → U(dbl(dbls(indx(z0, z1))))
SEL(s(y0), cons(y1, indx(z0, z1))) → U(indx(z0, z1))
SEL(s(dbl(sel(z0, z1))), cons(y1, y2)) → U(dbl(sel(z0, z1)))
U(dbl(dbls(dbl(indx(z0, z1))))) → U(dbls(dbl(indx(z0, z1))))
SEL(s(y0), cons(y1, dbl(sel(z0, z1)))) → U(dbl(sel(z0, z1)))
SEL(s(dbl(indx(z0, z1))), cons(y1, y2)) → U(dbl(indx(z0, z1)))
SEL(0, cons(indx(z0, z1), y1)) → U(indx(z0, z1))
U(dbl(indx(z0, z1))) → U(indx(z0, z1))
U(sel(X, Z)) → SEL(X, Z)
SEL(0, cons(dbl(z0), y1)) → U(dbl(z0))
U(dbls(dbl(dbls(sel(z0, z1))))) → U(dbl(dbls(sel(z0, z1))))
U(dbl(sel(z0, z1))) → U(sel(z0, z1))
U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) → U(dbls(dbl(dbls(dbl(dbls(z0))))))
U(dbls(dbl(sel(z0, z1)))) → U(dbl(sel(z0, z1)))
U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) → U(dbls(dbl(dbls(sel(z0, z1)))))
U(dbl(dbls(indx(z0, z1)))) → U(dbls(indx(z0, z1)))
U(sel(x_0, x_1)) → U(x_0)
U(dbls(indx(z0, z1))) → U(indx(z0, z1))
U(sel(x_0, x_1)) → U(x_1)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

Using the Context-Sensitive Forward Instantiation Processor
the pair U(dbls(dbl(dbls(dbl(dbls(z0)))))) → U(dbl(dbls(dbl(dbls(z0)))))
was transformed to the following new pairs:

U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) → U(dbl(dbls(dbl(dbls(sel(z0, z1))))))
U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) → U(dbl(dbls(dbl(dbls(indx(z0, z1))))))
U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) → U(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))
U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) → U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))
U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) → U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

        ↳ QCSDP

          ↳ QCSDependencyGraphProof

            ↳ QCSDP

              ↳ QCSDPForwardInstantiationProcessor

                ↳ QCSDP

                  ↳ QCSDPForwardInstantiationProcessor

                    ↳ QCSDP

                      ↳ QCSDependencyGraphProof

                        ↳ AND

                          ↳ QCSDP

                          ↳ QCSDP

                            ↳ QCSDPForwardInstantiationProcessor

                              ↳ QCSDP

                                ↳ QCSDPForwardInstantiationProcessor

                                  ↳ QCSDP

                                    ↳ QCSDependencyGraphProof

                                      ↳ AND

                                        ↳ QCSDP

                                        ↳ QCSDP

                                          ↳ QCSDPForwardInstantiationProcessor

                                            ↳ QCSDP

                                              ↳ QCSDPForwardInstantiationProcessor

                                                ↳ QCSDP

                                                  ↳ QCSDPForwardInstantiationProcessor

                                                    ↳ QCSDP

                                                      ↳ QCSDPForwardInstantiationProcessor

                                                        ↳ QCSDP

                                                          ↳ QCSDPForwardInstantiationProcessor

                                                            ↳ QCSDP

                                                              ↳ QCSDPForwardInstantiationProcessor

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

U(dbl(dbls(sel(z0, z1)))) → U(dbls(sel(z0, z1)))
U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) → U(dbls(dbl(dbls(indx(z0, z1)))))
U(dbls(sel(z0, z1))) → U(sel(z0, z1))
SEL(s(y0), cons(y1, dbls(sel(z0, z1)))) → U(dbls(sel(z0, z1)))
U(dbls(dbl(indx(z0, z1)))) → U(dbl(indx(z0, z1)))
SEL(s(indx(z0, z1)), cons(y1, y2)) → U(indx(z0, z1))
SEL(s(y0), cons(y1, dbl(dbls(z0)))) → U(dbl(dbls(z0)))
U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) → U(dbls(dbl(dbls(dbl(sel(z0, z1))))))
SEL(s(y0), cons(y1, dbls(indx(z0, z1)))) → U(dbls(indx(z0, z1)))
U(dbl(dbls(dbl(sel(z0, z1))))) → U(dbls(dbl(sel(z0, z1))))
SEL(s(y0), cons(y1, dbls(dbl(sel(z0, z1))))) → U(dbls(dbl(sel(z0, z1))))
U(indx(x_0, x_1)) → U(x_0)
U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) → U(dbl(dbls(dbl(indx(z0, z1)))))
U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) → U(dbls(dbl(dbls(dbl(indx(z0, z1))))))
SEL(s(sel(z0, z1)), cons(y1, y2)) → U(sel(z0, z1))
SEL(s(y0), cons(y1, sel(z0, z1))) → U(sel(z0, z1))
SEL(s(dbl(sel(z0, z1))), cons(y1, y2)) → U(dbl(sel(z0, z1)))
U(dbl(dbls(dbl(indx(z0, z1))))) → U(dbls(dbl(indx(z0, z1))))
SEL(s(y0), cons(y1, dbl(sel(z0, z1)))) → U(dbl(sel(z0, z1)))
SEL(0, cons(indx(z0, z1), y1)) → U(indx(z0, z1))
U(sel(X, Z)) → SEL(X, Z)
SEL(0, cons(dbl(z0), y1)) → U(dbl(z0))
U(dbl(sel(z0, z1))) → U(sel(z0, z1))
U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) → U(dbls(dbl(dbls(dbl(dbls(z0))))))
U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) → U(dbls(dbl(dbls(sel(z0, z1)))))
U(dbl(dbls(indx(z0, z1)))) → U(dbls(indx(z0, z1)))
U(sel(x_0, x_1)) → U(x_1)
SEL(0, cons(sel(z0, z1), y1)) → U(sel(z0, z1))
SEL(s(X), cons(Y, Z)) → SEL(X, Z)
U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) → U(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))
U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) → U(dbl(dbls(dbl(dbls(sel(z0, z1))))))
U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) → U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))
U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) → U(dbl(dbls(dbl(sel(z0, z1)))))
SEL(s(y0), cons(y1, dbls(dbl(indx(z0, z1))))) → U(dbls(dbl(indx(z0, z1))))
SEL(s(dbl(dbls(z0))), cons(y1, y2)) → U(dbl(dbls(z0)))
SEL(s(y0), cons(y1, dbl(indx(z0, z1)))) → U(dbl(indx(z0, z1)))
SEL(0, cons(dbls(z0), y1)) → U(dbls(z0))
SEL(s(dbls(z0)), cons(y1, y2)) → U(dbls(z0))
SEL(s(y0), cons(y1, dbls(dbl(dbls(z0))))) → U(dbls(dbl(dbls(z0))))
U(dbls(dbl(dbls(indx(z0, z1))))) → U(dbl(dbls(indx(z0, z1))))
SEL(s(y0), cons(y1, indx(z0, z1))) → U(indx(z0, z1))
U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) → U(dbl(dbls(dbl(dbls(indx(z0, z1))))))
SEL(s(dbl(indx(z0, z1))), cons(y1, y2)) → U(dbl(indx(z0, z1)))
U(dbl(indx(z0, z1))) → U(indx(z0, z1))
U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) → U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))
U(dbls(dbl(dbls(sel(z0, z1))))) → U(dbl(dbls(sel(z0, z1))))
U(dbls(dbl(sel(z0, z1)))) → U(dbl(sel(z0, z1)))
U(sel(x_0, x_1)) → U(x_0)
U(dbls(indx(z0, z1))) → U(indx(z0, z1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

Using the Context-Sensitive Forward Instantiation Processor
the pair SEL(s(y0), cons(y1, dbl(dbls(z0)))) → U(dbl(dbls(z0)))
was transformed to the following new pairs:

SEL(s(y0), cons(y1, dbl(dbls(sel(z0, z1))))) → U(dbl(dbls(sel(z0, z1))))
SEL(s(y0), cons(y1, dbl(dbls(indx(z0, z1))))) → U(dbl(dbls(indx(z0, z1))))
SEL(s(y0), cons(y1, dbl(dbls(dbl(sel(z0, z1)))))) → U(dbl(dbls(dbl(sel(z0, z1)))))
SEL(s(y0), cons(y1, dbl(dbls(dbl(indx(z0, z1)))))) → U(dbl(dbls(dbl(indx(z0, z1)))))
SEL(s(y0), cons(y1, dbl(dbls(dbl(dbls(sel(z0, z1))))))) → U(dbl(dbls(dbl(dbls(sel(z0, z1))))))
SEL(s(y0), cons(y1, dbl(dbls(dbl(dbls(indx(z0, z1))))))) → U(dbl(dbls(dbl(dbls(indx(z0, z1))))))
SEL(s(y0), cons(y1, dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) → U(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))
SEL(s(y0), cons(y1, dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) → U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))
SEL(s(y0), cons(y1, dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) → U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

        ↳ QCSDP

          ↳ QCSDependencyGraphProof

            ↳ QCSDP

              ↳ QCSDPForwardInstantiationProcessor

                ↳ QCSDP

                  ↳ QCSDPForwardInstantiationProcessor

                    ↳ QCSDP

                      ↳ QCSDependencyGraphProof

                        ↳ AND

                          ↳ QCSDP

                          ↳ QCSDP

                            ↳ QCSDPForwardInstantiationProcessor

                              ↳ QCSDP

                                ↳ QCSDPForwardInstantiationProcessor

                                  ↳ QCSDP

                                    ↳ QCSDependencyGraphProof

                                      ↳ AND

                                        ↳ QCSDP

                                        ↳ QCSDP

                                          ↳ QCSDPForwardInstantiationProcessor

                                            ↳ QCSDP

                                              ↳ QCSDPForwardInstantiationProcessor

                                                ↳ QCSDP

                                                  ↳ QCSDPForwardInstantiationProcessor

                                                    ↳ QCSDP

                                                      ↳ QCSDPForwardInstantiationProcessor

                                                        ↳ QCSDP

                                                          ↳ QCSDPForwardInstantiationProcessor

                                                            ↳ QCSDP

                                                              ↳ QCSDPForwardInstantiationProcessor

                                                                ↳ QCSDP

                                                                  ↳ QCSDPForwardInstantiationProcessor

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

U(dbl(dbls(sel(z0, z1)))) → U(dbls(sel(z0, z1)))
U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) → U(dbls(dbl(dbls(indx(z0, z1)))))
U(dbls(sel(z0, z1))) → U(sel(z0, z1))
SEL(s(y0), cons(y1, dbls(sel(z0, z1)))) → U(dbls(sel(z0, z1)))
U(dbls(dbl(indx(z0, z1)))) → U(dbl(indx(z0, z1)))
SEL(s(indx(z0, z1)), cons(y1, y2)) → U(indx(z0, z1))
U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) → U(dbls(dbl(dbls(dbl(sel(z0, z1))))))
SEL(s(y0), cons(y1, dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) → U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))
SEL(s(y0), cons(y1, dbls(indx(z0, z1)))) → U(dbls(indx(z0, z1)))
U(dbl(dbls(dbl(sel(z0, z1))))) → U(dbls(dbl(sel(z0, z1))))
SEL(s(y0), cons(y1, dbls(dbl(sel(z0, z1))))) → U(dbls(dbl(sel(z0, z1))))
U(indx(x_0, x_1)) → U(x_0)
U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) → U(dbl(dbls(dbl(indx(z0, z1)))))
U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) → U(dbls(dbl(dbls(dbl(indx(z0, z1))))))
SEL(s(sel(z0, z1)), cons(y1, y2)) → U(sel(z0, z1))
SEL(s(y0), cons(y1, sel(z0, z1))) → U(sel(z0, z1))
SEL(s(y0), cons(y1, dbl(dbls(sel(z0, z1))))) → U(dbl(dbls(sel(z0, z1))))
SEL(s(dbl(sel(z0, z1))), cons(y1, y2)) → U(dbl(sel(z0, z1)))
U(dbl(dbls(dbl(indx(z0, z1))))) → U(dbls(dbl(indx(z0, z1))))
SEL(s(y0), cons(y1, dbl(sel(z0, z1)))) → U(dbl(sel(z0, z1)))
SEL(0, cons(indx(z0, z1), y1)) → U(indx(z0, z1))
SEL(0, cons(dbl(z0), y1)) → U(dbl(z0))
U(sel(X, Z)) → SEL(X, Z)
U(dbl(sel(z0, z1))) → U(sel(z0, z1))
U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) → U(dbls(dbl(dbls(dbl(dbls(z0))))))
U(dbl(dbls(indx(z0, z1)))) → U(dbls(indx(z0, z1)))
U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) → U(dbls(dbl(dbls(sel(z0, z1)))))
U(sel(x_0, x_1)) → U(x_1)
SEL(s(y0), cons(y1, dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) → U(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))
SEL(0, cons(sel(z0, z1), y1)) → U(sel(z0, z1))
SEL(s(y0), cons(y1, dbl(dbls(dbl(dbls(indx(z0, z1))))))) → U(dbl(dbls(dbl(dbls(indx(z0, z1))))))
SEL(s(y0), cons(y1, dbl(dbls(indx(z0, z1))))) → U(dbl(dbls(indx(z0, z1))))
U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) → U(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))
SEL(s(X), cons(Y, Z)) → SEL(X, Z)
SEL(s(y0), cons(y1, dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) → U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))
U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) → U(dbl(dbls(dbl(dbls(sel(z0, z1))))))
SEL(s(y0), cons(y1, dbl(dbls(dbl(indx(z0, z1)))))) → U(dbl(dbls(dbl(indx(z0, z1)))))
SEL(s(y0), cons(y1, dbl(dbls(dbl(sel(z0, z1)))))) → U(dbl(dbls(dbl(sel(z0, z1)))))
U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) → U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))
U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) → U(dbl(dbls(dbl(sel(z0, z1)))))
SEL(s(y0), cons(y1, dbls(dbl(indx(z0, z1))))) → U(dbls(dbl(indx(z0, z1))))
SEL(s(dbl(dbls(z0))), cons(y1, y2)) → U(dbl(dbls(z0)))
SEL(s(y0), cons(y1, dbl(indx(z0, z1)))) → U(dbl(indx(z0, z1)))
SEL(0, cons(dbls(z0), y1)) → U(dbls(z0))
SEL(s(dbls(z0)), cons(y1, y2)) → U(dbls(z0))
SEL(s(y0), cons(y1, dbls(dbl(dbls(z0))))) → U(dbls(dbl(dbls(z0))))
U(dbls(dbl(dbls(indx(z0, z1))))) → U(dbl(dbls(indx(z0, z1))))
SEL(s(y0), cons(y1, indx(z0, z1))) → U(indx(z0, z1))
U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) → U(dbl(dbls(dbl(dbls(indx(z0, z1))))))
SEL(s(dbl(indx(z0, z1))), cons(y1, y2)) → U(dbl(indx(z0, z1)))
U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) → U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))
U(dbl(indx(z0, z1))) → U(indx(z0, z1))
U(dbls(dbl(dbls(sel(z0, z1))))) → U(dbl(dbls(sel(z0, z1))))
U(dbls(dbl(sel(z0, z1)))) → U(dbl(sel(z0, z1)))
SEL(s(y0), cons(y1, dbl(dbls(dbl(dbls(sel(z0, z1))))))) → U(dbl(dbls(dbl(dbls(sel(z0, z1))))))
U(sel(x_0, x_1)) → U(x_0)
U(dbls(indx(z0, z1))) → U(indx(z0, z1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

Using the Context-Sensitive Forward Instantiation Processor
the pair U(indx(x_0, x_1)) → U(x_0)
was transformed to the following new pairs:

U(indx(sel(z0, z1), y1)) → U(sel(z0, z1))
U(indx(indx(z0, z1), y1)) → U(indx(z0, z1))
U(indx(dbl(sel(z0, z1)), y1)) → U(dbl(sel(z0, z1)))
U(indx(dbl(indx(z0, z1)), y1)) → U(dbl(indx(z0, z1)))
U(indx(dbls(sel(z0, z1)), y1)) → U(dbls(sel(z0, z1)))
U(indx(dbls(indx(z0, z1)), y1)) → U(dbls(indx(z0, z1)))
U(indx(dbls(dbl(sel(z0, z1))), y1)) → U(dbls(dbl(sel(z0, z1))))
U(indx(dbls(dbl(indx(z0, z1))), y1)) → U(dbls(dbl(indx(z0, z1))))
U(indx(dbl(dbls(sel(z0, z1))), y1)) → U(dbl(dbls(sel(z0, z1))))
U(indx(dbl(dbls(indx(z0, z1))), y1)) → U(dbl(dbls(indx(z0, z1))))
U(indx(dbl(dbls(dbl(sel(z0, z1)))), y1)) → U(dbl(dbls(dbl(sel(z0, z1)))))
U(indx(dbl(dbls(dbl(indx(z0, z1)))), y1)) → U(dbl(dbls(dbl(indx(z0, z1)))))
U(indx(dbls(dbl(dbls(sel(z0, z1)))), y1)) → U(dbls(dbl(dbls(sel(z0, z1)))))
U(indx(dbls(dbl(dbls(indx(z0, z1)))), y1)) → U(dbls(dbl(dbls(indx(z0, z1)))))
U(indx(dbls(dbl(dbls(dbl(sel(z0, z1))))), y1)) → U(dbls(dbl(dbls(dbl(sel(z0, z1))))))
U(indx(dbls(dbl(dbls(dbl(indx(z0, z1))))), y1)) → U(dbls(dbl(dbls(dbl(indx(z0, z1))))))
U(indx(dbl(dbls(dbl(dbls(sel(z0, z1))))), y1)) → U(dbl(dbls(dbl(dbls(sel(z0, z1))))))
U(indx(dbl(dbls(dbl(dbls(indx(z0, z1))))), y1)) → U(dbl(dbls(dbl(dbls(indx(z0, z1))))))
U(indx(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))), y1)) → U(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))
U(indx(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))), y1)) → U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))
U(indx(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))), y1)) → U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))
U(indx(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))), y1)) → U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))))
U(indx(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))), y1)) → U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))))
U(indx(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))), y1)) → U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))))
U(indx(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))), y1)) → U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))))
U(indx(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))), y1)) → U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))))

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

        ↳ QCSDP

          ↳ QCSDependencyGraphProof

            ↳ QCSDP

              ↳ QCSDPForwardInstantiationProcessor

                ↳ QCSDP

                  ↳ QCSDPForwardInstantiationProcessor

                    ↳ QCSDP

                      ↳ QCSDependencyGraphProof

                        ↳ AND

                          ↳ QCSDP

                          ↳ QCSDP

                            ↳ QCSDPForwardInstantiationProcessor

                              ↳ QCSDP

                                ↳ QCSDPForwardInstantiationProcessor

                                  ↳ QCSDP

                                    ↳ QCSDependencyGraphProof

                                      ↳ AND

                                        ↳ QCSDP

                                        ↳ QCSDP

                                          ↳ QCSDPForwardInstantiationProcessor

                                            ↳ QCSDP

                                              ↳ QCSDPForwardInstantiationProcessor

                                                ↳ QCSDP

                                                  ↳ QCSDPForwardInstantiationProcessor

                                                    ↳ QCSDP

                                                      ↳ QCSDPForwardInstantiationProcessor

                                                        ↳ QCSDP

                                                          ↳ QCSDPForwardInstantiationProcessor

                                                            ↳ QCSDP

                                                              ↳ QCSDPForwardInstantiationProcessor

                                                                ↳ QCSDP

                                                                  ↳ QCSDPForwardInstantiationProcessor

                                                                    ↳ QCSDP

                                                                      ↳ QCSDependencyGraphProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

U(indx(dbls(dbl(indx(z0, z1))), y1)) → U(dbls(dbl(indx(z0, z1))))
U(dbl(dbls(sel(z0, z1)))) → U(dbls(sel(z0, z1)))
U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) → U(dbls(dbl(dbls(indx(z0, z1)))))
U(dbls(sel(z0, z1))) → U(sel(z0, z1))
U(indx(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))), y1)) → U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))
SEL(s(y0), cons(y1, dbls(sel(z0, z1)))) → U(dbls(sel(z0, z1)))
U(dbls(dbl(indx(z0, z1)))) → U(dbl(indx(z0, z1)))
SEL(s(indx(z0, z1)), cons(y1, y2)) → U(indx(z0, z1))
U(indx(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))), y1)) → U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))))
U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) → U(dbls(dbl(dbls(dbl(sel(z0, z1))))))
SEL(s(y0), cons(y1, dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) → U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))
SEL(s(y0), cons(y1, dbls(indx(z0, z1)))) → U(dbls(indx(z0, z1)))
U(dbl(dbls(dbl(sel(z0, z1))))) → U(dbls(dbl(sel(z0, z1))))
SEL(s(y0), cons(y1, dbls(dbl(sel(z0, z1))))) → U(dbls(dbl(sel(z0, z1))))
U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) → U(dbl(dbls(dbl(indx(z0, z1)))))
U(indx(dbl(dbls(dbl(dbls(indx(z0, z1))))), y1)) → U(dbl(dbls(dbl(dbls(indx(z0, z1))))))
U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) → U(dbls(dbl(dbls(dbl(indx(z0, z1))))))
SEL(s(sel(z0, z1)), cons(y1, y2)) → U(sel(z0, z1))
U(indx(dbls(indx(z0, z1)), y1)) → U(dbls(indx(z0, z1)))
U(indx(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))), y1)) → U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))))
SEL(s(y0), cons(y1, sel(z0, z1))) → U(sel(z0, z1))
U(indx(indx(z0, z1), y1)) → U(indx(z0, z1))
U(indx(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))), y1)) → U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))))
SEL(s(y0), cons(y1, dbl(dbls(sel(z0, z1))))) → U(dbl(dbls(sel(z0, z1))))
SEL(s(dbl(sel(z0, z1))), cons(y1, y2)) → U(dbl(sel(z0, z1)))
U(dbl(dbls(dbl(indx(z0, z1))))) → U(dbls(dbl(indx(z0, z1))))
U(indx(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))), y1)) → U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))))
SEL(s(y0), cons(y1, dbl(sel(z0, z1)))) → U(dbl(sel(z0, z1)))
U(indx(dbls(sel(z0, z1)), y1)) → U(dbls(sel(z0, z1)))
SEL(0, cons(indx(z0, z1), y1)) → U(indx(z0, z1))
U(sel(X, Z)) → SEL(X, Z)
SEL(0, cons(dbl(z0), y1)) → U(dbl(z0))
U(dbl(sel(z0, z1))) → U(sel(z0, z1))
U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) → U(dbls(dbl(dbls(dbl(dbls(z0))))))
U(indx(sel(z0, z1), y1)) → U(sel(z0, z1))
U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) → U(dbls(dbl(dbls(sel(z0, z1)))))
U(dbl(dbls(indx(z0, z1)))) → U(dbls(indx(z0, z1)))
U(sel(x_0, x_1)) → U(x_1)
U(indx(dbls(dbl(dbls(dbl(sel(z0, z1))))), y1)) → U(dbls(dbl(dbls(dbl(sel(z0, z1))))))
U(indx(dbls(dbl(dbls(sel(z0, z1)))), y1)) → U(dbls(dbl(dbls(sel(z0, z1)))))
SEL(0, cons(sel(z0, z1), y1)) → U(sel(z0, z1))
SEL(s(y0), cons(y1, dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) → U(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))
SEL(s(y0), cons(y1, dbl(dbls(dbl(dbls(indx(z0, z1))))))) → U(dbl(dbls(dbl(dbls(indx(z0, z1))))))
SEL(s(y0), cons(y1, dbl(dbls(indx(z0, z1))))) → U(dbl(dbls(indx(z0, z1))))
SEL(s(X), cons(Y, Z)) → SEL(X, Z)
U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) → U(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))
U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) → U(dbl(dbls(dbl(dbls(sel(z0, z1))))))
SEL(s(y0), cons(y1, dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) → U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))
U(indx(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))), y1)) → U(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))
SEL(s(y0), cons(y1, dbl(dbls(dbl(indx(z0, z1)))))) → U(dbl(dbls(dbl(indx(z0, z1)))))
U(indx(dbl(dbls(dbl(sel(z0, z1)))), y1)) → U(dbl(dbls(dbl(sel(z0, z1)))))
U(indx(dbl(indx(z0, z1)), y1)) → U(dbl(indx(z0, z1)))
U(indx(dbl(dbls(indx(z0, z1))), y1)) → U(dbl(dbls(indx(z0, z1))))
SEL(s(y0), cons(y1, dbl(dbls(dbl(sel(z0, z1)))))) → U(dbl(dbls(dbl(sel(z0, z1)))))
U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) → U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))
U(indx(dbls(dbl(dbls(dbl(indx(z0, z1))))), y1)) → U(dbls(dbl(dbls(dbl(indx(z0, z1))))))
U(indx(dbl(dbls(dbl(indx(z0, z1)))), y1)) → U(dbl(dbls(dbl(indx(z0, z1)))))
U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) → U(dbl(dbls(dbl(sel(z0, z1)))))
SEL(s(y0), cons(y1, dbls(dbl(indx(z0, z1))))) → U(dbls(dbl(indx(z0, z1))))
U(indx(dbls(dbl(dbls(indx(z0, z1)))), y1)) → U(dbls(dbl(dbls(indx(z0, z1)))))
SEL(s(dbl(dbls(z0))), cons(y1, y2)) → U(dbl(dbls(z0)))
U(indx(dbls(dbl(sel(z0, z1))), y1)) → U(dbls(dbl(sel(z0, z1))))
SEL(s(y0), cons(y1, dbl(indx(z0, z1)))) → U(dbl(indx(z0, z1)))
U(indx(dbl(dbls(sel(z0, z1))), y1)) → U(dbl(dbls(sel(z0, z1))))
U(indx(dbl(sel(z0, z1)), y1)) → U(dbl(sel(z0, z1)))
SEL(0, cons(dbls(z0), y1)) → U(dbls(z0))
U(indx(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))), y1)) → U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))
SEL(s(dbls(z0)), cons(y1, y2)) → U(dbls(z0))
SEL(s(y0), cons(y1, dbls(dbl(dbls(z0))))) → U(dbls(dbl(dbls(z0))))
U(dbls(dbl(dbls(indx(z0, z1))))) → U(dbl(dbls(indx(z0, z1))))
SEL(s(y0), cons(y1, indx(z0, z1))) → U(indx(z0, z1))
U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) → U(dbl(dbls(dbl(dbls(indx(z0, z1))))))
SEL(s(dbl(indx(z0, z1))), cons(y1, y2)) → U(dbl(indx(z0, z1)))
U(dbl(indx(z0, z1))) → U(indx(z0, z1))
U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) → U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))
U(indx(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))), y1)) → U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))))
U(indx(dbl(dbls(dbl(dbls(sel(z0, z1))))), y1)) → U(dbl(dbls(dbl(dbls(sel(z0, z1))))))
U(dbls(dbl(dbls(sel(z0, z1))))) → U(dbl(dbls(sel(z0, z1))))
U(dbls(dbl(sel(z0, z1)))) → U(dbl(sel(z0, z1)))
U(sel(x_0, x_1)) → U(x_0)
SEL(s(y0), cons(y1, dbl(dbls(dbl(dbls(sel(z0, z1))))))) → U(dbl(dbls(dbl(dbls(sel(z0, z1))))))
U(dbls(indx(z0, z1))) → U(indx(z0, z1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

The approximation of the Context-Sensitive Dependency Graph contains 2 SCCs.
The rules U(indx(sel(z0, z1), z2)) → U(sel(z0, z1)) and U(dbl(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(sel(z0, z1))) = U(sel(z0, z1)) does not unify with U(dbl(sel(x0, x1))). The rules U(indx(sel(z0, z1), z2)) → U(sel(z0, z1)) and U(dbl(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(sel(z0, z1))) = U(sel(z0, z1)) does not unify with U(dbl(indx(x0, x1))). The rules U(indx(sel(z0, z1), z2)) → U(sel(z0, z1)) and U(dbls(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(sel(z0, z1))) = U(sel(z0, z1)) does not unify with U(dbls(sel(x0, x1))). The rules U(indx(sel(z0, z1), z2)) → U(sel(z0, z1)) and U(dbls(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(sel(z0, z1))) = U(sel(z0, z1)) does not unify with U(dbls(indx(x0, x1))). The rules U(indx(sel(z0, z1), z2)) → U(sel(z0, z1)) and U(dbls(dbl(sel(x0, x1)))) → U(dbl(sel(x0, x1))) form no chain, because ECap^µ(U(sel(z0, z1))) = U(sel(z0, z1)) does not unify with U(dbls(dbl(sel(x0, x1)))). The rules U(indx(sel(z0, z1), z2)) → U(sel(z0, z1)) and U(dbls(dbl(indx(x0, x1)))) → U(dbl(indx(x0, x1))) form no chain, because ECap^µ(U(sel(z0, z1))) = U(sel(z0, z1)) does not unify with U(dbls(dbl(indx(x0, x1)))). The rules U(indx(sel(z0, z1), z2)) → U(sel(z0, z1)) and U(dbl(dbls(sel(x0, x1)))) → U(dbls(sel(x0, x1))) form no chain, because ECap^µ(U(sel(z0, z1))) = U(sel(z0, z1)) does not unify with U(dbl(dbls(sel(x0, x1)))). The rules U(indx(sel(z0, z1), z2)) → U(sel(z0, z1)) and U(dbl(dbls(indx(x0, x1)))) → U(dbls(indx(x0, x1))) form no chain, because ECap^µ(U(sel(z0, z1))) = U(sel(z0, z1)) does not unify with U(dbl(dbls(indx(x0, x1)))). The rules U(indx(sel(z0, z1), z2)) → U(sel(z0, z1)) and U(dbl(dbls(dbl(sel(x0, x1))))) → U(dbls(dbl(sel(x0, x1)))) form no chain, because ECap^µ(U(sel(z0, z1))) = U(sel(z0, z1)) does not unify with U(dbl(dbls(dbl(sel(x0, x1))))). The rules U(indx(sel(z0, z1), z2)) → U(sel(z0, z1)) and U(dbl(dbls(dbl(indx(x0, x1))))) → U(dbls(dbl(indx(x0, x1)))) form no chain, because ECap^µ(U(sel(z0, z1))) = U(sel(z0, z1)) does not unify with U(dbl(dbls(dbl(indx(x0, x1))))). The rules U(indx(sel(z0, z1), z2)) → U(sel(z0, z1)) and U(dbls(dbl(dbls(sel(x0, x1))))) → U(dbl(dbls(sel(x0, x1)))) form no chain, because ECap^µ(U(sel(z0, z1))) = U(sel(z0, z1)) does not unify with U(dbls(dbl(dbls(sel(x0, x1))))). The rules U(indx(sel(z0, z1), z2)) → U(sel(z0, z1)) and U(dbls(dbl(dbls(indx(x0, x1))))) → U(dbl(dbls(indx(x0, x1)))) form no chain, because ECap^µ(U(sel(z0, z1))) = U(sel(z0, z1)) does not unify with U(dbls(dbl(dbls(indx(x0, x1))))). The rules U(indx(sel(z0, z1), z2)) → U(sel(z0, z1)) and U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) → U(dbl(dbls(dbl(sel(x0, x1))))) form no chain, because ECap^µ(U(sel(z0, z1))) = U(sel(z0, z1)) does not unify with U(dbls(dbl(dbls(dbl(sel(x0, x1)))))). The rules U(indx(sel(z0, z1), z2)) → U(sel(z0, z1)) and U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) → U(dbl(dbls(dbl(indx(x0, x1))))) form no chain, because ECap^µ(U(sel(z0, z1))) = U(sel(z0, z1)) does not unify with U(dbls(dbl(dbls(dbl(indx(x0, x1)))))). The rules U(indx(sel(z0, z1), z2)) → U(sel(z0, z1)) and U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) → U(dbls(dbl(dbls(sel(x0, x1))))) form no chain, because ECap^µ(U(sel(z0, z1))) = U(sel(z0, z1)) does not unify with U(dbl(dbls(dbl(dbls(sel(x0, x1)))))). The rules U(indx(sel(z0, z1), z2)) → U(sel(z0, z1)) and U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) → U(dbls(dbl(dbls(indx(x0, x1))))) form no chain, because ECap^µ(U(sel(z0, z1))) = U(sel(z0, z1)) does not unify with U(dbl(dbls(dbl(dbls(indx(x0, x1)))))). The rules U(indx(sel(z0, z1), z2)) → U(sel(z0, z1)) and U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) → U(dbls(dbl(dbls(dbl(dbls(x0)))))) form no chain, because ECap^µ(U(sel(z0, z1))) = U(sel(z0, z1)) does not unify with U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))). The rules U(indx(sel(z0, z1), z2)) → U(sel(z0, z1)) and U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) → U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) form no chain, because ECap^µ(U(sel(z0, z1))) = U(sel(z0, z1)) does not unify with U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))). The rules U(indx(sel(z0, z1), z2)) → U(sel(z0, z1)) and U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) → U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) form no chain, because ECap^µ(U(sel(z0, z1))) = U(sel(z0, z1)) does not unify with U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))). The rules U(indx(sel(z0, z1), z2)) → U(sel(z0, z1)) and U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))) → U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) form no chain, because ECap^µ(U(sel(z0, z1))) = U(sel(z0, z1)) does not unify with U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))). The rules U(indx(sel(z0, z1), z2)) → U(sel(z0, z1)) and U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))) → U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) form no chain, because ECap^µ(U(sel(z0, z1))) = U(sel(z0, z1)) does not unify with U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))). The rules U(indx(sel(z0, z1), z2)) → U(sel(z0, z1)) and U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))) → U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) form no chain, because ECap^µ(U(sel(z0, z1))) = U(sel(z0, z1)) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))). The rules U(indx(sel(z0, z1), z2)) → U(sel(z0, z1)) and U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) form no chain, because ECap^µ(U(sel(z0, z1))) = U(sel(z0, z1)) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))). The rules U(indx(sel(z0, z1), z2)) → U(sel(z0, z1)) and U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) form no chain, because ECap^µ(U(sel(z0, z1))) = U(sel(z0, z1)) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))). The rules U(indx(sel(z0, z1), z2)) → U(sel(z0, z1)) and U(indx(sel(x0, x1), x2)) → U(sel(x0, x1)) form no chain, because ECap^µ(U(sel(z0, z1))) = U(sel(z0, z1)) does not unify with U(indx(sel(x0, x1), x2)). The rules U(indx(indx(z0, z1), z2)) → U(indx(z0, z1)) and U(sel(x0, x1)) → U(x0) form no chain, because ECap^µ(U(indx(z0, z1))) = U(indx(z0, z1)) does not unify with U(sel(x0, x1)). The rules U(indx(indx(z0, z1), z2)) → U(indx(z0, z1)) and U(sel(x0, x1)) → U(x1) form no chain, because ECap^µ(U(indx(z0, z1))) = U(indx(z0, z1)) does not unify with U(sel(x0, x1)). The rules U(indx(indx(z0, z1), z2)) → U(indx(z0, z1)) and U(sel(x0, x1)) → SEL(x0, x1) form no chain, because ECap^µ(U(indx(z0, z1))) = U(indx(z0, z1)) does not unify with U(sel(x0, x1)). The rules U(indx(indx(z0, z1), z2)) → U(indx(z0, z1)) and U(dbl(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(indx(z0, z1))) = U(indx(z0, z1)) does not unify with U(dbl(sel(x0, x1))). The rules U(indx(indx(z0, z1), z2)) → U(indx(z0, z1)) and U(dbl(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(indx(z0, z1))) = U(indx(z0, z1)) does not unify with U(dbl(indx(x0, x1))). The rules U(indx(indx(z0, z1), z2)) → U(indx(z0, z1)) and U(dbls(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(indx(z0, z1))) = U(indx(z0, z1)) does not unify with U(dbls(sel(x0, x1))). The rules U(indx(indx(z0, z1), z2)) → U(indx(z0, z1)) and U(dbls(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(indx(z0, z1))) = U(indx(z0, z1)) does not unify with U(dbls(indx(x0, x1))). The rules U(indx(indx(z0, z1), z2)) → U(indx(z0, z1)) and U(dbls(dbl(sel(x0, x1)))) → U(dbl(sel(x0, x1))) form no chain, because ECap^µ(U(indx(z0, z1))) = U(indx(z0, z1)) does not unify with U(dbls(dbl(sel(x0, x1)))). The rules U(indx(indx(z0, z1), z2)) → U(indx(z0, z1)) and U(dbls(dbl(indx(x0, x1)))) → U(dbl(indx(x0, x1))) form no chain, because ECap^µ(U(indx(z0, z1))) = U(indx(z0, z1)) does not unify with U(dbls(dbl(indx(x0, x1)))). The rules U(indx(indx(z0, z1), z2)) → U(indx(z0, z1)) and U(dbl(dbls(sel(x0, x1)))) → U(dbls(sel(x0, x1))) form no chain, because ECap^µ(U(indx(z0, z1))) = U(indx(z0, z1)) does not unify with U(dbl(dbls(sel(x0, x1)))). The rules U(indx(indx(z0, z1), z2)) → U(indx(z0, z1)) and U(dbl(dbls(indx(x0, x1)))) → U(dbls(indx(x0, x1))) form no chain, because ECap^µ(U(indx(z0, z1))) = U(indx(z0, z1)) does not unify with U(dbl(dbls(indx(x0, x1)))). The rules U(indx(indx(z0, z1), z2)) → U(indx(z0, z1)) and U(dbl(dbls(dbl(sel(x0, x1))))) → U(dbls(dbl(sel(x0, x1)))) form no chain, because ECap^µ(U(indx(z0, z1))) = U(indx(z0, z1)) does not unify with U(dbl(dbls(dbl(sel(x0, x1))))). The rules U(indx(indx(z0, z1), z2)) → U(indx(z0, z1)) and U(dbl(dbls(dbl(indx(x0, x1))))) → U(dbls(dbl(indx(x0, x1)))) form no chain, because ECap^µ(U(indx(z0, z1))) = U(indx(z0, z1)) does not unify with U(dbl(dbls(dbl(indx(x0, x1))))). The rules U(indx(indx(z0, z1), z2)) → U(indx(z0, z1)) and U(dbls(dbl(dbls(sel(x0, x1))))) → U(dbl(dbls(sel(x0, x1)))) form no chain, because ECap^µ(U(indx(z0, z1))) = U(indx(z0, z1)) does not unify with U(dbls(dbl(dbls(sel(x0, x1))))). The rules U(indx(indx(z0, z1), z2)) → U(indx(z0, z1)) and U(dbls(dbl(dbls(indx(x0, x1))))) → U(dbl(dbls(indx(x0, x1)))) form no chain, because ECap^µ(U(indx(z0, z1))) = U(indx(z0, z1)) does not unify with U(dbls(dbl(dbls(indx(x0, x1))))). The rules U(indx(indx(z0, z1), z2)) → U(indx(z0, z1)) and U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) → U(dbl(dbls(dbl(sel(x0, x1))))) form no chain, because ECap^µ(U(indx(z0, z1))) = U(indx(z0, z1)) does not unify with U(dbls(dbl(dbls(dbl(sel(x0, x1)))))). The rules U(indx(indx(z0, z1), z2)) → U(indx(z0, z1)) and U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) → U(dbl(dbls(dbl(indx(x0, x1))))) form no chain, because ECap^µ(U(indx(z0, z1))) = U(indx(z0, z1)) does not unify with U(dbls(dbl(dbls(dbl(indx(x0, x1)))))). The rules U(indx(indx(z0, z1), z2)) → U(indx(z0, z1)) and U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) → U(dbls(dbl(dbls(sel(x0, x1))))) form no chain, because ECap^µ(U(indx(z0, z1))) = U(indx(z0, z1)) does not unify with U(dbl(dbls(dbl(dbls(sel(x0, x1)))))). The rules U(indx(indx(z0, z1), z2)) → U(indx(z0, z1)) and U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) → U(dbls(dbl(dbls(indx(x0, x1))))) form no chain, because ECap^µ(U(indx(z0, z1))) = U(indx(z0, z1)) does not unify with U(dbl(dbls(dbl(dbls(indx(x0, x1)))))). The rules U(indx(indx(z0, z1), z2)) → U(indx(z0, z1)) and U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) → U(dbls(dbl(dbls(dbl(dbls(x0)))))) form no chain, because ECap^µ(U(indx(z0, z1))) = U(indx(z0, z1)) does not unify with U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))). The rules U(indx(indx(z0, z1), z2)) → U(indx(z0, z1)) and U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) → U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) form no chain, because ECap^µ(U(indx(z0, z1))) = U(indx(z0, z1)) does not unify with U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))). The rules U(indx(indx(z0, z1), z2)) → U(indx(z0, z1)) and U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) → U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) form no chain, because ECap^µ(U(indx(z0, z1))) = U(indx(z0, z1)) does not unify with U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))). The rules U(indx(indx(z0, z1), z2)) → U(indx(z0, z1)) and U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))) → U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) form no chain, because ECap^µ(U(indx(z0, z1))) = U(indx(z0, z1)) does not unify with U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))). The rules U(indx(indx(z0, z1), z2)) → U(indx(z0, z1)) and U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))) → U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) form no chain, because ECap^µ(U(indx(z0, z1))) = U(indx(z0, z1)) does not unify with U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))). The rules U(indx(indx(z0, z1), z2)) → U(indx(z0, z1)) and U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))) → U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) form no chain, because ECap^µ(U(indx(z0, z1))) = U(indx(z0, z1)) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))). The rules U(indx(indx(z0, z1), z2)) → U(indx(z0, z1)) and U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) form no chain, because ECap^µ(U(indx(z0, z1))) = U(indx(z0, z1)) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))). The rules U(indx(indx(z0, z1), z2)) → U(indx(z0, z1)) and U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) form no chain, because ECap^µ(U(indx(z0, z1))) = U(indx(z0, z1)) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))). The rules U(indx(dbl(sel(z0, z1)), z2)) → U(dbl(sel(z0, z1))) and U(sel(x0, x1)) → U(x0) form no chain, because ECap^µ(U(dbl(sel(z0, z1)))) = U(dbl(sel(z0, z1))) does not unify with U(sel(x0, x1)). The rules U(indx(dbl(sel(z0, z1)), z2)) → U(dbl(sel(z0, z1))) and U(sel(x0, x1)) → U(x1) form no chain, because ECap^µ(U(dbl(sel(z0, z1)))) = U(dbl(sel(z0, z1))) does not unify with U(sel(x0, x1)). The rules U(indx(dbl(sel(z0, z1)), z2)) → U(dbl(sel(z0, z1))) and U(sel(x0, x1)) → SEL(x0, x1) form no chain, because ECap^µ(U(dbl(sel(z0, z1)))) = U(dbl(sel(z0, z1))) does not unify with U(sel(x0, x1)). The rules U(indx(dbl(sel(z0, z1)), z2)) → U(dbl(sel(z0, z1))) and U(dbl(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(dbl(sel(z0, z1)))) = U(dbl(sel(z0, z1))) does not unify with U(dbl(indx(x0, x1))). The rules U(indx(dbl(sel(z0, z1)), z2)) → U(dbl(sel(z0, z1))) and U(dbls(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(dbl(sel(z0, z1)))) = U(dbl(sel(z0, z1))) does not unify with U(dbls(sel(x0, x1))). The rules U(indx(dbl(sel(z0, z1)), z2)) → U(dbl(sel(z0, z1))) and U(dbls(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(dbl(sel(z0, z1)))) = U(dbl(sel(z0, z1))) does not unify with U(dbls(indx(x0, x1))). The rules U(indx(dbl(sel(z0, z1)), z2)) → U(dbl(sel(z0, z1))) and U(dbls(dbl(sel(x0, x1)))) → U(dbl(sel(x0, x1))) form no chain, because ECap^µ(U(dbl(sel(z0, z1)))) = U(dbl(sel(z0, z1))) does not unify with U(dbls(dbl(sel(x0, x1)))). The rules U(indx(dbl(sel(z0, z1)), z2)) → U(dbl(sel(z0, z1))) and U(dbls(dbl(indx(x0, x1)))) → U(dbl(indx(x0, x1))) form no chain, because ECap^µ(U(dbl(sel(z0, z1)))) = U(dbl(sel(z0, z1))) does not unify with U(dbls(dbl(indx(x0, x1)))). The rules U(indx(dbl(sel(z0, z1)), z2)) → U(dbl(sel(z0, z1))) and U(dbl(dbls(sel(x0, x1)))) → U(dbls(sel(x0, x1))) form no chain, because ECap^µ(U(dbl(sel(z0, z1)))) = U(dbl(sel(z0, z1))) does not unify with U(dbl(dbls(sel(x0, x1)))). The rules U(indx(dbl(sel(z0, z1)), z2)) → U(dbl(sel(z0, z1))) and U(dbl(dbls(indx(x0, x1)))) → U(dbls(indx(x0, x1))) form no chain, because ECap^µ(U(dbl(sel(z0, z1)))) = U(dbl(sel(z0, z1))) does not unify with U(dbl(dbls(indx(x0, x1)))). The rules U(indx(dbl(sel(z0, z1)), z2)) → U(dbl(sel(z0, z1))) and U(dbl(dbls(dbl(sel(x0, x1))))) → U(dbls(dbl(sel(x0, x1)))) form no chain, because ECap^µ(U(dbl(sel(z0, z1)))) = U(dbl(sel(z0, z1))) does not unify with U(dbl(dbls(dbl(sel(x0, x1))))). The rules U(indx(dbl(sel(z0, z1)), z2)) → U(dbl(sel(z0, z1))) and U(dbl(dbls(dbl(indx(x0, x1))))) → U(dbls(dbl(indx(x0, x1)))) form no chain, because ECap^µ(U(dbl(sel(z0, z1)))) = U(dbl(sel(z0, z1))) does not unify with U(dbl(dbls(dbl(indx(x0, x1))))). The rules U(indx(dbl(sel(z0, z1)), z2)) → U(dbl(sel(z0, z1))) and U(dbls(dbl(dbls(sel(x0, x1))))) → U(dbl(dbls(sel(x0, x1)))) form no chain, because ECap^µ(U(dbl(sel(z0, z1)))) = U(dbl(sel(z0, z1))) does not unify with U(dbls(dbl(dbls(sel(x0, x1))))). The rules U(indx(dbl(sel(z0, z1)), z2)) → U(dbl(sel(z0, z1))) and U(dbls(dbl(dbls(indx(x0, x1))))) → U(dbl(dbls(indx(x0, x1)))) form no chain, because ECap^µ(U(dbl(sel(z0, z1)))) = U(dbl(sel(z0, z1))) does not unify with U(dbls(dbl(dbls(indx(x0, x1))))). The rules U(indx(dbl(sel(z0, z1)), z2)) → U(dbl(sel(z0, z1))) and U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) → U(dbl(dbls(dbl(sel(x0, x1))))) form no chain, because ECap^µ(U(dbl(sel(z0, z1)))) = U(dbl(sel(z0, z1))) does not unify with U(dbls(dbl(dbls(dbl(sel(x0, x1)))))). The rules U(indx(dbl(sel(z0, z1)), z2)) → U(dbl(sel(z0, z1))) and U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) → U(dbl(dbls(dbl(indx(x0, x1))))) form no chain, because ECap^µ(U(dbl(sel(z0, z1)))) = U(dbl(sel(z0, z1))) does not unify with U(dbls(dbl(dbls(dbl(indx(x0, x1)))))). The rules U(indx(dbl(sel(z0, z1)), z2)) → U(dbl(sel(z0, z1))) and U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) → U(dbls(dbl(dbls(sel(x0, x1))))) form no chain, because ECap^µ(U(dbl(sel(z0, z1)))) = U(dbl(sel(z0, z1))) does not unify with U(dbl(dbls(dbl(dbls(sel(x0, x1)))))). The rules U(indx(dbl(sel(z0, z1)), z2)) → U(dbl(sel(z0, z1))) and U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) → U(dbls(dbl(dbls(indx(x0, x1))))) form no chain, because ECap^µ(U(dbl(sel(z0, z1)))) = U(dbl(sel(z0, z1))) does not unify with U(dbl(dbls(dbl(dbls(indx(x0, x1)))))). The rules U(indx(dbl(sel(z0, z1)), z2)) → U(dbl(sel(z0, z1))) and U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) → U(dbls(dbl(dbls(dbl(dbls(x0)))))) form no chain, because ECap^µ(U(dbl(sel(z0, z1)))) = U(dbl(sel(z0, z1))) does not unify with U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))). The rules U(indx(dbl(sel(z0, z1)), z2)) → U(dbl(sel(z0, z1))) and U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) → U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) form no chain, because ECap^µ(U(dbl(sel(z0, z1)))) = U(dbl(sel(z0, z1))) does not unify with U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))). The rules U(indx(dbl(sel(z0, z1)), z2)) → U(dbl(sel(z0, z1))) and U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) → U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) form no chain, because ECap^µ(U(dbl(sel(z0, z1)))) = U(dbl(sel(z0, z1))) does not unify with U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))). The rules U(indx(dbl(sel(z0, z1)), z2)) → U(dbl(sel(z0, z1))) and U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))) → U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) form no chain, because ECap^µ(U(dbl(sel(z0, z1)))) = U(dbl(sel(z0, z1))) does not unify with U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))). The rules U(indx(dbl(sel(z0, z1)), z2)) → U(dbl(sel(z0, z1))) and U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))) → U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) form no chain, because ECap^µ(U(dbl(sel(z0, z1)))) = U(dbl(sel(z0, z1))) does not unify with U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))). The rules U(indx(dbl(sel(z0, z1)), z2)) → U(dbl(sel(z0, z1))) and U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))) → U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) form no chain, because ECap^µ(U(dbl(sel(z0, z1)))) = U(dbl(sel(z0, z1))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))). The rules U(indx(dbl(sel(z0, z1)), z2)) → U(dbl(sel(z0, z1))) and U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) form no chain, because ECap^µ(U(dbl(sel(z0, z1)))) = U(dbl(sel(z0, z1))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))). The rules U(indx(dbl(sel(z0, z1)), z2)) → U(dbl(sel(z0, z1))) and U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) form no chain, because ECap^µ(U(dbl(sel(z0, z1)))) = U(dbl(sel(z0, z1))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))). The rules U(indx(dbl(sel(z0, z1)), z2)) → U(dbl(sel(z0, z1))) and U(indx(dbl(sel(x0, x1)), x2)) → U(dbl(sel(x0, x1))) form no chain, because ECap^µ(U(dbl(sel(z0, z1)))) = U(dbl(sel(z0, z1))) does not unify with U(indx(dbl(sel(x0, x1)), x2)). The rules U(indx(dbl(indx(z0, z1)), z2)) → U(dbl(indx(z0, z1))) and U(sel(x0, x1)) → U(x0) form no chain, because ECap^µ(U(dbl(indx(z0, z1)))) = U(dbl(indx(z0, z1))) does not unify with U(sel(x0, x1)). The rules U(indx(dbl(indx(z0, z1)), z2)) → U(dbl(indx(z0, z1))) and U(sel(x0, x1)) → U(x1) form no chain, because ECap^µ(U(dbl(indx(z0, z1)))) = U(dbl(indx(z0, z1))) does not unify with U(sel(x0, x1)). The rules U(indx(dbl(indx(z0, z1)), z2)) → U(dbl(indx(z0, z1))) and U(sel(x0, x1)) → SEL(x0, x1) form no chain, because ECap^µ(U(dbl(indx(z0, z1)))) = U(dbl(indx(z0, z1))) does not unify with U(sel(x0, x1)). The rules U(indx(dbl(indx(z0, z1)), z2)) → U(dbl(indx(z0, z1))) and U(dbl(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(dbl(indx(z0, z1)))) = U(dbl(indx(z0, z1))) does not unify with U(dbl(sel(x0, x1))). The rules U(indx(dbl(indx(z0, z1)), z2)) → U(dbl(indx(z0, z1))) and U(dbls(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(dbl(indx(z0, z1)))) = U(dbl(indx(z0, z1))) does not unify with U(dbls(sel(x0, x1))). The rules U(indx(dbl(indx(z0, z1)), z2)) → U(dbl(indx(z0, z1))) and U(dbls(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(dbl(indx(z0, z1)))) = U(dbl(indx(z0, z1))) does not unify with U(dbls(indx(x0, x1))). The rules U(indx(dbl(indx(z0, z1)), z2)) → U(dbl(indx(z0, z1))) and U(dbls(dbl(sel(x0, x1)))) → U(dbl(sel(x0, x1))) form no chain, because ECap^µ(U(dbl(indx(z0, z1)))) = U(dbl(indx(z0, z1))) does not unify with U(dbls(dbl(sel(x0, x1)))). The rules U(indx(dbl(indx(z0, z1)), z2)) → U(dbl(indx(z0, z1))) and U(dbls(dbl(indx(x0, x1)))) → U(dbl(indx(x0, x1))) form no chain, because ECap^µ(U(dbl(indx(z0, z1)))) = U(dbl(indx(z0, z1))) does not unify with U(dbls(dbl(indx(x0, x1)))). The rules U(indx(dbl(indx(z0, z1)), z2)) → U(dbl(indx(z0, z1))) and U(dbl(dbls(sel(x0, x1)))) → U(dbls(sel(x0, x1))) form no chain, because ECap^µ(U(dbl(indx(z0, z1)))) = U(dbl(indx(z0, z1))) does not unify with U(dbl(dbls(sel(x0, x1)))). The rules U(indx(dbl(indx(z0, z1)), z2)) → U(dbl(indx(z0, z1))) and U(dbl(dbls(indx(x0, x1)))) → U(dbls(indx(x0, x1))) form no chain, because ECap^µ(U(dbl(indx(z0, z1)))) = U(dbl(indx(z0, z1))) does not unify with U(dbl(dbls(indx(x0, x1)))). The rules U(indx(dbl(indx(z0, z1)), z2)) → U(dbl(indx(z0, z1))) and U(dbl(dbls(dbl(sel(x0, x1))))) → U(dbls(dbl(sel(x0, x1)))) form no chain, because ECap^µ(U(dbl(indx(z0, z1)))) = U(dbl(indx(z0, z1))) does not unify with U(dbl(dbls(dbl(sel(x0, x1))))). The rules U(indx(dbl(indx(z0, z1)), z2)) → U(dbl(indx(z0, z1))) and U(dbl(dbls(dbl(indx(x0, x1))))) → U(dbls(dbl(indx(x0, x1)))) form no chain, because ECap^µ(U(dbl(indx(z0, z1)))) = U(dbl(indx(z0, z1))) does not unify with U(dbl(dbls(dbl(indx(x0, x1))))). The rules U(indx(dbl(indx(z0, z1)), z2)) → U(dbl(indx(z0, z1))) and U(dbls(dbl(dbls(sel(x0, x1))))) → U(dbl(dbls(sel(x0, x1)))) form no chain, because ECap^µ(U(dbl(indx(z0, z1)))) = U(dbl(indx(z0, z1))) does not unify with U(dbls(dbl(dbls(sel(x0, x1))))). The rules U(indx(dbl(indx(z0, z1)), z2)) → U(dbl(indx(z0, z1))) and U(dbls(dbl(dbls(indx(x0, x1))))) → U(dbl(dbls(indx(x0, x1)))) form no chain, because ECap^µ(U(dbl(indx(z0, z1)))) = U(dbl(indx(z0, z1))) does not unify with U(dbls(dbl(dbls(indx(x0, x1))))). The rules U(indx(dbl(indx(z0, z1)), z2)) → U(dbl(indx(z0, z1))) and U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) → U(dbl(dbls(dbl(sel(x0, x1))))) form no chain, because ECap^µ(U(dbl(indx(z0, z1)))) = U(dbl(indx(z0, z1))) does not unify with U(dbls(dbl(dbls(dbl(sel(x0, x1)))))). The rules U(indx(dbl(indx(z0, z1)), z2)) → U(dbl(indx(z0, z1))) and U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) → U(dbl(dbls(dbl(indx(x0, x1))))) form no chain, because ECap^µ(U(dbl(indx(z0, z1)))) = U(dbl(indx(z0, z1))) does not unify with U(dbls(dbl(dbls(dbl(indx(x0, x1)))))). The rules U(indx(dbl(indx(z0, z1)), z2)) → U(dbl(indx(z0, z1))) and U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) → U(dbls(dbl(dbls(sel(x0, x1))))) form no chain, because ECap^µ(U(dbl(indx(z0, z1)))) = U(dbl(indx(z0, z1))) does not unify with U(dbl(dbls(dbl(dbls(sel(x0, x1)))))). The rules U(indx(dbl(indx(z0, z1)), z2)) → U(dbl(indx(z0, z1))) and U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) → U(dbls(dbl(dbls(indx(x0, x1))))) form no chain, because ECap^µ(U(dbl(indx(z0, z1)))) = U(dbl(indx(z0, z1))) does not unify with U(dbl(dbls(dbl(dbls(indx(x0, x1)))))). The rules U(indx(dbl(indx(z0, z1)), z2)) → U(dbl(indx(z0, z1))) and U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) → U(dbls(dbl(dbls(dbl(dbls(x0)))))) form no chain, because ECap^µ(U(dbl(indx(z0, z1)))) = U(dbl(indx(z0, z1))) does not unify with U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))). The rules U(indx(dbl(indx(z0, z1)), z2)) → U(dbl(indx(z0, z1))) and U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) → U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) form no chain, because ECap^µ(U(dbl(indx(z0, z1)))) = U(dbl(indx(z0, z1))) does not unify with U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))). The rules U(indx(dbl(indx(z0, z1)), z2)) → U(dbl(indx(z0, z1))) and U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) → U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) form no chain, because ECap^µ(U(dbl(indx(z0, z1)))) = U(dbl(indx(z0, z1))) does not unify with U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))). The rules U(indx(dbl(indx(z0, z1)), z2)) → U(dbl(indx(z0, z1))) and U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))) → U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) form no chain, because ECap^µ(U(dbl(indx(z0, z1)))) = U(dbl(indx(z0, z1))) does not unify with U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))). The rules U(indx(dbl(indx(z0, z1)), z2)) → U(dbl(indx(z0, z1))) and U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))) → U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) form no chain, because ECap^µ(U(dbl(indx(z0, z1)))) = U(dbl(indx(z0, z1))) does not unify with U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))). The rules U(indx(dbl(indx(z0, z1)), z2)) → U(dbl(indx(z0, z1))) and U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))) → U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) form no chain, because ECap^µ(U(dbl(indx(z0, z1)))) = U(dbl(indx(z0, z1))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))). The rules U(indx(dbl(indx(z0, z1)), z2)) → U(dbl(indx(z0, z1))) and U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) form no chain, because ECap^µ(U(dbl(indx(z0, z1)))) = U(dbl(indx(z0, z1))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))). The rules U(indx(dbl(indx(z0, z1)), z2)) → U(dbl(indx(z0, z1))) and U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) form no chain, because ECap^µ(U(dbl(indx(z0, z1)))) = U(dbl(indx(z0, z1))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))). The rules U(indx(dbl(indx(z0, z1)), z2)) → U(dbl(indx(z0, z1))) and U(indx(dbl(indx(x0, x1)), x2)) → U(dbl(indx(x0, x1))) form no chain, because ECap^µ(U(dbl(indx(z0, z1)))) = U(dbl(indx(z0, z1))) does not unify with U(indx(dbl(indx(x0, x1)), x2)). The rules U(indx(dbls(sel(z0, z1)), z2)) → U(dbls(sel(z0, z1))) and U(sel(x0, x1)) → U(x0) form no chain, because ECap^µ(U(dbls(sel(z0, z1)))) = U(dbls(sel(z0, z1))) does not unify with U(sel(x0, x1)). The rules U(indx(dbls(sel(z0, z1)), z2)) → U(dbls(sel(z0, z1))) and U(sel(x0, x1)) → U(x1) form no chain, because ECap^µ(U(dbls(sel(z0, z1)))) = U(dbls(sel(z0, z1))) does not unify with U(sel(x0, x1)). The rules U(indx(dbls(sel(z0, z1)), z2)) → U(dbls(sel(z0, z1))) and U(sel(x0, x1)) → SEL(x0, x1) form no chain, because ECap^µ(U(dbls(sel(z0, z1)))) = U(dbls(sel(z0, z1))) does not unify with U(sel(x0, x1)). The rules U(indx(dbls(sel(z0, z1)), z2)) → U(dbls(sel(z0, z1))) and U(dbl(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(dbls(sel(z0, z1)))) = U(dbls(sel(z0, z1))) does not unify with U(dbl(sel(x0, x1))). The rules U(indx(dbls(sel(z0, z1)), z2)) → U(dbls(sel(z0, z1))) and U(dbl(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(dbls(sel(z0, z1)))) = U(dbls(sel(z0, z1))) does not unify with U(dbl(indx(x0, x1))). The rules U(indx(dbls(sel(z0, z1)), z2)) → U(dbls(sel(z0, z1))) and U(dbls(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(dbls(sel(z0, z1)))) = U(dbls(sel(z0, z1))) does not unify with U(dbls(indx(x0, x1))). The rules U(indx(dbls(sel(z0, z1)), z2)) → U(dbls(sel(z0, z1))) and U(dbls(dbl(sel(x0, x1)))) → U(dbl(sel(x0, x1))) form no chain, because ECap^µ(U(dbls(sel(z0, z1)))) = U(dbls(sel(z0, z1))) does not unify with U(dbls(dbl(sel(x0, x1)))). The rules U(indx(dbls(sel(z0, z1)), z2)) → U(dbls(sel(z0, z1))) and U(dbls(dbl(indx(x0, x1)))) → U(dbl(indx(x0, x1))) form no chain, because ECap^µ(U(dbls(sel(z0, z1)))) = U(dbls(sel(z0, z1))) does not unify with U(dbls(dbl(indx(x0, x1)))). The rules U(indx(dbls(sel(z0, z1)), z2)) → U(dbls(sel(z0, z1))) and U(dbl(dbls(sel(x0, x1)))) → U(dbls(sel(x0, x1))) form no chain, because ECap^µ(U(dbls(sel(z0, z1)))) = U(dbls(sel(z0, z1))) does not unify with U(dbl(dbls(sel(x0, x1)))). The rules U(indx(dbls(sel(z0, z1)), z2)) → U(dbls(sel(z0, z1))) and U(dbl(dbls(indx(x0, x1)))) → U(dbls(indx(x0, x1))) form no chain, because ECap^µ(U(dbls(sel(z0, z1)))) = U(dbls(sel(z0, z1))) does not unify with U(dbl(dbls(indx(x0, x1)))). The rules U(indx(dbls(sel(z0, z1)), z2)) → U(dbls(sel(z0, z1))) and U(dbl(dbls(dbl(sel(x0, x1))))) → U(dbls(dbl(sel(x0, x1)))) form no chain, because ECap^µ(U(dbls(sel(z0, z1)))) = U(dbls(sel(z0, z1))) does not unify with U(dbl(dbls(dbl(sel(x0, x1))))). The rules U(indx(dbls(sel(z0, z1)), z2)) → U(dbls(sel(z0, z1))) and U(dbl(dbls(dbl(indx(x0, x1))))) → U(dbls(dbl(indx(x0, x1)))) form no chain, because ECap^µ(U(dbls(sel(z0, z1)))) = U(dbls(sel(z0, z1))) does not unify with U(dbl(dbls(dbl(indx(x0, x1))))). The rules U(indx(dbls(sel(z0, z1)), z2)) → U(dbls(sel(z0, z1))) and U(dbls(dbl(dbls(sel(x0, x1))))) → U(dbl(dbls(sel(x0, x1)))) form no chain, because ECap^µ(U(dbls(sel(z0, z1)))) = U(dbls(sel(z0, z1))) does not unify with U(dbls(dbl(dbls(sel(x0, x1))))). The rules U(indx(dbls(sel(z0, z1)), z2)) → U(dbls(sel(z0, z1))) and U(dbls(dbl(dbls(indx(x0, x1))))) → U(dbl(dbls(indx(x0, x1)))) form no chain, because ECap^µ(U(dbls(sel(z0, z1)))) = U(dbls(sel(z0, z1))) does not unify with U(dbls(dbl(dbls(indx(x0, x1))))). The rules U(indx(dbls(sel(z0, z1)), z2)) → U(dbls(sel(z0, z1))) and U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) → U(dbl(dbls(dbl(sel(x0, x1))))) form no chain, because ECap^µ(U(dbls(sel(z0, z1)))) = U(dbls(sel(z0, z1))) does not unify with U(dbls(dbl(dbls(dbl(sel(x0, x1)))))). The rules U(indx(dbls(sel(z0, z1)), z2)) → U(dbls(sel(z0, z1))) and U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) → U(dbl(dbls(dbl(indx(x0, x1))))) form no chain, because ECap^µ(U(dbls(sel(z0, z1)))) = U(dbls(sel(z0, z1))) does not unify with U(dbls(dbl(dbls(dbl(indx(x0, x1)))))). The rules U(indx(dbls(sel(z0, z1)), z2)) → U(dbls(sel(z0, z1))) and U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) → U(dbls(dbl(dbls(sel(x0, x1))))) form no chain, because ECap^µ(U(dbls(sel(z0, z1)))) = U(dbls(sel(z0, z1))) does not unify with U(dbl(dbls(dbl(dbls(sel(x0, x1)))))). The rules U(indx(dbls(sel(z0, z1)), z2)) → U(dbls(sel(z0, z1))) and U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) → U(dbls(dbl(dbls(indx(x0, x1))))) form no chain, because ECap^µ(U(dbls(sel(z0, z1)))) = U(dbls(sel(z0, z1))) does not unify with U(dbl(dbls(dbl(dbls(indx(x0, x1)))))). The rules U(indx(dbls(sel(z0, z1)), z2)) → U(dbls(sel(z0, z1))) and U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) → U(dbls(dbl(dbls(dbl(dbls(x0)))))) form no chain, because ECap^µ(U(dbls(sel(z0, z1)))) = U(dbls(sel(z0, z1))) does not unify with U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))). The rules U(indx(dbls(sel(z0, z1)), z2)) → U(dbls(sel(z0, z1))) and U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) → U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) form no chain, because ECap^µ(U(dbls(sel(z0, z1)))) = U(dbls(sel(z0, z1))) does not unify with U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))). The rules U(indx(dbls(sel(z0, z1)), z2)) → U(dbls(sel(z0, z1))) and U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) → U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) form no chain, because ECap^µ(U(dbls(sel(z0, z1)))) = U(dbls(sel(z0, z1))) does not unify with U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))). The rules U(indx(dbls(sel(z0, z1)), z2)) → U(dbls(sel(z0, z1))) and U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))) → U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) form no chain, because ECap^µ(U(dbls(sel(z0, z1)))) = U(dbls(sel(z0, z1))) does not unify with U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))). The rules U(indx(dbls(sel(z0, z1)), z2)) → U(dbls(sel(z0, z1))) and U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))) → U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) form no chain, because ECap^µ(U(dbls(sel(z0, z1)))) = U(dbls(sel(z0, z1))) does not unify with U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))). The rules U(indx(dbls(sel(z0, z1)), z2)) → U(dbls(sel(z0, z1))) and U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))) → U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) form no chain, because ECap^µ(U(dbls(sel(z0, z1)))) = U(dbls(sel(z0, z1))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))). The rules U(indx(dbls(sel(z0, z1)), z2)) → U(dbls(sel(z0, z1))) and U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) form no chain, because ECap^µ(U(dbls(sel(z0, z1)))) = U(dbls(sel(z0, z1))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))). The rules U(indx(dbls(sel(z0, z1)), z2)) → U(dbls(sel(z0, z1))) and U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) form no chain, because ECap^µ(U(dbls(sel(z0, z1)))) = U(dbls(sel(z0, z1))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))). The rules U(indx(dbls(sel(z0, z1)), z2)) → U(dbls(sel(z0, z1))) and U(indx(dbls(sel(x0, x1)), x2)) → U(dbls(sel(x0, x1))) form no chain, because ECap^µ(U(dbls(sel(z0, z1)))) = U(dbls(sel(z0, z1))) does not unify with U(indx(dbls(sel(x0, x1)), x2)). The rules U(indx(dbls(indx(z0, z1)), z2)) → U(dbls(indx(z0, z1))) and U(sel(x0, x1)) → U(x0) form no chain, because ECap^µ(U(dbls(indx(z0, z1)))) = U(dbls(indx(z0, z1))) does not unify with U(sel(x0, x1)). The rules U(indx(dbls(indx(z0, z1)), z2)) → U(dbls(indx(z0, z1))) and U(sel(x0, x1)) → U(x1) form no chain, because ECap^µ(U(dbls(indx(z0, z1)))) = U(dbls(indx(z0, z1))) does not unify with U(sel(x0, x1)). The rules U(indx(dbls(indx(z0, z1)), z2)) → U(dbls(indx(z0, z1))) and U(sel(x0, x1)) → SEL(x0, x1) form no chain, because ECap^µ(U(dbls(indx(z0, z1)))) = U(dbls(indx(z0, z1))) does not unify with U(sel(x0, x1)). The rules U(indx(dbls(indx(z0, z1)), z2)) → U(dbls(indx(z0, z1))) and U(dbl(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(dbls(indx(z0, z1)))) = U(dbls(indx(z0, z1))) does not unify with U(dbl(sel(x0, x1))). The rules U(indx(dbls(indx(z0, z1)), z2)) → U(dbls(indx(z0, z1))) and U(dbl(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(dbls(indx(z0, z1)))) = U(dbls(indx(z0, z1))) does not unify with U(dbl(indx(x0, x1))). The rules U(indx(dbls(indx(z0, z1)), z2)) → U(dbls(indx(z0, z1))) and U(dbls(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(dbls(indx(z0, z1)))) = U(dbls(indx(z0, z1))) does not unify with U(dbls(sel(x0, x1))). The rules U(indx(dbls(indx(z0, z1)), z2)) → U(dbls(indx(z0, z1))) and U(dbls(dbl(sel(x0, x1)))) → U(dbl(sel(x0, x1))) form no chain, because ECap^µ(U(dbls(indx(z0, z1)))) = U(dbls(indx(z0, z1))) does not unify with U(dbls(dbl(sel(x0, x1)))). The rules U(indx(dbls(indx(z0, z1)), z2)) → U(dbls(indx(z0, z1))) and U(dbls(dbl(indx(x0, x1)))) → U(dbl(indx(x0, x1))) form no chain, because ECap^µ(U(dbls(indx(z0, z1)))) = U(dbls(indx(z0, z1))) does not unify with U(dbls(dbl(indx(x0, x1)))). The rules U(indx(dbls(indx(z0, z1)), z2)) → U(dbls(indx(z0, z1))) and U(dbl(dbls(sel(x0, x1)))) → U(dbls(sel(x0, x1))) form no chain, because ECap^µ(U(dbls(indx(z0, z1)))) = U(dbls(indx(z0, z1))) does not unify with U(dbl(dbls(sel(x0, x1)))). The rules U(indx(dbls(indx(z0, z1)), z2)) → U(dbls(indx(z0, z1))) and U(dbl(dbls(indx(x0, x1)))) → U(dbls(indx(x0, x1))) form no chain, because ECap^µ(U(dbls(indx(z0, z1)))) = U(dbls(indx(z0, z1))) does not unify with U(dbl(dbls(indx(x0, x1)))). The rules U(indx(dbls(indx(z0, z1)), z2)) → U(dbls(indx(z0, z1))) and U(dbl(dbls(dbl(sel(x0, x1))))) → U(dbls(dbl(sel(x0, x1)))) form no chain, because ECap^µ(U(dbls(indx(z0, z1)))) = U(dbls(indx(z0, z1))) does not unify with U(dbl(dbls(dbl(sel(x0, x1))))). The rules U(indx(dbls(indx(z0, z1)), z2)) → U(dbls(indx(z0, z1))) and U(dbl(dbls(dbl(indx(x0, x1))))) → U(dbls(dbl(indx(x0, x1)))) form no chain, because ECap^µ(U(dbls(indx(z0, z1)))) = U(dbls(indx(z0, z1))) does not unify with U(dbl(dbls(dbl(indx(x0, x1))))). The rules U(indx(dbls(indx(z0, z1)), z2)) → U(dbls(indx(z0, z1))) and U(dbls(dbl(dbls(sel(x0, x1))))) → U(dbl(dbls(sel(x0, x1)))) form no chain, because ECap^µ(U(dbls(indx(z0, z1)))) = U(dbls(indx(z0, z1))) does not unify with U(dbls(dbl(dbls(sel(x0, x1))))). The rules U(indx(dbls(indx(z0, z1)), z2)) → U(dbls(indx(z0, z1))) and U(dbls(dbl(dbls(indx(x0, x1))))) → U(dbl(dbls(indx(x0, x1)))) form no chain, because ECap^µ(U(dbls(indx(z0, z1)))) = U(dbls(indx(z0, z1))) does not unify with U(dbls(dbl(dbls(indx(x0, x1))))). The rules U(indx(dbls(indx(z0, z1)), z2)) → U(dbls(indx(z0, z1))) and U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) → U(dbl(dbls(dbl(sel(x0, x1))))) form no chain, because ECap^µ(U(dbls(indx(z0, z1)))) = U(dbls(indx(z0, z1))) does not unify with U(dbls(dbl(dbls(dbl(sel(x0, x1)))))). The rules U(indx(dbls(indx(z0, z1)), z2)) → U(dbls(indx(z0, z1))) and U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) → U(dbl(dbls(dbl(indx(x0, x1))))) form no chain, because ECap^µ(U(dbls(indx(z0, z1)))) = U(dbls(indx(z0, z1))) does not unify with U(dbls(dbl(dbls(dbl(indx(x0, x1)))))). The rules U(indx(dbls(indx(z0, z1)), z2)) → U(dbls(indx(z0, z1))) and U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) → U(dbls(dbl(dbls(sel(x0, x1))))) form no chain, because ECap^µ(U(dbls(indx(z0, z1)))) = U(dbls(indx(z0, z1))) does not unify with U(dbl(dbls(dbl(dbls(sel(x0, x1)))))). The rules U(indx(dbls(indx(z0, z1)), z2)) → U(dbls(indx(z0, z1))) and U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) → U(dbls(dbl(dbls(indx(x0, x1))))) form no chain, because ECap^µ(U(dbls(indx(z0, z1)))) = U(dbls(indx(z0, z1))) does not unify with U(dbl(dbls(dbl(dbls(indx(x0, x1)))))). The rules U(indx(dbls(indx(z0, z1)), z2)) → U(dbls(indx(z0, z1))) and U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) → U(dbls(dbl(dbls(dbl(dbls(x0)))))) form no chain, because ECap^µ(U(dbls(indx(z0, z1)))) = U(dbls(indx(z0, z1))) does not unify with U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))). The rules U(indx(dbls(indx(z0, z1)), z2)) → U(dbls(indx(z0, z1))) and U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) → U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) form no chain, because ECap^µ(U(dbls(indx(z0, z1)))) = U(dbls(indx(z0, z1))) does not unify with U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))). The rules U(indx(dbls(indx(z0, z1)), z2)) → U(dbls(indx(z0, z1))) and U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) → U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) form no chain, because ECap^µ(U(dbls(indx(z0, z1)))) = U(dbls(indx(z0, z1))) does not unify with U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))). The rules U(indx(dbls(indx(z0, z1)), z2)) → U(dbls(indx(z0, z1))) and U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))) → U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) form no chain, because ECap^µ(U(dbls(indx(z0, z1)))) = U(dbls(indx(z0, z1))) does not unify with U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))). The rules U(indx(dbls(indx(z0, z1)), z2)) → U(dbls(indx(z0, z1))) and U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))) → U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) form no chain, because ECap^µ(U(dbls(indx(z0, z1)))) = U(dbls(indx(z0, z1))) does not unify with U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))). The rules U(indx(dbls(indx(z0, z1)), z2)) → U(dbls(indx(z0, z1))) and U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))) → U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) form no chain, because ECap^µ(U(dbls(indx(z0, z1)))) = U(dbls(indx(z0, z1))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))). The rules U(indx(dbls(indx(z0, z1)), z2)) → U(dbls(indx(z0, z1))) and U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) form no chain, because ECap^µ(U(dbls(indx(z0, z1)))) = U(dbls(indx(z0, z1))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))). The rules U(indx(dbls(indx(z0, z1)), z2)) → U(dbls(indx(z0, z1))) and U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) form no chain, because ECap^µ(U(dbls(indx(z0, z1)))) = U(dbls(indx(z0, z1))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))). The rules U(indx(dbls(indx(z0, z1)), z2)) → U(dbls(indx(z0, z1))) and U(indx(dbls(indx(x0, x1)), x2)) → U(dbls(indx(x0, x1))) form no chain, because ECap^µ(U(dbls(indx(z0, z1)))) = U(dbls(indx(z0, z1))) does not unify with U(indx(dbls(indx(x0, x1)), x2)). The rules U(indx(dbls(dbl(sel(z0, z1))), z2)) → U(dbls(dbl(sel(z0, z1)))) and U(sel(x0, x1)) → U(x0) form no chain, because ECap^µ(U(dbls(dbl(sel(z0, z1))))) = U(dbls(dbl(sel(z0, z1)))) does not unify with U(sel(x0, x1)). The rules U(indx(dbls(dbl(sel(z0, z1))), z2)) → U(dbls(dbl(sel(z0, z1)))) and U(sel(x0, x1)) → U(x1) form no chain, because ECap^µ(U(dbls(dbl(sel(z0, z1))))) = U(dbls(dbl(sel(z0, z1)))) does not unify with U(sel(x0, x1)). The rules U(indx(dbls(dbl(sel(z0, z1))), z2)) → U(dbls(dbl(sel(z0, z1)))) and U(sel(x0, x1)) → SEL(x0, x1) form no chain, because ECap^µ(U(dbls(dbl(sel(z0, z1))))) = U(dbls(dbl(sel(z0, z1)))) does not unify with U(sel(x0, x1)). The rules U(indx(dbls(dbl(sel(z0, z1))), z2)) → U(dbls(dbl(sel(z0, z1)))) and U(dbl(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(dbls(dbl(sel(z0, z1))))) = U(dbls(dbl(sel(z0, z1)))) does not unify with U(dbl(sel(x0, x1))). The rules U(indx(dbls(dbl(sel(z0, z1))), z2)) → U(dbls(dbl(sel(z0, z1)))) and U(dbl(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(dbls(dbl(sel(z0, z1))))) = U(dbls(dbl(sel(z0, z1)))) does not unify with U(dbl(indx(x0, x1))). The rules U(indx(dbls(dbl(sel(z0, z1))), z2)) → U(dbls(dbl(sel(z0, z1)))) and U(dbls(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(dbls(dbl(sel(z0, z1))))) = U(dbls(dbl(sel(z0, z1)))) does not unify with U(dbls(sel(x0, x1))). The rules U(indx(dbls(dbl(sel(z0, z1))), z2)) → U(dbls(dbl(sel(z0, z1)))) and U(dbls(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(dbls(dbl(sel(z0, z1))))) = U(dbls(dbl(sel(z0, z1)))) does not unify with U(dbls(indx(x0, x1))). The rules U(indx(dbls(dbl(sel(z0, z1))), z2)) → U(dbls(dbl(sel(z0, z1)))) and U(dbls(dbl(indx(x0, x1)))) → U(dbl(indx(x0, x1))) form no chain, because ECap^µ(U(dbls(dbl(sel(z0, z1))))) = U(dbls(dbl(sel(z0, z1)))) does not unify with U(dbls(dbl(indx(x0, x1)))). The rules U(indx(dbls(dbl(sel(z0, z1))), z2)) → U(dbls(dbl(sel(z0, z1)))) and U(dbl(dbls(sel(x0, x1)))) → U(dbls(sel(x0, x1))) form no chain, because ECap^µ(U(dbls(dbl(sel(z0, z1))))) = U(dbls(dbl(sel(z0, z1)))) does not unify with U(dbl(dbls(sel(x0, x1)))). The rules U(indx(dbls(dbl(sel(z0, z1))), z2)) → U(dbls(dbl(sel(z0, z1)))) and U(dbl(dbls(indx(x0, x1)))) → U(dbls(indx(x0, x1))) form no chain, because ECap^µ(U(dbls(dbl(sel(z0, z1))))) = U(dbls(dbl(sel(z0, z1)))) does not unify with U(dbl(dbls(indx(x0, x1)))). The rules U(indx(dbls(dbl(sel(z0, z1))), z2)) → U(dbls(dbl(sel(z0, z1)))) and U(dbl(dbls(dbl(sel(x0, x1))))) → U(dbls(dbl(sel(x0, x1)))) form no chain, because ECap^µ(U(dbls(dbl(sel(z0, z1))))) = U(dbls(dbl(sel(z0, z1)))) does not unify with U(dbl(dbls(dbl(sel(x0, x1))))). The rules U(indx(dbls(dbl(sel(z0, z1))), z2)) → U(dbls(dbl(sel(z0, z1)))) and U(dbl(dbls(dbl(indx(x0, x1))))) → U(dbls(dbl(indx(x0, x1)))) form no chain, because ECap^µ(U(dbls(dbl(sel(z0, z1))))) = U(dbls(dbl(sel(z0, z1)))) does not unify with U(dbl(dbls(dbl(indx(x0, x1))))). The rules U(indx(dbls(dbl(sel(z0, z1))), z2)) → U(dbls(dbl(sel(z0, z1)))) and U(dbls(dbl(dbls(sel(x0, x1))))) → U(dbl(dbls(sel(x0, x1)))) form no chain, because ECap^µ(U(dbls(dbl(sel(z0, z1))))) = U(dbls(dbl(sel(z0, z1)))) does not unify with U(dbls(dbl(dbls(sel(x0, x1))))). The rules U(indx(dbls(dbl(sel(z0, z1))), z2)) → U(dbls(dbl(sel(z0, z1)))) and U(dbls(dbl(dbls(indx(x0, x1))))) → U(dbl(dbls(indx(x0, x1)))) form no chain, because ECap^µ(U(dbls(dbl(sel(z0, z1))))) = U(dbls(dbl(sel(z0, z1)))) does not unify with U(dbls(dbl(dbls(indx(x0, x1))))). The rules U(indx(dbls(dbl(sel(z0, z1))), z2)) → U(dbls(dbl(sel(z0, z1)))) and U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) → U(dbl(dbls(dbl(sel(x0, x1))))) form no chain, because ECap^µ(U(dbls(dbl(sel(z0, z1))))) = U(dbls(dbl(sel(z0, z1)))) does not unify with U(dbls(dbl(dbls(dbl(sel(x0, x1)))))). The rules U(indx(dbls(dbl(sel(z0, z1))), z2)) → U(dbls(dbl(sel(z0, z1)))) and U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) → U(dbl(dbls(dbl(indx(x0, x1))))) form no chain, because ECap^µ(U(dbls(dbl(sel(z0, z1))))) = U(dbls(dbl(sel(z0, z1)))) does not unify with U(dbls(dbl(dbls(dbl(indx(x0, x1)))))). The rules U(indx(dbls(dbl(sel(z0, z1))), z2)) → U(dbls(dbl(sel(z0, z1)))) and U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) → U(dbls(dbl(dbls(sel(x0, x1))))) form no chain, because ECap^µ(U(dbls(dbl(sel(z0, z1))))) = U(dbls(dbl(sel(z0, z1)))) does not unify with U(dbl(dbls(dbl(dbls(sel(x0, x1)))))). The rules U(indx(dbls(dbl(sel(z0, z1))), z2)) → U(dbls(dbl(sel(z0, z1)))) and U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) → U(dbls(dbl(dbls(indx(x0, x1))))) form no chain, because ECap^µ(U(dbls(dbl(sel(z0, z1))))) = U(dbls(dbl(sel(z0, z1)))) does not unify with U(dbl(dbls(dbl(dbls(indx(x0, x1)))))). The rules U(indx(dbls(dbl(sel(z0, z1))), z2)) → U(dbls(dbl(sel(z0, z1)))) and U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) → U(dbls(dbl(dbls(dbl(dbls(x0)))))) form no chain, because ECap^µ(U(dbls(dbl(sel(z0, z1))))) = U(dbls(dbl(sel(z0, z1)))) does not unify with U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))). The rules U(indx(dbls(dbl(sel(z0, z1))), z2)) → U(dbls(dbl(sel(z0, z1)))) and U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) → U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) form no chain, because ECap^µ(U(dbls(dbl(sel(z0, z1))))) = U(dbls(dbl(sel(z0, z1)))) does not unify with U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))). The rules U(indx(dbls(dbl(sel(z0, z1))), z2)) → U(dbls(dbl(sel(z0, z1)))) and U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) → U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) form no chain, because ECap^µ(U(dbls(dbl(sel(z0, z1))))) = U(dbls(dbl(sel(z0, z1)))) does not unify with U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))). The rules U(indx(dbls(dbl(sel(z0, z1))), z2)) → U(dbls(dbl(sel(z0, z1)))) and U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))) → U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) form no chain, because ECap^µ(U(dbls(dbl(sel(z0, z1))))) = U(dbls(dbl(sel(z0, z1)))) does not unify with U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))). The rules U(indx(dbls(dbl(sel(z0, z1))), z2)) → U(dbls(dbl(sel(z0, z1)))) and U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))) → U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) form no chain, because ECap^µ(U(dbls(dbl(sel(z0, z1))))) = U(dbls(dbl(sel(z0, z1)))) does not unify with U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))). The rules U(indx(dbls(dbl(sel(z0, z1))), z2)) → U(dbls(dbl(sel(z0, z1)))) and U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))) → U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) form no chain, because ECap^µ(U(dbls(dbl(sel(z0, z1))))) = U(dbls(dbl(sel(z0, z1)))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))). The rules U(indx(dbls(dbl(sel(z0, z1))), z2)) → U(dbls(dbl(sel(z0, z1)))) and U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) form no chain, because ECap^µ(U(dbls(dbl(sel(z0, z1))))) = U(dbls(dbl(sel(z0, z1)))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))). The rules U(indx(dbls(dbl(sel(z0, z1))), z2)) → U(dbls(dbl(sel(z0, z1)))) and U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) form no chain, because ECap^µ(U(dbls(dbl(sel(z0, z1))))) = U(dbls(dbl(sel(z0, z1)))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))). The rules U(indx(dbls(dbl(sel(z0, z1))), z2)) → U(dbls(dbl(sel(z0, z1)))) and U(indx(dbls(dbl(sel(x0, x1))), x2)) → U(dbls(dbl(sel(x0, x1)))) form no chain, because ECap^µ(U(dbls(dbl(sel(z0, z1))))) = U(dbls(dbl(sel(z0, z1)))) does not unify with U(indx(dbls(dbl(sel(x0, x1))), x2)). The rules U(indx(dbls(dbl(indx(z0, z1))), z2)) → U(dbls(dbl(indx(z0, z1)))) and U(sel(x0, x1)) → U(x0) form no chain, because ECap^µ(U(dbls(dbl(indx(z0, z1))))) = U(dbls(dbl(indx(z0, z1)))) does not unify with U(sel(x0, x1)). The rules U(indx(dbls(dbl(indx(z0, z1))), z2)) → U(dbls(dbl(indx(z0, z1)))) and U(sel(x0, x1)) → U(x1) form no chain, because ECap^µ(U(dbls(dbl(indx(z0, z1))))) = U(dbls(dbl(indx(z0, z1)))) does not unify with U(sel(x0, x1)). The rules U(indx(dbls(dbl(indx(z0, z1))), z2)) → U(dbls(dbl(indx(z0, z1)))) and U(sel(x0, x1)) → SEL(x0, x1) form no chain, because ECap^µ(U(dbls(dbl(indx(z0, z1))))) = U(dbls(dbl(indx(z0, z1)))) does not unify with U(sel(x0, x1)). The rules U(indx(dbls(dbl(indx(z0, z1))), z2)) → U(dbls(dbl(indx(z0, z1)))) and U(dbl(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(dbls(dbl(indx(z0, z1))))) = U(dbls(dbl(indx(z0, z1)))) does not unify with U(dbl(sel(x0, x1))). The rules U(indx(dbls(dbl(indx(z0, z1))), z2)) → U(dbls(dbl(indx(z0, z1)))) and U(dbl(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(dbls(dbl(indx(z0, z1))))) = U(dbls(dbl(indx(z0, z1)))) does not unify with U(dbl(indx(x0, x1))). The rules U(indx(dbls(dbl(indx(z0, z1))), z2)) → U(dbls(dbl(indx(z0, z1)))) and U(dbls(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(dbls(dbl(indx(z0, z1))))) = U(dbls(dbl(indx(z0, z1)))) does not unify with U(dbls(sel(x0, x1))). The rules U(indx(dbls(dbl(indx(z0, z1))), z2)) → U(dbls(dbl(indx(z0, z1)))) and U(dbls(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(dbls(dbl(indx(z0, z1))))) = U(dbls(dbl(indx(z0, z1)))) does not unify with U(dbls(indx(x0, x1))). The rules U(indx(dbls(dbl(indx(z0, z1))), z2)) → U(dbls(dbl(indx(z0, z1)))) and U(dbls(dbl(sel(x0, x1)))) → U(dbl(sel(x0, x1))) form no chain, because ECap^µ(U(dbls(dbl(indx(z0, z1))))) = U(dbls(dbl(indx(z0, z1)))) does not unify with U(dbls(dbl(sel(x0, x1)))). The rules U(indx(dbls(dbl(indx(z0, z1))), z2)) → U(dbls(dbl(indx(z0, z1)))) and U(dbl(dbls(sel(x0, x1)))) → U(dbls(sel(x0, x1))) form no chain, because ECap^µ(U(dbls(dbl(indx(z0, z1))))) = U(dbls(dbl(indx(z0, z1)))) does not unify with U(dbl(dbls(sel(x0, x1)))). The rules U(indx(dbls(dbl(indx(z0, z1))), z2)) → U(dbls(dbl(indx(z0, z1)))) and U(dbl(dbls(indx(x0, x1)))) → U(dbls(indx(x0, x1))) form no chain, because ECap^µ(U(dbls(dbl(indx(z0, z1))))) = U(dbls(dbl(indx(z0, z1)))) does not unify with U(dbl(dbls(indx(x0, x1)))). The rules U(indx(dbls(dbl(indx(z0, z1))), z2)) → U(dbls(dbl(indx(z0, z1)))) and U(dbl(dbls(dbl(sel(x0, x1))))) → U(dbls(dbl(sel(x0, x1)))) form no chain, because ECap^µ(U(dbls(dbl(indx(z0, z1))))) = U(dbls(dbl(indx(z0, z1)))) does not unify with U(dbl(dbls(dbl(sel(x0, x1))))). The rules U(indx(dbls(dbl(indx(z0, z1))), z2)) → U(dbls(dbl(indx(z0, z1)))) and U(dbl(dbls(dbl(indx(x0, x1))))) → U(dbls(dbl(indx(x0, x1)))) form no chain, because ECap^µ(U(dbls(dbl(indx(z0, z1))))) = U(dbls(dbl(indx(z0, z1)))) does not unify with U(dbl(dbls(dbl(indx(x0, x1))))). The rules U(indx(dbls(dbl(indx(z0, z1))), z2)) → U(dbls(dbl(indx(z0, z1)))) and U(dbls(dbl(dbls(sel(x0, x1))))) → U(dbl(dbls(sel(x0, x1)))) form no chain, because ECap^µ(U(dbls(dbl(indx(z0, z1))))) = U(dbls(dbl(indx(z0, z1)))) does not unify with U(dbls(dbl(dbls(sel(x0, x1))))). The rules U(indx(dbls(dbl(indx(z0, z1))), z2)) → U(dbls(dbl(indx(z0, z1)))) and U(dbls(dbl(dbls(indx(x0, x1))))) → U(dbl(dbls(indx(x0, x1)))) form no chain, because ECap^µ(U(dbls(dbl(indx(z0, z1))))) = U(dbls(dbl(indx(z0, z1)))) does not unify with U(dbls(dbl(dbls(indx(x0, x1))))). The rules U(indx(dbls(dbl(indx(z0, z1))), z2)) → U(dbls(dbl(indx(z0, z1)))) and U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) → U(dbl(dbls(dbl(sel(x0, x1))))) form no chain, because ECap^µ(U(dbls(dbl(indx(z0, z1))))) = U(dbls(dbl(indx(z0, z1)))) does not unify with U(dbls(dbl(dbls(dbl(sel(x0, x1)))))). The rules U(indx(dbls(dbl(indx(z0, z1))), z2)) → U(dbls(dbl(indx(z0, z1)))) and U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) → U(dbl(dbls(dbl(indx(x0, x1))))) form no chain, because ECap^µ(U(dbls(dbl(indx(z0, z1))))) = U(dbls(dbl(indx(z0, z1)))) does not unify with U(dbls(dbl(dbls(dbl(indx(x0, x1)))))). The rules U(indx(dbls(dbl(indx(z0, z1))), z2)) → U(dbls(dbl(indx(z0, z1)))) and U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) → U(dbls(dbl(dbls(sel(x0, x1))))) form no chain, because ECap^µ(U(dbls(dbl(indx(z0, z1))))) = U(dbls(dbl(indx(z0, z1)))) does not unify with U(dbl(dbls(dbl(dbls(sel(x0, x1)))))). The rules U(indx(dbls(dbl(indx(z0, z1))), z2)) → U(dbls(dbl(indx(z0, z1)))) and U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) → U(dbls(dbl(dbls(indx(x0, x1))))) form no chain, because ECap^µ(U(dbls(dbl(indx(z0, z1))))) = U(dbls(dbl(indx(z0, z1)))) does not unify with U(dbl(dbls(dbl(dbls(indx(x0, x1)))))). The rules U(indx(dbls(dbl(indx(z0, z1))), z2)) → U(dbls(dbl(indx(z0, z1)))) and U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) → U(dbls(dbl(dbls(dbl(dbls(x0)))))) form no chain, because ECap^µ(U(dbls(dbl(indx(z0, z1))))) = U(dbls(dbl(indx(z0, z1)))) does not unify with U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))). The rules U(indx(dbls(dbl(indx(z0, z1))), z2)) → U(dbls(dbl(indx(z0, z1)))) and U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) → U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) form no chain, because ECap^µ(U(dbls(dbl(indx(z0, z1))))) = U(dbls(dbl(indx(z0, z1)))) does not unify with U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))). The rules U(indx(dbls(dbl(indx(z0, z1))), z2)) → U(dbls(dbl(indx(z0, z1)))) and U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) → U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) form no chain, because ECap^µ(U(dbls(dbl(indx(z0, z1))))) = U(dbls(dbl(indx(z0, z1)))) does not unify with U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))). The rules U(indx(dbls(dbl(indx(z0, z1))), z2)) → U(dbls(dbl(indx(z0, z1)))) and U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))) → U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) form no chain, because ECap^µ(U(dbls(dbl(indx(z0, z1))))) = U(dbls(dbl(indx(z0, z1)))) does not unify with U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))). The rules U(indx(dbls(dbl(indx(z0, z1))), z2)) → U(dbls(dbl(indx(z0, z1)))) and U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))) → U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) form no chain, because ECap^µ(U(dbls(dbl(indx(z0, z1))))) = U(dbls(dbl(indx(z0, z1)))) does not unify with U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))). The rules U(indx(dbls(dbl(indx(z0, z1))), z2)) → U(dbls(dbl(indx(z0, z1)))) and U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))) → U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) form no chain, because ECap^µ(U(dbls(dbl(indx(z0, z1))))) = U(dbls(dbl(indx(z0, z1)))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))). The rules U(indx(dbls(dbl(indx(z0, z1))), z2)) → U(dbls(dbl(indx(z0, z1)))) and U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) form no chain, because ECap^µ(U(dbls(dbl(indx(z0, z1))))) = U(dbls(dbl(indx(z0, z1)))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))). The rules U(indx(dbls(dbl(indx(z0, z1))), z2)) → U(dbls(dbl(indx(z0, z1)))) and U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) form no chain, because ECap^µ(U(dbls(dbl(indx(z0, z1))))) = U(dbls(dbl(indx(z0, z1)))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))). The rules U(indx(dbls(dbl(indx(z0, z1))), z2)) → U(dbls(dbl(indx(z0, z1)))) and U(indx(dbls(dbl(indx(x0, x1))), x2)) → U(dbls(dbl(indx(x0, x1)))) form no chain, because ECap^µ(U(dbls(dbl(indx(z0, z1))))) = U(dbls(dbl(indx(z0, z1)))) does not unify with U(indx(dbls(dbl(indx(x0, x1))), x2)). The rules U(indx(dbl(dbls(sel(z0, z1))), z2)) → U(dbl(dbls(sel(z0, z1)))) and U(sel(x0, x1)) → U(x0) form no chain, because ECap^µ(U(dbl(dbls(sel(z0, z1))))) = U(dbl(dbls(sel(z0, z1)))) does not unify with U(sel(x0, x1)). The rules U(indx(dbl(dbls(sel(z0, z1))), z2)) → U(dbl(dbls(sel(z0, z1)))) and U(sel(x0, x1)) → U(x1) form no chain, because ECap^µ(U(dbl(dbls(sel(z0, z1))))) = U(dbl(dbls(sel(z0, z1)))) does not unify with U(sel(x0, x1)). The rules U(indx(dbl(dbls(sel(z0, z1))), z2)) → U(dbl(dbls(sel(z0, z1)))) and U(sel(x0, x1)) → SEL(x0, x1) form no chain, because ECap^µ(U(dbl(dbls(sel(z0, z1))))) = U(dbl(dbls(sel(z0, z1)))) does not unify with U(sel(x0, x1)). The rules U(indx(dbl(dbls(sel(z0, z1))), z2)) → U(dbl(dbls(sel(z0, z1)))) and U(dbl(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(dbl(dbls(sel(z0, z1))))) = U(dbl(dbls(sel(z0, z1)))) does not unify with U(dbl(sel(x0, x1))). The rules U(indx(dbl(dbls(sel(z0, z1))), z2)) → U(dbl(dbls(sel(z0, z1)))) and U(dbl(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(dbl(dbls(sel(z0, z1))))) = U(dbl(dbls(sel(z0, z1)))) does not unify with U(dbl(indx(x0, x1))). The rules U(indx(dbl(dbls(sel(z0, z1))), z2)) → U(dbl(dbls(sel(z0, z1)))) and U(dbls(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(dbl(dbls(sel(z0, z1))))) = U(dbl(dbls(sel(z0, z1)))) does not unify with U(dbls(sel(x0, x1))). The rules U(indx(dbl(dbls(sel(z0, z1))), z2)) → U(dbl(dbls(sel(z0, z1)))) and U(dbls(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(dbl(dbls(sel(z0, z1))))) = U(dbl(dbls(sel(z0, z1)))) does not unify with U(dbls(indx(x0, x1))). The rules U(indx(dbl(dbls(sel(z0, z1))), z2)) → U(dbl(dbls(sel(z0, z1)))) and U(dbls(dbl(sel(x0, x1)))) → U(dbl(sel(x0, x1))) form no chain, because ECap^µ(U(dbl(dbls(sel(z0, z1))))) = U(dbl(dbls(sel(z0, z1)))) does not unify with U(dbls(dbl(sel(x0, x1)))). The rules U(indx(dbl(dbls(sel(z0, z1))), z2)) → U(dbl(dbls(sel(z0, z1)))) and U(dbls(dbl(indx(x0, x1)))) → U(dbl(indx(x0, x1))) form no chain, because ECap^µ(U(dbl(dbls(sel(z0, z1))))) = U(dbl(dbls(sel(z0, z1)))) does not unify with U(dbls(dbl(indx(x0, x1)))). The rules U(indx(dbl(dbls(sel(z0, z1))), z2)) → U(dbl(dbls(sel(z0, z1)))) and U(dbl(dbls(indx(x0, x1)))) → U(dbls(indx(x0, x1))) form no chain, because ECap^µ(U(dbl(dbls(sel(z0, z1))))) = U(dbl(dbls(sel(z0, z1)))) does not unify with U(dbl(dbls(indx(x0, x1)))). The rules U(indx(dbl(dbls(sel(z0, z1))), z2)) → U(dbl(dbls(sel(z0, z1)))) and U(dbl(dbls(dbl(sel(x0, x1))))) → U(dbls(dbl(sel(x0, x1)))) form no chain, because ECap^µ(U(dbl(dbls(sel(z0, z1))))) = U(dbl(dbls(sel(z0, z1)))) does not unify with U(dbl(dbls(dbl(sel(x0, x1))))). The rules U(indx(dbl(dbls(sel(z0, z1))), z2)) → U(dbl(dbls(sel(z0, z1)))) and U(dbl(dbls(dbl(indx(x0, x1))))) → U(dbls(dbl(indx(x0, x1)))) form no chain, because ECap^µ(U(dbl(dbls(sel(z0, z1))))) = U(dbl(dbls(sel(z0, z1)))) does not unify with U(dbl(dbls(dbl(indx(x0, x1))))). The rules U(indx(dbl(dbls(sel(z0, z1))), z2)) → U(dbl(dbls(sel(z0, z1)))) and U(dbls(dbl(dbls(sel(x0, x1))))) → U(dbl(dbls(sel(x0, x1)))) form no chain, because ECap^µ(U(dbl(dbls(sel(z0, z1))))) = U(dbl(dbls(sel(z0, z1)))) does not unify with U(dbls(dbl(dbls(sel(x0, x1))))). The rules U(indx(dbl(dbls(sel(z0, z1))), z2)) → U(dbl(dbls(sel(z0, z1)))) and U(dbls(dbl(dbls(indx(x0, x1))))) → U(dbl(dbls(indx(x0, x1)))) form no chain, because ECap^µ(U(dbl(dbls(sel(z0, z1))))) = U(dbl(dbls(sel(z0, z1)))) does not unify with U(dbls(dbl(dbls(indx(x0, x1))))). The rules U(indx(dbl(dbls(sel(z0, z1))), z2)) → U(dbl(dbls(sel(z0, z1)))) and U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) → U(dbl(dbls(dbl(sel(x0, x1))))) form no chain, because ECap^µ(U(dbl(dbls(sel(z0, z1))))) = U(dbl(dbls(sel(z0, z1)))) does not unify with U(dbls(dbl(dbls(dbl(sel(x0, x1)))))). The rules U(indx(dbl(dbls(sel(z0, z1))), z2)) → U(dbl(dbls(sel(z0, z1)))) and U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) → U(dbl(dbls(dbl(indx(x0, x1))))) form no chain, because ECap^µ(U(dbl(dbls(sel(z0, z1))))) = U(dbl(dbls(sel(z0, z1)))) does not unify with U(dbls(dbl(dbls(dbl(indx(x0, x1)))))). The rules U(indx(dbl(dbls(sel(z0, z1))), z2)) → U(dbl(dbls(sel(z0, z1)))) and U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) → U(dbls(dbl(dbls(sel(x0, x1))))) form no chain, because ECap^µ(U(dbl(dbls(sel(z0, z1))))) = U(dbl(dbls(sel(z0, z1)))) does not unify with U(dbl(dbls(dbl(dbls(sel(x0, x1)))))). The rules U(indx(dbl(dbls(sel(z0, z1))), z2)) → U(dbl(dbls(sel(z0, z1)))) and U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) → U(dbls(dbl(dbls(indx(x0, x1))))) form no chain, because ECap^µ(U(dbl(dbls(sel(z0, z1))))) = U(dbl(dbls(sel(z0, z1)))) does not unify with U(dbl(dbls(dbl(dbls(indx(x0, x1)))))). The rules U(indx(dbl(dbls(sel(z0, z1))), z2)) → U(dbl(dbls(sel(z0, z1)))) and U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) → U(dbls(dbl(dbls(dbl(dbls(x0)))))) form no chain, because ECap^µ(U(dbl(dbls(sel(z0, z1))))) = U(dbl(dbls(sel(z0, z1)))) does not unify with U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))). The rules U(indx(dbl(dbls(sel(z0, z1))), z2)) → U(dbl(dbls(sel(z0, z1)))) and U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) → U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) form no chain, because ECap^µ(U(dbl(dbls(sel(z0, z1))))) = U(dbl(dbls(sel(z0, z1)))) does not unify with U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))). The rules U(indx(dbl(dbls(sel(z0, z1))), z2)) → U(dbl(dbls(sel(z0, z1)))) and U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) → U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) form no chain, because ECap^µ(U(dbl(dbls(sel(z0, z1))))) = U(dbl(dbls(sel(z0, z1)))) does not unify with U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))). The rules U(indx(dbl(dbls(sel(z0, z1))), z2)) → U(dbl(dbls(sel(z0, z1)))) and U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))) → U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) form no chain, because ECap^µ(U(dbl(dbls(sel(z0, z1))))) = U(dbl(dbls(sel(z0, z1)))) does not unify with U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))). The rules U(indx(dbl(dbls(sel(z0, z1))), z2)) → U(dbl(dbls(sel(z0, z1)))) and U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))) → U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) form no chain, because ECap^µ(U(dbl(dbls(sel(z0, z1))))) = U(dbl(dbls(sel(z0, z1)))) does not unify with U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))). The rules U(indx(dbl(dbls(sel(z0, z1))), z2)) → U(dbl(dbls(sel(z0, z1)))) and U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))) → U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) form no chain, because ECap^µ(U(dbl(dbls(sel(z0, z1))))) = U(dbl(dbls(sel(z0, z1)))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))). The rules U(indx(dbl(dbls(sel(z0, z1))), z2)) → U(dbl(dbls(sel(z0, z1)))) and U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) form no chain, because ECap^µ(U(dbl(dbls(sel(z0, z1))))) = U(dbl(dbls(sel(z0, z1)))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))). The rules U(indx(dbl(dbls(sel(z0, z1))), z2)) → U(dbl(dbls(sel(z0, z1)))) and U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) form no chain, because ECap^µ(U(dbl(dbls(sel(z0, z1))))) = U(dbl(dbls(sel(z0, z1)))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))). The rules U(indx(dbl(dbls(sel(z0, z1))), z2)) → U(dbl(dbls(sel(z0, z1)))) and U(indx(dbl(dbls(sel(x0, x1))), x2)) → U(dbl(dbls(sel(x0, x1)))) form no chain, because ECap^µ(U(dbl(dbls(sel(z0, z1))))) = U(dbl(dbls(sel(z0, z1)))) does not unify with U(indx(dbl(dbls(sel(x0, x1))), x2)). The rules U(indx(dbl(dbls(indx(z0, z1))), z2)) → U(dbl(dbls(indx(z0, z1)))) and U(sel(x0, x1)) → U(x0) form no chain, because ECap^µ(U(dbl(dbls(indx(z0, z1))))) = U(dbl(dbls(indx(z0, z1)))) does not unify with U(sel(x0, x1)). The rules U(indx(dbl(dbls(indx(z0, z1))), z2)) → U(dbl(dbls(indx(z0, z1)))) and U(sel(x0, x1)) → U(x1) form no chain, because ECap^µ(U(dbl(dbls(indx(z0, z1))))) = U(dbl(dbls(indx(z0, z1)))) does not unify with U(sel(x0, x1)). The rules U(indx(dbl(dbls(indx(z0, z1))), z2)) → U(dbl(dbls(indx(z0, z1)))) and U(sel(x0, x1)) → SEL(x0, x1) form no chain, because ECap^µ(U(dbl(dbls(indx(z0, z1))))) = U(dbl(dbls(indx(z0, z1)))) does not unify with U(sel(x0, x1)). The rules U(indx(dbl(dbls(indx(z0, z1))), z2)) → U(dbl(dbls(indx(z0, z1)))) and U(dbl(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(dbl(dbls(indx(z0, z1))))) = U(dbl(dbls(indx(z0, z1)))) does not unify with U(dbl(sel(x0, x1))). The rules U(indx(dbl(dbls(indx(z0, z1))), z2)) → U(dbl(dbls(indx(z0, z1)))) and U(dbl(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(dbl(dbls(indx(z0, z1))))) = U(dbl(dbls(indx(z0, z1)))) does not unify with U(dbl(indx(x0, x1))). The rules U(indx(dbl(dbls(indx(z0, z1))), z2)) → U(dbl(dbls(indx(z0, z1)))) and U(dbls(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(dbl(dbls(indx(z0, z1))))) = U(dbl(dbls(indx(z0, z1)))) does not unify with U(dbls(sel(x0, x1))). The rules U(indx(dbl(dbls(indx(z0, z1))), z2)) → U(dbl(dbls(indx(z0, z1)))) and U(dbls(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(dbl(dbls(indx(z0, z1))))) = U(dbl(dbls(indx(z0, z1)))) does not unify with U(dbls(indx(x0, x1))). The rules U(indx(dbl(dbls(indx(z0, z1))), z2)) → U(dbl(dbls(indx(z0, z1)))) and U(dbls(dbl(sel(x0, x1)))) → U(dbl(sel(x0, x1))) form no chain, because ECap^µ(U(dbl(dbls(indx(z0, z1))))) = U(dbl(dbls(indx(z0, z1)))) does not unify with U(dbls(dbl(sel(x0, x1)))). The rules U(indx(dbl(dbls(indx(z0, z1))), z2)) → U(dbl(dbls(indx(z0, z1)))) and U(dbls(dbl(indx(x0, x1)))) → U(dbl(indx(x0, x1))) form no chain, because ECap^µ(U(dbl(dbls(indx(z0, z1))))) = U(dbl(dbls(indx(z0, z1)))) does not unify with U(dbls(dbl(indx(x0, x1)))). The rules U(indx(dbl(dbls(indx(z0, z1))), z2)) → U(dbl(dbls(indx(z0, z1)))) and U(dbl(dbls(sel(x0, x1)))) → U(dbls(sel(x0, x1))) form no chain, because ECap^µ(U(dbl(dbls(indx(z0, z1))))) = U(dbl(dbls(indx(z0, z1)))) does not unify with U(dbl(dbls(sel(x0, x1)))). The rules U(indx(dbl(dbls(indx(z0, z1))), z2)) → U(dbl(dbls(indx(z0, z1)))) and U(dbl(dbls(dbl(sel(x0, x1))))) → U(dbls(dbl(sel(x0, x1)))) form no chain, because ECap^µ(U(dbl(dbls(indx(z0, z1))))) = U(dbl(dbls(indx(z0, z1)))) does not unify with U(dbl(dbls(dbl(sel(x0, x1))))). The rules U(indx(dbl(dbls(indx(z0, z1))), z2)) → U(dbl(dbls(indx(z0, z1)))) and U(dbl(dbls(dbl(indx(x0, x1))))) → U(dbls(dbl(indx(x0, x1)))) form no chain, because ECap^µ(U(dbl(dbls(indx(z0, z1))))) = U(dbl(dbls(indx(z0, z1)))) does not unify with U(dbl(dbls(dbl(indx(x0, x1))))). The rules U(indx(dbl(dbls(indx(z0, z1))), z2)) → U(dbl(dbls(indx(z0, z1)))) and U(dbls(dbl(dbls(sel(x0, x1))))) → U(dbl(dbls(sel(x0, x1)))) form no chain, because ECap^µ(U(dbl(dbls(indx(z0, z1))))) = U(dbl(dbls(indx(z0, z1)))) does not unify with U(dbls(dbl(dbls(sel(x0, x1))))). The rules U(indx(dbl(dbls(indx(z0, z1))), z2)) → U(dbl(dbls(indx(z0, z1)))) and U(dbls(dbl(dbls(indx(x0, x1))))) → U(dbl(dbls(indx(x0, x1)))) form no chain, because ECap^µ(U(dbl(dbls(indx(z0, z1))))) = U(dbl(dbls(indx(z0, z1)))) does not unify with U(dbls(dbl(dbls(indx(x0, x1))))). The rules U(indx(dbl(dbls(indx(z0, z1))), z2)) → U(dbl(dbls(indx(z0, z1)))) and U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) → U(dbl(dbls(dbl(sel(x0, x1))))) form no chain, because ECap^µ(U(dbl(dbls(indx(z0, z1))))) = U(dbl(dbls(indx(z0, z1)))) does not unify with U(dbls(dbl(dbls(dbl(sel(x0, x1)))))). The rules U(indx(dbl(dbls(indx(z0, z1))), z2)) → U(dbl(dbls(indx(z0, z1)))) and U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) → U(dbl(dbls(dbl(indx(x0, x1))))) form no chain, because ECap^µ(U(dbl(dbls(indx(z0, z1))))) = U(dbl(dbls(indx(z0, z1)))) does not unify with U(dbls(dbl(dbls(dbl(indx(x0, x1)))))). The rules U(indx(dbl(dbls(indx(z0, z1))), z2)) → U(dbl(dbls(indx(z0, z1)))) and U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) → U(dbls(dbl(dbls(sel(x0, x1))))) form no chain, because ECap^µ(U(dbl(dbls(indx(z0, z1))))) = U(dbl(dbls(indx(z0, z1)))) does not unify with U(dbl(dbls(dbl(dbls(sel(x0, x1)))))). The rules U(indx(dbl(dbls(indx(z0, z1))), z2)) → U(dbl(dbls(indx(z0, z1)))) and U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) → U(dbls(dbl(dbls(indx(x0, x1))))) form no chain, because ECap^µ(U(dbl(dbls(indx(z0, z1))))) = U(dbl(dbls(indx(z0, z1)))) does not unify with U(dbl(dbls(dbl(dbls(indx(x0, x1)))))). The rules U(indx(dbl(dbls(indx(z0, z1))), z2)) → U(dbl(dbls(indx(z0, z1)))) and U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) → U(dbls(dbl(dbls(dbl(dbls(x0)))))) form no chain, because ECap^µ(U(dbl(dbls(indx(z0, z1))))) = U(dbl(dbls(indx(z0, z1)))) does not unify with U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))). The rules U(indx(dbl(dbls(indx(z0, z1))), z2)) → U(dbl(dbls(indx(z0, z1)))) and U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) → U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) form no chain, because ECap^µ(U(dbl(dbls(indx(z0, z1))))) = U(dbl(dbls(indx(z0, z1)))) does not unify with U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))). The rules U(indx(dbl(dbls(indx(z0, z1))), z2)) → U(dbl(dbls(indx(z0, z1)))) and U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) → U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) form no chain, because ECap^µ(U(dbl(dbls(indx(z0, z1))))) = U(dbl(dbls(indx(z0, z1)))) does not unify with U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))). The rules U(indx(dbl(dbls(indx(z0, z1))), z2)) → U(dbl(dbls(indx(z0, z1)))) and U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))) → U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) form no chain, because ECap^µ(U(dbl(dbls(indx(z0, z1))))) = U(dbl(dbls(indx(z0, z1)))) does not unify with U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))). The rules U(indx(dbl(dbls(indx(z0, z1))), z2)) → U(dbl(dbls(indx(z0, z1)))) and U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))) → U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) form no chain, because ECap^µ(U(dbl(dbls(indx(z0, z1))))) = U(dbl(dbls(indx(z0, z1)))) does not unify with U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))). The rules U(indx(dbl(dbls(indx(z0, z1))), z2)) → U(dbl(dbls(indx(z0, z1)))) and U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))) → U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) form no chain, because ECap^µ(U(dbl(dbls(indx(z0, z1))))) = U(dbl(dbls(indx(z0, z1)))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))). The rules U(indx(dbl(dbls(indx(z0, z1))), z2)) → U(dbl(dbls(indx(z0, z1)))) and U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) form no chain, because ECap^µ(U(dbl(dbls(indx(z0, z1))))) = U(dbl(dbls(indx(z0, z1)))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))). The rules U(indx(dbl(dbls(indx(z0, z1))), z2)) → U(dbl(dbls(indx(z0, z1)))) and U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) form no chain, because ECap^µ(U(dbl(dbls(indx(z0, z1))))) = U(dbl(dbls(indx(z0, z1)))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))). The rules U(indx(dbl(dbls(indx(z0, z1))), z2)) → U(dbl(dbls(indx(z0, z1)))) and U(indx(dbl(dbls(indx(x0, x1))), x2)) → U(dbl(dbls(indx(x0, x1)))) form no chain, because ECap^µ(U(dbl(dbls(indx(z0, z1))))) = U(dbl(dbls(indx(z0, z1)))) does not unify with U(indx(dbl(dbls(indx(x0, x1))), x2)). The rules U(indx(dbl(dbls(dbl(sel(z0, z1)))), z2)) → U(dbl(dbls(dbl(sel(z0, z1))))) and U(sel(x0, x1)) → U(x0) form no chain, because ECap^µ(U(dbl(dbls(dbl(sel(z0, z1)))))) = U(dbl(dbls(dbl(sel(z0, z1))))) does not unify with U(sel(x0, x1)). The rules U(indx(dbl(dbls(dbl(sel(z0, z1)))), z2)) → U(dbl(dbls(dbl(sel(z0, z1))))) and U(sel(x0, x1)) → U(x1) form no chain, because ECap^µ(U(dbl(dbls(dbl(sel(z0, z1)))))) = U(dbl(dbls(dbl(sel(z0, z1))))) does not unify with U(sel(x0, x1)). The rules U(indx(dbl(dbls(dbl(sel(z0, z1)))), z2)) → U(dbl(dbls(dbl(sel(z0, z1))))) and U(sel(x0, x1)) → SEL(x0, x1) form no chain, because ECap^µ(U(dbl(dbls(dbl(sel(z0, z1)))))) = U(dbl(dbls(dbl(sel(z0, z1))))) does not unify with U(sel(x0, x1)). The rules U(indx(dbl(dbls(dbl(sel(z0, z1)))), z2)) → U(dbl(dbls(dbl(sel(z0, z1))))) and U(dbl(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(dbl(dbls(dbl(sel(z0, z1)))))) = U(dbl(dbls(dbl(sel(z0, z1))))) does not unify with U(dbl(sel(x0, x1))). The rules U(indx(dbl(dbls(dbl(sel(z0, z1)))), z2)) → U(dbl(dbls(dbl(sel(z0, z1))))) and U(dbl(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(dbl(dbls(dbl(sel(z0, z1)))))) = U(dbl(dbls(dbl(sel(z0, z1))))) does not unify with U(dbl(indx(x0, x1))). The rules U(indx(dbl(dbls(dbl(sel(z0, z1)))), z2)) → U(dbl(dbls(dbl(sel(z0, z1))))) and U(dbls(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(dbl(dbls(dbl(sel(z0, z1)))))) = U(dbl(dbls(dbl(sel(z0, z1))))) does not unify with U(dbls(sel(x0, x1))). The rules U(indx(dbl(dbls(dbl(sel(z0, z1)))), z2)) → U(dbl(dbls(dbl(sel(z0, z1))))) and U(dbls(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(dbl(dbls(dbl(sel(z0, z1)))))) = U(dbl(dbls(dbl(sel(z0, z1))))) does not unify with U(dbls(indx(x0, x1))). The rules U(indx(dbl(dbls(dbl(sel(z0, z1)))), z2)) → U(dbl(dbls(dbl(sel(z0, z1))))) and U(dbls(dbl(sel(x0, x1)))) → U(dbl(sel(x0, x1))) form no chain, because ECap^µ(U(dbl(dbls(dbl(sel(z0, z1)))))) = U(dbl(dbls(dbl(sel(z0, z1))))) does not unify with U(dbls(dbl(sel(x0, x1)))). The rules U(indx(dbl(dbls(dbl(sel(z0, z1)))), z2)) → U(dbl(dbls(dbl(sel(z0, z1))))) and U(dbls(dbl(indx(x0, x1)))) → U(dbl(indx(x0, x1))) form no chain, because ECap^µ(U(dbl(dbls(dbl(sel(z0, z1)))))) = U(dbl(dbls(dbl(sel(z0, z1))))) does not unify with U(dbls(dbl(indx(x0, x1)))). The rules U(indx(dbl(dbls(dbl(sel(z0, z1)))), z2)) → U(dbl(dbls(dbl(sel(z0, z1))))) and U(dbl(dbls(sel(x0, x1)))) → U(dbls(sel(x0, x1))) form no chain, because ECap^µ(U(dbl(dbls(dbl(sel(z0, z1)))))) = U(dbl(dbls(dbl(sel(z0, z1))))) does not unify with U(dbl(dbls(sel(x0, x1)))). The rules U(indx(dbl(dbls(dbl(sel(z0, z1)))), z2)) → U(dbl(dbls(dbl(sel(z0, z1))))) and U(dbl(dbls(indx(x0, x1)))) → U(dbls(indx(x0, x1))) form no chain, because ECap^µ(U(dbl(dbls(dbl(sel(z0, z1)))))) = U(dbl(dbls(dbl(sel(z0, z1))))) does not unify with U(dbl(dbls(indx(x0, x1)))). The rules U(indx(dbl(dbls(dbl(sel(z0, z1)))), z2)) → U(dbl(dbls(dbl(sel(z0, z1))))) and U(dbl(dbls(dbl(indx(x0, x1))))) → U(dbls(dbl(indx(x0, x1)))) form no chain, because ECap^µ(U(dbl(dbls(dbl(sel(z0, z1)))))) = U(dbl(dbls(dbl(sel(z0, z1))))) does not unify with U(dbl(dbls(dbl(indx(x0, x1))))). The rules U(indx(dbl(dbls(dbl(sel(z0, z1)))), z2)) → U(dbl(dbls(dbl(sel(z0, z1))))) and U(dbls(dbl(dbls(sel(x0, x1))))) → U(dbl(dbls(sel(x0, x1)))) form no chain, because ECap^µ(U(dbl(dbls(dbl(sel(z0, z1)))))) = U(dbl(dbls(dbl(sel(z0, z1))))) does not unify with U(dbls(dbl(dbls(sel(x0, x1))))). The rules U(indx(dbl(dbls(dbl(sel(z0, z1)))), z2)) → U(dbl(dbls(dbl(sel(z0, z1))))) and U(dbls(dbl(dbls(indx(x0, x1))))) → U(dbl(dbls(indx(x0, x1)))) form no chain, because ECap^µ(U(dbl(dbls(dbl(sel(z0, z1)))))) = U(dbl(dbls(dbl(sel(z0, z1))))) does not unify with U(dbls(dbl(dbls(indx(x0, x1))))). The rules U(indx(dbl(dbls(dbl(sel(z0, z1)))), z2)) → U(dbl(dbls(dbl(sel(z0, z1))))) and U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) → U(dbl(dbls(dbl(sel(x0, x1))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(sel(z0, z1)))))) = U(dbl(dbls(dbl(sel(z0, z1))))) does not unify with U(dbls(dbl(dbls(dbl(sel(x0, x1)))))). The rules U(indx(dbl(dbls(dbl(sel(z0, z1)))), z2)) → U(dbl(dbls(dbl(sel(z0, z1))))) and U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) → U(dbl(dbls(dbl(indx(x0, x1))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(sel(z0, z1)))))) = U(dbl(dbls(dbl(sel(z0, z1))))) does not unify with U(dbls(dbl(dbls(dbl(indx(x0, x1)))))). The rules U(indx(dbl(dbls(dbl(sel(z0, z1)))), z2)) → U(dbl(dbls(dbl(sel(z0, z1))))) and U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) → U(dbls(dbl(dbls(sel(x0, x1))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(sel(z0, z1)))))) = U(dbl(dbls(dbl(sel(z0, z1))))) does not unify with U(dbl(dbls(dbl(dbls(sel(x0, x1)))))). The rules U(indx(dbl(dbls(dbl(sel(z0, z1)))), z2)) → U(dbl(dbls(dbl(sel(z0, z1))))) and U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) → U(dbls(dbl(dbls(indx(x0, x1))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(sel(z0, z1)))))) = U(dbl(dbls(dbl(sel(z0, z1))))) does not unify with U(dbl(dbls(dbl(dbls(indx(x0, x1)))))). The rules U(indx(dbl(dbls(dbl(sel(z0, z1)))), z2)) → U(dbl(dbls(dbl(sel(z0, z1))))) and U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) → U(dbls(dbl(dbls(dbl(dbls(x0)))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(sel(z0, z1)))))) = U(dbl(dbls(dbl(sel(z0, z1))))) does not unify with U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))). The rules U(indx(dbl(dbls(dbl(sel(z0, z1)))), z2)) → U(dbl(dbls(dbl(sel(z0, z1))))) and U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) → U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(sel(z0, z1)))))) = U(dbl(dbls(dbl(sel(z0, z1))))) does not unify with U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))). The rules U(indx(dbl(dbls(dbl(sel(z0, z1)))), z2)) → U(dbl(dbls(dbl(sel(z0, z1))))) and U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) → U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(sel(z0, z1)))))) = U(dbl(dbls(dbl(sel(z0, z1))))) does not unify with U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))). The rules U(indx(dbl(dbls(dbl(sel(z0, z1)))), z2)) → U(dbl(dbls(dbl(sel(z0, z1))))) and U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))) → U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(sel(z0, z1)))))) = U(dbl(dbls(dbl(sel(z0, z1))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))). The rules U(indx(dbl(dbls(dbl(sel(z0, z1)))), z2)) → U(dbl(dbls(dbl(sel(z0, z1))))) and U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))) → U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(sel(z0, z1)))))) = U(dbl(dbls(dbl(sel(z0, z1))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))). The rules U(indx(dbl(dbls(dbl(sel(z0, z1)))), z2)) → U(dbl(dbls(dbl(sel(z0, z1))))) and U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))) → U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(sel(z0, z1)))))) = U(dbl(dbls(dbl(sel(z0, z1))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))). The rules U(indx(dbl(dbls(dbl(sel(z0, z1)))), z2)) → U(dbl(dbls(dbl(sel(z0, z1))))) and U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(sel(z0, z1)))))) = U(dbl(dbls(dbl(sel(z0, z1))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))). The rules U(indx(dbl(dbls(dbl(sel(z0, z1)))), z2)) → U(dbl(dbls(dbl(sel(z0, z1))))) and U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(sel(z0, z1)))))) = U(dbl(dbls(dbl(sel(z0, z1))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))). The rules U(indx(dbl(dbls(dbl(sel(z0, z1)))), z2)) → U(dbl(dbls(dbl(sel(z0, z1))))) and U(indx(dbl(dbls(dbl(sel(x0, x1)))), x2)) → U(dbl(dbls(dbl(sel(x0, x1))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(sel(z0, z1)))))) = U(dbl(dbls(dbl(sel(z0, z1))))) does not unify with U(indx(dbl(dbls(dbl(sel(x0, x1)))), x2)). The rules U(indx(dbl(dbls(dbl(indx(z0, z1)))), z2)) → U(dbl(dbls(dbl(indx(z0, z1))))) and U(sel(x0, x1)) → U(x0) form no chain, because ECap^µ(U(dbl(dbls(dbl(indx(z0, z1)))))) = U(dbl(dbls(dbl(indx(z0, z1))))) does not unify with U(sel(x0, x1)). The rules U(indx(dbl(dbls(dbl(indx(z0, z1)))), z2)) → U(dbl(dbls(dbl(indx(z0, z1))))) and U(sel(x0, x1)) → U(x1) form no chain, because ECap^µ(U(dbl(dbls(dbl(indx(z0, z1)))))) = U(dbl(dbls(dbl(indx(z0, z1))))) does not unify with U(sel(x0, x1)). The rules U(indx(dbl(dbls(dbl(indx(z0, z1)))), z2)) → U(dbl(dbls(dbl(indx(z0, z1))))) and U(sel(x0, x1)) → SEL(x0, x1) form no chain, because ECap^µ(U(dbl(dbls(dbl(indx(z0, z1)))))) = U(dbl(dbls(dbl(indx(z0, z1))))) does not unify with U(sel(x0, x1)). The rules U(indx(dbl(dbls(dbl(indx(z0, z1)))), z2)) → U(dbl(dbls(dbl(indx(z0, z1))))) and U(dbl(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(dbl(dbls(dbl(indx(z0, z1)))))) = U(dbl(dbls(dbl(indx(z0, z1))))) does not unify with U(dbl(sel(x0, x1))). The rules U(indx(dbl(dbls(dbl(indx(z0, z1)))), z2)) → U(dbl(dbls(dbl(indx(z0, z1))))) and U(dbl(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(dbl(dbls(dbl(indx(z0, z1)))))) = U(dbl(dbls(dbl(indx(z0, z1))))) does not unify with U(dbl(indx(x0, x1))). The rules U(indx(dbl(dbls(dbl(indx(z0, z1)))), z2)) → U(dbl(dbls(dbl(indx(z0, z1))))) and U(dbls(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(dbl(dbls(dbl(indx(z0, z1)))))) = U(dbl(dbls(dbl(indx(z0, z1))))) does not unify with U(dbls(sel(x0, x1))). The rules U(indx(dbl(dbls(dbl(indx(z0, z1)))), z2)) → U(dbl(dbls(dbl(indx(z0, z1))))) and U(dbls(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(dbl(dbls(dbl(indx(z0, z1)))))) = U(dbl(dbls(dbl(indx(z0, z1))))) does not unify with U(dbls(indx(x0, x1))). The rules U(indx(dbl(dbls(dbl(indx(z0, z1)))), z2)) → U(dbl(dbls(dbl(indx(z0, z1))))) and U(dbls(dbl(sel(x0, x1)))) → U(dbl(sel(x0, x1))) form no chain, because ECap^µ(U(dbl(dbls(dbl(indx(z0, z1)))))) = U(dbl(dbls(dbl(indx(z0, z1))))) does not unify with U(dbls(dbl(sel(x0, x1)))). The rules U(indx(dbl(dbls(dbl(indx(z0, z1)))), z2)) → U(dbl(dbls(dbl(indx(z0, z1))))) and U(dbls(dbl(indx(x0, x1)))) → U(dbl(indx(x0, x1))) form no chain, because ECap^µ(U(dbl(dbls(dbl(indx(z0, z1)))))) = U(dbl(dbls(dbl(indx(z0, z1))))) does not unify with U(dbls(dbl(indx(x0, x1)))). The rules U(indx(dbl(dbls(dbl(indx(z0, z1)))), z2)) → U(dbl(dbls(dbl(indx(z0, z1))))) and U(dbl(dbls(sel(x0, x1)))) → U(dbls(sel(x0, x1))) form no chain, because ECap^µ(U(dbl(dbls(dbl(indx(z0, z1)))))) = U(dbl(dbls(dbl(indx(z0, z1))))) does not unify with U(dbl(dbls(sel(x0, x1)))). The rules U(indx(dbl(dbls(dbl(indx(z0, z1)))), z2)) → U(dbl(dbls(dbl(indx(z0, z1))))) and U(dbl(dbls(indx(x0, x1)))) → U(dbls(indx(x0, x1))) form no chain, because ECap^µ(U(dbl(dbls(dbl(indx(z0, z1)))))) = U(dbl(dbls(dbl(indx(z0, z1))))) does not unify with U(dbl(dbls(indx(x0, x1)))). The rules U(indx(dbl(dbls(dbl(indx(z0, z1)))), z2)) → U(dbl(dbls(dbl(indx(z0, z1))))) and U(dbl(dbls(dbl(sel(x0, x1))))) → U(dbls(dbl(sel(x0, x1)))) form no chain, because ECap^µ(U(dbl(dbls(dbl(indx(z0, z1)))))) = U(dbl(dbls(dbl(indx(z0, z1))))) does not unify with U(dbl(dbls(dbl(sel(x0, x1))))). The rules U(indx(dbl(dbls(dbl(indx(z0, z1)))), z2)) → U(dbl(dbls(dbl(indx(z0, z1))))) and U(dbls(dbl(dbls(sel(x0, x1))))) → U(dbl(dbls(sel(x0, x1)))) form no chain, because ECap^µ(U(dbl(dbls(dbl(indx(z0, z1)))))) = U(dbl(dbls(dbl(indx(z0, z1))))) does not unify with U(dbls(dbl(dbls(sel(x0, x1))))). The rules U(indx(dbl(dbls(dbl(indx(z0, z1)))), z2)) → U(dbl(dbls(dbl(indx(z0, z1))))) and U(dbls(dbl(dbls(indx(x0, x1))))) → U(dbl(dbls(indx(x0, x1)))) form no chain, because ECap^µ(U(dbl(dbls(dbl(indx(z0, z1)))))) = U(dbl(dbls(dbl(indx(z0, z1))))) does not unify with U(dbls(dbl(dbls(indx(x0, x1))))). The rules U(indx(dbl(dbls(dbl(indx(z0, z1)))), z2)) → U(dbl(dbls(dbl(indx(z0, z1))))) and U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) → U(dbl(dbls(dbl(sel(x0, x1))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(indx(z0, z1)))))) = U(dbl(dbls(dbl(indx(z0, z1))))) does not unify with U(dbls(dbl(dbls(dbl(sel(x0, x1)))))). The rules U(indx(dbl(dbls(dbl(indx(z0, z1)))), z2)) → U(dbl(dbls(dbl(indx(z0, z1))))) and U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) → U(dbl(dbls(dbl(indx(x0, x1))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(indx(z0, z1)))))) = U(dbl(dbls(dbl(indx(z0, z1))))) does not unify with U(dbls(dbl(dbls(dbl(indx(x0, x1)))))). The rules U(indx(dbl(dbls(dbl(indx(z0, z1)))), z2)) → U(dbl(dbls(dbl(indx(z0, z1))))) and U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) → U(dbls(dbl(dbls(sel(x0, x1))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(indx(z0, z1)))))) = U(dbl(dbls(dbl(indx(z0, z1))))) does not unify with U(dbl(dbls(dbl(dbls(sel(x0, x1)))))). The rules U(indx(dbl(dbls(dbl(indx(z0, z1)))), z2)) → U(dbl(dbls(dbl(indx(z0, z1))))) and U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) → U(dbls(dbl(dbls(indx(x0, x1))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(indx(z0, z1)))))) = U(dbl(dbls(dbl(indx(z0, z1))))) does not unify with U(dbl(dbls(dbl(dbls(indx(x0, x1)))))). The rules U(indx(dbl(dbls(dbl(indx(z0, z1)))), z2)) → U(dbl(dbls(dbl(indx(z0, z1))))) and U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) → U(dbls(dbl(dbls(dbl(dbls(x0)))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(indx(z0, z1)))))) = U(dbl(dbls(dbl(indx(z0, z1))))) does not unify with U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))). The rules U(indx(dbl(dbls(dbl(indx(z0, z1)))), z2)) → U(dbl(dbls(dbl(indx(z0, z1))))) and U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) → U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(indx(z0, z1)))))) = U(dbl(dbls(dbl(indx(z0, z1))))) does not unify with U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))). The rules U(indx(dbl(dbls(dbl(indx(z0, z1)))), z2)) → U(dbl(dbls(dbl(indx(z0, z1))))) and U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) → U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(indx(z0, z1)))))) = U(dbl(dbls(dbl(indx(z0, z1))))) does not unify with U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))). The rules U(indx(dbl(dbls(dbl(indx(z0, z1)))), z2)) → U(dbl(dbls(dbl(indx(z0, z1))))) and U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))) → U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(indx(z0, z1)))))) = U(dbl(dbls(dbl(indx(z0, z1))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))). The rules U(indx(dbl(dbls(dbl(indx(z0, z1)))), z2)) → U(dbl(dbls(dbl(indx(z0, z1))))) and U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))) → U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(indx(z0, z1)))))) = U(dbl(dbls(dbl(indx(z0, z1))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))). The rules U(indx(dbl(dbls(dbl(indx(z0, z1)))), z2)) → U(dbl(dbls(dbl(indx(z0, z1))))) and U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))) → U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(indx(z0, z1)))))) = U(dbl(dbls(dbl(indx(z0, z1))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))). The rules U(indx(dbl(dbls(dbl(indx(z0, z1)))), z2)) → U(dbl(dbls(dbl(indx(z0, z1))))) and U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(indx(z0, z1)))))) = U(dbl(dbls(dbl(indx(z0, z1))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))). The rules U(indx(dbl(dbls(dbl(indx(z0, z1)))), z2)) → U(dbl(dbls(dbl(indx(z0, z1))))) and U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(indx(z0, z1)))))) = U(dbl(dbls(dbl(indx(z0, z1))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))). The rules U(indx(dbl(dbls(dbl(indx(z0, z1)))), z2)) → U(dbl(dbls(dbl(indx(z0, z1))))) and U(indx(dbl(dbls(dbl(indx(x0, x1)))), x2)) → U(dbl(dbls(dbl(indx(x0, x1))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(indx(z0, z1)))))) = U(dbl(dbls(dbl(indx(z0, z1))))) does not unify with U(indx(dbl(dbls(dbl(indx(x0, x1)))), x2)). The rules U(indx(dbls(dbl(dbls(sel(z0, z1)))), z2)) → U(dbls(dbl(dbls(sel(z0, z1))))) and U(sel(x0, x1)) → U(x0) form no chain, because ECap^µ(U(dbls(dbl(dbls(sel(z0, z1)))))) = U(dbls(dbl(dbls(sel(z0, z1))))) does not unify with U(sel(x0, x1)). The rules U(indx(dbls(dbl(dbls(sel(z0, z1)))), z2)) → U(dbls(dbl(dbls(sel(z0, z1))))) and U(sel(x0, x1)) → U(x1) form no chain, because ECap^µ(U(dbls(dbl(dbls(sel(z0, z1)))))) = U(dbls(dbl(dbls(sel(z0, z1))))) does not unify with U(sel(x0, x1)). The rules U(indx(dbls(dbl(dbls(sel(z0, z1)))), z2)) → U(dbls(dbl(dbls(sel(z0, z1))))) and U(sel(x0, x1)) → SEL(x0, x1) form no chain, because ECap^µ(U(dbls(dbl(dbls(sel(z0, z1)))))) = U(dbls(dbl(dbls(sel(z0, z1))))) does not unify with U(sel(x0, x1)). The rules U(indx(dbls(dbl(dbls(sel(z0, z1)))), z2)) → U(dbls(dbl(dbls(sel(z0, z1))))) and U(dbl(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(dbls(dbl(dbls(sel(z0, z1)))))) = U(dbls(dbl(dbls(sel(z0, z1))))) does not unify with U(dbl(sel(x0, x1))). The rules U(indx(dbls(dbl(dbls(sel(z0, z1)))), z2)) → U(dbls(dbl(dbls(sel(z0, z1))))) and U(dbl(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(dbls(dbl(dbls(sel(z0, z1)))))) = U(dbls(dbl(dbls(sel(z0, z1))))) does not unify with U(dbl(indx(x0, x1))). The rules U(indx(dbls(dbl(dbls(sel(z0, z1)))), z2)) → U(dbls(dbl(dbls(sel(z0, z1))))) and U(dbls(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(dbls(dbl(dbls(sel(z0, z1)))))) = U(dbls(dbl(dbls(sel(z0, z1))))) does not unify with U(dbls(sel(x0, x1))). The rules U(indx(dbls(dbl(dbls(sel(z0, z1)))), z2)) → U(dbls(dbl(dbls(sel(z0, z1))))) and U(dbls(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(dbls(dbl(dbls(sel(z0, z1)))))) = U(dbls(dbl(dbls(sel(z0, z1))))) does not unify with U(dbls(indx(x0, x1))). The rules U(indx(dbls(dbl(dbls(sel(z0, z1)))), z2)) → U(dbls(dbl(dbls(sel(z0, z1))))) and U(dbls(dbl(sel(x0, x1)))) → U(dbl(sel(x0, x1))) form no chain, because ECap^µ(U(dbls(dbl(dbls(sel(z0, z1)))))) = U(dbls(dbl(dbls(sel(z0, z1))))) does not unify with U(dbls(dbl(sel(x0, x1)))). The rules U(indx(dbls(dbl(dbls(sel(z0, z1)))), z2)) → U(dbls(dbl(dbls(sel(z0, z1))))) and U(dbls(dbl(indx(x0, x1)))) → U(dbl(indx(x0, x1))) form no chain, because ECap^µ(U(dbls(dbl(dbls(sel(z0, z1)))))) = U(dbls(dbl(dbls(sel(z0, z1))))) does not unify with U(dbls(dbl(indx(x0, x1)))). The rules U(indx(dbls(dbl(dbls(sel(z0, z1)))), z2)) → U(dbls(dbl(dbls(sel(z0, z1))))) and U(dbl(dbls(sel(x0, x1)))) → U(dbls(sel(x0, x1))) form no chain, because ECap^µ(U(dbls(dbl(dbls(sel(z0, z1)))))) = U(dbls(dbl(dbls(sel(z0, z1))))) does not unify with U(dbl(dbls(sel(x0, x1)))). The rules U(indx(dbls(dbl(dbls(sel(z0, z1)))), z2)) → U(dbls(dbl(dbls(sel(z0, z1))))) and U(dbl(dbls(indx(x0, x1)))) → U(dbls(indx(x0, x1))) form no chain, because ECap^µ(U(dbls(dbl(dbls(sel(z0, z1)))))) = U(dbls(dbl(dbls(sel(z0, z1))))) does not unify with U(dbl(dbls(indx(x0, x1)))). The rules U(indx(dbls(dbl(dbls(sel(z0, z1)))), z2)) → U(dbls(dbl(dbls(sel(z0, z1))))) and U(dbl(dbls(dbl(sel(x0, x1))))) → U(dbls(dbl(sel(x0, x1)))) form no chain, because ECap^µ(U(dbls(dbl(dbls(sel(z0, z1)))))) = U(dbls(dbl(dbls(sel(z0, z1))))) does not unify with U(dbl(dbls(dbl(sel(x0, x1))))). The rules U(indx(dbls(dbl(dbls(sel(z0, z1)))), z2)) → U(dbls(dbl(dbls(sel(z0, z1))))) and U(dbl(dbls(dbl(indx(x0, x1))))) → U(dbls(dbl(indx(x0, x1)))) form no chain, because ECap^µ(U(dbls(dbl(dbls(sel(z0, z1)))))) = U(dbls(dbl(dbls(sel(z0, z1))))) does not unify with U(dbl(dbls(dbl(indx(x0, x1))))). The rules U(indx(dbls(dbl(dbls(sel(z0, z1)))), z2)) → U(dbls(dbl(dbls(sel(z0, z1))))) and U(dbls(dbl(dbls(indx(x0, x1))))) → U(dbl(dbls(indx(x0, x1)))) form no chain, because ECap^µ(U(dbls(dbl(dbls(sel(z0, z1)))))) = U(dbls(dbl(dbls(sel(z0, z1))))) does not unify with U(dbls(dbl(dbls(indx(x0, x1))))). The rules U(indx(dbls(dbl(dbls(sel(z0, z1)))), z2)) → U(dbls(dbl(dbls(sel(z0, z1))))) and U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) → U(dbl(dbls(dbl(sel(x0, x1))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(sel(z0, z1)))))) = U(dbls(dbl(dbls(sel(z0, z1))))) does not unify with U(dbls(dbl(dbls(dbl(sel(x0, x1)))))). The rules U(indx(dbls(dbl(dbls(sel(z0, z1)))), z2)) → U(dbls(dbl(dbls(sel(z0, z1))))) and U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) → U(dbl(dbls(dbl(indx(x0, x1))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(sel(z0, z1)))))) = U(dbls(dbl(dbls(sel(z0, z1))))) does not unify with U(dbls(dbl(dbls(dbl(indx(x0, x1)))))). The rules U(indx(dbls(dbl(dbls(sel(z0, z1)))), z2)) → U(dbls(dbl(dbls(sel(z0, z1))))) and U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) → U(dbls(dbl(dbls(sel(x0, x1))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(sel(z0, z1)))))) = U(dbls(dbl(dbls(sel(z0, z1))))) does not unify with U(dbl(dbls(dbl(dbls(sel(x0, x1)))))). The rules U(indx(dbls(dbl(dbls(sel(z0, z1)))), z2)) → U(dbls(dbl(dbls(sel(z0, z1))))) and U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) → U(dbls(dbl(dbls(indx(x0, x1))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(sel(z0, z1)))))) = U(dbls(dbl(dbls(sel(z0, z1))))) does not unify with U(dbl(dbls(dbl(dbls(indx(x0, x1)))))). The rules U(indx(dbls(dbl(dbls(sel(z0, z1)))), z2)) → U(dbls(dbl(dbls(sel(z0, z1))))) and U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) → U(dbls(dbl(dbls(dbl(dbls(x0)))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(sel(z0, z1)))))) = U(dbls(dbl(dbls(sel(z0, z1))))) does not unify with U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))). The rules U(indx(dbls(dbl(dbls(sel(z0, z1)))), z2)) → U(dbls(dbl(dbls(sel(z0, z1))))) and U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) → U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(sel(z0, z1)))))) = U(dbls(dbl(dbls(sel(z0, z1))))) does not unify with U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))). The rules U(indx(dbls(dbl(dbls(sel(z0, z1)))), z2)) → U(dbls(dbl(dbls(sel(z0, z1))))) and U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) → U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(sel(z0, z1)))))) = U(dbls(dbl(dbls(sel(z0, z1))))) does not unify with U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))). The rules U(indx(dbls(dbl(dbls(sel(z0, z1)))), z2)) → U(dbls(dbl(dbls(sel(z0, z1))))) and U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))) → U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(sel(z0, z1)))))) = U(dbls(dbl(dbls(sel(z0, z1))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))). The rules U(indx(dbls(dbl(dbls(sel(z0, z1)))), z2)) → U(dbls(dbl(dbls(sel(z0, z1))))) and U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))) → U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(sel(z0, z1)))))) = U(dbls(dbl(dbls(sel(z0, z1))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))). The rules U(indx(dbls(dbl(dbls(sel(z0, z1)))), z2)) → U(dbls(dbl(dbls(sel(z0, z1))))) and U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))) → U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(sel(z0, z1)))))) = U(dbls(dbl(dbls(sel(z0, z1))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))). The rules U(indx(dbls(dbl(dbls(sel(z0, z1)))), z2)) → U(dbls(dbl(dbls(sel(z0, z1))))) and U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(sel(z0, z1)))))) = U(dbls(dbl(dbls(sel(z0, z1))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))). The rules U(indx(dbls(dbl(dbls(sel(z0, z1)))), z2)) → U(dbls(dbl(dbls(sel(z0, z1))))) and U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(sel(z0, z1)))))) = U(dbls(dbl(dbls(sel(z0, z1))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))). The rules U(indx(dbls(dbl(dbls(sel(z0, z1)))), z2)) → U(dbls(dbl(dbls(sel(z0, z1))))) and U(indx(dbls(dbl(dbls(sel(x0, x1)))), x2)) → U(dbls(dbl(dbls(sel(x0, x1))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(sel(z0, z1)))))) = U(dbls(dbl(dbls(sel(z0, z1))))) does not unify with U(indx(dbls(dbl(dbls(sel(x0, x1)))), x2)). The rules U(indx(dbls(dbl(dbls(indx(z0, z1)))), z2)) → U(dbls(dbl(dbls(indx(z0, z1))))) and U(sel(x0, x1)) → U(x0) form no chain, because ECap^µ(U(dbls(dbl(dbls(indx(z0, z1)))))) = U(dbls(dbl(dbls(indx(z0, z1))))) does not unify with U(sel(x0, x1)). The rules U(indx(dbls(dbl(dbls(indx(z0, z1)))), z2)) → U(dbls(dbl(dbls(indx(z0, z1))))) and U(sel(x0, x1)) → U(x1) form no chain, because ECap^µ(U(dbls(dbl(dbls(indx(z0, z1)))))) = U(dbls(dbl(dbls(indx(z0, z1))))) does not unify with U(sel(x0, x1)). The rules U(indx(dbls(dbl(dbls(indx(z0, z1)))), z2)) → U(dbls(dbl(dbls(indx(z0, z1))))) and U(sel(x0, x1)) → SEL(x0, x1) form no chain, because ECap^µ(U(dbls(dbl(dbls(indx(z0, z1)))))) = U(dbls(dbl(dbls(indx(z0, z1))))) does not unify with U(sel(x0, x1)). The rules U(indx(dbls(dbl(dbls(indx(z0, z1)))), z2)) → U(dbls(dbl(dbls(indx(z0, z1))))) and U(dbl(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(dbls(dbl(dbls(indx(z0, z1)))))) = U(dbls(dbl(dbls(indx(z0, z1))))) does not unify with U(dbl(sel(x0, x1))). The rules U(indx(dbls(dbl(dbls(indx(z0, z1)))), z2)) → U(dbls(dbl(dbls(indx(z0, z1))))) and U(dbl(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(dbls(dbl(dbls(indx(z0, z1)))))) = U(dbls(dbl(dbls(indx(z0, z1))))) does not unify with U(dbl(indx(x0, x1))). The rules U(indx(dbls(dbl(dbls(indx(z0, z1)))), z2)) → U(dbls(dbl(dbls(indx(z0, z1))))) and U(dbls(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(dbls(dbl(dbls(indx(z0, z1)))))) = U(dbls(dbl(dbls(indx(z0, z1))))) does not unify with U(dbls(sel(x0, x1))). The rules U(indx(dbls(dbl(dbls(indx(z0, z1)))), z2)) → U(dbls(dbl(dbls(indx(z0, z1))))) and U(dbls(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(dbls(dbl(dbls(indx(z0, z1)))))) = U(dbls(dbl(dbls(indx(z0, z1))))) does not unify with U(dbls(indx(x0, x1))). The rules U(indx(dbls(dbl(dbls(indx(z0, z1)))), z2)) → U(dbls(dbl(dbls(indx(z0, z1))))) and U(dbls(dbl(sel(x0, x1)))) → U(dbl(sel(x0, x1))) form no chain, because ECap^µ(U(dbls(dbl(dbls(indx(z0, z1)))))) = U(dbls(dbl(dbls(indx(z0, z1))))) does not unify with U(dbls(dbl(sel(x0, x1)))). The rules U(indx(dbls(dbl(dbls(indx(z0, z1)))), z2)) → U(dbls(dbl(dbls(indx(z0, z1))))) and U(dbls(dbl(indx(x0, x1)))) → U(dbl(indx(x0, x1))) form no chain, because ECap^µ(U(dbls(dbl(dbls(indx(z0, z1)))))) = U(dbls(dbl(dbls(indx(z0, z1))))) does not unify with U(dbls(dbl(indx(x0, x1)))). The rules U(indx(dbls(dbl(dbls(indx(z0, z1)))), z2)) → U(dbls(dbl(dbls(indx(z0, z1))))) and U(dbl(dbls(sel(x0, x1)))) → U(dbls(sel(x0, x1))) form no chain, because ECap^µ(U(dbls(dbl(dbls(indx(z0, z1)))))) = U(dbls(dbl(dbls(indx(z0, z1))))) does not unify with U(dbl(dbls(sel(x0, x1)))). The rules U(indx(dbls(dbl(dbls(indx(z0, z1)))), z2)) → U(dbls(dbl(dbls(indx(z0, z1))))) and U(dbl(dbls(indx(x0, x1)))) → U(dbls(indx(x0, x1))) form no chain, because ECap^µ(U(dbls(dbl(dbls(indx(z0, z1)))))) = U(dbls(dbl(dbls(indx(z0, z1))))) does not unify with U(dbl(dbls(indx(x0, x1)))). The rules U(indx(dbls(dbl(dbls(indx(z0, z1)))), z2)) → U(dbls(dbl(dbls(indx(z0, z1))))) and U(dbl(dbls(dbl(sel(x0, x1))))) → U(dbls(dbl(sel(x0, x1)))) form no chain, because ECap^µ(U(dbls(dbl(dbls(indx(z0, z1)))))) = U(dbls(dbl(dbls(indx(z0, z1))))) does not unify with U(dbl(dbls(dbl(sel(x0, x1))))). The rules U(indx(dbls(dbl(dbls(indx(z0, z1)))), z2)) → U(dbls(dbl(dbls(indx(z0, z1))))) and U(dbl(dbls(dbl(indx(x0, x1))))) → U(dbls(dbl(indx(x0, x1)))) form no chain, because ECap^µ(U(dbls(dbl(dbls(indx(z0, z1)))))) = U(dbls(dbl(dbls(indx(z0, z1))))) does not unify with U(dbl(dbls(dbl(indx(x0, x1))))). The rules U(indx(dbls(dbl(dbls(indx(z0, z1)))), z2)) → U(dbls(dbl(dbls(indx(z0, z1))))) and U(dbls(dbl(dbls(sel(x0, x1))))) → U(dbl(dbls(sel(x0, x1)))) form no chain, because ECap^µ(U(dbls(dbl(dbls(indx(z0, z1)))))) = U(dbls(dbl(dbls(indx(z0, z1))))) does not unify with U(dbls(dbl(dbls(sel(x0, x1))))). The rules U(indx(dbls(dbl(dbls(indx(z0, z1)))), z2)) → U(dbls(dbl(dbls(indx(z0, z1))))) and U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) → U(dbl(dbls(dbl(sel(x0, x1))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(indx(z0, z1)))))) = U(dbls(dbl(dbls(indx(z0, z1))))) does not unify with U(dbls(dbl(dbls(dbl(sel(x0, x1)))))). The rules U(indx(dbls(dbl(dbls(indx(z0, z1)))), z2)) → U(dbls(dbl(dbls(indx(z0, z1))))) and U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) → U(dbl(dbls(dbl(indx(x0, x1))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(indx(z0, z1)))))) = U(dbls(dbl(dbls(indx(z0, z1))))) does not unify with U(dbls(dbl(dbls(dbl(indx(x0, x1)))))). The rules U(indx(dbls(dbl(dbls(indx(z0, z1)))), z2)) → U(dbls(dbl(dbls(indx(z0, z1))))) and U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) → U(dbls(dbl(dbls(sel(x0, x1))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(indx(z0, z1)))))) = U(dbls(dbl(dbls(indx(z0, z1))))) does not unify with U(dbl(dbls(dbl(dbls(sel(x0, x1)))))). The rules U(indx(dbls(dbl(dbls(indx(z0, z1)))), z2)) → U(dbls(dbl(dbls(indx(z0, z1))))) and U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) → U(dbls(dbl(dbls(indx(x0, x1))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(indx(z0, z1)))))) = U(dbls(dbl(dbls(indx(z0, z1))))) does not unify with U(dbl(dbls(dbl(dbls(indx(x0, x1)))))). The rules U(indx(dbls(dbl(dbls(indx(z0, z1)))), z2)) → U(dbls(dbl(dbls(indx(z0, z1))))) and U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) → U(dbls(dbl(dbls(dbl(dbls(x0)))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(indx(z0, z1)))))) = U(dbls(dbl(dbls(indx(z0, z1))))) does not unify with U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))). The rules U(indx(dbls(dbl(dbls(indx(z0, z1)))), z2)) → U(dbls(dbl(dbls(indx(z0, z1))))) and U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) → U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(indx(z0, z1)))))) = U(dbls(dbl(dbls(indx(z0, z1))))) does not unify with U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))). The rules U(indx(dbls(dbl(dbls(indx(z0, z1)))), z2)) → U(dbls(dbl(dbls(indx(z0, z1))))) and U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) → U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(indx(z0, z1)))))) = U(dbls(dbl(dbls(indx(z0, z1))))) does not unify with U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))). The rules U(indx(dbls(dbl(dbls(indx(z0, z1)))), z2)) → U(dbls(dbl(dbls(indx(z0, z1))))) and U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))) → U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(indx(z0, z1)))))) = U(dbls(dbl(dbls(indx(z0, z1))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))). The rules U(indx(dbls(dbl(dbls(indx(z0, z1)))), z2)) → U(dbls(dbl(dbls(indx(z0, z1))))) and U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))) → U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(indx(z0, z1)))))) = U(dbls(dbl(dbls(indx(z0, z1))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))). The rules U(indx(dbls(dbl(dbls(indx(z0, z1)))), z2)) → U(dbls(dbl(dbls(indx(z0, z1))))) and U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))) → U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(indx(z0, z1)))))) = U(dbls(dbl(dbls(indx(z0, z1))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))). The rules U(indx(dbls(dbl(dbls(indx(z0, z1)))), z2)) → U(dbls(dbl(dbls(indx(z0, z1))))) and U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(indx(z0, z1)))))) = U(dbls(dbl(dbls(indx(z0, z1))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))). The rules U(indx(dbls(dbl(dbls(indx(z0, z1)))), z2)) → U(dbls(dbl(dbls(indx(z0, z1))))) and U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(indx(z0, z1)))))) = U(dbls(dbl(dbls(indx(z0, z1))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))). The rules U(indx(dbls(dbl(dbls(indx(z0, z1)))), z2)) → U(dbls(dbl(dbls(indx(z0, z1))))) and U(indx(dbls(dbl(dbls(indx(x0, x1)))), x2)) → U(dbls(dbl(dbls(indx(x0, x1))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(indx(z0, z1)))))) = U(dbls(dbl(dbls(indx(z0, z1))))) does not unify with U(indx(dbls(dbl(dbls(indx(x0, x1)))), x2)). The rules U(indx(dbls(dbl(dbls(dbl(sel(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) and U(sel(x0, x1)) → U(x0) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(sel(z0, z1))))))) = U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) does not unify with U(sel(x0, x1)). The rules U(indx(dbls(dbl(dbls(dbl(sel(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) and U(sel(x0, x1)) → U(x1) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(sel(z0, z1))))))) = U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) does not unify with U(sel(x0, x1)). The rules U(indx(dbls(dbl(dbls(dbl(sel(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) and U(sel(x0, x1)) → SEL(x0, x1) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(sel(z0, z1))))))) = U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) does not unify with U(sel(x0, x1)). The rules U(indx(dbls(dbl(dbls(dbl(sel(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) and U(dbl(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(sel(z0, z1))))))) = U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) does not unify with U(dbl(sel(x0, x1))). The rules U(indx(dbls(dbl(dbls(dbl(sel(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) and U(dbl(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(sel(z0, z1))))))) = U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) does not unify with U(dbl(indx(x0, x1))). The rules U(indx(dbls(dbl(dbls(dbl(sel(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) and U(dbls(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(sel(z0, z1))))))) = U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) does not unify with U(dbls(sel(x0, x1))). The rules U(indx(dbls(dbl(dbls(dbl(sel(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) and U(dbls(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(sel(z0, z1))))))) = U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) does not unify with U(dbls(indx(x0, x1))). The rules U(indx(dbls(dbl(dbls(dbl(sel(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) and U(dbls(dbl(sel(x0, x1)))) → U(dbl(sel(x0, x1))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(sel(z0, z1))))))) = U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) does not unify with U(dbls(dbl(sel(x0, x1)))). The rules U(indx(dbls(dbl(dbls(dbl(sel(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) and U(dbls(dbl(indx(x0, x1)))) → U(dbl(indx(x0, x1))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(sel(z0, z1))))))) = U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) does not unify with U(dbls(dbl(indx(x0, x1)))). The rules U(indx(dbls(dbl(dbls(dbl(sel(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) and U(dbl(dbls(sel(x0, x1)))) → U(dbls(sel(x0, x1))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(sel(z0, z1))))))) = U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) does not unify with U(dbl(dbls(sel(x0, x1)))). The rules U(indx(dbls(dbl(dbls(dbl(sel(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) and U(dbl(dbls(indx(x0, x1)))) → U(dbls(indx(x0, x1))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(sel(z0, z1))))))) = U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) does not unify with U(dbl(dbls(indx(x0, x1)))). The rules U(indx(dbls(dbl(dbls(dbl(sel(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) and U(dbl(dbls(dbl(sel(x0, x1))))) → U(dbls(dbl(sel(x0, x1)))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(sel(z0, z1))))))) = U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) does not unify with U(dbl(dbls(dbl(sel(x0, x1))))). The rules U(indx(dbls(dbl(dbls(dbl(sel(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) and U(dbl(dbls(dbl(indx(x0, x1))))) → U(dbls(dbl(indx(x0, x1)))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(sel(z0, z1))))))) = U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) does not unify with U(dbl(dbls(dbl(indx(x0, x1))))). The rules U(indx(dbls(dbl(dbls(dbl(sel(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) and U(dbls(dbl(dbls(sel(x0, x1))))) → U(dbl(dbls(sel(x0, x1)))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(sel(z0, z1))))))) = U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) does not unify with U(dbls(dbl(dbls(sel(x0, x1))))). The rules U(indx(dbls(dbl(dbls(dbl(sel(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) and U(dbls(dbl(dbls(indx(x0, x1))))) → U(dbl(dbls(indx(x0, x1)))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(sel(z0, z1))))))) = U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) does not unify with U(dbls(dbl(dbls(indx(x0, x1))))). The rules U(indx(dbls(dbl(dbls(dbl(sel(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) and U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) → U(dbl(dbls(dbl(indx(x0, x1))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(sel(z0, z1))))))) = U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) does not unify with U(dbls(dbl(dbls(dbl(indx(x0, x1)))))). The rules U(indx(dbls(dbl(dbls(dbl(sel(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) and U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) → U(dbls(dbl(dbls(sel(x0, x1))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(sel(z0, z1))))))) = U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) does not unify with U(dbl(dbls(dbl(dbls(sel(x0, x1)))))). The rules U(indx(dbls(dbl(dbls(dbl(sel(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) and U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) → U(dbls(dbl(dbls(indx(x0, x1))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(sel(z0, z1))))))) = U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) does not unify with U(dbl(dbls(dbl(dbls(indx(x0, x1)))))). The rules U(indx(dbls(dbl(dbls(dbl(sel(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) and U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) → U(dbls(dbl(dbls(dbl(dbls(x0)))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(sel(z0, z1))))))) = U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) does not unify with U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))). The rules U(indx(dbls(dbl(dbls(dbl(sel(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) and U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) → U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(sel(z0, z1))))))) = U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) does not unify with U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))). The rules U(indx(dbls(dbl(dbls(dbl(sel(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) and U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) → U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(sel(z0, z1))))))) = U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) does not unify with U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))). The rules U(indx(dbls(dbl(dbls(dbl(sel(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) and U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))) → U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(sel(z0, z1))))))) = U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))). The rules U(indx(dbls(dbl(dbls(dbl(sel(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) and U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))) → U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(sel(z0, z1))))))) = U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))). The rules U(indx(dbls(dbl(dbls(dbl(sel(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) and U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))) → U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(sel(z0, z1))))))) = U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))). The rules U(indx(dbls(dbl(dbls(dbl(sel(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) and U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(sel(z0, z1))))))) = U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))). The rules U(indx(dbls(dbl(dbls(dbl(sel(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) and U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(sel(z0, z1))))))) = U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))). The rules U(indx(dbls(dbl(dbls(dbl(sel(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) and U(indx(dbls(dbl(dbls(dbl(sel(x0, x1))))), x2)) → U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(sel(z0, z1))))))) = U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) does not unify with U(indx(dbls(dbl(dbls(dbl(sel(x0, x1))))), x2)). The rules U(indx(dbls(dbl(dbls(dbl(indx(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) and U(sel(x0, x1)) → U(x0) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(indx(z0, z1))))))) = U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) does not unify with U(sel(x0, x1)). The rules U(indx(dbls(dbl(dbls(dbl(indx(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) and U(sel(x0, x1)) → U(x1) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(indx(z0, z1))))))) = U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) does not unify with U(sel(x0, x1)). The rules U(indx(dbls(dbl(dbls(dbl(indx(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) and U(sel(x0, x1)) → SEL(x0, x1) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(indx(z0, z1))))))) = U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) does not unify with U(sel(x0, x1)). The rules U(indx(dbls(dbl(dbls(dbl(indx(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) and U(dbl(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(indx(z0, z1))))))) = U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) does not unify with U(dbl(sel(x0, x1))). The rules U(indx(dbls(dbl(dbls(dbl(indx(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) and U(dbl(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(indx(z0, z1))))))) = U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) does not unify with U(dbl(indx(x0, x1))). The rules U(indx(dbls(dbl(dbls(dbl(indx(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) and U(dbls(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(indx(z0, z1))))))) = U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) does not unify with U(dbls(sel(x0, x1))). The rules U(indx(dbls(dbl(dbls(dbl(indx(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) and U(dbls(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(indx(z0, z1))))))) = U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) does not unify with U(dbls(indx(x0, x1))). The rules U(indx(dbls(dbl(dbls(dbl(indx(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) and U(dbls(dbl(sel(x0, x1)))) → U(dbl(sel(x0, x1))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(indx(z0, z1))))))) = U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) does not unify with U(dbls(dbl(sel(x0, x1)))). The rules U(indx(dbls(dbl(dbls(dbl(indx(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) and U(dbls(dbl(indx(x0, x1)))) → U(dbl(indx(x0, x1))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(indx(z0, z1))))))) = U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) does not unify with U(dbls(dbl(indx(x0, x1)))). The rules U(indx(dbls(dbl(dbls(dbl(indx(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) and U(dbl(dbls(sel(x0, x1)))) → U(dbls(sel(x0, x1))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(indx(z0, z1))))))) = U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) does not unify with U(dbl(dbls(sel(x0, x1)))). The rules U(indx(dbls(dbl(dbls(dbl(indx(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) and U(dbl(dbls(indx(x0, x1)))) → U(dbls(indx(x0, x1))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(indx(z0, z1))))))) = U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) does not unify with U(dbl(dbls(indx(x0, x1)))). The rules U(indx(dbls(dbl(dbls(dbl(indx(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) and U(dbl(dbls(dbl(sel(x0, x1))))) → U(dbls(dbl(sel(x0, x1)))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(indx(z0, z1))))))) = U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) does not unify with U(dbl(dbls(dbl(sel(x0, x1))))). The rules U(indx(dbls(dbl(dbls(dbl(indx(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) and U(dbl(dbls(dbl(indx(x0, x1))))) → U(dbls(dbl(indx(x0, x1)))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(indx(z0, z1))))))) = U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) does not unify with U(dbl(dbls(dbl(indx(x0, x1))))). The rules U(indx(dbls(dbl(dbls(dbl(indx(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) and U(dbls(dbl(dbls(sel(x0, x1))))) → U(dbl(dbls(sel(x0, x1)))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(indx(z0, z1))))))) = U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) does not unify with U(dbls(dbl(dbls(sel(x0, x1))))). The rules U(indx(dbls(dbl(dbls(dbl(indx(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) and U(dbls(dbl(dbls(indx(x0, x1))))) → U(dbl(dbls(indx(x0, x1)))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(indx(z0, z1))))))) = U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) does not unify with U(dbls(dbl(dbls(indx(x0, x1))))). The rules U(indx(dbls(dbl(dbls(dbl(indx(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) and U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) → U(dbl(dbls(dbl(sel(x0, x1))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(indx(z0, z1))))))) = U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) does not unify with U(dbls(dbl(dbls(dbl(sel(x0, x1)))))). The rules U(indx(dbls(dbl(dbls(dbl(indx(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) and U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) → U(dbls(dbl(dbls(sel(x0, x1))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(indx(z0, z1))))))) = U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) does not unify with U(dbl(dbls(dbl(dbls(sel(x0, x1)))))). The rules U(indx(dbls(dbl(dbls(dbl(indx(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) and U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) → U(dbls(dbl(dbls(indx(x0, x1))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(indx(z0, z1))))))) = U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) does not unify with U(dbl(dbls(dbl(dbls(indx(x0, x1)))))). The rules U(indx(dbls(dbl(dbls(dbl(indx(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) and U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) → U(dbls(dbl(dbls(dbl(dbls(x0)))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(indx(z0, z1))))))) = U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) does not unify with U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))). The rules U(indx(dbls(dbl(dbls(dbl(indx(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) and U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) → U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(indx(z0, z1))))))) = U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) does not unify with U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))). The rules U(indx(dbls(dbl(dbls(dbl(indx(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) and U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) → U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(indx(z0, z1))))))) = U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) does not unify with U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))). The rules U(indx(dbls(dbl(dbls(dbl(indx(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) and U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))) → U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(indx(z0, z1))))))) = U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))). The rules U(indx(dbls(dbl(dbls(dbl(indx(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) and U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))) → U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(indx(z0, z1))))))) = U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))). The rules U(indx(dbls(dbl(dbls(dbl(indx(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) and U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))) → U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(indx(z0, z1))))))) = U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))). The rules U(indx(dbls(dbl(dbls(dbl(indx(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) and U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(indx(z0, z1))))))) = U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))). The rules U(indx(dbls(dbl(dbls(dbl(indx(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) and U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(indx(z0, z1))))))) = U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))). The rules U(indx(dbls(dbl(dbls(dbl(indx(z0, z1))))), z2)) → U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) and U(indx(dbls(dbl(dbls(dbl(indx(x0, x1))))), x2)) → U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(indx(z0, z1))))))) = U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) does not unify with U(indx(dbls(dbl(dbls(dbl(indx(x0, x1))))), x2)). The rules U(indx(dbl(dbls(dbl(dbls(sel(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) and U(sel(x0, x1)) → U(x0) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(sel(z0, z1))))))) = U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) does not unify with U(sel(x0, x1)). The rules U(indx(dbl(dbls(dbl(dbls(sel(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) and U(sel(x0, x1)) → U(x1) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(sel(z0, z1))))))) = U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) does not unify with U(sel(x0, x1)). The rules U(indx(dbl(dbls(dbl(dbls(sel(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) and U(sel(x0, x1)) → SEL(x0, x1) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(sel(z0, z1))))))) = U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) does not unify with U(sel(x0, x1)). The rules U(indx(dbl(dbls(dbl(dbls(sel(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) and U(dbl(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(sel(z0, z1))))))) = U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) does not unify with U(dbl(sel(x0, x1))). The rules U(indx(dbl(dbls(dbl(dbls(sel(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) and U(dbl(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(sel(z0, z1))))))) = U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) does not unify with U(dbl(indx(x0, x1))). The rules U(indx(dbl(dbls(dbl(dbls(sel(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) and U(dbls(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(sel(z0, z1))))))) = U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) does not unify with U(dbls(sel(x0, x1))). The rules U(indx(dbl(dbls(dbl(dbls(sel(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) and U(dbls(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(sel(z0, z1))))))) = U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) does not unify with U(dbls(indx(x0, x1))). The rules U(indx(dbl(dbls(dbl(dbls(sel(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) and U(dbls(dbl(sel(x0, x1)))) → U(dbl(sel(x0, x1))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(sel(z0, z1))))))) = U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) does not unify with U(dbls(dbl(sel(x0, x1)))). The rules U(indx(dbl(dbls(dbl(dbls(sel(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) and U(dbls(dbl(indx(x0, x1)))) → U(dbl(indx(x0, x1))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(sel(z0, z1))))))) = U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) does not unify with U(dbls(dbl(indx(x0, x1)))). The rules U(indx(dbl(dbls(dbl(dbls(sel(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) and U(dbl(dbls(sel(x0, x1)))) → U(dbls(sel(x0, x1))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(sel(z0, z1))))))) = U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) does not unify with U(dbl(dbls(sel(x0, x1)))). The rules U(indx(dbl(dbls(dbl(dbls(sel(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) and U(dbl(dbls(indx(x0, x1)))) → U(dbls(indx(x0, x1))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(sel(z0, z1))))))) = U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) does not unify with U(dbl(dbls(indx(x0, x1)))). The rules U(indx(dbl(dbls(dbl(dbls(sel(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) and U(dbl(dbls(dbl(sel(x0, x1))))) → U(dbls(dbl(sel(x0, x1)))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(sel(z0, z1))))))) = U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) does not unify with U(dbl(dbls(dbl(sel(x0, x1))))). The rules U(indx(dbl(dbls(dbl(dbls(sel(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) and U(dbl(dbls(dbl(indx(x0, x1))))) → U(dbls(dbl(indx(x0, x1)))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(sel(z0, z1))))))) = U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) does not unify with U(dbl(dbls(dbl(indx(x0, x1))))). The rules U(indx(dbl(dbls(dbl(dbls(sel(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) and U(dbls(dbl(dbls(sel(x0, x1))))) → U(dbl(dbls(sel(x0, x1)))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(sel(z0, z1))))))) = U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) does not unify with U(dbls(dbl(dbls(sel(x0, x1))))). The rules U(indx(dbl(dbls(dbl(dbls(sel(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) and U(dbls(dbl(dbls(indx(x0, x1))))) → U(dbl(dbls(indx(x0, x1)))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(sel(z0, z1))))))) = U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) does not unify with U(dbls(dbl(dbls(indx(x0, x1))))). The rules U(indx(dbl(dbls(dbl(dbls(sel(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) and U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) → U(dbl(dbls(dbl(sel(x0, x1))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(sel(z0, z1))))))) = U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) does not unify with U(dbls(dbl(dbls(dbl(sel(x0, x1)))))). The rules U(indx(dbl(dbls(dbl(dbls(sel(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) and U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) → U(dbl(dbls(dbl(indx(x0, x1))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(sel(z0, z1))))))) = U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) does not unify with U(dbls(dbl(dbls(dbl(indx(x0, x1)))))). The rules U(indx(dbl(dbls(dbl(dbls(sel(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) and U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) → U(dbls(dbl(dbls(indx(x0, x1))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(sel(z0, z1))))))) = U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) does not unify with U(dbl(dbls(dbl(dbls(indx(x0, x1)))))). The rules U(indx(dbl(dbls(dbl(dbls(sel(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) and U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) → U(dbls(dbl(dbls(dbl(dbls(x0)))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(sel(z0, z1))))))) = U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) does not unify with U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))). The rules U(indx(dbl(dbls(dbl(dbls(sel(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) and U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) → U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(sel(z0, z1))))))) = U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) does not unify with U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))). The rules U(indx(dbl(dbls(dbl(dbls(sel(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) and U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) → U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(sel(z0, z1))))))) = U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) does not unify with U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))). The rules U(indx(dbl(dbls(dbl(dbls(sel(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) and U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))) → U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(sel(z0, z1))))))) = U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))). The rules U(indx(dbl(dbls(dbl(dbls(sel(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) and U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))) → U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(sel(z0, z1))))))) = U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))). The rules U(indx(dbl(dbls(dbl(dbls(sel(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) and U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))) → U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(sel(z0, z1))))))) = U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))). The rules U(indx(dbl(dbls(dbl(dbls(sel(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) and U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(sel(z0, z1))))))) = U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))). The rules U(indx(dbl(dbls(dbl(dbls(sel(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) and U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(sel(z0, z1))))))) = U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))). The rules U(indx(dbl(dbls(dbl(dbls(sel(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) and U(indx(dbl(dbls(dbl(dbls(sel(x0, x1))))), x2)) → U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(sel(z0, z1))))))) = U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) does not unify with U(indx(dbl(dbls(dbl(dbls(sel(x0, x1))))), x2)). The rules U(indx(dbl(dbls(dbl(dbls(indx(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) and U(sel(x0, x1)) → U(x0) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(indx(z0, z1))))))) = U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) does not unify with U(sel(x0, x1)). The rules U(indx(dbl(dbls(dbl(dbls(indx(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) and U(sel(x0, x1)) → U(x1) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(indx(z0, z1))))))) = U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) does not unify with U(sel(x0, x1)). The rules U(indx(dbl(dbls(dbl(dbls(indx(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) and U(sel(x0, x1)) → SEL(x0, x1) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(indx(z0, z1))))))) = U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) does not unify with U(sel(x0, x1)). The rules U(indx(dbl(dbls(dbl(dbls(indx(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) and U(dbl(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(indx(z0, z1))))))) = U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) does not unify with U(dbl(sel(x0, x1))). The rules U(indx(dbl(dbls(dbl(dbls(indx(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) and U(dbl(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(indx(z0, z1))))))) = U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) does not unify with U(dbl(indx(x0, x1))). The rules U(indx(dbl(dbls(dbl(dbls(indx(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) and U(dbls(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(indx(z0, z1))))))) = U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) does not unify with U(dbls(sel(x0, x1))). The rules U(indx(dbl(dbls(dbl(dbls(indx(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) and U(dbls(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(indx(z0, z1))))))) = U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) does not unify with U(dbls(indx(x0, x1))). The rules U(indx(dbl(dbls(dbl(dbls(indx(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) and U(dbls(dbl(sel(x0, x1)))) → U(dbl(sel(x0, x1))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(indx(z0, z1))))))) = U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) does not unify with U(dbls(dbl(sel(x0, x1)))). The rules U(indx(dbl(dbls(dbl(dbls(indx(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) and U(dbls(dbl(indx(x0, x1)))) → U(dbl(indx(x0, x1))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(indx(z0, z1))))))) = U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) does not unify with U(dbls(dbl(indx(x0, x1)))). The rules U(indx(dbl(dbls(dbl(dbls(indx(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) and U(dbl(dbls(sel(x0, x1)))) → U(dbls(sel(x0, x1))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(indx(z0, z1))))))) = U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) does not unify with U(dbl(dbls(sel(x0, x1)))). The rules U(indx(dbl(dbls(dbl(dbls(indx(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) and U(dbl(dbls(indx(x0, x1)))) → U(dbls(indx(x0, x1))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(indx(z0, z1))))))) = U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) does not unify with U(dbl(dbls(indx(x0, x1)))). The rules U(indx(dbl(dbls(dbl(dbls(indx(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) and U(dbl(dbls(dbl(sel(x0, x1))))) → U(dbls(dbl(sel(x0, x1)))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(indx(z0, z1))))))) = U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) does not unify with U(dbl(dbls(dbl(sel(x0, x1))))). The rules U(indx(dbl(dbls(dbl(dbls(indx(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) and U(dbl(dbls(dbl(indx(x0, x1))))) → U(dbls(dbl(indx(x0, x1)))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(indx(z0, z1))))))) = U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) does not unify with U(dbl(dbls(dbl(indx(x0, x1))))). The rules U(indx(dbl(dbls(dbl(dbls(indx(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) and U(dbls(dbl(dbls(sel(x0, x1))))) → U(dbl(dbls(sel(x0, x1)))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(indx(z0, z1))))))) = U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) does not unify with U(dbls(dbl(dbls(sel(x0, x1))))). The rules U(indx(dbl(dbls(dbl(dbls(indx(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) and U(dbls(dbl(dbls(indx(x0, x1))))) → U(dbl(dbls(indx(x0, x1)))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(indx(z0, z1))))))) = U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) does not unify with U(dbls(dbl(dbls(indx(x0, x1))))). The rules U(indx(dbl(dbls(dbl(dbls(indx(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) and U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) → U(dbl(dbls(dbl(sel(x0, x1))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(indx(z0, z1))))))) = U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) does not unify with U(dbls(dbl(dbls(dbl(sel(x0, x1)))))). The rules U(indx(dbl(dbls(dbl(dbls(indx(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) and U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) → U(dbl(dbls(dbl(indx(x0, x1))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(indx(z0, z1))))))) = U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) does not unify with U(dbls(dbl(dbls(dbl(indx(x0, x1)))))). The rules U(indx(dbl(dbls(dbl(dbls(indx(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) and U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) → U(dbls(dbl(dbls(sel(x0, x1))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(indx(z0, z1))))))) = U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) does not unify with U(dbl(dbls(dbl(dbls(sel(x0, x1)))))). The rules U(indx(dbl(dbls(dbl(dbls(indx(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) and U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) → U(dbls(dbl(dbls(dbl(dbls(x0)))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(indx(z0, z1))))))) = U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) does not unify with U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))). The rules U(indx(dbl(dbls(dbl(dbls(indx(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) and U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) → U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(indx(z0, z1))))))) = U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) does not unify with U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))). The rules U(indx(dbl(dbls(dbl(dbls(indx(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) and U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) → U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(indx(z0, z1))))))) = U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) does not unify with U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))). The rules U(indx(dbl(dbls(dbl(dbls(indx(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) and U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))) → U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(indx(z0, z1))))))) = U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))). The rules U(indx(dbl(dbls(dbl(dbls(indx(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) and U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))) → U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(indx(z0, z1))))))) = U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))). The rules U(indx(dbl(dbls(dbl(dbls(indx(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) and U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))) → U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(indx(z0, z1))))))) = U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))). The rules U(indx(dbl(dbls(dbl(dbls(indx(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) and U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(indx(z0, z1))))))) = U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))). The rules U(indx(dbl(dbls(dbl(dbls(indx(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) and U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(indx(z0, z1))))))) = U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))). The rules U(indx(dbl(dbls(dbl(dbls(indx(z0, z1))))), z2)) → U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) and U(indx(dbl(dbls(dbl(dbls(indx(x0, x1))))), x2)) → U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(indx(z0, z1))))))) = U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) does not unify with U(indx(dbl(dbls(dbl(dbls(indx(x0, x1))))), x2)). The rules U(indx(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))), z1)) → U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) and U(sel(x0, x1)) → U(x0) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) = U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) does not unify with U(sel(x0, x1)). The rules U(indx(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))), z1)) → U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) and U(sel(x0, x1)) → U(x1) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) = U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) does not unify with U(sel(x0, x1)). The rules U(indx(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))), z1)) → U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) and U(sel(x0, x1)) → SEL(x0, x1) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) = U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) does not unify with U(sel(x0, x1)). The rules U(indx(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))), z1)) → U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) and U(dbl(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) = U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) does not unify with U(dbl(sel(x0, x1))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))), z1)) → U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) and U(dbl(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) = U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) does not unify with U(dbl(indx(x0, x1))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))), z1)) → U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) and U(dbls(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) = U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) does not unify with U(dbls(sel(x0, x1))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))), z1)) → U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) and U(dbls(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) = U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) does not unify with U(dbls(indx(x0, x1))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))), z1)) → U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) and U(dbls(dbl(sel(x0, x1)))) → U(dbl(sel(x0, x1))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) = U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) does not unify with U(dbls(dbl(sel(x0, x1)))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))), z1)) → U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) and U(dbls(dbl(indx(x0, x1)))) → U(dbl(indx(x0, x1))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) = U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) does not unify with U(dbls(dbl(indx(x0, x1)))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))), z1)) → U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) and U(dbl(dbls(sel(x0, x1)))) → U(dbls(sel(x0, x1))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) = U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) does not unify with U(dbl(dbls(sel(x0, x1)))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))), z1)) → U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) and U(dbl(dbls(indx(x0, x1)))) → U(dbls(indx(x0, x1))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) = U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) does not unify with U(dbl(dbls(indx(x0, x1)))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))), z1)) → U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) and U(dbl(dbls(dbl(sel(x0, x1))))) → U(dbls(dbl(sel(x0, x1)))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) = U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) does not unify with U(dbl(dbls(dbl(sel(x0, x1))))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))), z1)) → U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) and U(dbl(dbls(dbl(indx(x0, x1))))) → U(dbls(dbl(indx(x0, x1)))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) = U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) does not unify with U(dbl(dbls(dbl(indx(x0, x1))))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))), z1)) → U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) and U(dbls(dbl(dbls(sel(x0, x1))))) → U(dbl(dbls(sel(x0, x1)))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) = U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) does not unify with U(dbls(dbl(dbls(sel(x0, x1))))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))), z1)) → U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) and U(dbls(dbl(dbls(indx(x0, x1))))) → U(dbl(dbls(indx(x0, x1)))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) = U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) does not unify with U(dbls(dbl(dbls(indx(x0, x1))))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))), z1)) → U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) and U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) → U(dbl(dbls(dbl(sel(x0, x1))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) = U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) does not unify with U(dbls(dbl(dbls(dbl(sel(x0, x1)))))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))), z1)) → U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) and U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) → U(dbl(dbls(dbl(indx(x0, x1))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) = U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) does not unify with U(dbls(dbl(dbls(dbl(indx(x0, x1)))))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))), z1)) → U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) and U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) → U(dbls(dbl(dbls(sel(x0, x1))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) = U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) does not unify with U(dbl(dbls(dbl(dbls(sel(x0, x1)))))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))), z1)) → U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) and U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) → U(dbls(dbl(dbls(indx(x0, x1))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) = U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) does not unify with U(dbl(dbls(dbl(dbls(indx(x0, x1)))))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))), z1)) → U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) and U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) → U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) = U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) does not unify with U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))), z1)) → U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) and U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) → U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) = U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) does not unify with U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))), z1)) → U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) and U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))) → U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) = U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))), z1)) → U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) and U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))) → U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) = U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))), z1)) → U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) and U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))) → U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) = U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))), z1)) → U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) and U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) = U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))), z1)) → U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) and U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) = U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))), z1)) → U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) and U(indx(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))), x1)) → U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) = U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) does not unify with U(indx(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))), x1)). The rules U(indx(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) and U(sel(x0, x1)) → U(x0) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) does not unify with U(sel(x0, x1)). The rules U(indx(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) and U(sel(x0, x1)) → U(x1) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) does not unify with U(sel(x0, x1)). The rules U(indx(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) and U(sel(x0, x1)) → SEL(x0, x1) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) does not unify with U(sel(x0, x1)). The rules U(indx(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) and U(dbl(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) does not unify with U(dbl(sel(x0, x1))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) and U(dbl(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) does not unify with U(dbl(indx(x0, x1))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) and U(dbls(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) does not unify with U(dbls(sel(x0, x1))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) and U(dbls(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) does not unify with U(dbls(indx(x0, x1))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) and U(dbls(dbl(sel(x0, x1)))) → U(dbl(sel(x0, x1))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) does not unify with U(dbls(dbl(sel(x0, x1)))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) and U(dbls(dbl(indx(x0, x1)))) → U(dbl(indx(x0, x1))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) does not unify with U(dbls(dbl(indx(x0, x1)))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) and U(dbl(dbls(sel(x0, x1)))) → U(dbls(sel(x0, x1))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) does not unify with U(dbl(dbls(sel(x0, x1)))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) and U(dbl(dbls(indx(x0, x1)))) → U(dbls(indx(x0, x1))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) does not unify with U(dbl(dbls(indx(x0, x1)))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) and U(dbl(dbls(dbl(sel(x0, x1))))) → U(dbls(dbl(sel(x0, x1)))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) does not unify with U(dbl(dbls(dbl(sel(x0, x1))))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) and U(dbl(dbls(dbl(indx(x0, x1))))) → U(dbls(dbl(indx(x0, x1)))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) does not unify with U(dbl(dbls(dbl(indx(x0, x1))))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) and U(dbls(dbl(dbls(sel(x0, x1))))) → U(dbl(dbls(sel(x0, x1)))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) does not unify with U(dbls(dbl(dbls(sel(x0, x1))))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) and U(dbls(dbl(dbls(indx(x0, x1))))) → U(dbl(dbls(indx(x0, x1)))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) does not unify with U(dbls(dbl(dbls(indx(x0, x1))))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) and U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) → U(dbl(dbls(dbl(sel(x0, x1))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) does not unify with U(dbls(dbl(dbls(dbl(sel(x0, x1)))))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) and U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) → U(dbl(dbls(dbl(indx(x0, x1))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) does not unify with U(dbls(dbl(dbls(dbl(indx(x0, x1)))))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) and U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) → U(dbls(dbl(dbls(sel(x0, x1))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) does not unify with U(dbl(dbls(dbl(dbls(sel(x0, x1)))))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) and U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) → U(dbls(dbl(dbls(indx(x0, x1))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) does not unify with U(dbl(dbls(dbl(dbls(indx(x0, x1)))))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) and U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) → U(dbls(dbl(dbls(dbl(dbls(x0)))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) does not unify with U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) and U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) → U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) does not unify with U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) and U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))) → U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) and U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))) → U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) and U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))) → U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) and U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) and U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) and U(indx(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))), x2)) → U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) does not unify with U(indx(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))), x2)). The rules U(indx(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) and U(sel(x0, x1)) → U(x0) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) does not unify with U(sel(x0, x1)). The rules U(indx(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) and U(sel(x0, x1)) → U(x1) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) does not unify with U(sel(x0, x1)). The rules U(indx(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) and U(sel(x0, x1)) → SEL(x0, x1) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) does not unify with U(sel(x0, x1)). The rules U(indx(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) and U(dbl(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) does not unify with U(dbl(sel(x0, x1))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) and U(dbl(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) does not unify with U(dbl(indx(x0, x1))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) and U(dbls(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) does not unify with U(dbls(sel(x0, x1))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) and U(dbls(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) does not unify with U(dbls(indx(x0, x1))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) and U(dbls(dbl(sel(x0, x1)))) → U(dbl(sel(x0, x1))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) does not unify with U(dbls(dbl(sel(x0, x1)))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) and U(dbls(dbl(indx(x0, x1)))) → U(dbl(indx(x0, x1))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) does not unify with U(dbls(dbl(indx(x0, x1)))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) and U(dbl(dbls(sel(x0, x1)))) → U(dbls(sel(x0, x1))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) does not unify with U(dbl(dbls(sel(x0, x1)))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) and U(dbl(dbls(indx(x0, x1)))) → U(dbls(indx(x0, x1))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) does not unify with U(dbl(dbls(indx(x0, x1)))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) and U(dbl(dbls(dbl(sel(x0, x1))))) → U(dbls(dbl(sel(x0, x1)))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) does not unify with U(dbl(dbls(dbl(sel(x0, x1))))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) and U(dbl(dbls(dbl(indx(x0, x1))))) → U(dbls(dbl(indx(x0, x1)))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) does not unify with U(dbl(dbls(dbl(indx(x0, x1))))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) and U(dbls(dbl(dbls(sel(x0, x1))))) → U(dbl(dbls(sel(x0, x1)))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) does not unify with U(dbls(dbl(dbls(sel(x0, x1))))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) and U(dbls(dbl(dbls(indx(x0, x1))))) → U(dbl(dbls(indx(x0, x1)))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) does not unify with U(dbls(dbl(dbls(indx(x0, x1))))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) and U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) → U(dbl(dbls(dbl(sel(x0, x1))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) does not unify with U(dbls(dbl(dbls(dbl(sel(x0, x1)))))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) and U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) → U(dbl(dbls(dbl(indx(x0, x1))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) does not unify with U(dbls(dbl(dbls(dbl(indx(x0, x1)))))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) and U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) → U(dbls(dbl(dbls(sel(x0, x1))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) does not unify with U(dbl(dbls(dbl(dbls(sel(x0, x1)))))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) and U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) → U(dbls(dbl(dbls(indx(x0, x1))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) does not unify with U(dbl(dbls(dbl(dbls(indx(x0, x1)))))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) and U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) → U(dbls(dbl(dbls(dbl(dbls(x0)))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) does not unify with U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) and U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) → U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) does not unify with U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) and U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))) → U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) and U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))) → U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) and U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))) → U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) and U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) and U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))). The rules U(indx(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))), z2)) → U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) and U(indx(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))), x2)) → U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) form no chain, because ECap^µ(U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) = U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) does not unify with U(indx(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))), x2)). The rules U(indx(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) and U(sel(x0, x1)) → U(x0) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) does not unify with U(sel(x0, x1)). The rules U(indx(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) and U(sel(x0, x1)) → U(x1) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) does not unify with U(sel(x0, x1)). The rules U(indx(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) and U(sel(x0, x1)) → SEL(x0, x1) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) does not unify with U(sel(x0, x1)). The rules U(indx(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) and U(dbl(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) does not unify with U(dbl(sel(x0, x1))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) and U(dbl(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) does not unify with U(dbl(indx(x0, x1))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) and U(dbls(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) does not unify with U(dbls(sel(x0, x1))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) and U(dbls(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) does not unify with U(dbls(indx(x0, x1))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) and U(dbls(dbl(sel(x0, x1)))) → U(dbl(sel(x0, x1))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) does not unify with U(dbls(dbl(sel(x0, x1)))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) and U(dbls(dbl(indx(x0, x1)))) → U(dbl(indx(x0, x1))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) does not unify with U(dbls(dbl(indx(x0, x1)))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) and U(dbl(dbls(sel(x0, x1)))) → U(dbls(sel(x0, x1))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) does not unify with U(dbl(dbls(sel(x0, x1)))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) and U(dbl(dbls(indx(x0, x1)))) → U(dbls(indx(x0, x1))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) does not unify with U(dbl(dbls(indx(x0, x1)))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) and U(dbl(dbls(dbl(sel(x0, x1))))) → U(dbls(dbl(sel(x0, x1)))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) does not unify with U(dbl(dbls(dbl(sel(x0, x1))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) and U(dbl(dbls(dbl(indx(x0, x1))))) → U(dbls(dbl(indx(x0, x1)))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) does not unify with U(dbl(dbls(dbl(indx(x0, x1))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) and U(dbls(dbl(dbls(sel(x0, x1))))) → U(dbl(dbls(sel(x0, x1)))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) does not unify with U(dbls(dbl(dbls(sel(x0, x1))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) and U(dbls(dbl(dbls(indx(x0, x1))))) → U(dbl(dbls(indx(x0, x1)))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) does not unify with U(dbls(dbl(dbls(indx(x0, x1))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) and U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) → U(dbl(dbls(dbl(sel(x0, x1))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) does not unify with U(dbls(dbl(dbls(dbl(sel(x0, x1)))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) and U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) → U(dbl(dbls(dbl(indx(x0, x1))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) does not unify with U(dbls(dbl(dbls(dbl(indx(x0, x1)))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) and U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) → U(dbls(dbl(dbls(sel(x0, x1))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) does not unify with U(dbl(dbls(dbl(dbls(sel(x0, x1)))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) and U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) → U(dbls(dbl(dbls(indx(x0, x1))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) does not unify with U(dbl(dbls(dbl(dbls(indx(x0, x1)))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) and U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) → U(dbls(dbl(dbls(dbl(dbls(x0)))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) does not unify with U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) and U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) → U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) does not unify with U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) and U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) → U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) does not unify with U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) and U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))) → U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) and U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))) → U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) and U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) and U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) and U(indx(dbls(dbl(dbls(dbl(dbls(sel(x0, x1)))))), x2)) → U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) does not unify with U(indx(dbls(dbl(dbls(dbl(dbls(sel(x0, x1)))))), x2)). The rules U(indx(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) and U(sel(x0, x1)) → U(x0) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) does not unify with U(sel(x0, x1)). The rules U(indx(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) and U(sel(x0, x1)) → U(x1) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) does not unify with U(sel(x0, x1)). The rules U(indx(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) and U(sel(x0, x1)) → SEL(x0, x1) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) does not unify with U(sel(x0, x1)). The rules U(indx(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) and U(dbl(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) does not unify with U(dbl(sel(x0, x1))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) and U(dbl(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) does not unify with U(dbl(indx(x0, x1))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) and U(dbls(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) does not unify with U(dbls(sel(x0, x1))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) and U(dbls(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) does not unify with U(dbls(indx(x0, x1))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) and U(dbls(dbl(sel(x0, x1)))) → U(dbl(sel(x0, x1))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) does not unify with U(dbls(dbl(sel(x0, x1)))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) and U(dbls(dbl(indx(x0, x1)))) → U(dbl(indx(x0, x1))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) does not unify with U(dbls(dbl(indx(x0, x1)))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) and U(dbl(dbls(sel(x0, x1)))) → U(dbls(sel(x0, x1))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) does not unify with U(dbl(dbls(sel(x0, x1)))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) and U(dbl(dbls(indx(x0, x1)))) → U(dbls(indx(x0, x1))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) does not unify with U(dbl(dbls(indx(x0, x1)))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) and U(dbl(dbls(dbl(sel(x0, x1))))) → U(dbls(dbl(sel(x0, x1)))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) does not unify with U(dbl(dbls(dbl(sel(x0, x1))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) and U(dbl(dbls(dbl(indx(x0, x1))))) → U(dbls(dbl(indx(x0, x1)))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) does not unify with U(dbl(dbls(dbl(indx(x0, x1))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) and U(dbls(dbl(dbls(sel(x0, x1))))) → U(dbl(dbls(sel(x0, x1)))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) does not unify with U(dbls(dbl(dbls(sel(x0, x1))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) and U(dbls(dbl(dbls(indx(x0, x1))))) → U(dbl(dbls(indx(x0, x1)))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) does not unify with U(dbls(dbl(dbls(indx(x0, x1))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) and U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) → U(dbl(dbls(dbl(sel(x0, x1))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) does not unify with U(dbls(dbl(dbls(dbl(sel(x0, x1)))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) and U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) → U(dbl(dbls(dbl(indx(x0, x1))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) does not unify with U(dbls(dbl(dbls(dbl(indx(x0, x1)))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) and U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) → U(dbls(dbl(dbls(sel(x0, x1))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) does not unify with U(dbl(dbls(dbl(dbls(sel(x0, x1)))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) and U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) → U(dbls(dbl(dbls(indx(x0, x1))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) does not unify with U(dbl(dbls(dbl(dbls(indx(x0, x1)))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) and U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) → U(dbls(dbl(dbls(dbl(dbls(x0)))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) does not unify with U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) and U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) → U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) does not unify with U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) and U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) → U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) does not unify with U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) and U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))) → U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) and U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))) → U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) and U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) and U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) and U(indx(dbls(dbl(dbls(dbl(dbls(indx(x0, x1)))))), x2)) → U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))))) = U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) does not unify with U(indx(dbls(dbl(dbls(dbl(dbls(indx(x0, x1)))))), x2)). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))), z1)) → U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) and U(sel(x0, x1)) → U(x0) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) does not unify with U(sel(x0, x1)). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))), z1)) → U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) and U(sel(x0, x1)) → U(x1) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) does not unify with U(sel(x0, x1)). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))), z1)) → U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) and U(sel(x0, x1)) → SEL(x0, x1) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) does not unify with U(sel(x0, x1)). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))), z1)) → U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) and U(dbl(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) does not unify with U(dbl(sel(x0, x1))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))), z1)) → U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) and U(dbl(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) does not unify with U(dbl(indx(x0, x1))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))), z1)) → U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) and U(dbls(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) does not unify with U(dbls(sel(x0, x1))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))), z1)) → U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) and U(dbls(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) does not unify with U(dbls(indx(x0, x1))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))), z1)) → U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) and U(dbls(dbl(sel(x0, x1)))) → U(dbl(sel(x0, x1))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) does not unify with U(dbls(dbl(sel(x0, x1)))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))), z1)) → U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) and U(dbls(dbl(indx(x0, x1)))) → U(dbl(indx(x0, x1))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) does not unify with U(dbls(dbl(indx(x0, x1)))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))), z1)) → U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) and U(dbl(dbls(sel(x0, x1)))) → U(dbls(sel(x0, x1))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) does not unify with U(dbl(dbls(sel(x0, x1)))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))), z1)) → U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) and U(dbl(dbls(indx(x0, x1)))) → U(dbls(indx(x0, x1))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) does not unify with U(dbl(dbls(indx(x0, x1)))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))), z1)) → U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) and U(dbl(dbls(dbl(sel(x0, x1))))) → U(dbls(dbl(sel(x0, x1)))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) does not unify with U(dbl(dbls(dbl(sel(x0, x1))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))), z1)) → U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) and U(dbl(dbls(dbl(indx(x0, x1))))) → U(dbls(dbl(indx(x0, x1)))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) does not unify with U(dbl(dbls(dbl(indx(x0, x1))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))), z1)) → U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) and U(dbls(dbl(dbls(sel(x0, x1))))) → U(dbl(dbls(sel(x0, x1)))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) does not unify with U(dbls(dbl(dbls(sel(x0, x1))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))), z1)) → U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) and U(dbls(dbl(dbls(indx(x0, x1))))) → U(dbl(dbls(indx(x0, x1)))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) does not unify with U(dbls(dbl(dbls(indx(x0, x1))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))), z1)) → U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) and U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) → U(dbl(dbls(dbl(sel(x0, x1))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) does not unify with U(dbls(dbl(dbls(dbl(sel(x0, x1)))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))), z1)) → U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) and U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) → U(dbl(dbls(dbl(indx(x0, x1))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) does not unify with U(dbls(dbl(dbls(dbl(indx(x0, x1)))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))), z1)) → U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) and U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) → U(dbls(dbl(dbls(sel(x0, x1))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) does not unify with U(dbl(dbls(dbl(dbls(sel(x0, x1)))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))), z1)) → U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) and U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) → U(dbls(dbl(dbls(indx(x0, x1))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) does not unify with U(dbl(dbls(dbl(dbls(indx(x0, x1)))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))), z1)) → U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) and U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) → U(dbls(dbl(dbls(dbl(dbls(x0)))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) does not unify with U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))), z1)) → U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) and U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) → U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) does not unify with U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))), z1)) → U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) and U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) → U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) does not unify with U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))), z1)) → U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) and U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))) → U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))), z1)) → U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) and U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))) → U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))), z1)) → U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) and U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))), z1)) → U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) and U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))), z1)) → U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) and U(indx(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))), x1)) → U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) does not unify with U(indx(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))), x1)). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) and U(sel(x0, x1)) → U(x0) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) does not unify with U(sel(x0, x1)). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) and U(sel(x0, x1)) → U(x1) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) does not unify with U(sel(x0, x1)). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) and U(sel(x0, x1)) → SEL(x0, x1) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) does not unify with U(sel(x0, x1)). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) and U(dbl(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) does not unify with U(dbl(sel(x0, x1))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) and U(dbl(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) does not unify with U(dbl(indx(x0, x1))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) and U(dbls(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) does not unify with U(dbls(sel(x0, x1))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) and U(dbls(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) does not unify with U(dbls(indx(x0, x1))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) and U(dbls(dbl(sel(x0, x1)))) → U(dbl(sel(x0, x1))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) does not unify with U(dbls(dbl(sel(x0, x1)))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) and U(dbls(dbl(indx(x0, x1)))) → U(dbl(indx(x0, x1))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) does not unify with U(dbls(dbl(indx(x0, x1)))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) and U(dbl(dbls(sel(x0, x1)))) → U(dbls(sel(x0, x1))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) does not unify with U(dbl(dbls(sel(x0, x1)))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) and U(dbl(dbls(indx(x0, x1)))) → U(dbls(indx(x0, x1))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) does not unify with U(dbl(dbls(indx(x0, x1)))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) and U(dbl(dbls(dbl(sel(x0, x1))))) → U(dbls(dbl(sel(x0, x1)))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) does not unify with U(dbl(dbls(dbl(sel(x0, x1))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) and U(dbl(dbls(dbl(indx(x0, x1))))) → U(dbls(dbl(indx(x0, x1)))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) does not unify with U(dbl(dbls(dbl(indx(x0, x1))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) and U(dbls(dbl(dbls(sel(x0, x1))))) → U(dbl(dbls(sel(x0, x1)))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) does not unify with U(dbls(dbl(dbls(sel(x0, x1))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) and U(dbls(dbl(dbls(indx(x0, x1))))) → U(dbl(dbls(indx(x0, x1)))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) does not unify with U(dbls(dbl(dbls(indx(x0, x1))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) and U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) → U(dbl(dbls(dbl(sel(x0, x1))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) does not unify with U(dbls(dbl(dbls(dbl(sel(x0, x1)))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) and U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) → U(dbl(dbls(dbl(indx(x0, x1))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) does not unify with U(dbls(dbl(dbls(dbl(indx(x0, x1)))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) and U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) → U(dbls(dbl(dbls(sel(x0, x1))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) does not unify with U(dbl(dbls(dbl(dbls(sel(x0, x1)))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) and U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) → U(dbls(dbl(dbls(indx(x0, x1))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) does not unify with U(dbl(dbls(dbl(dbls(indx(x0, x1)))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) and U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) → U(dbls(dbl(dbls(dbl(dbls(x0)))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) does not unify with U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) and U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) → U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) does not unify with U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) and U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) → U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) does not unify with U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) and U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))) → U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) and U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))) → U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) and U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))) → U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) and U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) and U(indx(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))), x2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) does not unify with U(indx(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))), x2)). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) and U(sel(x0, x1)) → U(x0) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) does not unify with U(sel(x0, x1)). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) and U(sel(x0, x1)) → U(x1) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) does not unify with U(sel(x0, x1)). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) and U(sel(x0, x1)) → SEL(x0, x1) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) does not unify with U(sel(x0, x1)). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) and U(dbl(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) does not unify with U(dbl(sel(x0, x1))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) and U(dbl(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) does not unify with U(dbl(indx(x0, x1))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) and U(dbls(sel(x0, x1))) → U(sel(x0, x1)) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) does not unify with U(dbls(sel(x0, x1))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) and U(dbls(indx(x0, x1))) → U(indx(x0, x1)) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) does not unify with U(dbls(indx(x0, x1))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) and U(dbls(dbl(sel(x0, x1)))) → U(dbl(sel(x0, x1))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) does not unify with U(dbls(dbl(sel(x0, x1)))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) and U(dbls(dbl(indx(x0, x1)))) → U(dbl(indx(x0, x1))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) does not unify with U(dbls(dbl(indx(x0, x1)))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) and U(dbl(dbls(sel(x0, x1)))) → U(dbls(sel(x0, x1))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) does not unify with U(dbl(dbls(sel(x0, x1)))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) and U(dbl(dbls(indx(x0, x1)))) → U(dbls(indx(x0, x1))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) does not unify with U(dbl(dbls(indx(x0, x1)))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) and U(dbl(dbls(dbl(sel(x0, x1))))) → U(dbls(dbl(sel(x0, x1)))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) does not unify with U(dbl(dbls(dbl(sel(x0, x1))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) and U(dbl(dbls(dbl(indx(x0, x1))))) → U(dbls(dbl(indx(x0, x1)))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) does not unify with U(dbl(dbls(dbl(indx(x0, x1))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) and U(dbls(dbl(dbls(sel(x0, x1))))) → U(dbl(dbls(sel(x0, x1)))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) does not unify with U(dbls(dbl(dbls(sel(x0, x1))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) and U(dbls(dbl(dbls(indx(x0, x1))))) → U(dbl(dbls(indx(x0, x1)))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) does not unify with U(dbls(dbl(dbls(indx(x0, x1))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) and U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) → U(dbl(dbls(dbl(sel(x0, x1))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) does not unify with U(dbls(dbl(dbls(dbl(sel(x0, x1)))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) and U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) → U(dbl(dbls(dbl(indx(x0, x1))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) does not unify with U(dbls(dbl(dbls(dbl(indx(x0, x1)))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) and U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) → U(dbls(dbl(dbls(sel(x0, x1))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) does not unify with U(dbl(dbls(dbl(dbls(sel(x0, x1)))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) and U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) → U(dbls(dbl(dbls(indx(x0, x1))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) does not unify with U(dbl(dbls(dbl(dbls(indx(x0, x1)))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) and U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) → U(dbls(dbl(dbls(dbl(dbls(x0)))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) does not unify with U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) and U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) → U(dbls(dbl(dbls(dbl(sel(x0, x1)))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) does not unify with U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) and U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))) → U(dbls(dbl(dbls(dbl(indx(x0, x1)))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) does not unify with U(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) and U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))) → U(dbl(dbls(dbl(dbls(sel(x0, x1)))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(sel(x0, x1))))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) and U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))) → U(dbl(dbls(dbl(dbls(indx(x0, x1)))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(indx(x0, x1))))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) and U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))) → U(dbl(dbls(dbl(dbls(dbl(dbls(x0))))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(x0)))))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) and U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))) → U(dbl(dbls(dbl(dbls(dbl(sel(x0, x1))))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) does not unify with U(dbls(dbl(dbls(dbl(dbls(dbl(sel(x0, x1)))))))). The rules U(indx(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))), z2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) and U(indx(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))), x2)) → U(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1)))))))) form no chain, because ECap^µ(U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))))) = U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) does not unify with U(indx(dbls(dbl(dbls(dbl(dbls(dbl(indx(x0, x1))))))), x2)).

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

        ↳ QCSDP

          ↳ QCSDependencyGraphProof

            ↳ QCSDP

              ↳ QCSDPForwardInstantiationProcessor

                ↳ QCSDP

                  ↳ QCSDPForwardInstantiationProcessor

                    ↳ QCSDP

                      ↳ QCSDependencyGraphProof

                        ↳ AND

                          ↳ QCSDP

                          ↳ QCSDP

                            ↳ QCSDPForwardInstantiationProcessor

                              ↳ QCSDP

                                ↳ QCSDPForwardInstantiationProcessor

                                  ↳ QCSDP

                                    ↳ QCSDependencyGraphProof

                                      ↳ AND

                                        ↳ QCSDP

                                        ↳ QCSDP

                                          ↳ QCSDPForwardInstantiationProcessor

                                            ↳ QCSDP

                                              ↳ QCSDPForwardInstantiationProcessor

                                                ↳ QCSDP

                                                  ↳ QCSDPForwardInstantiationProcessor

                                                    ↳ QCSDP

                                                      ↳ QCSDPForwardInstantiationProcessor

                                                        ↳ QCSDP

                                                          ↳ QCSDPForwardInstantiationProcessor

                                                            ↳ QCSDP

                                                              ↳ QCSDPForwardInstantiationProcessor

                                                                ↳ QCSDP

                                                                  ↳ QCSDPForwardInstantiationProcessor

                                                                    ↳ QCSDP

                                                                      ↳ QCSDependencyGraphProof

                                                                        ↳ AND

                                                                          ↳ QCSDP

                                                                            ↳ QCSDPSubtermProof

                                                                          ↳ QCSDP

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

U(indx(indx(z0, z1), y1)) → U(indx(z0, z1))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We use the subterm processor [20].

The following pairs can be oriented strictly and are deleted.

U(indx(indx(z0, z1), y1)) → U(indx(z0, z1))

The remaining pairs can at least be oriented weakly.
none
Used ordering: Combined order from the following AFS and order.
U(x1) = x1

Subterm Order

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

        ↳ QCSDP

          ↳ QCSDependencyGraphProof

            ↳ QCSDP

              ↳ QCSDPForwardInstantiationProcessor

                ↳ QCSDP

                  ↳ QCSDPForwardInstantiationProcessor

                    ↳ QCSDP

                      ↳ QCSDependencyGraphProof

                        ↳ AND

                          ↳ QCSDP

                          ↳ QCSDP

                            ↳ QCSDPForwardInstantiationProcessor

                              ↳ QCSDP

                                ↳ QCSDPForwardInstantiationProcessor

                                  ↳ QCSDP

                                    ↳ QCSDependencyGraphProof

                                      ↳ AND

                                        ↳ QCSDP

                                        ↳ QCSDP

                                          ↳ QCSDPForwardInstantiationProcessor

                                            ↳ QCSDP

                                              ↳ QCSDPForwardInstantiationProcessor

                                                ↳ QCSDP

                                                  ↳ QCSDPForwardInstantiationProcessor

                                                    ↳ QCSDP

                                                      ↳ QCSDPForwardInstantiationProcessor

                                                        ↳ QCSDP

                                                          ↳ QCSDPForwardInstantiationProcessor

                                                            ↳ QCSDP

                                                              ↳ QCSDPForwardInstantiationProcessor

                                                                ↳ QCSDP

                                                                  ↳ QCSDPForwardInstantiationProcessor

                                                                    ↳ QCSDP

                                                                      ↳ QCSDependencyGraphProof

                                                                        ↳ AND

                                                                          ↳ QCSDP

                                                                            ↳ QCSDPSubtermProof

                                                                              ↳ QCSDP

                                                                                ↳ PIsEmptyProof

                                                                          ↳ QCSDP

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

Q-restricted context-sensitive dependency pair problem:
The symbols in {dbl, dbls, sel} 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:
none

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

The TRS P is empty. Hence, there is no (P,Q,R,µ)-chain.

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

        ↳ QCSDP

          ↳ QCSDependencyGraphProof

            ↳ QCSDP

              ↳ QCSDPForwardInstantiationProcessor

                ↳ QCSDP

                  ↳ QCSDPForwardInstantiationProcessor

                    ↳ QCSDP

                      ↳ QCSDependencyGraphProof

                        ↳ AND

                          ↳ QCSDP

                          ↳ QCSDP

                            ↳ QCSDPForwardInstantiationProcessor

                              ↳ QCSDP

                                ↳ QCSDPForwardInstantiationProcessor

                                  ↳ QCSDP

                                    ↳ QCSDependencyGraphProof

                                      ↳ AND

                                        ↳ QCSDP

                                        ↳ QCSDP

                                          ↳ QCSDPForwardInstantiationProcessor

                                            ↳ QCSDP

                                              ↳ QCSDPForwardInstantiationProcessor

                                                ↳ QCSDP

                                                  ↳ QCSDPForwardInstantiationProcessor

                                                    ↳ QCSDP

                                                      ↳ QCSDPForwardInstantiationProcessor

                                                        ↳ QCSDP

                                                          ↳ QCSDPForwardInstantiationProcessor

                                                            ↳ QCSDP

                                                              ↳ QCSDPForwardInstantiationProcessor

                                                                ↳ QCSDP

                                                                  ↳ QCSDPForwardInstantiationProcessor

                                                                    ↳ QCSDP

                                                                      ↳ QCSDependencyGraphProof

                                                                        ↳ AND

                                                                          ↳ QCSDP

                                                                          ↳ QCSDP

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

SEL(s(X), cons(Y, Z)) → SEL(X, Z)
SEL(0, cons(dbl(z0), y1)) → U(dbl(z0))
U(dbl(sel(z0, z1))) → U(sel(z0, z1))
U(sel(x_0, x_1)) → U(x_0)
U(sel(x_0, x_1)) → U(x_1)
U(sel(X, Z)) → SEL(X, Z)
SEL(0, cons(dbls(z0), y1)) → U(dbls(z0))
U(dbls(sel(z0, z1))) → U(sel(z0, z1))
U(dbls(indx(z0, z1))) → U(indx(z0, z1))
U(indx(sel(z0, z1), y1)) → U(sel(z0, z1))
U(indx(dbl(sel(z0, z1)), y1)) → U(dbl(sel(z0, z1)))
U(indx(dbl(indx(z0, z1)), y1)) → U(dbl(indx(z0, z1)))
U(dbl(indx(z0, z1))) → U(indx(z0, z1))
U(indx(dbls(sel(z0, z1)), y1)) → U(dbls(sel(z0, z1)))
U(indx(dbls(indx(z0, z1)), y1)) → U(dbls(indx(z0, z1)))
U(indx(dbls(dbl(sel(z0, z1))), y1)) → U(dbls(dbl(sel(z0, z1))))
U(dbls(dbl(sel(z0, z1)))) → U(dbl(sel(z0, z1)))
U(indx(dbls(dbl(indx(z0, z1))), y1)) → U(dbls(dbl(indx(z0, z1))))
U(dbls(dbl(indx(z0, z1)))) → U(dbl(indx(z0, z1)))
U(indx(dbl(dbls(sel(z0, z1))), y1)) → U(dbl(dbls(sel(z0, z1))))
U(dbl(dbls(sel(z0, z1)))) → U(dbls(sel(z0, z1)))
U(indx(dbl(dbls(indx(z0, z1))), y1)) → U(dbl(dbls(indx(z0, z1))))
U(dbl(dbls(indx(z0, z1)))) → U(dbls(indx(z0, z1)))
U(indx(dbl(dbls(dbl(sel(z0, z1)))), y1)) → U(dbl(dbls(dbl(sel(z0, z1)))))
U(dbl(dbls(dbl(sel(z0, z1))))) → U(dbls(dbl(sel(z0, z1))))
U(indx(dbl(dbls(dbl(indx(z0, z1)))), y1)) → U(dbl(dbls(dbl(indx(z0, z1)))))
U(dbl(dbls(dbl(indx(z0, z1))))) → U(dbls(dbl(indx(z0, z1))))
U(indx(dbls(dbl(dbls(sel(z0, z1)))), y1)) → U(dbls(dbl(dbls(sel(z0, z1)))))
U(dbls(dbl(dbls(sel(z0, z1))))) → U(dbl(dbls(sel(z0, z1))))
U(indx(dbls(dbl(dbls(indx(z0, z1)))), y1)) → U(dbls(dbl(dbls(indx(z0, z1)))))
U(dbls(dbl(dbls(indx(z0, z1))))) → U(dbl(dbls(indx(z0, z1))))
U(indx(dbls(dbl(dbls(dbl(sel(z0, z1))))), y1)) → U(dbls(dbl(dbls(dbl(sel(z0, z1))))))
U(dbls(dbl(dbls(dbl(sel(z0, z1)))))) → U(dbl(dbls(dbl(sel(z0, z1)))))
U(indx(dbls(dbl(dbls(dbl(indx(z0, z1))))), y1)) → U(dbls(dbl(dbls(dbl(indx(z0, z1))))))
U(dbls(dbl(dbls(dbl(indx(z0, z1)))))) → U(dbl(dbls(dbl(indx(z0, z1)))))
U(indx(dbl(dbls(dbl(dbls(sel(z0, z1))))), y1)) → U(dbl(dbls(dbl(dbls(sel(z0, z1))))))
U(dbl(dbls(dbl(dbls(sel(z0, z1)))))) → U(dbls(dbl(dbls(sel(z0, z1)))))
U(indx(dbl(dbls(dbl(dbls(indx(z0, z1))))), y1)) → U(dbl(dbls(dbl(dbls(indx(z0, z1))))))
U(dbl(dbls(dbl(dbls(indx(z0, z1)))))) → U(dbls(dbl(dbls(indx(z0, z1)))))
U(indx(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))), y1)) → U(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))
U(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))) → U(dbls(dbl(dbls(dbl(dbls(z0))))))
U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1))))))) → U(dbl(dbls(dbl(dbls(sel(z0, z1))))))
U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1))))))) → U(dbl(dbls(dbl(dbls(indx(z0, z1))))))
U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) → U(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))
U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) → U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))
U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))) → U(dbls(dbl(dbls(dbl(sel(z0, z1))))))
U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) → U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))
U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))) → U(dbls(dbl(dbls(dbl(indx(z0, z1))))))
U(indx(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))), y1)) → U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))
U(indx(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))), y1)) → U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))
U(indx(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))), y1)) → U(dbls(dbl(dbls(dbl(dbls(sel(z0, z1)))))))
U(indx(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))), y1)) → U(dbls(dbl(dbls(dbl(dbls(indx(z0, z1)))))))
U(indx(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))), y1)) → U(dbls(dbl(dbls(dbl(dbls(dbl(dbls(z0))))))))
U(indx(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))), y1)) → U(dbls(dbl(dbls(dbl(dbls(dbl(sel(z0, z1))))))))
U(indx(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))), y1)) → U(dbls(dbl(dbls(dbl(dbls(dbl(indx(z0, z1))))))))
SEL(0, cons(sel(z0, z1), y1)) → U(sel(z0, z1))
SEL(0, cons(indx(z0, z1), y1)) → U(indx(z0, z1))
SEL(s(dbls(z0)), cons(y1, y2)) → U(dbls(z0))
SEL(s(sel(z0, z1)), cons(y1, y2)) → U(sel(z0, z1))
SEL(s(indx(z0, z1)), cons(y1, y2)) → U(indx(z0, z1))
SEL(s(dbl(dbls(z0))), cons(y1, y2)) → U(dbl(dbls(z0)))
SEL(s(dbl(sel(z0, z1))), cons(y1, y2)) → U(dbl(sel(z0, z1)))
SEL(s(dbl(indx(z0, z1))), cons(y1, y2)) → U(dbl(indx(z0, z1)))
SEL(s(y0), cons(y1, sel(z0, z1))) → U(sel(z0, z1))
SEL(s(y0), cons(y1, indx(z0, z1))) → U(indx(z0, z1))
SEL(s(y0), cons(y1, dbl(sel(z0, z1)))) → U(dbl(sel(z0, z1)))
SEL(s(y0), cons(y1, dbl(indx(z0, z1)))) → U(dbl(indx(z0, z1)))
SEL(s(y0), cons(y1, dbls(sel(z0, z1)))) → U(dbls(sel(z0, z1)))
SEL(s(y0), cons(y1, dbls(indx(z0, z1)))) → U(dbls(indx(z0, z1)))
SEL(s(y0), cons(y1, dbls(dbl(dbls(z0))))) → U(dbls(dbl(dbls(z0))))
SEL(s(y0), cons(y1, dbls(dbl(sel(z0, z1))))) → U(dbls(dbl(sel(z0, z1))))
SEL(s(y0), cons(y1, dbls(dbl(indx(z0, z1))))) → U(dbls(dbl(indx(z0, z1))))
SEL(s(y0), cons(y1, dbl(dbls(sel(z0, z1))))) → U(dbl(dbls(sel(z0, z1))))
SEL(s(y0), cons(y1, dbl(dbls(indx(z0, z1))))) → U(dbl(dbls(indx(z0, z1))))
SEL(s(y0), cons(y1, dbl(dbls(dbl(sel(z0, z1)))))) → U(dbl(dbls(dbl(sel(z0, z1)))))
SEL(s(y0), cons(y1, dbl(dbls(dbl(indx(z0, z1)))))) → U(dbl(dbls(dbl(indx(z0, z1)))))
SEL(s(y0), cons(y1, dbl(dbls(dbl(dbls(sel(z0, z1))))))) → U(dbl(dbls(dbl(dbls(sel(z0, z1))))))
SEL(s(y0), cons(y1, dbl(dbls(dbl(dbls(indx(z0, z1))))))) → U(dbl(dbls(dbl(dbls(indx(z0, z1))))))
SEL(s(y0), cons(y1, dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))) → U(dbl(dbls(dbl(dbls(dbl(dbls(z0)))))))
SEL(s(y0), cons(y1, dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))) → U(dbl(dbls(dbl(dbls(dbl(sel(z0, z1)))))))
SEL(s(y0), cons(y1, dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))) → U(dbl(dbls(dbl(dbls(dbl(indx(z0, z1)))))))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

We applied the Zantema transformation [34] to transform the context-sensitive TRS to a usual TRS.

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)

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:

A(fromInact(x1)) → FROM(x1)
A(dblInact(x1)) → DBL(x1)
DBL(s(X)) → S(sInact(dblInact(a(X))))
DBLS(cons(X, Y)) → A(Y)
INDX(cons(X, Y), Z) → A(Z)
SEL(s(X), cons(Y, Z)) → A(X)
A(selInact(x1, x2)) → SEL(x1, x2)
INDX(cons(X, Y), Z) → A(X)
DBL(s(X)) → A(X)
SEL(0, cons(X, Y)) → A(X)
FROM(X) → A(X)
DBLS(cons(X, Y)) → A(X)
A(dblsInact(x1)) → DBLS(x1)
SEL(s(X), cons(Y, Z)) → A(Z)
INDX(cons(X, Y), Z) → A(Y)
A(indxInact(x1, x2)) → INDX(x1, x2)
A(sInact(x1)) → S(x1)
SEL(s(X), cons(Y, Z)) → SEL(a(X), a(Z))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

A(fromInact(x1)) → FROM(x1)
A(dblInact(x1)) → DBL(x1)
DBL(s(X)) → S(sInact(dblInact(a(X))))
DBLS(cons(X, Y)) → A(Y)
INDX(cons(X, Y), Z) → A(Z)
SEL(s(X), cons(Y, Z)) → A(X)
A(selInact(x1, x2)) → SEL(x1, x2)
INDX(cons(X, Y), Z) → A(X)
DBL(s(X)) → A(X)
SEL(0, cons(X, Y)) → A(X)
FROM(X) → A(X)
DBLS(cons(X, Y)) → A(X)
A(dblsInact(x1)) → DBLS(x1)
SEL(s(X), cons(Y, Z)) → A(Z)
INDX(cons(X, Y), Z) → A(Y)
A(indxInact(x1, x2)) → INDX(x1, x2)
A(sInact(x1)) → S(x1)
SEL(s(X), cons(Y, Z)) → SEL(a(X), a(Z))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

A(fromInact(x1)) → FROM(x1)
A(dblInact(x1)) → DBL(x1)
INDX(cons(X, Y), Z) → A(Z)
DBLS(cons(X, Y)) → A(Y)
SEL(s(X), cons(Y, Z)) → A(X)
A(selInact(x1, x2)) → SEL(x1, x2)
INDX(cons(X, Y), Z) → A(X)
DBL(s(X)) → A(X)
SEL(0, cons(X, Y)) → A(X)
FROM(X) → A(X)
DBLS(cons(X, Y)) → A(X)
SEL(s(X), cons(Y, Z)) → A(Z)
A(dblsInact(x1)) → DBLS(x1)
INDX(cons(X, Y), Z) → A(Y)
A(indxInact(x1, x2)) → INDX(x1, x2)
SEL(s(X), cons(Y, Z)) → SEL(a(X), a(Z))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)

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

The following pairs can be oriented strictly and are deleted.

SEL(0, cons(X, Y)) → A(X)
A(indxInact(x1, x2)) → INDX(x1, x2)

The remaining pairs can at least be oriented weakly.

A(fromInact(x1)) → FROM(x1)
A(dblInact(x1)) → DBL(x1)
INDX(cons(X, Y), Z) → A(Z)
DBLS(cons(X, Y)) → A(Y)
SEL(s(X), cons(Y, Z)) → A(X)
A(selInact(x1, x2)) → SEL(x1, x2)
INDX(cons(X, Y), Z) → A(X)
DBL(s(X)) → A(X)
FROM(X) → A(X)
DBLS(cons(X, Y)) → A(X)
SEL(s(X), cons(Y, Z)) → A(Z)
A(dblsInact(x1)) → DBLS(x1)
INDX(cons(X, Y), Z) → A(Y)
SEL(s(X), cons(Y, Z)) → SEL(a(X), a(Z))

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

POL(0) = 1
POL(A(x₁)) = x₁
POL(DBL(x₁)) = x₁
POL(DBLS(x₁)) = x₁
POL(FROM(x₁)) = x₁
POL(INDX(x₁, x₂)) = x₁ + x₂
POL(SEL(x₁, x₂)) = x₁ + x₂
POL(a(x₁)) = x₁
POL(cons(x₁, x₂)) = max(x₁, x₂)
POL(dbl(x₁)) = x₁
POL(dblInact(x₁)) = x₁
POL(dbls(x₁)) = x₁
POL(dblsInact(x₁)) = x₁
POL(from(x₁)) = x₁
POL(fromInact(x₁)) = x₁
POL(indx(x₁, x₂)) = 1 + x₁ + x₂
POL(indxInact(x₁, x₂)) = 1 + x₁ + x₂
POL(nil) = 1
POL(s(x₁)) = x₁
POL(sInact(x₁)) = x₁
POL(sel(x₁, x₂)) = x₁ + x₂
POL(selInact(x₁, x₂)) = x₁ + x₂

The following usable rules [17] were oriented:

indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
sel(x1, x2) → selInact(x1, x2)
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
a(selInact(x1, x2)) → sel(x1, x2)
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
a(dblsInact(x1)) → dbls(x1)
dbls(x1) → dblsInact(x1)
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
dbls(nil) → nil
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbl(0) → 0

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

A(fromInact(x1)) → FROM(x1)
A(dblInact(x1)) → DBL(x1)
INDX(cons(X, Y), Z) → A(Z)
DBLS(cons(X, Y)) → A(Y)
SEL(s(X), cons(Y, Z)) → A(X)
A(selInact(x1, x2)) → SEL(x1, x2)
INDX(cons(X, Y), Z) → A(X)
DBL(s(X)) → A(X)
FROM(X) → A(X)
DBLS(cons(X, Y)) → A(X)
SEL(s(X), cons(Y, Z)) → A(Z)
A(dblsInact(x1)) → DBLS(x1)
INDX(cons(X, Y), Z) → A(Y)
SEL(s(X), cons(Y, Z)) → SEL(a(X), a(Z))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ QDPOrderProof

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

SEL(s(X), cons(Y, Z)) → A(Z)
A(dblsInact(x1)) → DBLS(x1)
A(fromInact(x1)) → FROM(x1)
A(dblInact(x1)) → DBL(x1)
DBLS(cons(X, Y)) → A(Y)
SEL(s(X), cons(Y, Z)) → A(X)
A(selInact(x1, x2)) → SEL(x1, x2)
SEL(s(X), cons(Y, Z)) → SEL(a(X), a(Z))
DBL(s(X)) → A(X)
FROM(X) → A(X)
DBLS(cons(X, Y)) → A(X)

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)

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

The following pairs can be oriented strictly and are deleted.

FROM(X) → A(X)

The remaining pairs can at least be oriented weakly.

SEL(s(X), cons(Y, Z)) → A(Z)
A(dblsInact(x1)) → DBLS(x1)
A(fromInact(x1)) → FROM(x1)
A(dblInact(x1)) → DBL(x1)
DBLS(cons(X, Y)) → A(Y)
SEL(s(X), cons(Y, Z)) → A(X)
A(selInact(x1, x2)) → SEL(x1, x2)
SEL(s(X), cons(Y, Z)) → SEL(a(X), a(Z))
DBL(s(X)) → A(X)
DBLS(cons(X, Y)) → A(X)

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

POL(0) = 1
POL(A(x₁)) = x₁
POL(DBL(x₁)) = x₁
POL(DBLS(x₁)) = x₁
POL(FROM(x₁)) = 1 + x₁
POL(SEL(x₁, x₂)) = x₁ + x₂
POL(a(x₁)) = x₁
POL(cons(x₁, x₂)) = max(x₁, x₂)
POL(dbl(x₁)) = x₁
POL(dblInact(x₁)) = x₁
POL(dbls(x₁)) = x₁
POL(dblsInact(x₁)) = x₁
POL(from(x₁)) = 1 + x₁
POL(fromInact(x₁)) = 1 + x₁
POL(indx(x₁, x₂)) = x₁ + x₂
POL(indxInact(x₁, x₂)) = x₁ + x₂
POL(nil) = 0
POL(s(x₁)) = x₁
POL(sInact(x₁)) = x₁
POL(sel(x₁, x₂)) = x₁ + x₂
POL(selInact(x₁, x₂)) = x₁ + x₂

The following usable rules [17] were oriented:

indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
sel(x1, x2) → selInact(x1, x2)
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
a(selInact(x1, x2)) → sel(x1, x2)
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
a(dblsInact(x1)) → dbls(x1)
dbls(x1) → dblsInact(x1)
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
dbls(nil) → nil
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbl(0) → 0

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

A(fromInact(x1)) → FROM(x1)
A(dblsInact(x1)) → DBLS(x1)
SEL(s(X), cons(Y, Z)) → A(Z)
A(dblInact(x1)) → DBL(x1)
DBLS(cons(X, Y)) → A(Y)
SEL(s(X), cons(Y, Z)) → A(X)
A(selInact(x1, x2)) → SEL(x1, x2)
SEL(s(X), cons(Y, Z)) → SEL(a(X), a(Z))
DBL(s(X)) → A(X)
DBLS(cons(X, Y)) → A(X)

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ QDPOrderProof

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

SEL(s(X), cons(Y, Z)) → A(Z)
A(dblsInact(x1)) → DBLS(x1)
A(dblInact(x1)) → DBL(x1)
DBLS(cons(X, Y)) → A(Y)
SEL(s(X), cons(Y, Z)) → A(X)
A(selInact(x1, x2)) → SEL(x1, x2)
SEL(s(X), cons(Y, Z)) → SEL(a(X), a(Z))
DBL(s(X)) → A(X)
DBLS(cons(X, Y)) → A(X)

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)

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

The following pairs can be oriented strictly and are deleted.

DBLS(cons(X, Y)) → A(Y)
DBLS(cons(X, Y)) → A(X)

The remaining pairs can at least be oriented weakly.

SEL(s(X), cons(Y, Z)) → A(Z)
A(dblsInact(x1)) → DBLS(x1)
A(dblInact(x1)) → DBL(x1)
SEL(s(X), cons(Y, Z)) → A(X)
A(selInact(x1, x2)) → SEL(x1, x2)
SEL(s(X), cons(Y, Z)) → SEL(a(X), a(Z))
DBL(s(X)) → A(X)

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

POL(0) = 0
POL(A(x₁)) = x₁
POL(DBL(x₁)) = x₁
POL(DBLS(x₁)) = 1 + x₁
POL(SEL(x₁, x₂)) = x₁ + x₂
POL(a(x₁)) = x₁
POL(cons(x₁, x₂)) = max(x₁, x₂)
POL(dbl(x₁)) = x₁
POL(dblInact(x₁)) = x₁
POL(dbls(x₁)) = 1 + x₁
POL(dblsInact(x₁)) = 1 + x₁
POL(from(x₁)) = x₁
POL(fromInact(x₁)) = x₁
POL(indx(x₁, x₂)) = x₁ + x₂
POL(indxInact(x₁, x₂)) = x₁ + x₂
POL(nil) = 0
POL(s(x₁)) = x₁
POL(sInact(x₁)) = x₁
POL(sel(x₁, x₂)) = x₁ + x₂
POL(selInact(x₁, x₂)) = x₁ + x₂

The following usable rules [17] were oriented:

indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
sel(x1, x2) → selInact(x1, x2)
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
a(selInact(x1, x2)) → sel(x1, x2)
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
a(dblsInact(x1)) → dbls(x1)
dbls(x1) → dblsInact(x1)
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
dbls(nil) → nil
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbl(0) → 0

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

A(dblsInact(x1)) → DBLS(x1)
SEL(s(X), cons(Y, Z)) → A(Z)
A(dblInact(x1)) → DBL(x1)
SEL(s(X), cons(Y, Z)) → A(X)
A(selInact(x1, x2)) → SEL(x1, x2)
SEL(s(X), cons(Y, Z)) → SEL(a(X), a(Z))
DBL(s(X)) → A(X)

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ QDPOrderProof

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

SEL(s(X), cons(Y, Z)) → A(Z)
A(dblInact(x1)) → DBL(x1)
SEL(s(X), cons(Y, Z)) → A(X)
A(selInact(x1, x2)) → SEL(x1, x2)
SEL(s(X), cons(Y, Z)) → SEL(a(X), a(Z))
DBL(s(X)) → A(X)

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)

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

The following pairs can be oriented strictly and are deleted.

A(dblInact(x1)) → DBL(x1)

The remaining pairs can at least be oriented weakly.

SEL(s(X), cons(Y, Z)) → A(Z)
SEL(s(X), cons(Y, Z)) → A(X)
A(selInact(x1, x2)) → SEL(x1, x2)
SEL(s(X), cons(Y, Z)) → SEL(a(X), a(Z))
DBL(s(X)) → A(X)

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

POL(0) = 0
POL(A(x₁)) = x₁
POL(DBL(x₁)) = x₁
POL(SEL(x₁, x₂)) = x₁ + x₂
POL(a(x₁)) = x₁
POL(cons(x₁, x₂)) = max(x₁, x₂)
POL(dbl(x₁)) = 1 + x₁
POL(dblInact(x₁)) = 1 + x₁
POL(dbls(x₁)) = 1 + x₁
POL(dblsInact(x₁)) = 1 + x₁
POL(from(x₁)) = x₁
POL(fromInact(x₁)) = x₁
POL(indx(x₁, x₂)) = x₁ + x₂
POL(indxInact(x₁, x₂)) = x₁ + x₂
POL(nil) = 0
POL(s(x₁)) = x₁
POL(sInact(x₁)) = x₁
POL(sel(x₁, x₂)) = x₁ + x₂
POL(selInact(x₁, x₂)) = x₁ + x₂

The following usable rules [17] were oriented:

indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
sel(x1, x2) → selInact(x1, x2)
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
a(selInact(x1, x2)) → sel(x1, x2)
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
a(dblsInact(x1)) → dbls(x1)
dbls(x1) → dblsInact(x1)
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
dbls(nil) → nil
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbl(0) → 0

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ QDPOrderProof

                                            ↳ QDP

                                              ↳ DependencyGraphProof

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

SEL(s(X), cons(Y, Z)) → A(Z)
SEL(s(X), cons(Y, Z)) → A(X)
A(selInact(x1, x2)) → SEL(x1, x2)
SEL(s(X), cons(Y, Z)) → SEL(a(X), a(Z))
DBL(s(X)) → A(X)

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ QDPOrderProof

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

SEL(s(X), cons(Y, Z)) → A(Z)
SEL(s(X), cons(Y, Z)) → A(X)
A(selInact(x1, x2)) → SEL(x1, x2)
SEL(s(X), cons(Y, Z)) → SEL(a(X), a(Z))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)

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

The following pairs can be oriented strictly and are deleted.

A(selInact(x1, x2)) → SEL(x1, x2)

The remaining pairs can at least be oriented weakly.

SEL(s(X), cons(Y, Z)) → A(Z)
SEL(s(X), cons(Y, Z)) → A(X)
SEL(s(X), cons(Y, Z)) → SEL(a(X), a(Z))

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

POL(0) = 0
POL(A(x₁)) = x₁
POL(SEL(x₁, x₂)) = x₁ + x₂
POL(a(x₁)) = x₁
POL(cons(x₁, x₂)) = max(x₁, x₂)
POL(dbl(x₁)) = 0
POL(dblInact(x₁)) = 0
POL(dbls(x₁)) = 0
POL(dblsInact(x₁)) = 0
POL(from(x₁)) = x₁
POL(fromInact(x₁)) = x₁
POL(indx(x₁, x₂)) = 1 + x₁ + x₂
POL(indxInact(x₁, x₂)) = 1 + x₁ + x₂
POL(nil) = 0
POL(s(x₁)) = x₁
POL(sInact(x₁)) = x₁
POL(sel(x₁, x₂)) = 1 + x₁ + x₂
POL(selInact(x₁, x₂)) = 1 + x₁ + x₂

The following usable rules [17] were oriented:

indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
sel(x1, x2) → selInact(x1, x2)
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
a(selInact(x1, x2)) → sel(x1, x2)
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
a(dblsInact(x1)) → dbls(x1)
dbls(x1) → dblsInact(x1)
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
dbls(nil) → nil
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbl(0) → 0

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ QDPOrderProof

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

SEL(s(X), cons(Y, Z)) → A(Z)
SEL(s(X), cons(Y, Z)) → A(X)
SEL(s(X), cons(Y, Z)) → SEL(a(X), a(Z))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ QDPOrderProof

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

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

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)

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

SEL(s(selInact(x0, x1)), cons(y1, y2)) → SEL(sel(x0, x1), a(y2))
SEL(s(indxInact(x0, x1)), cons(y1, y2)) → SEL(indx(x0, x1), a(y2))
SEL(s(fromInact(x0)), cons(y1, y2)) → SEL(from(x0), a(y2))
SEL(s(sInact(x0)), cons(y1, y2)) → SEL(s(x0), a(y2))
SEL(s(dblsInact(x0)), cons(y1, y2)) → SEL(dbls(x0), a(y2))
SEL(s(x0), cons(y1, y2)) → SEL(x0, a(y2))
SEL(s(dblInact(x0)), cons(y1, y2)) → SEL(dbl(x0), a(y2))

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ QDPOrderProof

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ Narrowing

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

SEL(s(selInact(x0, x1)), cons(y1, y2)) → SEL(sel(x0, x1), a(y2))
SEL(s(fromInact(x0)), cons(y1, y2)) → SEL(from(x0), a(y2))
SEL(s(indxInact(x0, x1)), cons(y1, y2)) → SEL(indx(x0, x1), a(y2))
SEL(s(sInact(x0)), cons(y1, y2)) → SEL(s(x0), a(y2))
SEL(s(dblsInact(x0)), cons(y1, y2)) → SEL(dbls(x0), a(y2))
SEL(s(x0), cons(y1, y2)) → SEL(x0, a(y2))
SEL(s(dblInact(x0)), cons(y1, y2)) → SEL(dbl(x0), a(y2))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)

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

SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(y0, y1), dbl(x0))
SEL(s(selInact(y0, y1)), cons(y2, sInact(x0))) → SEL(sel(y0, y1), s(x0))
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(y0, y1), from(x0))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(y0, y1), dbls(x0))

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ QDPOrderProof

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ Narrowing

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

SEL(s(indxInact(x0, x1)), cons(y1, y2)) → SEL(indx(x0, x1), a(y2))
SEL(s(selInact(y0, y1)), cons(y2, sInact(x0))) → SEL(sel(y0, y1), s(x0))
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
SEL(s(fromInact(x0)), cons(y1, y2)) → SEL(from(x0), a(y2))
SEL(s(sInact(x0)), cons(y1, y2)) → SEL(s(x0), a(y2))
SEL(s(dblsInact(x0)), cons(y1, y2)) → SEL(dbls(x0), a(y2))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(y0, y1), dbl(x0))
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(y0, y1), from(x0))
SEL(s(x0), cons(y1, y2)) → SEL(x0, a(y2))
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(y0, y1), dbls(x0))
SEL(s(dblInact(x0)), cons(y1, y2)) → SEL(dbl(x0), a(y2))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)

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

SEL(s(indxInact(cons(x0, x1), x2)), cons(y2, y3)) → SEL(cons(selInact(a(x0), a(x2)), indxInact(a(x1), a(x2))), a(y3))
SEL(s(indxInact(nil, x0)), cons(y2, y3)) → SEL(nil, a(y3))
SEL(s(indxInact(x0, x1)), cons(y2, y3)) → SEL(indxInact(x0, x1), a(y3))

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ QDPOrderProof

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

SEL(s(indxInact(nil, x0)), cons(y2, y3)) → SEL(nil, a(y3))
SEL(s(selInact(y0, y1)), cons(y2, sInact(x0))) → SEL(sel(y0, y1), s(x0))
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
SEL(s(fromInact(x0)), cons(y1, y2)) → SEL(from(x0), a(y2))
SEL(s(indxInact(cons(x0, x1), x2)), cons(y2, y3)) → SEL(cons(selInact(a(x0), a(x2)), indxInact(a(x1), a(x2))), a(y3))
SEL(s(sInact(x0)), cons(y1, y2)) → SEL(s(x0), a(y2))
SEL(s(dblsInact(x0)), cons(y1, y2)) → SEL(dbls(x0), a(y2))
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(y0, y1), dbl(x0))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
SEL(s(indxInact(x0, x1)), cons(y2, y3)) → SEL(indxInact(x0, x1), a(y3))
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(y0, y1), from(x0))
SEL(s(x0), cons(y1, y2)) → SEL(x0, a(y2))
SEL(s(dblInact(x0)), cons(y1, y2)) → SEL(dbl(x0), a(y2))
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(y0, y1), dbls(x0))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ QDPOrderProof

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

SEL(s(fromInact(x0)), cons(y1, y2)) → SEL(from(x0), a(y2))
SEL(s(sInact(x0)), cons(y1, y2)) → SEL(s(x0), a(y2))
SEL(s(dblsInact(x0)), cons(y1, y2)) → SEL(dbls(x0), a(y2))
SEL(s(selInact(y0, y1)), cons(y2, sInact(x0))) → SEL(sel(y0, y1), s(x0))
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(y0, y1), dbl(x0))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(y0, y1), from(x0))
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))
SEL(s(x0), cons(y1, y2)) → SEL(x0, a(y2))
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(y0, y1), dbls(x0))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
SEL(s(dblInact(x0)), cons(y1, y2)) → SEL(dbl(x0), a(y2))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)

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

SEL(s(fromInact(x0)), cons(y1, y2)) → SEL(fromInact(x0), a(y2))
SEL(s(fromInact(x0)), cons(y1, y2)) → SEL(cons(a(x0), fromInact(sInact(a(x0)))), a(y2))

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ QDPOrderProof

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

SEL(s(fromInact(x0)), cons(y1, y2)) → SEL(fromInact(x0), a(y2))
SEL(s(selInact(y0, y1)), cons(y2, sInact(x0))) → SEL(sel(y0, y1), s(x0))
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
SEL(s(fromInact(x0)), cons(y1, y2)) → SEL(cons(a(x0), fromInact(sInact(a(x0)))), a(y2))
SEL(s(sInact(x0)), cons(y1, y2)) → SEL(s(x0), a(y2))
SEL(s(dblsInact(x0)), cons(y1, y2)) → SEL(dbls(x0), a(y2))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(y0, y1), dbl(x0))
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(y0, y1), from(x0))
SEL(s(x0), cons(y1, y2)) → SEL(x0, a(y2))
SEL(s(dblInact(x0)), cons(y1, y2)) → SEL(dbl(x0), a(y2))
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(y0, y1), dbls(x0))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ QDPOrderProof

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

SEL(s(sInact(x0)), cons(y1, y2)) → SEL(s(x0), a(y2))
SEL(s(dblsInact(x0)), cons(y1, y2)) → SEL(dbls(x0), a(y2))
SEL(s(selInact(y0, y1)), cons(y2, sInact(x0))) → SEL(sel(y0, y1), s(x0))
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(y0, y1), dbl(x0))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(y0, y1), from(x0))
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))
SEL(s(x0), cons(y1, y2)) → SEL(x0, a(y2))
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(y0, y1), dbls(x0))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
SEL(s(dblInact(x0)), cons(y1, y2)) → SEL(dbl(x0), a(y2))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)

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

SEL(s(sInact(y0)), cons(y1, dblInact(x0))) → SEL(s(y0), dbl(x0))
SEL(s(sInact(y0)), cons(y1, fromInact(x0))) → SEL(s(y0), from(x0))
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(x0, x1))
SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(x0))
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(x0, x1))
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)
SEL(s(sInact(y0)), cons(y1, sInact(x0))) → SEL(s(y0), s(x0))

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ QDPOrderProof

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

SEL(s(sInact(y0)), cons(y1, dblInact(x0))) → SEL(s(y0), dbl(x0))
SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(x0))
SEL(s(selInact(y0, y1)), cons(y2, sInact(x0))) → SEL(sel(y0, y1), s(x0))
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))
SEL(s(sInact(y0)), cons(y1, sInact(x0))) → SEL(s(y0), s(x0))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
SEL(s(sInact(y0)), cons(y1, fromInact(x0))) → SEL(s(y0), from(x0))
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(x0, x1))
SEL(s(dblsInact(x0)), cons(y1, y2)) → SEL(dbls(x0), a(y2))
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(y0, y1), dbl(x0))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(x0, x1))
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(y0, y1), from(x0))
SEL(s(x0), cons(y1, y2)) → SEL(x0, a(y2))
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(y0, y1), dbls(x0))
SEL(s(dblInact(x0)), cons(y1, y2)) → SEL(dbl(x0), a(y2))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)

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

SEL(s(dblsInact(x0)), cons(y1, y2)) → SEL(dblsInact(x0), a(y2))
SEL(s(dblsInact(cons(x0, x1))), cons(y1, y2)) → SEL(cons(dblInact(a(x0)), dblsInact(a(x1))), a(y2))
SEL(s(dblsInact(nil)), cons(y1, y2)) → SEL(nil, a(y2))

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ QDPOrderProof

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

SEL(s(sInact(y0)), cons(y1, dblInact(x0))) → SEL(s(y0), dbl(x0))
SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(x0))
SEL(s(selInact(y0, y1)), cons(y2, sInact(x0))) → SEL(sel(y0, y1), s(x0))
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))
SEL(s(dblsInact(nil)), cons(y1, y2)) → SEL(nil, a(y2))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
SEL(s(sInact(y0)), cons(y1, sInact(x0))) → SEL(s(y0), s(x0))
SEL(s(dblsInact(x0)), cons(y1, y2)) → SEL(dblsInact(x0), a(y2))
SEL(s(sInact(y0)), cons(y1, fromInact(x0))) → SEL(s(y0), from(x0))
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(x0, x1))
SEL(s(dblsInact(cons(x0, x1))), cons(y1, y2)) → SEL(cons(dblInact(a(x0)), dblsInact(a(x1))), a(y2))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(y0, y1), dbl(x0))
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(y0, y1), from(x0))
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(x0, x1))
SEL(s(x0), cons(y1, y2)) → SEL(x0, a(y2))
SEL(s(dblInact(x0)), cons(y1, y2)) → SEL(dbl(x0), a(y2))
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(y0, y1), dbls(x0))
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ QDPOrderProof

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

SEL(s(sInact(y0)), cons(y1, dblInact(x0))) → SEL(s(y0), dbl(x0))
SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(x0))
SEL(s(selInact(y0, y1)), cons(y2, sInact(x0))) → SEL(sel(y0, y1), s(x0))
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))
SEL(s(sInact(y0)), cons(y1, sInact(x0))) → SEL(s(y0), s(x0))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
SEL(s(sInact(y0)), cons(y1, fromInact(x0))) → SEL(s(y0), from(x0))
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(x0, x1))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(y0, y1), dbl(x0))
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(x0, x1))
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(y0, y1), from(x0))
SEL(s(x0), cons(y1, y2)) → SEL(x0, a(y2))
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)
SEL(s(dblInact(x0)), cons(y1, y2)) → SEL(dbl(x0), a(y2))
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(y0, y1), dbls(x0))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)

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

SEL(s(y0), cons(y1, indxInact(x0, x1))) → SEL(y0, indx(x0, x1))
SEL(s(y0), cons(y1, selInact(x0, x1))) → SEL(y0, sel(x0, x1))
SEL(s(y0), cons(y1, dblInact(x0))) → SEL(y0, dbl(x0))
SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(y0), cons(y1, sInact(x0))) → SEL(y0, s(x0))
SEL(s(y0), cons(y1, fromInact(x0))) → SEL(y0, from(x0))
SEL(s(y0), cons(y1, dblsInact(x0))) → SEL(y0, dbls(x0))

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ QDPOrderProof

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

SEL(s(sInact(y0)), cons(y1, dblInact(x0))) → SEL(s(y0), dbl(x0))
SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(x0))
SEL(s(y0), cons(y1, sInact(x0))) → SEL(y0, s(x0))
SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(selInact(y0, y1)), cons(y2, sInact(x0))) → SEL(sel(y0, y1), s(x0))
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))
SEL(s(y0), cons(y1, dblsInact(x0))) → SEL(y0, dbls(x0))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
SEL(s(sInact(y0)), cons(y1, sInact(x0))) → SEL(s(y0), s(x0))
SEL(s(y0), cons(y1, indxInact(x0, x1))) → SEL(y0, indx(x0, x1))
SEL(s(sInact(y0)), cons(y1, fromInact(x0))) → SEL(s(y0), from(x0))
SEL(s(y0), cons(y1, selInact(x0, x1))) → SEL(y0, sel(x0, x1))
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(x0, x1))
SEL(s(y0), cons(y1, dblInact(x0))) → SEL(y0, dbl(x0))
SEL(s(y0), cons(y1, fromInact(x0))) → SEL(y0, from(x0))
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(y0, y1), dbl(x0))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(y0, y1), from(x0))
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(x0, x1))
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(y0, y1), dbls(x0))
SEL(s(dblInact(x0)), cons(y1, y2)) → SEL(dbl(x0), a(y2))
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)

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

SEL(s(dblInact(y0)), cons(y1, sInact(x0))) → SEL(dbl(y0), s(x0))
SEL(s(dblInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbl(y0), indx(x0, x1))
SEL(s(dblInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbl(y0), sel(x0, x1))
SEL(s(dblInact(y0)), cons(y1, x0)) → SEL(dbl(y0), x0)
SEL(s(dblInact(y0)), cons(y1, dblInact(x0))) → SEL(dbl(y0), dbl(x0))
SEL(s(dblInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbl(y0), dbls(x0))
SEL(s(dblInact(y0)), cons(y1, fromInact(x0))) → SEL(dbl(y0), from(x0))

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ QDPOrderProof

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

SEL(s(y0), cons(y1, sInact(x0))) → SEL(y0, s(x0))
SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(selInact(y0, y1)), cons(y2, sInact(x0))) → SEL(sel(y0, y1), s(x0))
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))
SEL(s(y0), cons(y1, dblsInact(x0))) → SEL(y0, dbls(x0))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
SEL(s(sInact(y0)), cons(y1, fromInact(x0))) → SEL(s(y0), from(x0))
SEL(s(dblInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbl(y0), indx(x0, x1))
SEL(s(dblInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbl(y0), sel(x0, x1))
SEL(s(y0), cons(y1, dblInact(x0))) → SEL(y0, dbl(x0))
SEL(s(y0), cons(y1, fromInact(x0))) → SEL(y0, from(x0))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(y0, y1), dbls(x0))
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)
SEL(s(sInact(y0)), cons(y1, dblInact(x0))) → SEL(s(y0), dbl(x0))
SEL(s(dblInact(y0)), cons(y1, sInact(x0))) → SEL(dbl(y0), s(x0))
SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(x0))
SEL(s(dblInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbl(y0), dbls(x0))
SEL(s(sInact(y0)), cons(y1, sInact(x0))) → SEL(s(y0), s(x0))
SEL(s(y0), cons(y1, indxInact(x0, x1))) → SEL(y0, indx(x0, x1))
SEL(s(dblInact(y0)), cons(y1, dblInact(x0))) → SEL(dbl(y0), dbl(x0))
SEL(s(dblInact(y0)), cons(y1, x0)) → SEL(dbl(y0), x0)
SEL(s(y0), cons(y1, selInact(x0, x1))) → SEL(y0, sel(x0, x1))
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(x0, x1))
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(y0, y1), dbl(x0))
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(y0, y1), from(x0))
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(x0, x1))
SEL(s(dblInact(y0)), cons(y1, fromInact(x0))) → SEL(dbl(y0), from(x0))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)

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

SEL(s(selInact(y0, y1)), cons(y2, sInact(x0))) → SEL(sel(y0, y1), sInact(x0))

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ QDPOrderProof

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(y0), cons(y1, sInact(x0))) → SEL(y0, s(x0))
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
SEL(s(y0), cons(y1, dblsInact(x0))) → SEL(y0, dbls(x0))
SEL(s(dblInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbl(y0), indx(x0, x1))
SEL(s(sInact(y0)), cons(y1, fromInact(x0))) → SEL(s(y0), from(x0))
SEL(s(dblInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbl(y0), sel(x0, x1))
SEL(s(y0), cons(y1, dblInact(x0))) → SEL(y0, dbl(x0))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
SEL(s(y0), cons(y1, fromInact(x0))) → SEL(y0, from(x0))
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(y0, y1), dbls(x0))
SEL(s(dblInact(y0)), cons(y1, sInact(x0))) → SEL(dbl(y0), s(x0))
SEL(s(sInact(y0)), cons(y1, dblInact(x0))) → SEL(s(y0), dbl(x0))
SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(x0))
SEL(s(dblInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbl(y0), dbls(x0))
SEL(s(sInact(y0)), cons(y1, sInact(x0))) → SEL(s(y0), s(x0))
SEL(s(y0), cons(y1, indxInact(x0, x1))) → SEL(y0, indx(x0, x1))
SEL(s(selInact(y0, y1)), cons(y2, sInact(x0))) → SEL(sel(y0, y1), sInact(x0))
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(x0, x1))
SEL(s(y0), cons(y1, selInact(x0, x1))) → SEL(y0, sel(x0, x1))
SEL(s(dblInact(y0)), cons(y1, x0)) → SEL(dbl(y0), x0)
SEL(s(dblInact(y0)), cons(y1, dblInact(x0))) → SEL(dbl(y0), dbl(x0))
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(y0, y1), dbl(x0))
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(x0, x1))
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(y0, y1), from(x0))
SEL(s(dblInact(y0)), cons(y1, fromInact(x0))) → SEL(dbl(y0), from(x0))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ QDPOrderProof

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(y0), cons(y1, sInact(x0))) → SEL(y0, s(x0))
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
SEL(s(y0), cons(y1, dblsInact(x0))) → SEL(y0, dbls(x0))
SEL(s(sInact(y0)), cons(y1, fromInact(x0))) → SEL(s(y0), from(x0))
SEL(s(dblInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbl(y0), indx(x0, x1))
SEL(s(dblInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbl(y0), sel(x0, x1))
SEL(s(y0), cons(y1, dblInact(x0))) → SEL(y0, dbl(x0))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
SEL(s(y0), cons(y1, fromInact(x0))) → SEL(y0, from(x0))
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(y0, y1), dbls(x0))
SEL(s(sInact(y0)), cons(y1, dblInact(x0))) → SEL(s(y0), dbl(x0))
SEL(s(dblInact(y0)), cons(y1, sInact(x0))) → SEL(dbl(y0), s(x0))
SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(x0))
SEL(s(dblInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbl(y0), dbls(x0))
SEL(s(sInact(y0)), cons(y1, sInact(x0))) → SEL(s(y0), s(x0))
SEL(s(y0), cons(y1, indxInact(x0, x1))) → SEL(y0, indx(x0, x1))
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(x0, x1))
SEL(s(y0), cons(y1, selInact(x0, x1))) → SEL(y0, sel(x0, x1))
SEL(s(dblInact(y0)), cons(y1, x0)) → SEL(dbl(y0), x0)
SEL(s(dblInact(y0)), cons(y1, dblInact(x0))) → SEL(dbl(y0), dbl(x0))
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(y0, y1), dbl(x0))
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(x0, x1))
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(y0, y1), from(x0))
SEL(s(dblInact(y0)), cons(y1, fromInact(x0))) → SEL(dbl(y0), from(x0))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)

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

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ QDPOrderProof

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

SEL(s(y0), cons(y1, sInact(x0))) → SEL(y0, s(x0))
SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))
SEL(s(y0), cons(y1, dblsInact(x0))) → SEL(y0, dbls(x0))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
SEL(s(dblInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbl(y0), indx(x0, x1))
SEL(s(sInact(y0)), cons(y1, fromInact(x0))) → SEL(s(y0), from(x0))
SEL(s(dblInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbl(y0), sel(x0, x1))
SEL(s(y0), cons(y1, dblInact(x0))) → SEL(y0, dbl(x0))
SEL(s(sInact(y0)), cons(y1, sInact(x0))) → SEL(s(y0), sInact(x0))
SEL(s(y0), cons(y1, fromInact(x0))) → SEL(y0, from(x0))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(y0, y1), dbls(x0))
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)
SEL(s(dblInact(y0)), cons(y1, sInact(x0))) → SEL(dbl(y0), s(x0))
SEL(s(sInact(y0)), cons(y1, dblInact(x0))) → SEL(s(y0), dbl(x0))
SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(x0))
SEL(s(dblInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbl(y0), dbls(x0))
SEL(s(y0), cons(y1, indxInact(x0, x1))) → SEL(y0, indx(x0, x1))
SEL(s(dblInact(y0)), cons(y1, dblInact(x0))) → SEL(dbl(y0), dbl(x0))
SEL(s(dblInact(y0)), cons(y1, x0)) → SEL(dbl(y0), x0)
SEL(s(y0), cons(y1, selInact(x0, x1))) → SEL(y0, sel(x0, x1))
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(x0, x1))
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(y0, y1), dbl(x0))
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(y0, y1), from(x0))
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(x0, x1))
SEL(s(dblInact(y0)), cons(y1, fromInact(x0))) → SEL(dbl(y0), from(x0))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ QDPOrderProof

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(y0), cons(y1, sInact(x0))) → SEL(y0, s(x0))
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
SEL(s(y0), cons(y1, dblsInact(x0))) → SEL(y0, dbls(x0))
SEL(s(sInact(y0)), cons(y1, fromInact(x0))) → SEL(s(y0), from(x0))
SEL(s(dblInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbl(y0), indx(x0, x1))
SEL(s(dblInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbl(y0), sel(x0, x1))
SEL(s(y0), cons(y1, dblInact(x0))) → SEL(y0, dbl(x0))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
SEL(s(y0), cons(y1, fromInact(x0))) → SEL(y0, from(x0))
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(y0, y1), dbls(x0))
SEL(s(sInact(y0)), cons(y1, dblInact(x0))) → SEL(s(y0), dbl(x0))
SEL(s(dblInact(y0)), cons(y1, sInact(x0))) → SEL(dbl(y0), s(x0))
SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(x0))
SEL(s(dblInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbl(y0), dbls(x0))
SEL(s(y0), cons(y1, indxInact(x0, x1))) → SEL(y0, indx(x0, x1))
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(x0, x1))
SEL(s(y0), cons(y1, selInact(x0, x1))) → SEL(y0, sel(x0, x1))
SEL(s(dblInact(y0)), cons(y1, x0)) → SEL(dbl(y0), x0)
SEL(s(dblInact(y0)), cons(y1, dblInact(x0))) → SEL(dbl(y0), dbl(x0))
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(y0, y1), dbl(x0))
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(x0, x1))
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(y0, y1), from(x0))
SEL(s(dblInact(y0)), cons(y1, fromInact(x0))) → SEL(dbl(y0), from(x0))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)

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

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ QDPOrderProof

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))
SEL(s(y0), cons(y1, dblsInact(x0))) → SEL(y0, dbls(x0))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
SEL(s(dblInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbl(y0), indx(x0, x1))
SEL(s(sInact(y0)), cons(y1, fromInact(x0))) → SEL(s(y0), from(x0))
SEL(s(dblInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbl(y0), sel(x0, x1))
SEL(s(y0), cons(y1, sInact(x0))) → SEL(y0, sInact(x0))
SEL(s(y0), cons(y1, dblInact(x0))) → SEL(y0, dbl(x0))
SEL(s(y0), cons(y1, fromInact(x0))) → SEL(y0, from(x0))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(y0, y1), dbls(x0))
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)
SEL(s(dblInact(y0)), cons(y1, sInact(x0))) → SEL(dbl(y0), s(x0))
SEL(s(sInact(y0)), cons(y1, dblInact(x0))) → SEL(s(y0), dbl(x0))
SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(x0))
SEL(s(dblInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbl(y0), dbls(x0))
SEL(s(y0), cons(y1, indxInact(x0, x1))) → SEL(y0, indx(x0, x1))
SEL(s(dblInact(y0)), cons(y1, dblInact(x0))) → SEL(dbl(y0), dbl(x0))
SEL(s(dblInact(y0)), cons(y1, x0)) → SEL(dbl(y0), x0)
SEL(s(y0), cons(y1, selInact(x0, x1))) → SEL(y0, sel(x0, x1))
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(x0, x1))
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(y0, y1), dbl(x0))
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(y0, y1), from(x0))
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(x0, x1))
SEL(s(dblInact(y0)), cons(y1, fromInact(x0))) → SEL(dbl(y0), from(x0))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ QDPOrderProof

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Narrowing

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
SEL(s(y0), cons(y1, dblsInact(x0))) → SEL(y0, dbls(x0))
SEL(s(sInact(y0)), cons(y1, fromInact(x0))) → SEL(s(y0), from(x0))
SEL(s(dblInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbl(y0), indx(x0, x1))
SEL(s(dblInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbl(y0), sel(x0, x1))
SEL(s(y0), cons(y1, dblInact(x0))) → SEL(y0, dbl(x0))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
SEL(s(y0), cons(y1, fromInact(x0))) → SEL(y0, from(x0))
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(y0, y1), dbls(x0))
SEL(s(sInact(y0)), cons(y1, dblInact(x0))) → SEL(s(y0), dbl(x0))
SEL(s(dblInact(y0)), cons(y1, sInact(x0))) → SEL(dbl(y0), s(x0))
SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(x0))
SEL(s(dblInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbl(y0), dbls(x0))
SEL(s(y0), cons(y1, indxInact(x0, x1))) → SEL(y0, indx(x0, x1))
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(x0, x1))
SEL(s(y0), cons(y1, selInact(x0, x1))) → SEL(y0, sel(x0, x1))
SEL(s(dblInact(y0)), cons(y1, x0)) → SEL(dbl(y0), x0)
SEL(s(dblInact(y0)), cons(y1, dblInact(x0))) → SEL(dbl(y0), dbl(x0))
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(y0, y1), dbl(x0))
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(x0, x1))
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(y0, y1), from(x0))
SEL(s(dblInact(y0)), cons(y1, fromInact(x0))) → SEL(dbl(y0), from(x0))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)

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

SEL(s(dblInact(y0)), cons(y1, sInact(x0))) → SEL(dbl(y0), sInact(x0))

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ QDPOrderProof

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Narrowing

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ DependencyGraphProof

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))
SEL(s(y0), cons(y1, dblsInact(x0))) → SEL(y0, dbls(x0))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
SEL(s(dblInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbl(y0), indx(x0, x1))
SEL(s(sInact(y0)), cons(y1, fromInact(x0))) → SEL(s(y0), from(x0))
SEL(s(dblInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbl(y0), sel(x0, x1))
SEL(s(y0), cons(y1, dblInact(x0))) → SEL(y0, dbl(x0))
SEL(s(dblInact(y0)), cons(y1, sInact(x0))) → SEL(dbl(y0), sInact(x0))
SEL(s(y0), cons(y1, fromInact(x0))) → SEL(y0, from(x0))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(y0, y1), dbls(x0))
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)
SEL(s(sInact(y0)), cons(y1, dblInact(x0))) → SEL(s(y0), dbl(x0))
SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(x0))
SEL(s(dblInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbl(y0), dbls(x0))
SEL(s(y0), cons(y1, indxInact(x0, x1))) → SEL(y0, indx(x0, x1))
SEL(s(dblInact(y0)), cons(y1, dblInact(x0))) → SEL(dbl(y0), dbl(x0))
SEL(s(dblInact(y0)), cons(y1, x0)) → SEL(dbl(y0), x0)
SEL(s(y0), cons(y1, selInact(x0, x1))) → SEL(y0, sel(x0, x1))
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(x0, x1))
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(y0, y1), dbl(x0))
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(y0, y1), from(x0))
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(x0, x1))
SEL(s(dblInact(y0)), cons(y1, fromInact(x0))) → SEL(dbl(y0), from(x0))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ DependencyGraphProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ QDPOrderProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ QDPOrderProof

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ QDPOrderProof

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ Narrowing

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ Narrowing

                                                                                        ↳ QDP

                                                                                          ↳ DependencyGraphProof

                                                                                            ↳ QDP

                                                                                              ↳ Narrowing

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ Narrowing

                                                                                                        ↳ QDP

                                                                                                          ↳ DependencyGraphProof

                                                                                                            ↳ QDP

                                                                                                              ↳ Narrowing

                                                                                                                ↳ QDP

                                                                                                                  ↳ DependencyGraphProof

                                                                                                                    ↳ QDP

                                                                                                                      ↳ Narrowing

                                                                                                                        ↳ QDP

                                                                                                                          ↳ DependencyGraphProof

                                                                                                                            ↳ QDP

                                                                                                                              ↳ Narrowing

                                                                                                                                ↳ QDP

                                                                                                                                  ↳ DependencyGraphProof

                                                                                                                                    ↳ QDP

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

SEL(s(sInact(y0)), cons(y1, dblInact(x0))) → SEL(s(y0), dbl(x0))
SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(x0))
SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(dblInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbl(y0), dbls(x0))
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))
SEL(s(y0), cons(y1, dblsInact(x0))) → SEL(y0, dbls(x0))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
SEL(s(y0), cons(y1, indxInact(x0, x1))) → SEL(y0, indx(x0, x1))
SEL(s(dblInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbl(y0), indx(x0, x1))
SEL(s(sInact(y0)), cons(y1, fromInact(x0))) → SEL(s(y0), from(x0))
SEL(s(dblInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbl(y0), sel(x0, x1))
SEL(s(dblInact(y0)), cons(y1, dblInact(x0))) → SEL(dbl(y0), dbl(x0))
SEL(s(dblInact(y0)), cons(y1, x0)) → SEL(dbl(y0), x0)
SEL(s(y0), cons(y1, selInact(x0, x1))) → SEL(y0, sel(x0, x1))
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(x0, x1))
SEL(s(y0), cons(y1, dblInact(x0))) → SEL(y0, dbl(x0))
SEL(s(y0), cons(y1, fromInact(x0))) → SEL(y0, from(x0))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(y0, y1), dbl(x0))
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(y0, y1), from(x0))
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(x0, x1))
SEL(s(dblInact(y0)), cons(y1, fromInact(x0))) → SEL(dbl(y0), from(x0))
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(y0, y1), dbls(x0))
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We applied the Innermost Giesl Middeldorp transformation [10] to transform the context-sensitive TRS to a usual TRS.

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
mark(dbl(x1)) → active(dbl(mark(x1)))
dbl(active(x1)) → dbl(x1)
dbl(mark(x1)) → dbl(x1)
mark(0) → active(0)
mark(s(x1)) → active(s(x1))
mark(dbls(x1)) → active(dbls(mark(x1)))
dbls(active(x1)) → dbls(x1)
dbls(mark(x1)) → dbls(x1)
mark(nil) → active(nil)
mark(cons(x1, x2)) → active(cons(x1, x2))
mark(sel(x1, x2)) → active(sel(mark(x1), mark(x2)))
sel(active(x1), x2) → sel(x1, x2)
sel(mark(x1), x2) → sel(x1, x2)
sel(x1, active(x2)) → sel(x1, x2)
sel(x1, mark(x2)) → sel(x1, x2)
mark(indx(x1, x2)) → active(indx(mark(x1), x2))
indx(active(x1), x2) → indx(x1, x2)
indx(mark(x1), x2) → indx(x1, x2)
mark(from(x1)) → active(from(x1))

The set Q consists of the following terms:

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

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

ACTIVE(dbls(cons(X, Y))) → DBLS(Y)
SEL(mark(x1), x2) → SEL(x1, x2)
ACTIVE(dbls(cons(X, Y))) → MARK(cons(dbl(X), dbls(Y)))
MARK(indx(x1, x2)) → MARK(x1)
ACTIVE(dbl(0)) → MARK(0)
SEL(x1, active(x2)) → SEL(x1, x2)
ACTIVE(indx(cons(X, Y), Z)) → SEL(X, Z)
DBL(active(x1)) → DBL(x1)
MARK(dbls(x1)) → MARK(x1)
ACTIVE(indx(cons(X, Y), Z)) → MARK(cons(sel(X, Z), indx(Y, Z)))
DBL(mark(x1)) → DBL(x1)
MARK(dbls(x1)) → DBLS(mark(x1))
ACTIVE(sel(s(X), cons(Y, Z))) → MARK(sel(X, Z))
ACTIVE(indx(cons(X, Y), Z)) → INDX(Y, Z)
MARK(indx(x1, x2)) → INDX(mark(x1), x2)
SEL(x1, mark(x2)) → SEL(x1, x2)
SEL(active(x1), x2) → SEL(x1, x2)
ACTIVE(sel(0, cons(X, Y))) → MARK(X)
ACTIVE(from(X)) → MARK(cons(X, from(s(X))))
MARK(dbl(x1)) → DBL(mark(x1))
MARK(dbls(x1)) → ACTIVE(dbls(mark(x1)))
DBLS(mark(x1)) → DBLS(x1)
MARK(sel(x1, x2)) → MARK(x1)
DBLS(active(x1)) → DBLS(x1)
ACTIVE(dbls(nil)) → MARK(nil)
MARK(dbl(x1)) → MARK(x1)
MARK(s(x1)) → ACTIVE(s(x1))
INDX(active(x1), x2) → INDX(x1, x2)
INDX(mark(x1), x2) → INDX(x1, x2)
MARK(sel(x1, x2)) → ACTIVE(sel(mark(x1), mark(x2)))
MARK(dbl(x1)) → ACTIVE(dbl(mark(x1)))
MARK(sel(x1, x2)) → SEL(mark(x1), mark(x2))
ACTIVE(dbl(s(X))) → DBL(X)
MARK(0) → ACTIVE(0)
ACTIVE(dbl(s(X))) → MARK(s(s(dbl(X))))
MARK(from(x1)) → ACTIVE(from(x1))
MARK(indx(x1, x2)) → ACTIVE(indx(mark(x1), x2))
ACTIVE(indx(nil, X)) → MARK(nil)
MARK(cons(x1, x2)) → ACTIVE(cons(x1, x2))
ACTIVE(sel(s(X), cons(Y, Z))) → SEL(X, Z)
MARK(nil) → ACTIVE(nil)
MARK(sel(x1, x2)) → MARK(x2)
ACTIVE(dbls(cons(X, Y))) → DBL(X)

The TRS R consists of the following rules:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
mark(dbl(x1)) → active(dbl(mark(x1)))
dbl(active(x1)) → dbl(x1)
dbl(mark(x1)) → dbl(x1)
mark(0) → active(0)
mark(s(x1)) → active(s(x1))
mark(dbls(x1)) → active(dbls(mark(x1)))
dbls(active(x1)) → dbls(x1)
dbls(mark(x1)) → dbls(x1)
mark(nil) → active(nil)
mark(cons(x1, x2)) → active(cons(x1, x2))
mark(sel(x1, x2)) → active(sel(mark(x1), mark(x2)))
sel(active(x1), x2) → sel(x1, x2)
sel(mark(x1), x2) → sel(x1, x2)
sel(x1, active(x2)) → sel(x1, x2)
sel(x1, mark(x2)) → sel(x1, x2)
mark(indx(x1, x2)) → active(indx(mark(x1), x2))
indx(active(x1), x2) → indx(x1, x2)
indx(mark(x1), x2) → indx(x1, x2)
mark(from(x1)) → active(from(x1))

The set Q consists of the following terms:

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

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

ACTIVE(dbls(cons(X, Y))) → DBLS(Y)
SEL(mark(x1), x2) → SEL(x1, x2)
ACTIVE(dbls(cons(X, Y))) → MARK(cons(dbl(X), dbls(Y)))
MARK(indx(x1, x2)) → MARK(x1)
ACTIVE(dbl(0)) → MARK(0)
SEL(x1, active(x2)) → SEL(x1, x2)
ACTIVE(indx(cons(X, Y), Z)) → SEL(X, Z)
DBL(active(x1)) → DBL(x1)
MARK(dbls(x1)) → MARK(x1)
ACTIVE(indx(cons(X, Y), Z)) → MARK(cons(sel(X, Z), indx(Y, Z)))
DBL(mark(x1)) → DBL(x1)
MARK(dbls(x1)) → DBLS(mark(x1))
ACTIVE(sel(s(X), cons(Y, Z))) → MARK(sel(X, Z))
ACTIVE(indx(cons(X, Y), Z)) → INDX(Y, Z)
MARK(indx(x1, x2)) → INDX(mark(x1), x2)
SEL(x1, mark(x2)) → SEL(x1, x2)
SEL(active(x1), x2) → SEL(x1, x2)
ACTIVE(sel(0, cons(X, Y))) → MARK(X)
ACTIVE(from(X)) → MARK(cons(X, from(s(X))))
MARK(dbl(x1)) → DBL(mark(x1))
MARK(dbls(x1)) → ACTIVE(dbls(mark(x1)))
DBLS(mark(x1)) → DBLS(x1)
MARK(sel(x1, x2)) → MARK(x1)
DBLS(active(x1)) → DBLS(x1)
ACTIVE(dbls(nil)) → MARK(nil)
MARK(dbl(x1)) → MARK(x1)
MARK(s(x1)) → ACTIVE(s(x1))
INDX(active(x1), x2) → INDX(x1, x2)
INDX(mark(x1), x2) → INDX(x1, x2)
MARK(sel(x1, x2)) → ACTIVE(sel(mark(x1), mark(x2)))
MARK(dbl(x1)) → ACTIVE(dbl(mark(x1)))
MARK(sel(x1, x2)) → SEL(mark(x1), mark(x2))
ACTIVE(dbl(s(X))) → DBL(X)
MARK(0) → ACTIVE(0)
ACTIVE(dbl(s(X))) → MARK(s(s(dbl(X))))
MARK(from(x1)) → ACTIVE(from(x1))
MARK(indx(x1, x2)) → ACTIVE(indx(mark(x1), x2))
ACTIVE(indx(nil, X)) → MARK(nil)
MARK(cons(x1, x2)) → ACTIVE(cons(x1, x2))
ACTIVE(sel(s(X), cons(Y, Z))) → SEL(X, Z)
MARK(nil) → ACTIVE(nil)
MARK(sel(x1, x2)) → MARK(x2)
ACTIVE(dbls(cons(X, Y))) → DBL(X)

The TRS R consists of the following rules:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
mark(dbl(x1)) → active(dbl(mark(x1)))
dbl(active(x1)) → dbl(x1)
dbl(mark(x1)) → dbl(x1)
mark(0) → active(0)
mark(s(x1)) → active(s(x1))
mark(dbls(x1)) → active(dbls(mark(x1)))
dbls(active(x1)) → dbls(x1)
dbls(mark(x1)) → dbls(x1)
mark(nil) → active(nil)
mark(cons(x1, x2)) → active(cons(x1, x2))
mark(sel(x1, x2)) → active(sel(mark(x1), mark(x2)))
sel(active(x1), x2) → sel(x1, x2)
sel(mark(x1), x2) → sel(x1, x2)
sel(x1, active(x2)) → sel(x1, x2)
sel(x1, mark(x2)) → sel(x1, x2)
mark(indx(x1, x2)) → active(indx(mark(x1), x2))
indx(active(x1), x2) → indx(x1, x2)
indx(mark(x1), x2) → indx(x1, x2)
mark(from(x1)) → active(from(x1))

The set Q consists of the following terms:

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

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ UsableRulesProof

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

INDX(active(x1), x2) → INDX(x1, x2)
INDX(mark(x1), x2) → INDX(x1, x2)

The TRS R consists of the following rules:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
mark(dbl(x1)) → active(dbl(mark(x1)))
dbl(active(x1)) → dbl(x1)
dbl(mark(x1)) → dbl(x1)
mark(0) → active(0)
mark(s(x1)) → active(s(x1))
mark(dbls(x1)) → active(dbls(mark(x1)))
dbls(active(x1)) → dbls(x1)
dbls(mark(x1)) → dbls(x1)
mark(nil) → active(nil)
mark(cons(x1, x2)) → active(cons(x1, x2))
mark(sel(x1, x2)) → active(sel(mark(x1), mark(x2)))
sel(active(x1), x2) → sel(x1, x2)
sel(mark(x1), x2) → sel(x1, x2)
sel(x1, active(x2)) → sel(x1, x2)
sel(x1, mark(x2)) → sel(x1, x2)
mark(indx(x1, x2)) → active(indx(mark(x1), x2))
indx(active(x1), x2) → indx(x1, x2)
indx(mark(x1), x2) → indx(x1, x2)
mark(from(x1)) → active(from(x1))

The set Q consists of the following terms:

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

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

INDX(active(x1), x2) → INDX(x1, x2)
INDX(mark(x1), x2) → INDX(x1, x2)

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

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

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

dbl(active(x0))
dbl(mark(x0))
dbls(active(x0))
dbls(mark(x0))
sel(active(x0), x1)
sel(mark(x0), x1)
sel(x0, active(x1))
sel(x0, mark(x1))
indx(active(x0), x1)
indx(mark(x0), x1)

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

                          ↳ QDP

                            ↳ QDPSizeChangeProof

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

INDX(active(x1), x2) → INDX(x1, x2)
INDX(mark(x1), x2) → INDX(x1, x2)

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

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

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:

INDX(active(x1), x2) → INDX(x1, x2)
The graph contains the following edges 1 > 1, 2 >= 2
INDX(mark(x1), x2) → INDX(x1, x2)
The graph contains the following edges 1 > 1, 2 >= 2

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

SEL(mark(x1), x2) → SEL(x1, x2)
SEL(x1, active(x2)) → SEL(x1, x2)
SEL(x1, mark(x2)) → SEL(x1, x2)
SEL(active(x1), x2) → SEL(x1, x2)

The TRS R consists of the following rules:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
mark(dbl(x1)) → active(dbl(mark(x1)))
dbl(active(x1)) → dbl(x1)
dbl(mark(x1)) → dbl(x1)
mark(0) → active(0)
mark(s(x1)) → active(s(x1))
mark(dbls(x1)) → active(dbls(mark(x1)))
dbls(active(x1)) → dbls(x1)
dbls(mark(x1)) → dbls(x1)
mark(nil) → active(nil)
mark(cons(x1, x2)) → active(cons(x1, x2))
mark(sel(x1, x2)) → active(sel(mark(x1), mark(x2)))
sel(active(x1), x2) → sel(x1, x2)
sel(mark(x1), x2) → sel(x1, x2)
sel(x1, active(x2)) → sel(x1, x2)
sel(x1, mark(x2)) → sel(x1, x2)
mark(indx(x1, x2)) → active(indx(mark(x1), x2))
indx(active(x1), x2) → indx(x1, x2)
indx(mark(x1), x2) → indx(x1, x2)
mark(from(x1)) → active(from(x1))

The set Q consists of the following terms:

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

SEL(mark(x1), x2) → SEL(x1, x2)
SEL(x1, active(x2)) → SEL(x1, x2)
SEL(x1, mark(x2)) → SEL(x1, x2)
SEL(active(x1), x2) → SEL(x1, x2)

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

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

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

dbl(active(x0))
dbl(mark(x0))
dbls(active(x0))
dbls(mark(x0))
sel(active(x0), x1)
sel(mark(x0), x1)
sel(x0, active(x1))
sel(x0, mark(x1))
indx(active(x0), x1)
indx(mark(x0), x1)

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

                          ↳ QDP

                            ↳ QDPSizeChangeProof

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

SEL(mark(x1), x2) → SEL(x1, x2)
SEL(x1, active(x2)) → SEL(x1, x2)
SEL(x1, mark(x2)) → SEL(x1, x2)
SEL(active(x1), x2) → SEL(x1, x2)

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

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

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

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

SEL(mark(x1), x2) → SEL(x1, x2)
The graph contains the following edges 1 > 1, 2 >= 2
SEL(x1, active(x2)) → SEL(x1, x2)
The graph contains the following edges 1 >= 1, 2 > 2
SEL(x1, mark(x2)) → SEL(x1, x2)
The graph contains the following edges 1 >= 1, 2 > 2
SEL(active(x1), x2) → SEL(x1, x2)
The graph contains the following edges 1 > 1, 2 >= 2

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                  ↳ QDP

                  ↳ QDP

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

DBLS(mark(x1)) → DBLS(x1)
DBLS(active(x1)) → DBLS(x1)

The TRS R consists of the following rules:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
mark(dbl(x1)) → active(dbl(mark(x1)))
dbl(active(x1)) → dbl(x1)
dbl(mark(x1)) → dbl(x1)
mark(0) → active(0)
mark(s(x1)) → active(s(x1))
mark(dbls(x1)) → active(dbls(mark(x1)))
dbls(active(x1)) → dbls(x1)
dbls(mark(x1)) → dbls(x1)
mark(nil) → active(nil)
mark(cons(x1, x2)) → active(cons(x1, x2))
mark(sel(x1, x2)) → active(sel(mark(x1), mark(x2)))
sel(active(x1), x2) → sel(x1, x2)
sel(mark(x1), x2) → sel(x1, x2)
sel(x1, active(x2)) → sel(x1, x2)
sel(x1, mark(x2)) → sel(x1, x2)
mark(indx(x1, x2)) → active(indx(mark(x1), x2))
indx(active(x1), x2) → indx(x1, x2)
indx(mark(x1), x2) → indx(x1, x2)
mark(from(x1)) → active(from(x1))

The set Q consists of the following terms:

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

                  ↳ QDP

                  ↳ QDP

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

DBLS(mark(x1)) → DBLS(x1)
DBLS(active(x1)) → DBLS(x1)

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

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

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

dbl(active(x0))
dbl(mark(x0))
dbls(active(x0))
dbls(mark(x0))
sel(active(x0), x1)
sel(mark(x0), x1)
sel(x0, active(x1))
sel(x0, mark(x1))
indx(active(x0), x1)
indx(mark(x0), x1)

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

                          ↳ QDP

                            ↳ QDPSizeChangeProof

                  ↳ QDP

                  ↳ QDP

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

DBLS(mark(x1)) → DBLS(x1)
DBLS(active(x1)) → DBLS(x1)

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

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

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:

DBLS(mark(x1)) → DBLS(x1)
The graph contains the following edges 1 > 1
DBLS(active(x1)) → DBLS(x1)
The graph contains the following edges 1 > 1

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                  ↳ QDP

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

DBL(mark(x1)) → DBL(x1)
DBL(active(x1)) → DBL(x1)

The TRS R consists of the following rules:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
mark(dbl(x1)) → active(dbl(mark(x1)))
dbl(active(x1)) → dbl(x1)
dbl(mark(x1)) → dbl(x1)
mark(0) → active(0)
mark(s(x1)) → active(s(x1))
mark(dbls(x1)) → active(dbls(mark(x1)))
dbls(active(x1)) → dbls(x1)
dbls(mark(x1)) → dbls(x1)
mark(nil) → active(nil)
mark(cons(x1, x2)) → active(cons(x1, x2))
mark(sel(x1, x2)) → active(sel(mark(x1), mark(x2)))
sel(active(x1), x2) → sel(x1, x2)
sel(mark(x1), x2) → sel(x1, x2)
sel(x1, active(x2)) → sel(x1, x2)
sel(x1, mark(x2)) → sel(x1, x2)
mark(indx(x1, x2)) → active(indx(mark(x1), x2))
indx(active(x1), x2) → indx(x1, x2)
indx(mark(x1), x2) → indx(x1, x2)
mark(from(x1)) → active(from(x1))

The set Q consists of the following terms:

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

                  ↳ QDP

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

DBL(mark(x1)) → DBL(x1)
DBL(active(x1)) → DBL(x1)

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

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

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

dbl(active(x0))
dbl(mark(x0))
dbls(active(x0))
dbls(mark(x0))
sel(active(x0), x1)
sel(mark(x0), x1)
sel(x0, active(x1))
sel(x0, mark(x1))
indx(active(x0), x1)
indx(mark(x0), x1)

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ UsableRulesProof

                      ↳ QDP

                        ↳ QReductionProof

                          ↳ QDP

                            ↳ QDPSizeChangeProof

                  ↳ QDP

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

DBL(mark(x1)) → DBL(x1)
DBL(active(x1)) → DBL(x1)

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

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

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:

DBL(mark(x1)) → DBL(x1)
The graph contains the following edges 1 > 1
DBL(active(x1)) → DBL(x1)
The graph contains the following edges 1 > 1

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPOrderProof

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

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

The TRS R consists of the following rules:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
mark(dbl(x1)) → active(dbl(mark(x1)))
dbl(active(x1)) → dbl(x1)
dbl(mark(x1)) → dbl(x1)
mark(0) → active(0)
mark(s(x1)) → active(s(x1))
mark(dbls(x1)) → active(dbls(mark(x1)))
dbls(active(x1)) → dbls(x1)
dbls(mark(x1)) → dbls(x1)
mark(nil) → active(nil)
mark(cons(x1, x2)) → active(cons(x1, x2))
mark(sel(x1, x2)) → active(sel(mark(x1), mark(x2)))
sel(active(x1), x2) → sel(x1, x2)
sel(mark(x1), x2) → sel(x1, x2)
sel(x1, active(x2)) → sel(x1, x2)
sel(x1, mark(x2)) → sel(x1, x2)
mark(indx(x1, x2)) → active(indx(mark(x1), x2))
indx(active(x1), x2) → indx(x1, x2)
indx(mark(x1), x2) → indx(x1, x2)
mark(from(x1)) → active(from(x1))

The set Q consists of the following terms:

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

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

The following pairs can be oriented strictly and are deleted.

MARK(dbl(x1)) → ACTIVE(dbl(mark(x1)))
MARK(indx(x1, x2)) → ACTIVE(indx(mark(x1), x2))
MARK(dbls(x1)) → ACTIVE(dbls(mark(x1)))

The remaining pairs can at least be oriented weakly.

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

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = 1
POL(active(x₁)) = 0
POL(cons(x₁, x₂)) = 0
POL(dbl(x₁)) = 0
POL(dbls(x₁)) = 0
POL(from(x₁)) = 0
POL(indx(x₁, x₂)) = 0
POL(mark(x₁)) = 0
POL(nil) = 0
POL(s(x₁)) = 0
POL(sel(x₁, x₂)) = 1

The following usable rules [17] were oriented:

dbls(mark(x1)) → dbls(x1)
dbls(active(x1)) → dbls(x1)
sel(x1, mark(x2)) → sel(x1, x2)
sel(mark(x1), x2) → sel(x1, x2)
sel(x1, active(x2)) → sel(x1, x2)
sel(active(x1), x2) → sel(x1, x2)
dbl(mark(x1)) → dbl(x1)
dbl(active(x1)) → dbl(x1)
indx(active(x1), x2) → indx(x1, x2)
indx(mark(x1), x2) → indx(x1, x2)

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

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

The TRS R consists of the following rules:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
mark(dbl(x1)) → active(dbl(mark(x1)))
dbl(active(x1)) → dbl(x1)
dbl(mark(x1)) → dbl(x1)
mark(0) → active(0)
mark(s(x1)) → active(s(x1))
mark(dbls(x1)) → active(dbls(mark(x1)))
dbls(active(x1)) → dbls(x1)
dbls(mark(x1)) → dbls(x1)
mark(nil) → active(nil)
mark(cons(x1, x2)) → active(cons(x1, x2))
mark(sel(x1, x2)) → active(sel(mark(x1), mark(x2)))
sel(active(x1), x2) → sel(x1, x2)
sel(mark(x1), x2) → sel(x1, x2)
sel(x1, active(x2)) → sel(x1, x2)
sel(x1, mark(x2)) → sel(x1, x2)
mark(indx(x1, x2)) → active(indx(mark(x1), x2))
indx(active(x1), x2) → indx(x1, x2)
indx(mark(x1), x2) → indx(x1, x2)
mark(from(x1)) → active(from(x1))

The set Q consists of the following terms:

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

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

The following pairs can be oriented strictly and are deleted.

ACTIVE(sel(0, cons(X, Y))) → MARK(X)

The remaining pairs can at least be oriented weakly.

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

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

POL(0) = 1
POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(active(x₁)) = x₁
POL(cons(x₁, x₂)) = max(x₁, x₂)
POL(dbl(x₁)) = x₁
POL(dbls(x₁)) = x₁
POL(from(x₁)) = x₁
POL(indx(x₁, x₂)) = x₁ + x₂
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(sel(x₁, x₂)) = x₁ + x₂

The following usable rules [17] were oriented:

mark(cons(x1, x2)) → active(cons(x1, x2))
mark(nil) → active(nil)
dbls(mark(x1)) → dbls(x1)
dbls(active(x1)) → dbls(x1)
sel(x1, mark(x2)) → sel(x1, x2)
sel(mark(x1), x2) → sel(x1, x2)
sel(x1, active(x2)) → sel(x1, x2)
sel(active(x1), x2) → sel(x1, x2)
dbl(mark(x1)) → dbl(x1)
dbl(active(x1)) → dbl(x1)
active(from(X)) → mark(cons(X, from(s(X))))
mark(s(x1)) → active(s(x1))
mark(0) → active(0)
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
mark(indx(x1, x2)) → active(indx(mark(x1), x2))
mark(sel(x1, x2)) → active(sel(mark(x1), mark(x2)))
active(sel(0, cons(X, Y))) → mark(X)
mark(dbl(x1)) → active(dbl(mark(x1)))
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
mark(dbls(x1)) → active(dbls(mark(x1)))
indx(active(x1), x2) → indx(x1, x2)
indx(mark(x1), x2) → indx(x1, x2)
mark(from(x1)) → active(from(x1))

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

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

The TRS R consists of the following rules:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
mark(dbl(x1)) → active(dbl(mark(x1)))
dbl(active(x1)) → dbl(x1)
dbl(mark(x1)) → dbl(x1)
mark(0) → active(0)
mark(s(x1)) → active(s(x1))
mark(dbls(x1)) → active(dbls(mark(x1)))
dbls(active(x1)) → dbls(x1)
dbls(mark(x1)) → dbls(x1)
mark(nil) → active(nil)
mark(cons(x1, x2)) → active(cons(x1, x2))
mark(sel(x1, x2)) → active(sel(mark(x1), mark(x2)))
sel(active(x1), x2) → sel(x1, x2)
sel(mark(x1), x2) → sel(x1, x2)
sel(x1, active(x2)) → sel(x1, x2)
sel(x1, mark(x2)) → sel(x1, x2)
mark(indx(x1, x2)) → active(indx(mark(x1), x2))
indx(active(x1), x2) → indx(x1, x2)
indx(mark(x1), x2) → indx(x1, x2)
mark(from(x1)) → active(from(x1))

The set Q consists of the following terms:

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

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

The following pairs can be oriented strictly and are deleted.

MARK(dbls(x1)) → MARK(x1)

The remaining pairs can at least be oriented weakly.

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

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

POL(0) = 0
POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(active(x₁)) = x₁
POL(cons(x₁, x₂)) = max(x₁, x₂)
POL(dbl(x₁)) = x₁
POL(dbls(x₁)) = 1 + x₁
POL(from(x₁)) = 1 + x₁
POL(indx(x₁, x₂)) = x₁ + x₂
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(sel(x₁, x₂)) = x₁ + x₂

The following usable rules [17] were oriented:

mark(cons(x1, x2)) → active(cons(x1, x2))
mark(nil) → active(nil)
dbls(mark(x1)) → dbls(x1)
dbls(active(x1)) → dbls(x1)
sel(x1, mark(x2)) → sel(x1, x2)
sel(mark(x1), x2) → sel(x1, x2)
sel(x1, active(x2)) → sel(x1, x2)
sel(active(x1), x2) → sel(x1, x2)
dbl(mark(x1)) → dbl(x1)
dbl(active(x1)) → dbl(x1)
active(from(X)) → mark(cons(X, from(s(X))))
mark(s(x1)) → active(s(x1))
mark(0) → active(0)
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
mark(indx(x1, x2)) → active(indx(mark(x1), x2))
mark(sel(x1, x2)) → active(sel(mark(x1), mark(x2)))
active(sel(0, cons(X, Y))) → mark(X)
mark(dbl(x1)) → active(dbl(mark(x1)))
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
mark(dbls(x1)) → active(dbls(mark(x1)))
indx(active(x1), x2) → indx(x1, x2)
indx(mark(x1), x2) → indx(x1, x2)
mark(from(x1)) → active(from(x1))

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

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

The TRS R consists of the following rules:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
mark(dbl(x1)) → active(dbl(mark(x1)))
dbl(active(x1)) → dbl(x1)
dbl(mark(x1)) → dbl(x1)
mark(0) → active(0)
mark(s(x1)) → active(s(x1))
mark(dbls(x1)) → active(dbls(mark(x1)))
dbls(active(x1)) → dbls(x1)
dbls(mark(x1)) → dbls(x1)
mark(nil) → active(nil)
mark(cons(x1, x2)) → active(cons(x1, x2))
mark(sel(x1, x2)) → active(sel(mark(x1), mark(x2)))
sel(active(x1), x2) → sel(x1, x2)
sel(mark(x1), x2) → sel(x1, x2)
sel(x1, active(x2)) → sel(x1, x2)
sel(x1, mark(x2)) → sel(x1, x2)
mark(indx(x1, x2)) → active(indx(mark(x1), x2))
indx(active(x1), x2) → indx(x1, x2)
indx(mark(x1), x2) → indx(x1, x2)
mark(from(x1)) → active(from(x1))

The set Q consists of the following terms:

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

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

The following pairs can be oriented strictly and are deleted.

MARK(sel(x1, x2)) → MARK(x1)
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel(x1, x2)) → MARK(x2)

The remaining pairs can at least be oriented weakly.

ACTIVE(sel(s(X), cons(Y, Z))) → MARK(sel(X, Z))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(x1, x2)) → ACTIVE(sel(mark(x1), mark(x2)))

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

POL(0) = 0
POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(active(x₁)) = x₁
POL(cons(x₁, x₂)) = max(x₁, x₂)
POL(dbl(x₁)) = x₁
POL(dbls(x₁)) = x₁
POL(from(x₁)) = x₁
POL(indx(x₁, x₂)) = 1 + x₁ + x₂
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(sel(x₁, x₂)) = 1 + x₁ + x₂

The following usable rules [17] were oriented:

mark(cons(x1, x2)) → active(cons(x1, x2))
mark(nil) → active(nil)
dbls(mark(x1)) → dbls(x1)
dbls(active(x1)) → dbls(x1)
sel(x1, mark(x2)) → sel(x1, x2)
sel(mark(x1), x2) → sel(x1, x2)
sel(x1, active(x2)) → sel(x1, x2)
sel(active(x1), x2) → sel(x1, x2)
dbl(mark(x1)) → dbl(x1)
dbl(active(x1)) → dbl(x1)
active(from(X)) → mark(cons(X, from(s(X))))
mark(s(x1)) → active(s(x1))
mark(0) → active(0)
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
mark(indx(x1, x2)) → active(indx(mark(x1), x2))
mark(sel(x1, x2)) → active(sel(mark(x1), mark(x2)))
active(sel(0, cons(X, Y))) → mark(X)
mark(dbl(x1)) → active(dbl(mark(x1)))
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
mark(dbls(x1)) → active(dbls(mark(x1)))
indx(active(x1), x2) → indx(x1, x2)
indx(mark(x1), x2) → indx(x1, x2)
mark(from(x1)) → active(from(x1))

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

ACTIVE(sel(s(X), cons(Y, Z))) → MARK(sel(X, Z))
MARK(dbl(x1)) → MARK(x1)
MARK(sel(x1, x2)) → ACTIVE(sel(mark(x1), mark(x2)))

The TRS R consists of the following rules:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
mark(dbl(x1)) → active(dbl(mark(x1)))
dbl(active(x1)) → dbl(x1)
dbl(mark(x1)) → dbl(x1)
mark(0) → active(0)
mark(s(x1)) → active(s(x1))
mark(dbls(x1)) → active(dbls(mark(x1)))
dbls(active(x1)) → dbls(x1)
dbls(mark(x1)) → dbls(x1)
mark(nil) → active(nil)
mark(cons(x1, x2)) → active(cons(x1, x2))
mark(sel(x1, x2)) → active(sel(mark(x1), mark(x2)))
sel(active(x1), x2) → sel(x1, x2)
sel(mark(x1), x2) → sel(x1, x2)
sel(x1, active(x2)) → sel(x1, x2)
sel(x1, mark(x2)) → sel(x1, x2)
mark(indx(x1, x2)) → active(indx(mark(x1), x2))
indx(active(x1), x2) → indx(x1, x2)
indx(mark(x1), x2) → indx(x1, x2)
mark(from(x1)) → active(from(x1))

The set Q consists of the following terms:

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

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                          ↳ Narrowing

                                        ↳ QDP

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

ACTIVE(sel(s(X), cons(Y, Z))) → MARK(sel(X, Z))
MARK(sel(x1, x2)) → ACTIVE(sel(mark(x1), mark(x2)))

The TRS R consists of the following rules:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
mark(dbl(x1)) → active(dbl(mark(x1)))
dbl(active(x1)) → dbl(x1)
dbl(mark(x1)) → dbl(x1)
mark(0) → active(0)
mark(s(x1)) → active(s(x1))
mark(dbls(x1)) → active(dbls(mark(x1)))
dbls(active(x1)) → dbls(x1)
dbls(mark(x1)) → dbls(x1)
mark(nil) → active(nil)
mark(cons(x1, x2)) → active(cons(x1, x2))
mark(sel(x1, x2)) → active(sel(mark(x1), mark(x2)))
sel(active(x1), x2) → sel(x1, x2)
sel(mark(x1), x2) → sel(x1, x2)
sel(x1, active(x2)) → sel(x1, x2)
sel(x1, mark(x2)) → sel(x1, x2)
mark(indx(x1, x2)) → active(indx(mark(x1), x2))
indx(active(x1), x2) → indx(x1, x2)
indx(mark(x1), x2) → indx(x1, x2)
mark(from(x1)) → active(from(x1))

The set Q consists of the following terms:

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

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

MARK(sel(y0, cons(x0, x1))) → ACTIVE(sel(mark(y0), active(cons(x0, x1))))
MARK(sel(sel(x0, x1), y1)) → ACTIVE(sel(active(sel(mark(x0), mark(x1))), mark(y1)))
MARK(sel(dbl(x0), y1)) → ACTIVE(sel(active(dbl(mark(x0))), mark(y1)))
MARK(sel(cons(x0, x1), y1)) → ACTIVE(sel(active(cons(x0, x1)), mark(y1)))
MARK(sel(y0, dbls(x0))) → ACTIVE(sel(mark(y0), active(dbls(mark(x0)))))
MARK(sel(from(x0), y1)) → ACTIVE(sel(active(from(x0)), mark(y1)))
MARK(sel(indx(x0, x1), y1)) → ACTIVE(sel(active(indx(mark(x0), x1)), mark(y1)))
MARK(sel(y0, s(x0))) → ACTIVE(sel(mark(y0), active(s(x0))))
MARK(sel(y0, dbl(x0))) → ACTIVE(sel(mark(y0), active(dbl(mark(x0)))))
MARK(sel(y0, x1)) → ACTIVE(sel(mark(y0), x1))
MARK(sel(s(x0), y1)) → ACTIVE(sel(active(s(x0)), mark(y1)))
MARK(sel(x0, y1)) → ACTIVE(sel(x0, mark(y1)))
MARK(sel(y0, indx(x0, x1))) → ACTIVE(sel(mark(y0), active(indx(mark(x0), x1))))
MARK(sel(y0, from(x0))) → ACTIVE(sel(mark(y0), active(from(x0))))
MARK(sel(y0, sel(x0, x1))) → ACTIVE(sel(mark(y0), active(sel(mark(x0), mark(x1)))))
MARK(sel(y0, nil)) → ACTIVE(sel(mark(y0), active(nil)))
MARK(sel(nil, y1)) → ACTIVE(sel(active(nil), mark(y1)))
MARK(sel(0, y1)) → ACTIVE(sel(active(0), mark(y1)))
MARK(sel(y0, 0)) → ACTIVE(sel(mark(y0), active(0)))
MARK(sel(dbls(x0), y1)) → ACTIVE(sel(active(dbls(mark(x0))), mark(y1)))

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Rewriting

                                        ↳ QDP

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

MARK(sel(y0, cons(x0, x1))) → ACTIVE(sel(mark(y0), active(cons(x0, x1))))
MARK(sel(sel(x0, x1), y1)) → ACTIVE(sel(active(sel(mark(x0), mark(x1))), mark(y1)))
MARK(sel(dbl(x0), y1)) → ACTIVE(sel(active(dbl(mark(x0))), mark(y1)))
MARK(sel(cons(x0, x1), y1)) → ACTIVE(sel(active(cons(x0, x1)), mark(y1)))
MARK(sel(y0, dbls(x0))) → ACTIVE(sel(mark(y0), active(dbls(mark(x0)))))
MARK(sel(from(x0), y1)) → ACTIVE(sel(active(from(x0)), mark(y1)))
MARK(sel(indx(x0, x1), y1)) → ACTIVE(sel(active(indx(mark(x0), x1)), mark(y1)))
MARK(sel(y0, s(x0))) → ACTIVE(sel(mark(y0), active(s(x0))))
MARK(sel(y0, dbl(x0))) → ACTIVE(sel(mark(y0), active(dbl(mark(x0)))))
MARK(sel(y0, x1)) → ACTIVE(sel(mark(y0), x1))
MARK(sel(x0, y1)) → ACTIVE(sel(x0, mark(y1)))
MARK(sel(s(x0), y1)) → ACTIVE(sel(active(s(x0)), mark(y1)))
MARK(sel(y0, indx(x0, x1))) → ACTIVE(sel(mark(y0), active(indx(mark(x0), x1))))
ACTIVE(sel(s(X), cons(Y, Z))) → MARK(sel(X, Z))
MARK(sel(y0, from(x0))) → ACTIVE(sel(mark(y0), active(from(x0))))
MARK(sel(y0, sel(x0, x1))) → ACTIVE(sel(mark(y0), active(sel(mark(x0), mark(x1)))))
MARK(sel(y0, 0)) → ACTIVE(sel(mark(y0), active(0)))
MARK(sel(0, y1)) → ACTIVE(sel(active(0), mark(y1)))
MARK(sel(nil, y1)) → ACTIVE(sel(active(nil), mark(y1)))
MARK(sel(y0, nil)) → ACTIVE(sel(mark(y0), active(nil)))
MARK(sel(dbls(x0), y1)) → ACTIVE(sel(active(dbls(mark(x0))), mark(y1)))

The TRS R consists of the following rules:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
mark(dbl(x1)) → active(dbl(mark(x1)))
dbl(active(x1)) → dbl(x1)
dbl(mark(x1)) → dbl(x1)
mark(0) → active(0)
mark(s(x1)) → active(s(x1))
mark(dbls(x1)) → active(dbls(mark(x1)))
dbls(active(x1)) → dbls(x1)
dbls(mark(x1)) → dbls(x1)
mark(nil) → active(nil)
mark(cons(x1, x2)) → active(cons(x1, x2))
mark(sel(x1, x2)) → active(sel(mark(x1), mark(x2)))
sel(active(x1), x2) → sel(x1, x2)
sel(mark(x1), x2) → sel(x1, x2)
sel(x1, active(x2)) → sel(x1, x2)
sel(x1, mark(x2)) → sel(x1, x2)
mark(indx(x1, x2)) → active(indx(mark(x1), x2))
indx(active(x1), x2) → indx(x1, x2)
indx(mark(x1), x2) → indx(x1, x2)
mark(from(x1)) → active(from(x1))

The set Q consists of the following terms:

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

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

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Rewriting

                                                ↳ QDP

                                                  ↳ Rewriting

                                        ↳ QDP

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

MARK(sel(y0, cons(x0, x1))) → ACTIVE(sel(mark(y0), active(cons(x0, x1))))
MARK(sel(sel(x0, x1), y1)) → ACTIVE(sel(active(sel(mark(x0), mark(x1))), mark(y1)))
MARK(sel(from(x0), y1)) → ACTIVE(sel(mark(cons(x0, from(s(x0)))), mark(y1)))
MARK(sel(dbl(x0), y1)) → ACTIVE(sel(active(dbl(mark(x0))), mark(y1)))
MARK(sel(cons(x0, x1), y1)) → ACTIVE(sel(active(cons(x0, x1)), mark(y1)))
MARK(sel(y0, dbls(x0))) → ACTIVE(sel(mark(y0), active(dbls(mark(x0)))))
MARK(sel(indx(x0, x1), y1)) → ACTIVE(sel(active(indx(mark(x0), x1)), mark(y1)))
MARK(sel(y0, s(x0))) → ACTIVE(sel(mark(y0), active(s(x0))))
MARK(sel(y0, dbl(x0))) → ACTIVE(sel(mark(y0), active(dbl(mark(x0)))))
MARK(sel(y0, x1)) → ACTIVE(sel(mark(y0), x1))
MARK(sel(s(x0), y1)) → ACTIVE(sel(active(s(x0)), mark(y1)))
MARK(sel(x0, y1)) → ACTIVE(sel(x0, mark(y1)))
ACTIVE(sel(s(X), cons(Y, Z))) → MARK(sel(X, Z))
MARK(sel(y0, indx(x0, x1))) → ACTIVE(sel(mark(y0), active(indx(mark(x0), x1))))
MARK(sel(y0, from(x0))) → ACTIVE(sel(mark(y0), active(from(x0))))
MARK(sel(y0, sel(x0, x1))) → ACTIVE(sel(mark(y0), active(sel(mark(x0), mark(x1)))))
MARK(sel(y0, nil)) → ACTIVE(sel(mark(y0), active(nil)))
MARK(sel(nil, y1)) → ACTIVE(sel(active(nil), mark(y1)))
MARK(sel(0, y1)) → ACTIVE(sel(active(0), mark(y1)))
MARK(sel(y0, 0)) → ACTIVE(sel(mark(y0), active(0)))
MARK(sel(dbls(x0), y1)) → ACTIVE(sel(active(dbls(mark(x0))), mark(y1)))

The TRS R consists of the following rules:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
mark(dbl(x1)) → active(dbl(mark(x1)))
dbl(active(x1)) → dbl(x1)
dbl(mark(x1)) → dbl(x1)
mark(0) → active(0)
mark(s(x1)) → active(s(x1))
mark(dbls(x1)) → active(dbls(mark(x1)))
dbls(active(x1)) → dbls(x1)
dbls(mark(x1)) → dbls(x1)
mark(nil) → active(nil)
mark(cons(x1, x2)) → active(cons(x1, x2))
mark(sel(x1, x2)) → active(sel(mark(x1), mark(x2)))
sel(active(x1), x2) → sel(x1, x2)
sel(mark(x1), x2) → sel(x1, x2)
sel(x1, active(x2)) → sel(x1, x2)
sel(x1, mark(x2)) → sel(x1, x2)
mark(indx(x1, x2)) → active(indx(mark(x1), x2))
indx(active(x1), x2) → indx(x1, x2)
indx(mark(x1), x2) → indx(x1, x2)
mark(from(x1)) → active(from(x1))

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule MARK(sel(y0, from(x0))) → ACTIVE(sel(mark(y0), active(from(x0)))) at position [0,1] we obtained the following new rules:

MARK(sel(y0, from(x0))) → ACTIVE(sel(mark(y0), mark(cons(x0, from(s(x0))))))

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Rewriting

                                                ↳ QDP

                                                  ↳ Rewriting

                                                    ↳ QDP

                                                      ↳ Rewriting

                                        ↳ QDP

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

MARK(sel(y0, cons(x0, x1))) → ACTIVE(sel(mark(y0), active(cons(x0, x1))))
MARK(sel(sel(x0, x1), y1)) → ACTIVE(sel(active(sel(mark(x0), mark(x1))), mark(y1)))
MARK(sel(dbl(x0), y1)) → ACTIVE(sel(active(dbl(mark(x0))), mark(y1)))
MARK(sel(from(x0), y1)) → ACTIVE(sel(mark(cons(x0, from(s(x0)))), mark(y1)))
MARK(sel(cons(x0, x1), y1)) → ACTIVE(sel(active(cons(x0, x1)), mark(y1)))
MARK(sel(y0, dbls(x0))) → ACTIVE(sel(mark(y0), active(dbls(mark(x0)))))
MARK(sel(indx(x0, x1), y1)) → ACTIVE(sel(active(indx(mark(x0), x1)), mark(y1)))
MARK(sel(y0, s(x0))) → ACTIVE(sel(mark(y0), active(s(x0))))
MARK(sel(y0, dbl(x0))) → ACTIVE(sel(mark(y0), active(dbl(mark(x0)))))
MARK(sel(y0, x1)) → ACTIVE(sel(mark(y0), x1))
MARK(sel(y0, from(x0))) → ACTIVE(sel(mark(y0), mark(cons(x0, from(s(x0))))))
MARK(sel(x0, y1)) → ACTIVE(sel(x0, mark(y1)))
MARK(sel(s(x0), y1)) → ACTIVE(sel(active(s(x0)), mark(y1)))
MARK(sel(y0, indx(x0, x1))) → ACTIVE(sel(mark(y0), active(indx(mark(x0), x1))))
ACTIVE(sel(s(X), cons(Y, Z))) → MARK(sel(X, Z))
MARK(sel(y0, sel(x0, x1))) → ACTIVE(sel(mark(y0), active(sel(mark(x0), mark(x1)))))
MARK(sel(y0, 0)) → ACTIVE(sel(mark(y0), active(0)))
MARK(sel(0, y1)) → ACTIVE(sel(active(0), mark(y1)))
MARK(sel(nil, y1)) → ACTIVE(sel(active(nil), mark(y1)))
MARK(sel(y0, nil)) → ACTIVE(sel(mark(y0), active(nil)))
MARK(sel(dbls(x0), y1)) → ACTIVE(sel(active(dbls(mark(x0))), mark(y1)))

The TRS R consists of the following rules:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
mark(dbl(x1)) → active(dbl(mark(x1)))
dbl(active(x1)) → dbl(x1)
dbl(mark(x1)) → dbl(x1)
mark(0) → active(0)
mark(s(x1)) → active(s(x1))
mark(dbls(x1)) → active(dbls(mark(x1)))
dbls(active(x1)) → dbls(x1)
dbls(mark(x1)) → dbls(x1)
mark(nil) → active(nil)
mark(cons(x1, x2)) → active(cons(x1, x2))
mark(sel(x1, x2)) → active(sel(mark(x1), mark(x2)))
sel(active(x1), x2) → sel(x1, x2)
sel(mark(x1), x2) → sel(x1, x2)
sel(x1, active(x2)) → sel(x1, x2)
sel(x1, mark(x2)) → sel(x1, x2)
mark(indx(x1, x2)) → active(indx(mark(x1), x2))
indx(active(x1), x2) → indx(x1, x2)
indx(mark(x1), x2) → indx(x1, x2)
mark(from(x1)) → active(from(x1))

The set Q consists of the following terms:

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

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

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Rewriting

                                                ↳ QDP

                                                  ↳ Rewriting

                                                    ↳ QDP

                                                      ↳ Rewriting

                                                        ↳ QDP

                                                          ↳ Rewriting

                                        ↳ QDP

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

MARK(sel(y0, cons(x0, x1))) → ACTIVE(sel(mark(y0), active(cons(x0, x1))))
MARK(sel(sel(x0, x1), y1)) → ACTIVE(sel(active(sel(mark(x0), mark(x1))), mark(y1)))
MARK(sel(dbl(x0), y1)) → ACTIVE(sel(active(dbl(mark(x0))), mark(y1)))
MARK(sel(cons(x0, x1), y1)) → ACTIVE(sel(active(cons(x0, x1)), mark(y1)))
MARK(sel(y0, dbls(x0))) → ACTIVE(sel(mark(y0), active(dbls(mark(x0)))))
MARK(sel(indx(x0, x1), y1)) → ACTIVE(sel(active(indx(mark(x0), x1)), mark(y1)))
MARK(sel(y0, s(x0))) → ACTIVE(sel(mark(y0), active(s(x0))))
MARK(sel(y0, dbl(x0))) → ACTIVE(sel(mark(y0), active(dbl(mark(x0)))))
MARK(sel(y0, x1)) → ACTIVE(sel(mark(y0), x1))
MARK(sel(s(x0), y1)) → ACTIVE(sel(active(s(x0)), mark(y1)))
MARK(sel(x0, y1)) → ACTIVE(sel(x0, mark(y1)))
MARK(sel(y0, from(x0))) → ACTIVE(sel(mark(y0), mark(cons(x0, from(s(x0))))))
ACTIVE(sel(s(X), cons(Y, Z))) → MARK(sel(X, Z))
MARK(sel(y0, indx(x0, x1))) → ACTIVE(sel(mark(y0), active(indx(mark(x0), x1))))
MARK(sel(from(x0), y1)) → ACTIVE(sel(active(cons(x0, from(s(x0)))), mark(y1)))
MARK(sel(y0, sel(x0, x1))) → ACTIVE(sel(mark(y0), active(sel(mark(x0), mark(x1)))))
MARK(sel(y0, nil)) → ACTIVE(sel(mark(y0), active(nil)))
MARK(sel(nil, y1)) → ACTIVE(sel(active(nil), mark(y1)))
MARK(sel(0, y1)) → ACTIVE(sel(active(0), mark(y1)))
MARK(sel(y0, 0)) → ACTIVE(sel(mark(y0), active(0)))
MARK(sel(dbls(x0), y1)) → ACTIVE(sel(active(dbls(mark(x0))), mark(y1)))

The TRS R consists of the following rules:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
mark(dbl(x1)) → active(dbl(mark(x1)))
dbl(active(x1)) → dbl(x1)
dbl(mark(x1)) → dbl(x1)
mark(0) → active(0)
mark(s(x1)) → active(s(x1))
mark(dbls(x1)) → active(dbls(mark(x1)))
dbls(active(x1)) → dbls(x1)
dbls(mark(x1)) → dbls(x1)
mark(nil) → active(nil)
mark(cons(x1, x2)) → active(cons(x1, x2))
mark(sel(x1, x2)) → active(sel(mark(x1), mark(x2)))
sel(active(x1), x2) → sel(x1, x2)
sel(mark(x1), x2) → sel(x1, x2)
sel(x1, active(x2)) → sel(x1, x2)
sel(x1, mark(x2)) → sel(x1, x2)
mark(indx(x1, x2)) → active(indx(mark(x1), x2))
indx(active(x1), x2) → indx(x1, x2)
indx(mark(x1), x2) → indx(x1, x2)
mark(from(x1)) → active(from(x1))

The set Q consists of the following terms:

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

We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule MARK(sel(y0, from(x0))) → ACTIVE(sel(mark(y0), mark(cons(x0, from(s(x0)))))) at position [0,1] we obtained the following new rules:

MARK(sel(y0, from(x0))) → ACTIVE(sel(mark(y0), active(cons(x0, from(s(x0))))))

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ Rewriting

                                                ↳ QDP

                                                  ↳ Rewriting

                                                    ↳ QDP

                                                      ↳ Rewriting

                                                        ↳ QDP

                                                          ↳ Rewriting

                                                            ↳ QDP

                                        ↳ QDP

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

MARK(sel(y0, cons(x0, x1))) → ACTIVE(sel(mark(y0), active(cons(x0, x1))))
MARK(sel(sel(x0, x1), y1)) → ACTIVE(sel(active(sel(mark(x0), mark(x1))), mark(y1)))
MARK(sel(dbl(x0), y1)) → ACTIVE(sel(active(dbl(mark(x0))), mark(y1)))
MARK(sel(cons(x0, x1), y1)) → ACTIVE(sel(active(cons(x0, x1)), mark(y1)))
MARK(sel(y0, dbls(x0))) → ACTIVE(sel(mark(y0), active(dbls(mark(x0)))))
MARK(sel(indx(x0, x1), y1)) → ACTIVE(sel(active(indx(mark(x0), x1)), mark(y1)))
MARK(sel(y0, from(x0))) → ACTIVE(sel(mark(y0), active(cons(x0, from(s(x0))))))
MARK(sel(y0, s(x0))) → ACTIVE(sel(mark(y0), active(s(x0))))
MARK(sel(y0, dbl(x0))) → ACTIVE(sel(mark(y0), active(dbl(mark(x0)))))
MARK(sel(y0, x1)) → ACTIVE(sel(mark(y0), x1))
MARK(sel(x0, y1)) → ACTIVE(sel(x0, mark(y1)))
MARK(sel(s(x0), y1)) → ACTIVE(sel(active(s(x0)), mark(y1)))
MARK(sel(y0, indx(x0, x1))) → ACTIVE(sel(mark(y0), active(indx(mark(x0), x1))))
ACTIVE(sel(s(X), cons(Y, Z))) → MARK(sel(X, Z))
MARK(sel(from(x0), y1)) → ACTIVE(sel(active(cons(x0, from(s(x0)))), mark(y1)))
MARK(sel(y0, sel(x0, x1))) → ACTIVE(sel(mark(y0), active(sel(mark(x0), mark(x1)))))
MARK(sel(y0, 0)) → ACTIVE(sel(mark(y0), active(0)))
MARK(sel(0, y1)) → ACTIVE(sel(active(0), mark(y1)))
MARK(sel(nil, y1)) → ACTIVE(sel(active(nil), mark(y1)))
MARK(sel(y0, nil)) → ACTIVE(sel(mark(y0), active(nil)))
MARK(sel(dbls(x0), y1)) → ACTIVE(sel(active(dbls(mark(x0))), mark(y1)))

The TRS R consists of the following rules:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
mark(dbl(x1)) → active(dbl(mark(x1)))
dbl(active(x1)) → dbl(x1)
dbl(mark(x1)) → dbl(x1)
mark(0) → active(0)
mark(s(x1)) → active(s(x1))
mark(dbls(x1)) → active(dbls(mark(x1)))
dbls(active(x1)) → dbls(x1)
dbls(mark(x1)) → dbls(x1)
mark(nil) → active(nil)
mark(cons(x1, x2)) → active(cons(x1, x2))
mark(sel(x1, x2)) → active(sel(mark(x1), mark(x2)))
sel(active(x1), x2) → sel(x1, x2)
sel(mark(x1), x2) → sel(x1, x2)
sel(x1, active(x2)) → sel(x1, x2)
sel(x1, mark(x2)) → sel(x1, x2)
mark(indx(x1, x2)) → active(indx(mark(x1), x2))
indx(active(x1), x2) → indx(x1, x2)
indx(mark(x1), x2) → indx(x1, x2)
mark(from(x1)) → active(from(x1))

The set Q consists of the following terms:

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

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ AND

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                  ↳ QDP

                    ↳ QDPOrderProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ DependencyGraphProof

                                      ↳ AND

                                        ↳ QDP

                                        ↳ QDP

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

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

MARK(dbl(x1)) → MARK(x1)

The TRS R consists of the following rules:

active(dbl(0)) → mark(0)
active(dbl(s(X))) → mark(s(s(dbl(X))))
active(dbls(nil)) → mark(nil)
active(dbls(cons(X, Y))) → mark(cons(dbl(X), dbls(Y)))
active(sel(0, cons(X, Y))) → mark(X)
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(indx(nil, X)) → mark(nil)
active(indx(cons(X, Y), Z)) → mark(cons(sel(X, Z), indx(Y, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
mark(dbl(x1)) → active(dbl(mark(x1)))
dbl(active(x1)) → dbl(x1)
dbl(mark(x1)) → dbl(x1)
mark(0) → active(0)
mark(s(x1)) → active(s(x1))
mark(dbls(x1)) → active(dbls(mark(x1)))
dbls(active(x1)) → dbls(x1)
dbls(mark(x1)) → dbls(x1)
mark(nil) → active(nil)
mark(cons(x1, x2)) → active(cons(x1, x2))
mark(sel(x1, x2)) → active(sel(mark(x1), mark(x2)))
sel(active(x1), x2) → sel(x1, x2)
sel(mark(x1), x2) → sel(x1, x2)
sel(x1, active(x2)) → sel(x1, x2)
sel(x1, mark(x2)) → sel(x1, x2)
mark(indx(x1, x2)) → active(indx(mark(x1), x2))
indx(active(x1), x2) → indx(x1, x2)
indx(mark(x1), x2) → indx(x1, x2)
mark(from(x1)) → active(from(x1))

The set Q consists of the following terms:

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

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

        ↳ QTRS

          ↳ Overlay + Local Confluence

      ↳ Improved Ferreira Ribeiro-Transformation

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

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

Q is empty.

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

        ↳ QTRS

          ↳ Overlay + Local Confluence

            ↳ QTRS

              ↳ DependencyPairsProof

      ↳ Improved Ferreira Ribeiro-Transformation

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

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

The set Q consists of the following terms:

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

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

        ↳ QTRS

          ↳ Overlay + Local Confluence

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

      ↳ Improved Ferreira Ribeiro-Transformation

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

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

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

        ↳ QTRS

          ↳ Overlay + Local Confluence

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                        ↳ UsableRulesProof

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

      ↳ Improved Ferreira Ribeiro-Transformation

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

FROM(X) → FROM(s(X))

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

        ↳ QTRS

          ↳ Overlay + Local Confluence

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                        ↳ UsableRulesProof

                          ↳ QDP

                            ↳ QReductionProof

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

      ↳ Improved Ferreira Ribeiro-Transformation

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

FROM(X) → FROM(s(X))

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

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

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

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

        ↳ QTRS

          ↳ Overlay + Local Confluence

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                        ↳ UsableRulesProof

                          ↳ QDP

                            ↳ QReductionProof

                              ↳ QDP

                                ↳ Instantiation

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

      ↳ Improved Ferreira Ribeiro-Transformation

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.
By instantiating [15] the rule FROM(X) → FROM(s(X)) we obtained the following new rules:

FROM(s(z0)) → FROM(s(s(z0)))

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

        ↳ QTRS

          ↳ Overlay + Local Confluence

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                        ↳ UsableRulesProof

                          ↳ QDP

                            ↳ QReductionProof

                              ↳ QDP

                                ↳ Instantiation

                                  ↳ QDP

                                    ↳ Instantiation

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

      ↳ Improved Ferreira Ribeiro-Transformation

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

FROM(s(z0)) → FROM(s(s(z0)))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule FROM(s(z0)) → FROM(s(s(z0))) we obtained the following new rules:

FROM(s(s(z0))) → FROM(s(s(s(z0))))

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

        ↳ QTRS

          ↳ Overlay + Local Confluence

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                        ↳ UsableRulesProof

                          ↳ QDP

                            ↳ QReductionProof

                              ↳ QDP

                                ↳ Instantiation

                                  ↳ QDP

                                    ↳ Instantiation

                                      ↳ QDP

                                        ↳ NonTerminationProof

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

      ↳ Improved Ferreira Ribeiro-Transformation

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

FROM(s(s(z0))) → FROM(s(s(s(z0))))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We used the non-termination processor [17] to show that the DP problem is infinite.
Found a loop by semiunifying a rule from P directly.

The TRS P consists of the following rules:

FROM(s(s(z0))) → FROM(s(s(s(z0))))

The TRS R consists of the following rules:none

s = FROM(s(s(z0))) evaluates to t =FROM(s(s(s(z0))))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [z0 / s(z0)]

Rewriting sequence

The DP semiunifies directly so there is only one rewrite step from FROM(s(s(z0))) to FROM(s(s(s(z0)))).

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

        ↳ QTRS

          ↳ Overlay + Local Confluence

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ UsableRulesProof

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

      ↳ Improved Ferreira Ribeiro-Transformation

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

SEL(s(X), cons(Y, Z)) → SEL(X, Z)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

        ↳ QTRS

          ↳ Overlay + Local Confluence

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ UsableRulesProof

                          ↳ QDP

                            ↳ QReductionProof

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

      ↳ Improved Ferreira Ribeiro-Transformation

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

SEL(s(X), cons(Y, Z)) → SEL(X, Z)

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

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

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

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

        ↳ QTRS

          ↳ Overlay + Local Confluence

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                        ↳ UsableRulesProof

                          ↳ QDP

                            ↳ QReductionProof

                              ↳ QDP

                                ↳ QDPSizeChangeProof

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

      ↳ Improved Ferreira Ribeiro-Transformation

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

SEL(s(X), cons(Y, Z)) → SEL(X, Z)

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

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

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

        ↳ QTRS

          ↳ Overlay + Local Confluence

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ UsableRulesProof

                      ↳ QDP

                      ↳ QDP

      ↳ Improved Ferreira Ribeiro-Transformation

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

INDX(cons(X, Y), Z) → INDX(Y, Z)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

        ↳ QTRS

          ↳ Overlay + Local Confluence

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ UsableRulesProof

                          ↳ QDP

                            ↳ QReductionProof

                      ↳ QDP

                      ↳ QDP

      ↳ Improved Ferreira Ribeiro-Transformation

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

INDX(cons(X, Y), Z) → INDX(Y, Z)

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

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

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

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

        ↳ QTRS

          ↳ Overlay + Local Confluence

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ UsableRulesProof

                          ↳ QDP

                            ↳ QReductionProof

                              ↳ QDP

                                ↳ QDPSizeChangeProof

                      ↳ QDP

                      ↳ QDP

      ↳ Improved Ferreira Ribeiro-Transformation

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

INDX(cons(X, Y), Z) → INDX(Y, Z)

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

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

        ↳ QTRS

          ↳ Overlay + Local Confluence

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ UsableRulesProof

                      ↳ QDP

      ↳ Improved Ferreira Ribeiro-Transformation

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

DBL(s(X)) → DBL(X)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

        ↳ QTRS

          ↳ Overlay + Local Confluence

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ UsableRulesProof

                          ↳ QDP

                            ↳ QReductionProof

                      ↳ QDP

      ↳ Improved Ferreira Ribeiro-Transformation

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

DBL(s(X)) → DBL(X)

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

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

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

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

        ↳ QTRS

          ↳ Overlay + Local Confluence

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ UsableRulesProof

                          ↳ QDP

                            ↳ QReductionProof

                              ↳ QDP

                                ↳ QDPSizeChangeProof

                      ↳ QDP

      ↳ Improved Ferreira Ribeiro-Transformation

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

DBL(s(X)) → DBL(X)

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

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

        ↳ QTRS

          ↳ Overlay + Local Confluence

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ UsableRulesProof

      ↳ Improved Ferreira Ribeiro-Transformation

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

DBLS(cons(X, Y)) → DBLS(Y)

The TRS R consists of the following rules:

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

The set Q consists of the following terms:

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

        ↳ QTRS

          ↳ Overlay + Local Confluence

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ UsableRulesProof

                          ↳ QDP

                            ↳ QReductionProof

      ↳ Improved Ferreira Ribeiro-Transformation

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

DBLS(cons(X, Y)) → DBLS(Y)

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

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

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

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

        ↳ QTRS

          ↳ Overlay + Local Confluence

            ↳ QTRS

              ↳ DependencyPairsProof

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ AND

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                      ↳ QDP

                        ↳ UsableRulesProof

                          ↳ QDP

                            ↳ QReductionProof

                              ↳ QDP

                                ↳ QDPSizeChangeProof

      ↳ Improved Ferreira Ribeiro-Transformation

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

We applied the Improved Ferreira Ribeiro transformation [5,11] to transform the context-sensitive TRS to a usual TRS.

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

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

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(a(x1), a(x2))
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(a(x1))

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:

A(fromInact(x1)) → FROM(x1)
DBL(s(X)) → S(sInact(dblInact(a(X))))
DBLS(cons(X, Y)) → A(Y)
INDX(cons(X, Y), Z) → A(Z)
A(indxInact(x1, x2)) → A(x1)
A(dblInact(x1)) → A(x1)
A(selInact(x1, x2)) → SEL(a(x1), a(x2))
SEL(s(X), cons(Y, Z)) → A(X)
A(dblsInact(x1)) → DBLS(a(x1))
INDX(cons(X, Y), Z) → A(X)
DBL(s(X)) → A(X)
SEL(0, cons(X, Y)) → A(X)
FROM(X) → A(X)
A(indxInact(x1, x2)) → INDX(a(x1), x2)
DBLS(cons(X, Y)) → A(X)
A(dblsInact(x1)) → A(x1)
A(dblInact(x1)) → DBL(a(x1))
SEL(s(X), cons(Y, Z)) → A(Z)
A(selInact(x1, x2)) → A(x2)
INDX(cons(X, Y), Z) → A(Y)
A(selInact(x1, x2)) → A(x1)
A(sInact(x1)) → S(x1)
SEL(s(X), cons(Y, Z)) → SEL(a(X), a(Z))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(a(x1), a(x2))
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(a(x1))

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

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

A(fromInact(x1)) → FROM(x1)
DBL(s(X)) → S(sInact(dblInact(a(X))))
DBLS(cons(X, Y)) → A(Y)
INDX(cons(X, Y), Z) → A(Z)
A(indxInact(x1, x2)) → A(x1)
A(dblInact(x1)) → A(x1)
A(selInact(x1, x2)) → SEL(a(x1), a(x2))
SEL(s(X), cons(Y, Z)) → A(X)
A(dblsInact(x1)) → DBLS(a(x1))
INDX(cons(X, Y), Z) → A(X)
DBL(s(X)) → A(X)
SEL(0, cons(X, Y)) → A(X)
FROM(X) → A(X)
A(indxInact(x1, x2)) → INDX(a(x1), x2)
DBLS(cons(X, Y)) → A(X)
A(dblsInact(x1)) → A(x1)
A(dblInact(x1)) → DBL(a(x1))
SEL(s(X), cons(Y, Z)) → A(Z)
A(selInact(x1, x2)) → A(x2)
INDX(cons(X, Y), Z) → A(Y)
A(selInact(x1, x2)) → A(x1)
A(sInact(x1)) → S(x1)
SEL(s(X), cons(Y, Z)) → SEL(a(X), a(Z))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(a(x1), a(x2))
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(a(x1))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

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

A(fromInact(x1)) → FROM(x1)
INDX(cons(X, Y), Z) → A(Z)
DBLS(cons(X, Y)) → A(Y)
A(indxInact(x1, x2)) → A(x1)
SEL(s(X), cons(Y, Z)) → A(X)
A(selInact(x1, x2)) → SEL(a(x1), a(x2))
A(dblInact(x1)) → A(x1)
A(dblsInact(x1)) → DBLS(a(x1))
INDX(cons(X, Y), Z) → A(X)
DBL(s(X)) → A(X)
A(indxInact(x1, x2)) → INDX(a(x1), x2)
SEL(0, cons(X, Y)) → A(X)
FROM(X) → A(X)
DBLS(cons(X, Y)) → A(X)
A(dblsInact(x1)) → A(x1)
A(dblInact(x1)) → DBL(a(x1))
SEL(s(X), cons(Y, Z)) → A(Z)
INDX(cons(X, Y), Z) → A(Y)
A(selInact(x1, x2)) → A(x2)
A(selInact(x1, x2)) → A(x1)
SEL(s(X), cons(Y, Z)) → SEL(a(X), a(Z))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(a(x1), a(x2))
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(a(x1))

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

The following pairs can be oriented strictly and are deleted.

SEL(0, cons(X, Y)) → A(X)

The remaining pairs can at least be oriented weakly.

A(fromInact(x1)) → FROM(x1)
INDX(cons(X, Y), Z) → A(Z)
DBLS(cons(X, Y)) → A(Y)
A(indxInact(x1, x2)) → A(x1)
SEL(s(X), cons(Y, Z)) → A(X)
A(selInact(x1, x2)) → SEL(a(x1), a(x2))
A(dblInact(x1)) → A(x1)
A(dblsInact(x1)) → DBLS(a(x1))
INDX(cons(X, Y), Z) → A(X)
DBL(s(X)) → A(X)
A(indxInact(x1, x2)) → INDX(a(x1), x2)
FROM(X) → A(X)
DBLS(cons(X, Y)) → A(X)
A(dblsInact(x1)) → A(x1)
A(dblInact(x1)) → DBL(a(x1))
SEL(s(X), cons(Y, Z)) → A(Z)
INDX(cons(X, Y), Z) → A(Y)
A(selInact(x1, x2)) → A(x2)
A(selInact(x1, x2)) → A(x1)
SEL(s(X), cons(Y, Z)) → SEL(a(X), a(Z))

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

POL(0) = 1
POL(A(x₁)) = x₁
POL(DBL(x₁)) = x₁
POL(DBLS(x₁)) = x₁
POL(FROM(x₁)) = x₁
POL(INDX(x₁, x₂)) = x₁ + x₂
POL(SEL(x₁, x₂)) = x₁ + x₂
POL(a(x₁)) = x₁
POL(cons(x₁, x₂)) = max(x₁, x₂)
POL(dbl(x₁)) = x₁
POL(dblInact(x₁)) = x₁
POL(dbls(x₁)) = x₁
POL(dblsInact(x₁)) = x₁
POL(from(x₁)) = x₁
POL(fromInact(x₁)) = x₁
POL(indx(x₁, x₂)) = x₁ + x₂
POL(indxInact(x₁, x₂)) = x₁ + x₂
POL(nil) = 0
POL(s(x₁)) = x₁
POL(sInact(x₁)) = x₁
POL(sel(x₁, x₂)) = x₁ + x₂
POL(selInact(x₁, x₂)) = x₁ + x₂

The following usable rules [17] were oriented:

a(dblsInact(x1)) → dbls(a(x1))
a(dblInact(x1)) → dbl(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
a(selInact(x1, x2)) → sel(a(x1), a(x2))
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
sel(x1, x2) → selInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
indx(x1, x2) → indxInact(x1, x2)
a(x) → x
from(X) → cons(a(X), fromInact(sInact(a(X))))
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
indx(nil, X) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
dbls(nil) → nil
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbl(0) → 0

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

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

A(fromInact(x1)) → FROM(x1)
INDX(cons(X, Y), Z) → A(Z)
DBLS(cons(X, Y)) → A(Y)
A(indxInact(x1, x2)) → A(x1)
SEL(s(X), cons(Y, Z)) → A(X)
A(selInact(x1, x2)) → SEL(a(x1), a(x2))
A(dblInact(x1)) → A(x1)
A(dblsInact(x1)) → DBLS(a(x1))
INDX(cons(X, Y), Z) → A(X)
DBL(s(X)) → A(X)
A(indxInact(x1, x2)) → INDX(a(x1), x2)
FROM(X) → A(X)
DBLS(cons(X, Y)) → A(X)
A(dblsInact(x1)) → A(x1)
A(dblInact(x1)) → DBL(a(x1))
SEL(s(X), cons(Y, Z)) → A(Z)
A(selInact(x1, x2)) → A(x2)
INDX(cons(X, Y), Z) → A(Y)
A(selInact(x1, x2)) → A(x1)
SEL(s(X), cons(Y, Z)) → SEL(a(X), a(Z))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(a(x1), a(x2))
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(a(x1))

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

The following pairs can be oriented strictly and are deleted.

A(indxInact(x1, x2)) → A(x1)
A(indxInact(x1, x2)) → INDX(a(x1), x2)

The remaining pairs can at least be oriented weakly.

A(fromInact(x1)) → FROM(x1)
INDX(cons(X, Y), Z) → A(Z)
DBLS(cons(X, Y)) → A(Y)
SEL(s(X), cons(Y, Z)) → A(X)
A(selInact(x1, x2)) → SEL(a(x1), a(x2))
A(dblInact(x1)) → A(x1)
A(dblsInact(x1)) → DBLS(a(x1))
INDX(cons(X, Y), Z) → A(X)
DBL(s(X)) → A(X)
FROM(X) → A(X)
DBLS(cons(X, Y)) → A(X)
A(dblsInact(x1)) → A(x1)
A(dblInact(x1)) → DBL(a(x1))
SEL(s(X), cons(Y, Z)) → A(Z)
A(selInact(x1, x2)) → A(x2)
INDX(cons(X, Y), Z) → A(Y)
A(selInact(x1, x2)) → A(x1)
SEL(s(X), cons(Y, Z)) → SEL(a(X), a(Z))

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

POL(0) = 0
POL(A(x₁)) = x₁
POL(DBL(x₁)) = x₁
POL(DBLS(x₁)) = x₁
POL(FROM(x₁)) = x₁
POL(INDX(x₁, x₂)) = x₁ + x₂
POL(SEL(x₁, x₂)) = x₁ + x₂
POL(a(x₁)) = x₁
POL(cons(x₁, x₂)) = max(x₁, x₂)
POL(dbl(x₁)) = x₁
POL(dblInact(x₁)) = x₁
POL(dbls(x₁)) = x₁
POL(dblsInact(x₁)) = x₁
POL(from(x₁)) = x₁
POL(fromInact(x₁)) = x₁
POL(indx(x₁, x₂)) = 1 + x₁ + x₂
POL(indxInact(x₁, x₂)) = 1 + x₁ + x₂
POL(nil) = 1
POL(s(x₁)) = x₁
POL(sInact(x₁)) = x₁
POL(sel(x₁, x₂)) = x₁ + x₂
POL(selInact(x₁, x₂)) = x₁ + x₂

The following usable rules [17] were oriented:

a(dblsInact(x1)) → dbls(a(x1))
a(dblInact(x1)) → dbl(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
a(selInact(x1, x2)) → sel(a(x1), a(x2))
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
sel(x1, x2) → selInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
indx(x1, x2) → indxInact(x1, x2)
a(x) → x
from(X) → cons(a(X), fromInact(sInact(a(X))))
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
indx(nil, X) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
dbls(nil) → nil
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbl(0) → 0

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

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

A(fromInact(x1)) → FROM(x1)
INDX(cons(X, Y), Z) → A(Z)
DBLS(cons(X, Y)) → A(Y)
SEL(s(X), cons(Y, Z)) → A(X)
A(selInact(x1, x2)) → SEL(a(x1), a(x2))
A(dblInact(x1)) → A(x1)
A(dblsInact(x1)) → DBLS(a(x1))
INDX(cons(X, Y), Z) → A(X)
DBL(s(X)) → A(X)
FROM(X) → A(X)
DBLS(cons(X, Y)) → A(X)
A(dblsInact(x1)) → A(x1)
A(dblInact(x1)) → DBL(a(x1))
SEL(s(X), cons(Y, Z)) → A(Z)
INDX(cons(X, Y), Z) → A(Y)
A(selInact(x1, x2)) → A(x2)
A(selInact(x1, x2)) → A(x1)
SEL(s(X), cons(Y, Z)) → SEL(a(X), a(Z))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(a(x1), a(x2))
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(a(x1))

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ QDPOrderProof

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

A(fromInact(x1)) → FROM(x1)
DBLS(cons(X, Y)) → A(Y)
A(selInact(x1, x2)) → SEL(a(x1), a(x2))
SEL(s(X), cons(Y, Z)) → A(X)
A(dblInact(x1)) → A(x1)
A(dblsInact(x1)) → DBLS(a(x1))
FROM(X) → A(X)
DBL(s(X)) → A(X)
DBLS(cons(X, Y)) → A(X)
A(dblsInact(x1)) → A(x1)
A(dblInact(x1)) → DBL(a(x1))
SEL(s(X), cons(Y, Z)) → A(Z)
A(selInact(x1, x2)) → A(x2)
A(selInact(x1, x2)) → A(x1)
SEL(s(X), cons(Y, Z)) → SEL(a(X), a(Z))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(a(x1), a(x2))
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(a(x1))

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

The following pairs can be oriented strictly and are deleted.

DBLS(cons(X, Y)) → A(Y)
DBLS(cons(X, Y)) → A(X)
A(dblsInact(x1)) → A(x1)

The remaining pairs can at least be oriented weakly.

A(fromInact(x1)) → FROM(x1)
A(selInact(x1, x2)) → SEL(a(x1), a(x2))
SEL(s(X), cons(Y, Z)) → A(X)
A(dblInact(x1)) → A(x1)
A(dblsInact(x1)) → DBLS(a(x1))
FROM(X) → A(X)
DBL(s(X)) → A(X)
A(dblInact(x1)) → DBL(a(x1))
SEL(s(X), cons(Y, Z)) → A(Z)
A(selInact(x1, x2)) → A(x2)
A(selInact(x1, x2)) → A(x1)
SEL(s(X), cons(Y, Z)) → SEL(a(X), a(Z))

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

POL(0) = 0
POL(A(x₁)) = x₁
POL(DBL(x₁)) = x₁
POL(DBLS(x₁)) = 1 + x₁
POL(FROM(x₁)) = x₁
POL(SEL(x₁, x₂)) = x₁ + x₂
POL(a(x₁)) = x₁
POL(cons(x₁, x₂)) = max(x₁, x₂)
POL(dbl(x₁)) = x₁
POL(dblInact(x₁)) = x₁
POL(dbls(x₁)) = 1 + x₁
POL(dblsInact(x₁)) = 1 + x₁
POL(from(x₁)) = x₁
POL(fromInact(x₁)) = x₁
POL(indx(x₁, x₂)) = x₁ + x₂
POL(indxInact(x₁, x₂)) = x₁ + x₂
POL(nil) = 0
POL(s(x₁)) = x₁
POL(sInact(x₁)) = x₁
POL(sel(x₁, x₂)) = x₁ + x₂
POL(selInact(x₁, x₂)) = x₁ + x₂

The following usable rules [17] were oriented:

a(dblsInact(x1)) → dbls(a(x1))
a(dblInact(x1)) → dbl(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
a(selInact(x1, x2)) → sel(a(x1), a(x2))
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
sel(x1, x2) → selInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
indx(x1, x2) → indxInact(x1, x2)
a(x) → x
from(X) → cons(a(X), fromInact(sInact(a(X))))
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
indx(nil, X) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
dbls(nil) → nil
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbl(0) → 0

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ QDPOrderProof

                                ↳ QDP

                                  ↳ DependencyGraphProof

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

A(dblInact(x1)) → DBL(a(x1))
SEL(s(X), cons(Y, Z)) → A(Z)
A(fromInact(x1)) → FROM(x1)
A(selInact(x1, x2)) → A(x2)
A(dblInact(x1)) → A(x1)
SEL(s(X), cons(Y, Z)) → A(X)
A(selInact(x1, x2)) → SEL(a(x1), a(x2))
A(selInact(x1, x2)) → A(x1)
A(dblsInact(x1)) → DBLS(a(x1))
SEL(s(X), cons(Y, Z)) → SEL(a(X), a(Z))
DBL(s(X)) → A(X)
FROM(X) → A(X)

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(a(x1), a(x2))
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(a(x1))

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ QDPOrderProof

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ QDPOrderProof

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

A(dblInact(x1)) → DBL(a(x1))
SEL(s(X), cons(Y, Z)) → A(Z)
A(fromInact(x1)) → FROM(x1)
A(selInact(x1, x2)) → A(x2)
A(dblInact(x1)) → A(x1)
SEL(s(X), cons(Y, Z)) → A(X)
A(selInact(x1, x2)) → SEL(a(x1), a(x2))
A(selInact(x1, x2)) → A(x1)
SEL(s(X), cons(Y, Z)) → SEL(a(X), a(Z))
DBL(s(X)) → A(X)
FROM(X) → A(X)

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(a(x1), a(x2))
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(a(x1))

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

The following pairs can be oriented strictly and are deleted.

SEL(s(X), cons(Y, Z)) → A(Z)
A(selInact(x1, x2)) → A(x2)
SEL(s(X), cons(Y, Z)) → A(X)
A(selInact(x1, x2)) → A(x1)

The remaining pairs can at least be oriented weakly.

A(dblInact(x1)) → DBL(a(x1))
A(fromInact(x1)) → FROM(x1)
A(dblInact(x1)) → A(x1)
A(selInact(x1, x2)) → SEL(a(x1), a(x2))
SEL(s(X), cons(Y, Z)) → SEL(a(X), a(Z))
DBL(s(X)) → A(X)
FROM(X) → A(X)

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

POL(0) = 0
POL(A(x₁)) = x₁
POL(DBL(x₁)) = x₁
POL(FROM(x₁)) = x₁
POL(SEL(x₁, x₂)) = 1 + x₁ + x₂
POL(a(x₁)) = x₁
POL(cons(x₁, x₂)) = max(x₁, x₂)
POL(dbl(x₁)) = x₁
POL(dblInact(x₁)) = x₁
POL(dbls(x₁)) = x₁
POL(dblsInact(x₁)) = x₁
POL(from(x₁)) = x₁
POL(fromInact(x₁)) = x₁
POL(indx(x₁, x₂)) = 1 + x₁ + x₂
POL(indxInact(x₁, x₂)) = 1 + x₁ + x₂
POL(nil) = 0
POL(s(x₁)) = x₁
POL(sInact(x₁)) = x₁
POL(sel(x₁, x₂)) = 1 + x₁ + x₂
POL(selInact(x₁, x₂)) = 1 + x₁ + x₂

The following usable rules [17] were oriented:

a(dblsInact(x1)) → dbls(a(x1))
a(dblInact(x1)) → dbl(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
a(selInact(x1, x2)) → sel(a(x1), a(x2))
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
sel(x1, x2) → selInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
indx(x1, x2) → indxInact(x1, x2)
a(x) → x
from(X) → cons(a(X), fromInact(sInact(a(X))))
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
indx(nil, X) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
dbls(nil) → nil
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbl(0) → 0

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ QDPOrderProof

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ QDPOrderProof

                                        ↳ QDP

                                          ↳ DependencyGraphProof

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

A(dblInact(x1)) → DBL(a(x1))
A(fromInact(x1)) → FROM(x1)
A(selInact(x1, x2)) → SEL(a(x1), a(x2))
A(dblInact(x1)) → A(x1)
SEL(s(X), cons(Y, Z)) → SEL(a(X), a(Z))
FROM(X) → A(X)
DBL(s(X)) → A(X)

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(a(x1), a(x2))
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(a(x1))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 1 less node.

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ QDPOrderProof

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ QDPOrderProof

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ AND

                                              ↳ QDP

                                                ↳ Narrowing

                                              ↳ QDP

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

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

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(a(x1), a(x2))
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(a(x1))

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

SEL(s(selInact(x0, x1)), cons(y1, y2)) → SEL(sel(a(x0), a(x1)), a(y2))
SEL(s(dblsInact(x0)), cons(y1, y2)) → SEL(dbls(a(x0)), a(y2))
SEL(s(fromInact(x0)), cons(y1, y2)) → SEL(from(x0), a(y2))
SEL(s(sInact(x0)), cons(y1, y2)) → SEL(s(x0), a(y2))
SEL(s(dblInact(x0)), cons(y1, y2)) → SEL(dbl(a(x0)), a(y2))
SEL(s(x0), cons(y1, y2)) → SEL(x0, a(y2))
SEL(s(indxInact(x0, x1)), cons(y1, y2)) → SEL(indx(a(x0), x1), a(y2))

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ QDPOrderProof

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ QDPOrderProof

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ AND

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                              ↳ QDP

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

SEL(s(selInact(x0, x1)), cons(y1, y2)) → SEL(sel(a(x0), a(x1)), a(y2))
SEL(s(dblsInact(x0)), cons(y1, y2)) → SEL(dbls(a(x0)), a(y2))
SEL(s(fromInact(x0)), cons(y1, y2)) → SEL(from(x0), a(y2))
SEL(s(sInact(x0)), cons(y1, y2)) → SEL(s(x0), a(y2))
SEL(s(dblInact(x0)), cons(y1, y2)) → SEL(dbl(a(x0)), a(y2))
SEL(s(indxInact(x0, x1)), cons(y1, y2)) → SEL(indx(a(x0), x1), a(y2))
SEL(s(x0), cons(y1, y2)) → SEL(x0, a(y2))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(a(x1), a(x2))
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(a(x1))

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

SEL(s(selInact(y0, y1)), cons(y2, sInact(x0))) → SEL(sel(a(y0), a(y1)), s(x0))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(a(y0), a(y1)), indx(a(x0), x1))
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(a(y0), a(y1)), dbl(a(x0)))
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(a(y0), a(y1)), from(x0))
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(a(y0), a(y1)), dbls(a(x0)))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(a(y0), a(y1)), x0)
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(a(y0), a(y1)), sel(a(x0), a(x1)))

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ QDPOrderProof

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ QDPOrderProof

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ AND

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                              ↳ QDP

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

SEL(s(selInact(y0, y1)), cons(y2, sInact(x0))) → SEL(sel(a(y0), a(y1)), s(x0))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(a(y0), a(y1)), indx(a(x0), x1))
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(a(y0), a(y1)), dbl(a(x0)))
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(a(y0), a(y1)), from(x0))
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(a(y0), a(y1)), dbls(a(x0)))
SEL(s(indxInact(x0, x1)), cons(y1, y2)) → SEL(indx(a(x0), x1), a(y2))
SEL(s(dblsInact(x0)), cons(y1, y2)) → SEL(dbls(a(x0)), a(y2))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(a(y0), a(y1)), x0)
SEL(s(fromInact(x0)), cons(y1, y2)) → SEL(from(x0), a(y2))
SEL(s(sInact(x0)), cons(y1, y2)) → SEL(s(x0), a(y2))
SEL(s(dblInact(x0)), cons(y1, y2)) → SEL(dbl(a(x0)), a(y2))
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(a(y0), a(y1)), sel(a(x0), a(x1)))
SEL(s(x0), cons(y1, y2)) → SEL(x0, a(y2))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(a(x1), a(x2))
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(a(x1))

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

SEL(s(dblsInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbls(a(y0)), sel(a(x0), a(x1)))
SEL(s(dblsInact(y0)), cons(y1, fromInact(x0))) → SEL(dbls(a(y0)), from(x0))
SEL(s(dblsInact(y0)), cons(y1, sInact(x0))) → SEL(dbls(a(y0)), s(x0))
SEL(s(dblsInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbls(a(y0)), indx(a(x0), x1))
SEL(s(dblsInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbls(a(y0)), dbls(a(x0)))
SEL(s(dblsInact(y0)), cons(y1, dblInact(x0))) → SEL(dbls(a(y0)), dbl(a(x0)))
SEL(s(dblsInact(y0)), cons(y1, x0)) → SEL(dbls(a(y0)), x0)

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ QDPOrderProof

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ QDPOrderProof

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ AND

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Narrowing

                                              ↳ QDP

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

SEL(s(selInact(y0, y1)), cons(y2, sInact(x0))) → SEL(sel(a(y0), a(y1)), s(x0))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(a(y0), a(y1)), indx(a(x0), x1))
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(a(y0), a(y1)), dbl(a(x0)))
SEL(s(dblsInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbls(a(y0)), sel(a(x0), a(x1)))
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(a(y0), a(y1)), from(x0))
SEL(s(dblsInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbls(a(y0)), indx(a(x0), x1))
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(a(y0), a(y1)), dbls(a(x0)))
SEL(s(dblsInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbls(a(y0)), dbls(a(x0)))
SEL(s(dblsInact(y0)), cons(y1, x0)) → SEL(dbls(a(y0)), x0)
SEL(s(indxInact(x0, x1)), cons(y1, y2)) → SEL(indx(a(x0), x1), a(y2))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(a(y0), a(y1)), x0)
SEL(s(fromInact(x0)), cons(y1, y2)) → SEL(from(x0), a(y2))
SEL(s(dblsInact(y0)), cons(y1, fromInact(x0))) → SEL(dbls(a(y0)), from(x0))
SEL(s(sInact(x0)), cons(y1, y2)) → SEL(s(x0), a(y2))
SEL(s(dblsInact(y0)), cons(y1, sInact(x0))) → SEL(dbls(a(y0)), s(x0))
SEL(s(dblInact(x0)), cons(y1, y2)) → SEL(dbl(a(x0)), a(y2))
SEL(s(dblsInact(y0)), cons(y1, dblInact(x0))) → SEL(dbls(a(y0)), dbl(a(x0)))
SEL(s(x0), cons(y1, y2)) → SEL(x0, a(y2))
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(a(y0), a(y1)), sel(a(x0), a(x1)))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(a(x1), a(x2))
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(a(x1))

SEL(s(fromInact(x0)), cons(y1, y2)) → SEL(fromInact(x0), a(y2))
SEL(s(fromInact(x0)), cons(y1, y2)) → SEL(cons(a(x0), fromInact(sInact(a(x0)))), a(y2))

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ QDPOrderProof

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ QDPOrderProof

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ AND

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                              ↳ QDP

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

SEL(s(selInact(y0, y1)), cons(y2, sInact(x0))) → SEL(sel(a(y0), a(y1)), s(x0))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(a(y0), a(y1)), indx(a(x0), x1))
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(a(y0), a(y1)), dbl(a(x0)))
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(a(y0), a(y1)), from(x0))
SEL(s(dblsInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbls(a(y0)), sel(a(x0), a(x1)))
SEL(s(fromInact(x0)), cons(y1, y2)) → SEL(fromInact(x0), a(y2))
SEL(s(dblsInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbls(a(y0)), indx(a(x0), x1))
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(a(y0), a(y1)), dbls(a(x0)))
SEL(s(dblsInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbls(a(y0)), dbls(a(x0)))
SEL(s(indxInact(x0, x1)), cons(y1, y2)) → SEL(indx(a(x0), x1), a(y2))
SEL(s(dblsInact(y0)), cons(y1, x0)) → SEL(dbls(a(y0)), x0)
SEL(s(fromInact(x0)), cons(y1, y2)) → SEL(cons(a(x0), fromInact(sInact(a(x0)))), a(y2))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(a(y0), a(y1)), x0)
SEL(s(sInact(x0)), cons(y1, y2)) → SEL(s(x0), a(y2))
SEL(s(dblsInact(y0)), cons(y1, fromInact(x0))) → SEL(dbls(a(y0)), from(x0))
SEL(s(dblsInact(y0)), cons(y1, sInact(x0))) → SEL(dbls(a(y0)), s(x0))
SEL(s(dblInact(x0)), cons(y1, y2)) → SEL(dbl(a(x0)), a(y2))
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(a(y0), a(y1)), sel(a(x0), a(x1)))
SEL(s(x0), cons(y1, y2)) → SEL(x0, a(y2))
SEL(s(dblsInact(y0)), cons(y1, dblInact(x0))) → SEL(dbls(a(y0)), dbl(a(x0)))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(a(x1), a(x2))
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(a(x1))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ QDPOrderProof

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ QDPOrderProof

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ AND

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                              ↳ QDP

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

SEL(s(selInact(y0, y1)), cons(y2, sInact(x0))) → SEL(sel(a(y0), a(y1)), s(x0))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(a(y0), a(y1)), indx(a(x0), x1))
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(a(y0), a(y1)), dbl(a(x0)))
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(a(y0), a(y1)), from(x0))
SEL(s(dblsInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbls(a(y0)), sel(a(x0), a(x1)))
SEL(s(dblsInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbls(a(y0)), indx(a(x0), x1))
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(a(y0), a(y1)), dbls(a(x0)))
SEL(s(dblsInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbls(a(y0)), dbls(a(x0)))
SEL(s(dblsInact(y0)), cons(y1, x0)) → SEL(dbls(a(y0)), x0)
SEL(s(indxInact(x0, x1)), cons(y1, y2)) → SEL(indx(a(x0), x1), a(y2))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(a(y0), a(y1)), x0)
SEL(s(sInact(x0)), cons(y1, y2)) → SEL(s(x0), a(y2))
SEL(s(dblsInact(y0)), cons(y1, fromInact(x0))) → SEL(dbls(a(y0)), from(x0))
SEL(s(dblsInact(y0)), cons(y1, sInact(x0))) → SEL(dbls(a(y0)), s(x0))
SEL(s(dblInact(x0)), cons(y1, y2)) → SEL(dbl(a(x0)), a(y2))
SEL(s(dblsInact(y0)), cons(y1, dblInact(x0))) → SEL(dbls(a(y0)), dbl(a(x0)))
SEL(s(x0), cons(y1, y2)) → SEL(x0, a(y2))
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(a(y0), a(y1)), sel(a(x0), a(x1)))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(a(x1), a(x2))
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(a(x1))

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

SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(a(x0), a(x1)))
SEL(s(sInact(y0)), cons(y1, fromInact(x0))) → SEL(s(y0), from(x0))
SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(a(x0)))
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(a(x0), x1))
SEL(s(sInact(y0)), cons(y1, dblInact(x0))) → SEL(s(y0), dbl(a(x0)))
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)
SEL(s(sInact(y0)), cons(y1, sInact(x0))) → SEL(s(y0), s(x0))

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ QDPOrderProof

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ QDPOrderProof

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ AND

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                              ↳ QDP

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

SEL(s(selInact(y0, y1)), cons(y2, sInact(x0))) → SEL(sel(a(y0), a(y1)), s(x0))
SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(a(x0)))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(a(y0), a(y1)), indx(a(x0), x1))
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(a(y0), a(y1)), dbl(a(x0)))
SEL(s(dblsInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbls(a(y0)), indx(a(x0), x1))
SEL(s(dblsInact(y0)), cons(y1, x0)) → SEL(dbls(a(y0)), x0)
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(a(x0), a(x1)))
SEL(s(sInact(y0)), cons(y1, fromInact(x0))) → SEL(s(y0), from(x0))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(a(y0), a(y1)), x0)
SEL(s(dblsInact(y0)), cons(y1, fromInact(x0))) → SEL(dbls(a(y0)), from(x0))
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(a(x0), x1))
SEL(s(dblsInact(y0)), cons(y1, dblInact(x0))) → SEL(dbls(a(y0)), dbl(a(x0)))
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(a(y0), a(y1)), from(x0))
SEL(s(dblsInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbls(a(y0)), sel(a(x0), a(x1)))
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(a(y0), a(y1)), dbls(a(x0)))
SEL(s(dblsInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbls(a(y0)), dbls(a(x0)))
SEL(s(sInact(y0)), cons(y1, dblInact(x0))) → SEL(s(y0), dbl(a(x0)))
SEL(s(indxInact(x0, x1)), cons(y1, y2)) → SEL(indx(a(x0), x1), a(y2))
SEL(s(sInact(y0)), cons(y1, sInact(x0))) → SEL(s(y0), s(x0))
SEL(s(dblsInact(y0)), cons(y1, sInact(x0))) → SEL(dbls(a(y0)), s(x0))
SEL(s(dblInact(x0)), cons(y1, y2)) → SEL(dbl(a(x0)), a(y2))
SEL(s(x0), cons(y1, y2)) → SEL(x0, a(y2))
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(a(y0), a(y1)), sel(a(x0), a(x1)))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(a(x1), a(x2))
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(a(x1))

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

SEL(s(dblInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbl(a(y0)), sel(a(x0), a(x1)))
SEL(s(dblInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbl(a(y0)), indx(a(x0), x1))
SEL(s(dblInact(y0)), cons(y1, sInact(x0))) → SEL(dbl(a(y0)), s(x0))
SEL(s(dblInact(y0)), cons(y1, dblInact(x0))) → SEL(dbl(a(y0)), dbl(a(x0)))
SEL(s(dblInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbl(a(y0)), dbls(a(x0)))
SEL(s(dblInact(y0)), cons(y1, x0)) → SEL(dbl(a(y0)), x0)
SEL(s(dblInact(y0)), cons(y1, fromInact(x0))) → SEL(dbl(a(y0)), from(x0))

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ QDPOrderProof

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ QDPOrderProof

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ AND

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                              ↳ QDP

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

SEL(s(selInact(y0, y1)), cons(y2, sInact(x0))) → SEL(sel(a(y0), a(y1)), s(x0))
SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(a(x0)))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(a(y0), a(y1)), indx(a(x0), x1))
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(a(y0), a(y1)), dbl(a(x0)))
SEL(s(dblsInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbls(a(y0)), indx(a(x0), x1))
SEL(s(dblsInact(y0)), cons(y1, x0)) → SEL(dbls(a(y0)), x0)
SEL(s(dblInact(y0)), cons(y1, fromInact(x0))) → SEL(dbl(a(y0)), from(x0))
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(a(x0), a(x1)))
SEL(s(sInact(y0)), cons(y1, fromInact(x0))) → SEL(s(y0), from(x0))
SEL(s(dblInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbl(a(y0)), sel(a(x0), a(x1)))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(a(y0), a(y1)), x0)
SEL(s(dblInact(y0)), cons(y1, sInact(x0))) → SEL(dbl(a(y0)), s(x0))
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(a(x0), x1))
SEL(s(dblsInact(y0)), cons(y1, fromInact(x0))) → SEL(dbls(a(y0)), from(x0))
SEL(s(dblInact(y0)), cons(y1, x0)) → SEL(dbl(a(y0)), x0)
SEL(s(dblInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbl(a(y0)), dbls(a(x0)))
SEL(s(dblsInact(y0)), cons(y1, dblInact(x0))) → SEL(dbls(a(y0)), dbl(a(x0)))
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)
SEL(s(dblInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbl(a(y0)), indx(a(x0), x1))
SEL(s(dblsInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbls(a(y0)), sel(a(x0), a(x1)))
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(a(y0), a(y1)), from(x0))
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(a(y0), a(y1)), dbls(a(x0)))
SEL(s(sInact(y0)), cons(y1, dblInact(x0))) → SEL(s(y0), dbl(a(x0)))
SEL(s(dblsInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbls(a(y0)), dbls(a(x0)))
SEL(s(indxInact(x0, x1)), cons(y1, y2)) → SEL(indx(a(x0), x1), a(y2))
SEL(s(sInact(y0)), cons(y1, sInact(x0))) → SEL(s(y0), s(x0))
SEL(s(dblsInact(y0)), cons(y1, sInact(x0))) → SEL(dbls(a(y0)), s(x0))
SEL(s(dblInact(y0)), cons(y1, dblInact(x0))) → SEL(dbl(a(y0)), dbl(a(x0)))
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(a(y0), a(y1)), sel(a(x0), a(x1)))
SEL(s(x0), cons(y1, y2)) → SEL(x0, a(y2))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(a(x1), a(x2))
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(a(x1))

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

SEL(s(y0), cons(y1, dblsInact(x0))) → SEL(y0, dbls(a(x0)))
SEL(s(y0), cons(y1, indxInact(x0, x1))) → SEL(y0, indx(a(x0), x1))
SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(y0), cons(y1, sInact(x0))) → SEL(y0, s(x0))
SEL(s(y0), cons(y1, selInact(x0, x1))) → SEL(y0, sel(a(x0), a(x1)))
SEL(s(y0), cons(y1, fromInact(x0))) → SEL(y0, from(x0))
SEL(s(y0), cons(y1, dblInact(x0))) → SEL(y0, dbl(a(x0)))

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ QDPOrderProof

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ QDPOrderProof

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ AND

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                              ↳ QDP

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

SEL(s(selInact(y0, y1)), cons(y2, sInact(x0))) → SEL(sel(a(y0), a(y1)), s(x0))
SEL(s(y0), cons(y1, indxInact(x0, x1))) → SEL(y0, indx(a(x0), x1))
SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(a(x0)))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(a(y0), a(y1)), indx(a(x0), x1))
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(a(y0), a(y1)), dbl(a(x0)))
SEL(s(y0), cons(y1, sInact(x0))) → SEL(y0, s(x0))
SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(dblsInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbls(a(y0)), indx(a(x0), x1))
SEL(s(dblsInact(y0)), cons(y1, x0)) → SEL(dbls(a(y0)), x0)
SEL(s(dblInact(y0)), cons(y1, fromInact(x0))) → SEL(dbl(a(y0)), from(x0))
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(a(x0), a(x1)))
SEL(s(sInact(y0)), cons(y1, fromInact(x0))) → SEL(s(y0), from(x0))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(a(y0), a(y1)), x0)
SEL(s(dblInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbl(a(y0)), sel(a(x0), a(x1)))
SEL(s(dblInact(y0)), cons(y1, sInact(x0))) → SEL(dbl(a(y0)), s(x0))
SEL(s(dblsInact(y0)), cons(y1, fromInact(x0))) → SEL(dbls(a(y0)), from(x0))
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(a(x0), x1))
SEL(s(y0), cons(y1, fromInact(x0))) → SEL(y0, from(x0))
SEL(s(dblsInact(y0)), cons(y1, dblInact(x0))) → SEL(dbls(a(y0)), dbl(a(x0)))
SEL(s(dblInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbl(a(y0)), dbls(a(x0)))
SEL(s(dblInact(y0)), cons(y1, x0)) → SEL(dbl(a(y0)), x0)
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)
SEL(s(y0), cons(y1, dblsInact(x0))) → SEL(y0, dbls(a(x0)))
SEL(s(dblInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbl(a(y0)), indx(a(x0), x1))
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(a(y0), a(y1)), from(x0))
SEL(s(dblsInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbls(a(y0)), sel(a(x0), a(x1)))
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(a(y0), a(y1)), dbls(a(x0)))
SEL(s(dblsInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbls(a(y0)), dbls(a(x0)))
SEL(s(sInact(y0)), cons(y1, dblInact(x0))) → SEL(s(y0), dbl(a(x0)))
SEL(s(indxInact(x0, x1)), cons(y1, y2)) → SEL(indx(a(x0), x1), a(y2))
SEL(s(sInact(y0)), cons(y1, sInact(x0))) → SEL(s(y0), s(x0))
SEL(s(dblsInact(y0)), cons(y1, sInact(x0))) → SEL(dbls(a(y0)), s(x0))
SEL(s(y0), cons(y1, selInact(x0, x1))) → SEL(y0, sel(a(x0), a(x1)))
SEL(s(dblInact(y0)), cons(y1, dblInact(x0))) → SEL(dbl(a(y0)), dbl(a(x0)))
SEL(s(y0), cons(y1, dblInact(x0))) → SEL(y0, dbl(a(x0)))
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(a(y0), a(y1)), sel(a(x0), a(x1)))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(a(x1), a(x2))
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(a(x1))

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

SEL(s(indxInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(indx(a(y0), y1), dbls(a(x0)))
SEL(s(indxInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(indx(a(y0), y1), sel(a(x0), a(x1)))
SEL(s(indxInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(indx(a(y0), y1), indx(a(x0), x1))
SEL(s(indxInact(y0, y1)), cons(y2, sInact(x0))) → SEL(indx(a(y0), y1), s(x0))
SEL(s(indxInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(indx(a(y0), y1), dbl(a(x0)))
SEL(s(indxInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(indx(a(y0), y1), from(x0))
SEL(s(indxInact(y0, y1)), cons(y2, x0)) → SEL(indx(a(y0), y1), x0)

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ QDPOrderProof

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ QDPOrderProof

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ AND

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                              ↳ QDP

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

SEL(s(selInact(y0, y1)), cons(y2, sInact(x0))) → SEL(sel(a(y0), a(y1)), s(x0))
SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(a(x0)))
SEL(s(y0), cons(y1, indxInact(x0, x1))) → SEL(y0, indx(a(x0), x1))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(a(y0), a(y1)), indx(a(x0), x1))
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(a(y0), a(y1)), dbl(a(x0)))
SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(y0), cons(y1, sInact(x0))) → SEL(y0, s(x0))
SEL(s(dblsInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbls(a(y0)), indx(a(x0), x1))
SEL(s(indxInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(indx(a(y0), y1), indx(a(x0), x1))
SEL(s(indxInact(y0, y1)), cons(y2, sInact(x0))) → SEL(indx(a(y0), y1), s(x0))
SEL(s(dblsInact(y0)), cons(y1, x0)) → SEL(dbls(a(y0)), x0)
SEL(s(dblInact(y0)), cons(y1, fromInact(x0))) → SEL(dbl(a(y0)), from(x0))
SEL(s(indxInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(indx(a(y0), y1), dbls(a(x0)))
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(a(x0), a(x1)))
SEL(s(sInact(y0)), cons(y1, fromInact(x0))) → SEL(s(y0), from(x0))
SEL(s(dblInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbl(a(y0)), sel(a(x0), a(x1)))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(a(y0), a(y1)), x0)
SEL(s(dblInact(y0)), cons(y1, sInact(x0))) → SEL(dbl(a(y0)), s(x0))
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(a(x0), x1))
SEL(s(dblsInact(y0)), cons(y1, fromInact(x0))) → SEL(dbls(a(y0)), from(x0))
SEL(s(y0), cons(y1, fromInact(x0))) → SEL(y0, from(x0))
SEL(s(indxInact(y0, y1)), cons(y2, x0)) → SEL(indx(a(y0), y1), x0)
SEL(s(dblInact(y0)), cons(y1, x0)) → SEL(dbl(a(y0)), x0)
SEL(s(dblInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbl(a(y0)), dbls(a(x0)))
SEL(s(dblsInact(y0)), cons(y1, dblInact(x0))) → SEL(dbls(a(y0)), dbl(a(x0)))
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)
SEL(s(y0), cons(y1, dblsInact(x0))) → SEL(y0, dbls(a(x0)))
SEL(s(indxInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(indx(a(y0), y1), sel(a(x0), a(x1)))
SEL(s(dblInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbl(a(y0)), indx(a(x0), x1))
SEL(s(dblsInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbls(a(y0)), sel(a(x0), a(x1)))
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(a(y0), a(y1)), from(x0))
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(a(y0), a(y1)), dbls(a(x0)))
SEL(s(sInact(y0)), cons(y1, dblInact(x0))) → SEL(s(y0), dbl(a(x0)))
SEL(s(dblsInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbls(a(y0)), dbls(a(x0)))
SEL(s(indxInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(indx(a(y0), y1), from(x0))
SEL(s(sInact(y0)), cons(y1, sInact(x0))) → SEL(s(y0), s(x0))
SEL(s(dblsInact(y0)), cons(y1, sInact(x0))) → SEL(dbls(a(y0)), s(x0))
SEL(s(y0), cons(y1, selInact(x0, x1))) → SEL(y0, sel(a(x0), a(x1)))
SEL(s(indxInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(indx(a(y0), y1), dbl(a(x0)))
SEL(s(dblInact(y0)), cons(y1, dblInact(x0))) → SEL(dbl(a(y0)), dbl(a(x0)))
SEL(s(y0), cons(y1, dblInact(x0))) → SEL(y0, dbl(a(x0)))
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(a(y0), a(y1)), sel(a(x0), a(x1)))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(a(x1), a(x2))
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(a(x1))

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

SEL(s(selInact(y0, y1)), cons(y2, sInact(x0))) → SEL(sel(a(y0), a(y1)), sInact(x0))

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ QDPOrderProof

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ QDPOrderProof

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ AND

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                              ↳ QDP

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

SEL(s(y0), cons(y1, indxInact(x0, x1))) → SEL(y0, indx(a(x0), x1))
SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(a(x0)))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(a(y0), a(y1)), indx(a(x0), x1))
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(a(y0), a(y1)), dbl(a(x0)))
SEL(s(y0), cons(y1, sInact(x0))) → SEL(y0, s(x0))
SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(dblsInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbls(a(y0)), indx(a(x0), x1))
SEL(s(indxInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(indx(a(y0), y1), indx(a(x0), x1))
SEL(s(indxInact(y0, y1)), cons(y2, sInact(x0))) → SEL(indx(a(y0), y1), s(x0))
SEL(s(dblsInact(y0)), cons(y1, x0)) → SEL(dbls(a(y0)), x0)
SEL(s(dblInact(y0)), cons(y1, fromInact(x0))) → SEL(dbl(a(y0)), from(x0))
SEL(s(indxInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(indx(a(y0), y1), dbls(a(x0)))
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(a(x0), a(x1)))
SEL(s(sInact(y0)), cons(y1, fromInact(x0))) → SEL(s(y0), from(x0))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(a(y0), a(y1)), x0)
SEL(s(dblInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbl(a(y0)), sel(a(x0), a(x1)))
SEL(s(dblInact(y0)), cons(y1, sInact(x0))) → SEL(dbl(a(y0)), s(x0))
SEL(s(dblsInact(y0)), cons(y1, fromInact(x0))) → SEL(dbls(a(y0)), from(x0))
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(a(x0), x1))
SEL(s(selInact(y0, y1)), cons(y2, sInact(x0))) → SEL(sel(a(y0), a(y1)), sInact(x0))
SEL(s(y0), cons(y1, fromInact(x0))) → SEL(y0, from(x0))
SEL(s(dblsInact(y0)), cons(y1, dblInact(x0))) → SEL(dbls(a(y0)), dbl(a(x0)))
SEL(s(dblInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbl(a(y0)), dbls(a(x0)))
SEL(s(dblInact(y0)), cons(y1, x0)) → SEL(dbl(a(y0)), x0)
SEL(s(indxInact(y0, y1)), cons(y2, x0)) → SEL(indx(a(y0), y1), x0)
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)
SEL(s(y0), cons(y1, dblsInact(x0))) → SEL(y0, dbls(a(x0)))
SEL(s(dblInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbl(a(y0)), indx(a(x0), x1))
SEL(s(indxInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(indx(a(y0), y1), sel(a(x0), a(x1)))
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(a(y0), a(y1)), from(x0))
SEL(s(dblsInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbls(a(y0)), sel(a(x0), a(x1)))
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(a(y0), a(y1)), dbls(a(x0)))
SEL(s(dblsInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbls(a(y0)), dbls(a(x0)))
SEL(s(sInact(y0)), cons(y1, dblInact(x0))) → SEL(s(y0), dbl(a(x0)))
SEL(s(indxInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(indx(a(y0), y1), from(x0))
SEL(s(sInact(y0)), cons(y1, sInact(x0))) → SEL(s(y0), s(x0))
SEL(s(dblsInact(y0)), cons(y1, sInact(x0))) → SEL(dbls(a(y0)), s(x0))
SEL(s(y0), cons(y1, selInact(x0, x1))) → SEL(y0, sel(a(x0), a(x1)))
SEL(s(dblInact(y0)), cons(y1, dblInact(x0))) → SEL(dbl(a(y0)), dbl(a(x0)))
SEL(s(indxInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(indx(a(y0), y1), dbl(a(x0)))
SEL(s(y0), cons(y1, dblInact(x0))) → SEL(y0, dbl(a(x0)))
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(a(y0), a(y1)), sel(a(x0), a(x1)))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(a(x1), a(x2))
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(a(x1))

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ QDPOrderProof

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ QDPOrderProof

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ AND

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                              ↳ QDP

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

SEL(s(y0), cons(y1, indxInact(x0, x1))) → SEL(y0, indx(a(x0), x1))
SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(a(x0)))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(a(y0), a(y1)), indx(a(x0), x1))
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(a(y0), a(y1)), dbl(a(x0)))
SEL(s(y0), cons(y1, sInact(x0))) → SEL(y0, s(x0))
SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(dblsInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbls(a(y0)), indx(a(x0), x1))
SEL(s(indxInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(indx(a(y0), y1), indx(a(x0), x1))
SEL(s(indxInact(y0, y1)), cons(y2, sInact(x0))) → SEL(indx(a(y0), y1), s(x0))
SEL(s(dblsInact(y0)), cons(y1, x0)) → SEL(dbls(a(y0)), x0)
SEL(s(dblInact(y0)), cons(y1, fromInact(x0))) → SEL(dbl(a(y0)), from(x0))
SEL(s(indxInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(indx(a(y0), y1), dbls(a(x0)))
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(a(x0), a(x1)))
SEL(s(sInact(y0)), cons(y1, fromInact(x0))) → SEL(s(y0), from(x0))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(a(y0), a(y1)), x0)
SEL(s(dblInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbl(a(y0)), sel(a(x0), a(x1)))
SEL(s(dblInact(y0)), cons(y1, sInact(x0))) → SEL(dbl(a(y0)), s(x0))
SEL(s(dblsInact(y0)), cons(y1, fromInact(x0))) → SEL(dbls(a(y0)), from(x0))
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(a(x0), x1))
SEL(s(y0), cons(y1, fromInact(x0))) → SEL(y0, from(x0))
SEL(s(indxInact(y0, y1)), cons(y2, x0)) → SEL(indx(a(y0), y1), x0)
SEL(s(dblInact(y0)), cons(y1, x0)) → SEL(dbl(a(y0)), x0)
SEL(s(dblsInact(y0)), cons(y1, dblInact(x0))) → SEL(dbls(a(y0)), dbl(a(x0)))
SEL(s(dblInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbl(a(y0)), dbls(a(x0)))
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)
SEL(s(y0), cons(y1, dblsInact(x0))) → SEL(y0, dbls(a(x0)))
SEL(s(indxInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(indx(a(y0), y1), sel(a(x0), a(x1)))
SEL(s(dblInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbl(a(y0)), indx(a(x0), x1))
SEL(s(dblsInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbls(a(y0)), sel(a(x0), a(x1)))
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(a(y0), a(y1)), from(x0))
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(a(y0), a(y1)), dbls(a(x0)))
SEL(s(dblsInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbls(a(y0)), dbls(a(x0)))
SEL(s(sInact(y0)), cons(y1, dblInact(x0))) → SEL(s(y0), dbl(a(x0)))
SEL(s(indxInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(indx(a(y0), y1), from(x0))
SEL(s(sInact(y0)), cons(y1, sInact(x0))) → SEL(s(y0), s(x0))
SEL(s(dblsInact(y0)), cons(y1, sInact(x0))) → SEL(dbls(a(y0)), s(x0))
SEL(s(y0), cons(y1, selInact(x0, x1))) → SEL(y0, sel(a(x0), a(x1)))
SEL(s(indxInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(indx(a(y0), y1), dbl(a(x0)))
SEL(s(dblInact(y0)), cons(y1, dblInact(x0))) → SEL(dbl(a(y0)), dbl(a(x0)))
SEL(s(y0), cons(y1, dblInact(x0))) → SEL(y0, dbl(a(x0)))
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(a(y0), a(y1)), sel(a(x0), a(x1)))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(a(x1), a(x2))
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(a(x1))

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

SEL(s(dblsInact(y0)), cons(y1, sInact(x0))) → SEL(dbls(a(y0)), sInact(x0))

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ QDPOrderProof

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ QDPOrderProof

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ AND

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                              ↳ QDP

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

SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(a(x0)))
SEL(s(y0), cons(y1, indxInact(x0, x1))) → SEL(y0, indx(a(x0), x1))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(a(y0), a(y1)), indx(a(x0), x1))
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(a(y0), a(y1)), dbl(a(x0)))
SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(y0), cons(y1, sInact(x0))) → SEL(y0, s(x0))
SEL(s(dblsInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbls(a(y0)), indx(a(x0), x1))
SEL(s(indxInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(indx(a(y0), y1), indx(a(x0), x1))
SEL(s(indxInact(y0, y1)), cons(y2, sInact(x0))) → SEL(indx(a(y0), y1), s(x0))
SEL(s(dblsInact(y0)), cons(y1, x0)) → SEL(dbls(a(y0)), x0)
SEL(s(dblInact(y0)), cons(y1, fromInact(x0))) → SEL(dbl(a(y0)), from(x0))
SEL(s(indxInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(indx(a(y0), y1), dbls(a(x0)))
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(a(x0), a(x1)))
SEL(s(sInact(y0)), cons(y1, fromInact(x0))) → SEL(s(y0), from(x0))
SEL(s(dblInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbl(a(y0)), sel(a(x0), a(x1)))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(a(y0), a(y1)), x0)
SEL(s(dblInact(y0)), cons(y1, sInact(x0))) → SEL(dbl(a(y0)), s(x0))
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(a(x0), x1))
SEL(s(dblsInact(y0)), cons(y1, fromInact(x0))) → SEL(dbls(a(y0)), from(x0))
SEL(s(y0), cons(y1, fromInact(x0))) → SEL(y0, from(x0))
SEL(s(dblsInact(y0)), cons(y1, sInact(x0))) → SEL(dbls(a(y0)), sInact(x0))
SEL(s(dblInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbl(a(y0)), dbls(a(x0)))
SEL(s(dblsInact(y0)), cons(y1, dblInact(x0))) → SEL(dbls(a(y0)), dbl(a(x0)))
SEL(s(dblInact(y0)), cons(y1, x0)) → SEL(dbl(a(y0)), x0)
SEL(s(indxInact(y0, y1)), cons(y2, x0)) → SEL(indx(a(y0), y1), x0)
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)
SEL(s(y0), cons(y1, dblsInact(x0))) → SEL(y0, dbls(a(x0)))
SEL(s(dblInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbl(a(y0)), indx(a(x0), x1))
SEL(s(indxInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(indx(a(y0), y1), sel(a(x0), a(x1)))
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(a(y0), a(y1)), from(x0))
SEL(s(dblsInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbls(a(y0)), sel(a(x0), a(x1)))
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(a(y0), a(y1)), dbls(a(x0)))
SEL(s(sInact(y0)), cons(y1, dblInact(x0))) → SEL(s(y0), dbl(a(x0)))
SEL(s(dblsInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbls(a(y0)), dbls(a(x0)))
SEL(s(indxInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(indx(a(y0), y1), from(x0))
SEL(s(sInact(y0)), cons(y1, sInact(x0))) → SEL(s(y0), s(x0))
SEL(s(y0), cons(y1, selInact(x0, x1))) → SEL(y0, sel(a(x0), a(x1)))
SEL(s(dblInact(y0)), cons(y1, dblInact(x0))) → SEL(dbl(a(y0)), dbl(a(x0)))
SEL(s(indxInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(indx(a(y0), y1), dbl(a(x0)))
SEL(s(y0), cons(y1, dblInact(x0))) → SEL(y0, dbl(a(x0)))
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(a(y0), a(y1)), sel(a(x0), a(x1)))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(a(x1), a(x2))
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(a(x1))

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ QDPOrderProof

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ QDPOrderProof

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ AND

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                              ↳ QDP

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

SEL(s(y0), cons(y1, indxInact(x0, x1))) → SEL(y0, indx(a(x0), x1))
SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(a(x0)))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(a(y0), a(y1)), indx(a(x0), x1))
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(a(y0), a(y1)), dbl(a(x0)))
SEL(s(y0), cons(y1, sInact(x0))) → SEL(y0, s(x0))
SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(dblsInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbls(a(y0)), indx(a(x0), x1))
SEL(s(indxInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(indx(a(y0), y1), indx(a(x0), x1))
SEL(s(indxInact(y0, y1)), cons(y2, sInact(x0))) → SEL(indx(a(y0), y1), s(x0))
SEL(s(dblsInact(y0)), cons(y1, x0)) → SEL(dbls(a(y0)), x0)
SEL(s(dblInact(y0)), cons(y1, fromInact(x0))) → SEL(dbl(a(y0)), from(x0))
SEL(s(indxInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(indx(a(y0), y1), dbls(a(x0)))
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(a(x0), a(x1)))
SEL(s(sInact(y0)), cons(y1, fromInact(x0))) → SEL(s(y0), from(x0))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(a(y0), a(y1)), x0)
SEL(s(dblInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbl(a(y0)), sel(a(x0), a(x1)))
SEL(s(dblInact(y0)), cons(y1, sInact(x0))) → SEL(dbl(a(y0)), s(x0))
SEL(s(dblsInact(y0)), cons(y1, fromInact(x0))) → SEL(dbls(a(y0)), from(x0))
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(a(x0), x1))
SEL(s(y0), cons(y1, fromInact(x0))) → SEL(y0, from(x0))
SEL(s(indxInact(y0, y1)), cons(y2, x0)) → SEL(indx(a(y0), y1), x0)
SEL(s(dblsInact(y0)), cons(y1, dblInact(x0))) → SEL(dbls(a(y0)), dbl(a(x0)))
SEL(s(dblInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbl(a(y0)), dbls(a(x0)))
SEL(s(dblInact(y0)), cons(y1, x0)) → SEL(dbl(a(y0)), x0)
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)
SEL(s(y0), cons(y1, dblsInact(x0))) → SEL(y0, dbls(a(x0)))
SEL(s(indxInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(indx(a(y0), y1), sel(a(x0), a(x1)))
SEL(s(dblInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbl(a(y0)), indx(a(x0), x1))
SEL(s(dblsInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbls(a(y0)), sel(a(x0), a(x1)))
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(a(y0), a(y1)), from(x0))
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(a(y0), a(y1)), dbls(a(x0)))
SEL(s(dblsInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbls(a(y0)), dbls(a(x0)))
SEL(s(sInact(y0)), cons(y1, dblInact(x0))) → SEL(s(y0), dbl(a(x0)))
SEL(s(indxInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(indx(a(y0), y1), from(x0))
SEL(s(sInact(y0)), cons(y1, sInact(x0))) → SEL(s(y0), s(x0))
SEL(s(y0), cons(y1, selInact(x0, x1))) → SEL(y0, sel(a(x0), a(x1)))
SEL(s(indxInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(indx(a(y0), y1), dbl(a(x0)))
SEL(s(dblInact(y0)), cons(y1, dblInact(x0))) → SEL(dbl(a(y0)), dbl(a(x0)))
SEL(s(y0), cons(y1, dblInact(x0))) → SEL(y0, dbl(a(x0)))
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(a(y0), a(y1)), sel(a(x0), a(x1)))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(a(x1), a(x2))
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(a(x1))

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ QDPOrderProof

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ QDPOrderProof

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ AND

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                              ↳ QDP

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

SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(a(x0)))
SEL(s(y0), cons(y1, indxInact(x0, x1))) → SEL(y0, indx(a(x0), x1))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(a(y0), a(y1)), indx(a(x0), x1))
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(a(y0), a(y1)), dbl(a(x0)))
SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(y0), cons(y1, sInact(x0))) → SEL(y0, s(x0))
SEL(s(dblsInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbls(a(y0)), indx(a(x0), x1))
SEL(s(indxInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(indx(a(y0), y1), indx(a(x0), x1))
SEL(s(indxInact(y0, y1)), cons(y2, sInact(x0))) → SEL(indx(a(y0), y1), s(x0))
SEL(s(dblsInact(y0)), cons(y1, x0)) → SEL(dbls(a(y0)), x0)
SEL(s(dblInact(y0)), cons(y1, fromInact(x0))) → SEL(dbl(a(y0)), from(x0))
SEL(s(indxInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(indx(a(y0), y1), dbls(a(x0)))
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(a(x0), a(x1)))
SEL(s(sInact(y0)), cons(y1, fromInact(x0))) → SEL(s(y0), from(x0))
SEL(s(dblInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbl(a(y0)), sel(a(x0), a(x1)))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(a(y0), a(y1)), x0)
SEL(s(dblInact(y0)), cons(y1, sInact(x0))) → SEL(dbl(a(y0)), s(x0))
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(a(x0), x1))
SEL(s(dblsInact(y0)), cons(y1, fromInact(x0))) → SEL(dbls(a(y0)), from(x0))
SEL(s(sInact(y0)), cons(y1, sInact(x0))) → SEL(s(y0), sInact(x0))
SEL(s(y0), cons(y1, fromInact(x0))) → SEL(y0, from(x0))
SEL(s(dblInact(y0)), cons(y1, x0)) → SEL(dbl(a(y0)), x0)
SEL(s(dblInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbl(a(y0)), dbls(a(x0)))
SEL(s(dblsInact(y0)), cons(y1, dblInact(x0))) → SEL(dbls(a(y0)), dbl(a(x0)))
SEL(s(indxInact(y0, y1)), cons(y2, x0)) → SEL(indx(a(y0), y1), x0)
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)
SEL(s(y0), cons(y1, dblsInact(x0))) → SEL(y0, dbls(a(x0)))
SEL(s(dblInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbl(a(y0)), indx(a(x0), x1))
SEL(s(indxInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(indx(a(y0), y1), sel(a(x0), a(x1)))
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(a(y0), a(y1)), from(x0))
SEL(s(dblsInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbls(a(y0)), sel(a(x0), a(x1)))
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(a(y0), a(y1)), dbls(a(x0)))
SEL(s(sInact(y0)), cons(y1, dblInact(x0))) → SEL(s(y0), dbl(a(x0)))
SEL(s(dblsInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbls(a(y0)), dbls(a(x0)))
SEL(s(indxInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(indx(a(y0), y1), from(x0))
SEL(s(y0), cons(y1, selInact(x0, x1))) → SEL(y0, sel(a(x0), a(x1)))
SEL(s(dblInact(y0)), cons(y1, dblInact(x0))) → SEL(dbl(a(y0)), dbl(a(x0)))
SEL(s(indxInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(indx(a(y0), y1), dbl(a(x0)))
SEL(s(y0), cons(y1, dblInact(x0))) → SEL(y0, dbl(a(x0)))
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(a(y0), a(y1)), sel(a(x0), a(x1)))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(a(x1), a(x2))
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(a(x1))

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ QDPOrderProof

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ QDPOrderProof

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ AND

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                              ↳ QDP

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

SEL(s(y0), cons(y1, indxInact(x0, x1))) → SEL(y0, indx(a(x0), x1))
SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(a(x0)))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(a(y0), a(y1)), indx(a(x0), x1))
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(a(y0), a(y1)), dbl(a(x0)))
SEL(s(y0), cons(y1, sInact(x0))) → SEL(y0, s(x0))
SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(dblsInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbls(a(y0)), indx(a(x0), x1))
SEL(s(indxInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(indx(a(y0), y1), indx(a(x0), x1))
SEL(s(indxInact(y0, y1)), cons(y2, sInact(x0))) → SEL(indx(a(y0), y1), s(x0))
SEL(s(dblsInact(y0)), cons(y1, x0)) → SEL(dbls(a(y0)), x0)
SEL(s(dblInact(y0)), cons(y1, fromInact(x0))) → SEL(dbl(a(y0)), from(x0))
SEL(s(indxInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(indx(a(y0), y1), dbls(a(x0)))
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(a(x0), a(x1)))
SEL(s(sInact(y0)), cons(y1, fromInact(x0))) → SEL(s(y0), from(x0))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(a(y0), a(y1)), x0)
SEL(s(dblInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbl(a(y0)), sel(a(x0), a(x1)))
SEL(s(dblInact(y0)), cons(y1, sInact(x0))) → SEL(dbl(a(y0)), s(x0))
SEL(s(dblsInact(y0)), cons(y1, fromInact(x0))) → SEL(dbls(a(y0)), from(x0))
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(a(x0), x1))
SEL(s(y0), cons(y1, fromInact(x0))) → SEL(y0, from(x0))
SEL(s(indxInact(y0, y1)), cons(y2, x0)) → SEL(indx(a(y0), y1), x0)
SEL(s(dblsInact(y0)), cons(y1, dblInact(x0))) → SEL(dbls(a(y0)), dbl(a(x0)))
SEL(s(dblInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbl(a(y0)), dbls(a(x0)))
SEL(s(dblInact(y0)), cons(y1, x0)) → SEL(dbl(a(y0)), x0)
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)
SEL(s(y0), cons(y1, dblsInact(x0))) → SEL(y0, dbls(a(x0)))
SEL(s(indxInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(indx(a(y0), y1), sel(a(x0), a(x1)))
SEL(s(dblInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbl(a(y0)), indx(a(x0), x1))
SEL(s(dblsInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbls(a(y0)), sel(a(x0), a(x1)))
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(a(y0), a(y1)), from(x0))
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(a(y0), a(y1)), dbls(a(x0)))
SEL(s(dblsInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbls(a(y0)), dbls(a(x0)))
SEL(s(sInact(y0)), cons(y1, dblInact(x0))) → SEL(s(y0), dbl(a(x0)))
SEL(s(indxInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(indx(a(y0), y1), from(x0))
SEL(s(y0), cons(y1, selInact(x0, x1))) → SEL(y0, sel(a(x0), a(x1)))
SEL(s(indxInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(indx(a(y0), y1), dbl(a(x0)))
SEL(s(dblInact(y0)), cons(y1, dblInact(x0))) → SEL(dbl(a(y0)), dbl(a(x0)))
SEL(s(y0), cons(y1, dblInact(x0))) → SEL(y0, dbl(a(x0)))
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(a(y0), a(y1)), sel(a(x0), a(x1)))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(a(x1), a(x2))
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(a(x1))

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

SEL(s(dblInact(y0)), cons(y1, sInact(x0))) → SEL(dbl(a(y0)), sInact(x0))

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ QDPOrderProof

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ QDPOrderProof

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ AND

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                              ↳ QDP

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

SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(a(x0)))
SEL(s(y0), cons(y1, indxInact(x0, x1))) → SEL(y0, indx(a(x0), x1))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(a(y0), a(y1)), indx(a(x0), x1))
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(a(y0), a(y1)), dbl(a(x0)))
SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(y0), cons(y1, sInact(x0))) → SEL(y0, s(x0))
SEL(s(dblsInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbls(a(y0)), indx(a(x0), x1))
SEL(s(indxInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(indx(a(y0), y1), indx(a(x0), x1))
SEL(s(indxInact(y0, y1)), cons(y2, sInact(x0))) → SEL(indx(a(y0), y1), s(x0))
SEL(s(dblsInact(y0)), cons(y1, x0)) → SEL(dbls(a(y0)), x0)
SEL(s(dblInact(y0)), cons(y1, fromInact(x0))) → SEL(dbl(a(y0)), from(x0))
SEL(s(indxInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(indx(a(y0), y1), dbls(a(x0)))
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(a(x0), a(x1)))
SEL(s(sInact(y0)), cons(y1, fromInact(x0))) → SEL(s(y0), from(x0))
SEL(s(dblInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbl(a(y0)), sel(a(x0), a(x1)))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(a(y0), a(y1)), x0)
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(a(x0), x1))
SEL(s(dblsInact(y0)), cons(y1, fromInact(x0))) → SEL(dbls(a(y0)), from(x0))
SEL(s(y0), cons(y1, fromInact(x0))) → SEL(y0, from(x0))
SEL(s(dblInact(y0)), cons(y1, x0)) → SEL(dbl(a(y0)), x0)
SEL(s(dblInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbl(a(y0)), dbls(a(x0)))
SEL(s(dblsInact(y0)), cons(y1, dblInact(x0))) → SEL(dbls(a(y0)), dbl(a(x0)))
SEL(s(indxInact(y0, y1)), cons(y2, x0)) → SEL(indx(a(y0), y1), x0)
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)
SEL(s(dblInact(y0)), cons(y1, sInact(x0))) → SEL(dbl(a(y0)), sInact(x0))
SEL(s(y0), cons(y1, dblsInact(x0))) → SEL(y0, dbls(a(x0)))
SEL(s(dblInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbl(a(y0)), indx(a(x0), x1))
SEL(s(indxInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(indx(a(y0), y1), sel(a(x0), a(x1)))
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(a(y0), a(y1)), from(x0))
SEL(s(dblsInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbls(a(y0)), sel(a(x0), a(x1)))
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(a(y0), a(y1)), dbls(a(x0)))
SEL(s(sInact(y0)), cons(y1, dblInact(x0))) → SEL(s(y0), dbl(a(x0)))
SEL(s(dblsInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbls(a(y0)), dbls(a(x0)))
SEL(s(indxInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(indx(a(y0), y1), from(x0))
SEL(s(y0), cons(y1, selInact(x0, x1))) → SEL(y0, sel(a(x0), a(x1)))
SEL(s(dblInact(y0)), cons(y1, dblInact(x0))) → SEL(dbl(a(y0)), dbl(a(x0)))
SEL(s(indxInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(indx(a(y0), y1), dbl(a(x0)))
SEL(s(y0), cons(y1, dblInact(x0))) → SEL(y0, dbl(a(x0)))
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(a(y0), a(y1)), sel(a(x0), a(x1)))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(a(x1), a(x2))
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(a(x1))

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ QDPOrderProof

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ QDPOrderProof

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ AND

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Narrowing

                                              ↳ QDP

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

SEL(s(y0), cons(y1, indxInact(x0, x1))) → SEL(y0, indx(a(x0), x1))
SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(a(x0)))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(a(y0), a(y1)), indx(a(x0), x1))
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(a(y0), a(y1)), dbl(a(x0)))
SEL(s(y0), cons(y1, sInact(x0))) → SEL(y0, s(x0))
SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(dblsInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbls(a(y0)), indx(a(x0), x1))
SEL(s(indxInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(indx(a(y0), y1), indx(a(x0), x1))
SEL(s(indxInact(y0, y1)), cons(y2, sInact(x0))) → SEL(indx(a(y0), y1), s(x0))
SEL(s(dblsInact(y0)), cons(y1, x0)) → SEL(dbls(a(y0)), x0)
SEL(s(dblInact(y0)), cons(y1, fromInact(x0))) → SEL(dbl(a(y0)), from(x0))
SEL(s(indxInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(indx(a(y0), y1), dbls(a(x0)))
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(a(x0), a(x1)))
SEL(s(sInact(y0)), cons(y1, fromInact(x0))) → SEL(s(y0), from(x0))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(a(y0), a(y1)), x0)
SEL(s(dblInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbl(a(y0)), sel(a(x0), a(x1)))
SEL(s(dblsInact(y0)), cons(y1, fromInact(x0))) → SEL(dbls(a(y0)), from(x0))
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(a(x0), x1))
SEL(s(y0), cons(y1, fromInact(x0))) → SEL(y0, from(x0))
SEL(s(indxInact(y0, y1)), cons(y2, x0)) → SEL(indx(a(y0), y1), x0)
SEL(s(dblsInact(y0)), cons(y1, dblInact(x0))) → SEL(dbls(a(y0)), dbl(a(x0)))
SEL(s(dblInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbl(a(y0)), dbls(a(x0)))
SEL(s(dblInact(y0)), cons(y1, x0)) → SEL(dbl(a(y0)), x0)
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)
SEL(s(y0), cons(y1, dblsInact(x0))) → SEL(y0, dbls(a(x0)))
SEL(s(indxInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(indx(a(y0), y1), sel(a(x0), a(x1)))
SEL(s(dblInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbl(a(y0)), indx(a(x0), x1))
SEL(s(dblsInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbls(a(y0)), sel(a(x0), a(x1)))
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(a(y0), a(y1)), from(x0))
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(a(y0), a(y1)), dbls(a(x0)))
SEL(s(dblsInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbls(a(y0)), dbls(a(x0)))
SEL(s(sInact(y0)), cons(y1, dblInact(x0))) → SEL(s(y0), dbl(a(x0)))
SEL(s(indxInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(indx(a(y0), y1), from(x0))
SEL(s(y0), cons(y1, selInact(x0, x1))) → SEL(y0, sel(a(x0), a(x1)))
SEL(s(indxInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(indx(a(y0), y1), dbl(a(x0)))
SEL(s(dblInact(y0)), cons(y1, dblInact(x0))) → SEL(dbl(a(y0)), dbl(a(x0)))
SEL(s(y0), cons(y1, dblInact(x0))) → SEL(y0, dbl(a(x0)))
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(a(y0), a(y1)), sel(a(x0), a(x1)))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(a(x1), a(x2))
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(a(x1))

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

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ QDPOrderProof

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ QDPOrderProof

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ AND

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Narrowing

                                                                                                                      ↳ QDP

                                                                                                                        ↳ DependencyGraphProof

                                              ↳ QDP

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

SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(a(x0)))
SEL(s(y0), cons(y1, indxInact(x0, x1))) → SEL(y0, indx(a(x0), x1))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(a(y0), a(y1)), indx(a(x0), x1))
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(a(y0), a(y1)), dbl(a(x0)))
SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(dblsInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbls(a(y0)), indx(a(x0), x1))
SEL(s(indxInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(indx(a(y0), y1), indx(a(x0), x1))
SEL(s(indxInact(y0, y1)), cons(y2, sInact(x0))) → SEL(indx(a(y0), y1), s(x0))
SEL(s(dblsInact(y0)), cons(y1, x0)) → SEL(dbls(a(y0)), x0)
SEL(s(dblInact(y0)), cons(y1, fromInact(x0))) → SEL(dbl(a(y0)), from(x0))
SEL(s(indxInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(indx(a(y0), y1), dbls(a(x0)))
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(a(x0), a(x1)))
SEL(s(sInact(y0)), cons(y1, fromInact(x0))) → SEL(s(y0), from(x0))
SEL(s(y0), cons(y1, sInact(x0))) → SEL(y0, sInact(x0))
SEL(s(dblInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbl(a(y0)), sel(a(x0), a(x1)))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(a(y0), a(y1)), x0)
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(a(x0), x1))
SEL(s(dblsInact(y0)), cons(y1, fromInact(x0))) → SEL(dbls(a(y0)), from(x0))
SEL(s(y0), cons(y1, fromInact(x0))) → SEL(y0, from(x0))
SEL(s(dblInact(y0)), cons(y1, x0)) → SEL(dbl(a(y0)), x0)
SEL(s(dblInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbl(a(y0)), dbls(a(x0)))
SEL(s(dblsInact(y0)), cons(y1, dblInact(x0))) → SEL(dbls(a(y0)), dbl(a(x0)))
SEL(s(indxInact(y0, y1)), cons(y2, x0)) → SEL(indx(a(y0), y1), x0)
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)
SEL(s(y0), cons(y1, dblsInact(x0))) → SEL(y0, dbls(a(x0)))
SEL(s(dblInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbl(a(y0)), indx(a(x0), x1))
SEL(s(indxInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(indx(a(y0), y1), sel(a(x0), a(x1)))
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(a(y0), a(y1)), from(x0))
SEL(s(dblsInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbls(a(y0)), sel(a(x0), a(x1)))
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(a(y0), a(y1)), dbls(a(x0)))
SEL(s(sInact(y0)), cons(y1, dblInact(x0))) → SEL(s(y0), dbl(a(x0)))
SEL(s(dblsInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbls(a(y0)), dbls(a(x0)))
SEL(s(indxInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(indx(a(y0), y1), from(x0))
SEL(s(y0), cons(y1, selInact(x0, x1))) → SEL(y0, sel(a(x0), a(x1)))
SEL(s(dblInact(y0)), cons(y1, dblInact(x0))) → SEL(dbl(a(y0)), dbl(a(x0)))
SEL(s(indxInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(indx(a(y0), y1), dbl(a(x0)))
SEL(s(y0), cons(y1, dblInact(x0))) → SEL(y0, dbl(a(x0)))
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(a(y0), a(y1)), sel(a(x0), a(x1)))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(a(x1), a(x2))
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(a(x1))

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ QDPOrderProof

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ QDPOrderProof

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ AND

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Narrowing

                                                                                                                      ↳ QDP

                                                                                                                        ↳ DependencyGraphProof

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Narrowing

                                              ↳ QDP

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

SEL(s(y0), cons(y1, indxInact(x0, x1))) → SEL(y0, indx(a(x0), x1))
SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(a(x0)))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(a(y0), a(y1)), indx(a(x0), x1))
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(a(y0), a(y1)), dbl(a(x0)))
SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(dblsInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbls(a(y0)), indx(a(x0), x1))
SEL(s(indxInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(indx(a(y0), y1), indx(a(x0), x1))
SEL(s(indxInact(y0, y1)), cons(y2, sInact(x0))) → SEL(indx(a(y0), y1), s(x0))
SEL(s(dblsInact(y0)), cons(y1, x0)) → SEL(dbls(a(y0)), x0)
SEL(s(dblInact(y0)), cons(y1, fromInact(x0))) → SEL(dbl(a(y0)), from(x0))
SEL(s(indxInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(indx(a(y0), y1), dbls(a(x0)))
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(a(x0), a(x1)))
SEL(s(sInact(y0)), cons(y1, fromInact(x0))) → SEL(s(y0), from(x0))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(a(y0), a(y1)), x0)
SEL(s(dblInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbl(a(y0)), sel(a(x0), a(x1)))
SEL(s(dblsInact(y0)), cons(y1, fromInact(x0))) → SEL(dbls(a(y0)), from(x0))
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(a(x0), x1))
SEL(s(y0), cons(y1, fromInact(x0))) → SEL(y0, from(x0))
SEL(s(indxInact(y0, y1)), cons(y2, x0)) → SEL(indx(a(y0), y1), x0)
SEL(s(dblsInact(y0)), cons(y1, dblInact(x0))) → SEL(dbls(a(y0)), dbl(a(x0)))
SEL(s(dblInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbl(a(y0)), dbls(a(x0)))
SEL(s(dblInact(y0)), cons(y1, x0)) → SEL(dbl(a(y0)), x0)
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)
SEL(s(y0), cons(y1, dblsInact(x0))) → SEL(y0, dbls(a(x0)))
SEL(s(indxInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(indx(a(y0), y1), sel(a(x0), a(x1)))
SEL(s(dblInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbl(a(y0)), indx(a(x0), x1))
SEL(s(dblsInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbls(a(y0)), sel(a(x0), a(x1)))
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(a(y0), a(y1)), from(x0))
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(a(y0), a(y1)), dbls(a(x0)))
SEL(s(dblsInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbls(a(y0)), dbls(a(x0)))
SEL(s(sInact(y0)), cons(y1, dblInact(x0))) → SEL(s(y0), dbl(a(x0)))
SEL(s(indxInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(indx(a(y0), y1), from(x0))
SEL(s(y0), cons(y1, selInact(x0, x1))) → SEL(y0, sel(a(x0), a(x1)))
SEL(s(indxInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(indx(a(y0), y1), dbl(a(x0)))
SEL(s(dblInact(y0)), cons(y1, dblInact(x0))) → SEL(dbl(a(y0)), dbl(a(x0)))
SEL(s(y0), cons(y1, dblInact(x0))) → SEL(y0, dbl(a(x0)))
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(a(y0), a(y1)), sel(a(x0), a(x1)))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(a(x1), a(x2))
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(a(x1))

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

SEL(s(indxInact(y0, y1)), cons(y2, sInact(x0))) → SEL(indx(a(y0), y1), sInact(x0))

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ QDPOrderProof

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ QDPOrderProof

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ AND

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Narrowing

                                                                                                                      ↳ QDP

                                                                                                                        ↳ DependencyGraphProof

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Narrowing

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                              ↳ QDP

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

SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(a(x0)))
SEL(s(y0), cons(y1, indxInact(x0, x1))) → SEL(y0, indx(a(x0), x1))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(a(y0), a(y1)), indx(a(x0), x1))
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(a(y0), a(y1)), dbl(a(x0)))
SEL(s(indxInact(y0, y1)), cons(y2, sInact(x0))) → SEL(indx(a(y0), y1), sInact(x0))
SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(dblsInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbls(a(y0)), indx(a(x0), x1))
SEL(s(indxInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(indx(a(y0), y1), indx(a(x0), x1))
SEL(s(dblsInact(y0)), cons(y1, x0)) → SEL(dbls(a(y0)), x0)
SEL(s(dblInact(y0)), cons(y1, fromInact(x0))) → SEL(dbl(a(y0)), from(x0))
SEL(s(indxInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(indx(a(y0), y1), dbls(a(x0)))
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(a(x0), a(x1)))
SEL(s(sInact(y0)), cons(y1, fromInact(x0))) → SEL(s(y0), from(x0))
SEL(s(dblInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbl(a(y0)), sel(a(x0), a(x1)))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(a(y0), a(y1)), x0)
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(a(x0), x1))
SEL(s(dblsInact(y0)), cons(y1, fromInact(x0))) → SEL(dbls(a(y0)), from(x0))
SEL(s(y0), cons(y1, fromInact(x0))) → SEL(y0, from(x0))
SEL(s(dblInact(y0)), cons(y1, x0)) → SEL(dbl(a(y0)), x0)
SEL(s(dblInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbl(a(y0)), dbls(a(x0)))
SEL(s(dblsInact(y0)), cons(y1, dblInact(x0))) → SEL(dbls(a(y0)), dbl(a(x0)))
SEL(s(indxInact(y0, y1)), cons(y2, x0)) → SEL(indx(a(y0), y1), x0)
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)
SEL(s(y0), cons(y1, dblsInact(x0))) → SEL(y0, dbls(a(x0)))
SEL(s(dblInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbl(a(y0)), indx(a(x0), x1))
SEL(s(indxInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(indx(a(y0), y1), sel(a(x0), a(x1)))
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(a(y0), a(y1)), from(x0))
SEL(s(dblsInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbls(a(y0)), sel(a(x0), a(x1)))
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(a(y0), a(y1)), dbls(a(x0)))
SEL(s(sInact(y0)), cons(y1, dblInact(x0))) → SEL(s(y0), dbl(a(x0)))
SEL(s(dblsInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbls(a(y0)), dbls(a(x0)))
SEL(s(indxInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(indx(a(y0), y1), from(x0))
SEL(s(y0), cons(y1, selInact(x0, x1))) → SEL(y0, sel(a(x0), a(x1)))
SEL(s(dblInact(y0)), cons(y1, dblInact(x0))) → SEL(dbl(a(y0)), dbl(a(x0)))
SEL(s(indxInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(indx(a(y0), y1), dbl(a(x0)))
SEL(s(y0), cons(y1, dblInact(x0))) → SEL(y0, dbl(a(x0)))
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(a(y0), a(y1)), sel(a(x0), a(x1)))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(a(x1), a(x2))
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(a(x1))

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ QDPOrderProof

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ QDPOrderProof

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ AND

                                              ↳ QDP

                                                ↳ Narrowing

                                                  ↳ QDP

                                                    ↳ Narrowing

                                                      ↳ QDP

                                                        ↳ Narrowing

                                                          ↳ QDP

                                                            ↳ Narrowing

                                                              ↳ QDP

                                                                ↳ DependencyGraphProof

                                                                  ↳ QDP

                                                                    ↳ Narrowing

                                                                      ↳ QDP

                                                                        ↳ Narrowing

                                                                          ↳ QDP

                                                                            ↳ Narrowing

                                                                              ↳ QDP

                                                                                ↳ Narrowing

                                                                                  ↳ QDP

                                                                                    ↳ Narrowing

                                                                                      ↳ QDP

                                                                                        ↳ DependencyGraphProof

                                                                                          ↳ QDP

                                                                                            ↳ Narrowing

                                                                                              ↳ QDP

                                                                                                ↳ DependencyGraphProof

                                                                                                  ↳ QDP

                                                                                                    ↳ Narrowing

                                                                                                      ↳ QDP

                                                                                                        ↳ DependencyGraphProof

                                                                                                          ↳ QDP

                                                                                                            ↳ Narrowing

                                                                                                              ↳ QDP

                                                                                                                ↳ DependencyGraphProof

                                                                                                                  ↳ QDP

                                                                                                                    ↳ Narrowing

                                                                                                                      ↳ QDP

                                                                                                                        ↳ DependencyGraphProof

                                                                                                                          ↳ QDP

                                                                                                                            ↳ Narrowing

                                                                                                                              ↳ QDP

                                                                                                                                ↳ DependencyGraphProof

                                                                                                                                  ↳ QDP

                                              ↳ QDP

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

SEL(s(y0), cons(y1, indxInact(x0, x1))) → SEL(y0, indx(a(x0), x1))
SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(a(x0)))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(a(y0), a(y1)), indx(a(x0), x1))
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(a(y0), a(y1)), dbl(a(x0)))
SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(dblsInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbls(a(y0)), indx(a(x0), x1))
SEL(s(indxInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(indx(a(y0), y1), indx(a(x0), x1))
SEL(s(dblsInact(y0)), cons(y1, x0)) → SEL(dbls(a(y0)), x0)
SEL(s(dblInact(y0)), cons(y1, fromInact(x0))) → SEL(dbl(a(y0)), from(x0))
SEL(s(indxInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(indx(a(y0), y1), dbls(a(x0)))
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(a(x0), a(x1)))
SEL(s(sInact(y0)), cons(y1, fromInact(x0))) → SEL(s(y0), from(x0))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(a(y0), a(y1)), x0)
SEL(s(dblInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbl(a(y0)), sel(a(x0), a(x1)))
SEL(s(dblsInact(y0)), cons(y1, fromInact(x0))) → SEL(dbls(a(y0)), from(x0))
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(a(x0), x1))
SEL(s(y0), cons(y1, fromInact(x0))) → SEL(y0, from(x0))
SEL(s(indxInact(y0, y1)), cons(y2, x0)) → SEL(indx(a(y0), y1), x0)
SEL(s(dblsInact(y0)), cons(y1, dblInact(x0))) → SEL(dbls(a(y0)), dbl(a(x0)))
SEL(s(dblInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbl(a(y0)), dbls(a(x0)))
SEL(s(dblInact(y0)), cons(y1, x0)) → SEL(dbl(a(y0)), x0)
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)
SEL(s(y0), cons(y1, dblsInact(x0))) → SEL(y0, dbls(a(x0)))
SEL(s(indxInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(indx(a(y0), y1), sel(a(x0), a(x1)))
SEL(s(dblInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbl(a(y0)), indx(a(x0), x1))
SEL(s(dblsInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbls(a(y0)), sel(a(x0), a(x1)))
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(a(y0), a(y1)), from(x0))
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(a(y0), a(y1)), dbls(a(x0)))
SEL(s(dblsInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbls(a(y0)), dbls(a(x0)))
SEL(s(sInact(y0)), cons(y1, dblInact(x0))) → SEL(s(y0), dbl(a(x0)))
SEL(s(indxInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(indx(a(y0), y1), from(x0))
SEL(s(y0), cons(y1, selInact(x0, x1))) → SEL(y0, sel(a(x0), a(x1)))
SEL(s(indxInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(indx(a(y0), y1), dbl(a(x0)))
SEL(s(dblInact(y0)), cons(y1, dblInact(x0))) → SEL(dbl(a(y0)), dbl(a(x0)))
SEL(s(y0), cons(y1, dblInact(x0))) → SEL(y0, dbl(a(x0)))
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(a(y0), a(y1)), sel(a(x0), a(x1)))

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(a(x1), a(x2))
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(a(x1))

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

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ QDPOrderProof

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ QDPOrderProof

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ AND

                                              ↳ QDP

                                              ↳ QDP

                                                ↳ QDPOrderProof

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

A(dblInact(x1)) → DBL(a(x1))
A(fromInact(x1)) → FROM(x1)
A(dblInact(x1)) → A(x1)
FROM(X) → A(X)
DBL(s(X)) → A(X)

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(a(x1), a(x2))
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(a(x1))

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

The following pairs can be oriented strictly and are deleted.

A(dblInact(x1)) → DBL(a(x1))
A(dblInact(x1)) → A(x1)
FROM(X) → A(X)

The remaining pairs can at least be oriented weakly.

A(fromInact(x1)) → FROM(x1)
DBL(s(X)) → A(X)

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

POL(0) = 0
POL(A(x₁)) = x₁
POL(DBL(x₁)) = x₁
POL(FROM(x₁)) = 1 + x₁
POL(a(x₁)) = x₁
POL(cons(x₁, x₂)) = max(x₁, x₂)
POL(dbl(x₁)) = 1 + x₁
POL(dblInact(x₁)) = 1 + x₁
POL(dbls(x₁)) = 1 + x₁
POL(dblsInact(x₁)) = 1 + x₁
POL(from(x₁)) = 1 + x₁
POL(fromInact(x₁)) = 1 + x₁
POL(indx(x₁, x₂)) = x₂
POL(indxInact(x₁, x₂)) = x₂
POL(nil) = 0
POL(s(x₁)) = x₁
POL(sInact(x₁)) = x₁
POL(sel(x₁, x₂)) = x₂
POL(selInact(x₁, x₂)) = x₂

The following usable rules [17] were oriented:

a(dblsInact(x1)) → dbls(a(x1))
a(dblInact(x1)) → dbl(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
a(selInact(x1, x2)) → sel(a(x1), a(x2))
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
sel(x1, x2) → selInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
indx(x1, x2) → indxInact(x1, x2)
a(x) → x
from(X) → cons(a(X), fromInact(sInact(a(X))))
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
indx(nil, X) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
dbls(nil) → nil
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbl(0) → 0

↳ CSR

  ↳ CSRInnermostProof

    ↳ CSR

      ↳ CSDependencyPairsProof

      ↳ Zantema-Transformation

      ↳ Innermost Giesl Middeldorp-Transformation

      ↳ Trivial-Transformation

      ↳ Improved Ferreira Ribeiro-Transformation

        ↳ QTRS

          ↳ DependencyPairsProof

            ↳ QDP

              ↳ DependencyGraphProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ DependencyGraphProof

                            ↳ QDP

                              ↳ QDPOrderProof

                                ↳ QDP

                                  ↳ DependencyGraphProof

                                    ↳ QDP

                                      ↳ QDPOrderProof

                                        ↳ QDP

                                          ↳ DependencyGraphProof

                                            ↳ AND

                                              ↳ QDP

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ DependencyGraphProof

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

A(fromInact(x1)) → FROM(x1)
DBL(s(X)) → A(X)

The TRS R consists of the following rules:

dbl(0) → 0
dbl(s(X)) → s(sInact(dblInact(a(X))))
dbls(nil) → nil
dbls(cons(X, Y)) → cons(dblInact(a(X)), dblsInact(a(Y)))
sel(0, cons(X, Y)) → a(X)
sel(s(X), cons(Y, Z)) → sel(a(X), a(Z))
indx(nil, X) → nil
indx(cons(X, Y), Z) → cons(selInact(a(X), a(Z)), indxInact(a(Y), a(Z)))
from(X) → cons(a(X), fromInact(sInact(a(X))))
a(x) → x
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(a(x1), a(x2))
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(a(x1))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 2 less nodes.