MAYBE Termination proof of ../tpdb/TRS/CSR/Ex3_2_Luc97.trs
Termination of the following Term Rewriting System could not be shown:

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

dbl(0) → 0
dbl(s(X)) → s(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
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-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 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

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

↳ CSR
  ↳ Zantema-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
  ↳ Incomplete Giesl Middeldorp-Transformation
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-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
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(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:

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

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

↳ CSR
  ↳ Zantema-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
  ↳ Incomplete Giesl Middeldorp-Transformation
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

DBLS(cons(X, Y)) → A(X)
SEL(s(X), cons(Y, Z)) → SEL(a(X), a(Z))
A(sInact(x1)) → S(x1)
FROM(X) → A(X)
SEL(0, cons(X, Y)) → A(X)
INDX(cons(X, Y), Z) → A(X)
A(indxInact(x1, x2)) → INDX(x1, x2)
DBLS(cons(X, Y)) → A(Y)
DBL(s(X)) → A(X)
INDX(cons(X, Y), Z) → A(Y)
A(fromInact(x1)) → FROM(x1)
SEL(s(X), cons(Y, Z)) → A(Z)
A(selInact(x1, x2)) → SEL(x1, x2)
A(dblInact(x1)) → DBL(x1)
SEL(s(X), cons(Y, Z)) → A(X)
INDX(cons(X, Y), Z) → A(Z)
A(dblsInact(x1)) → DBLS(x1)
DBL(s(X)) → S(sInact(dblInact(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
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(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
  ↳ Zantema-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
  ↳ Incomplete Giesl Middeldorp-Transformation
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

SEL(s(X), cons(Y, Z)) → SEL(a(X), a(Z))
DBLS(cons(X, Y)) → A(X)
FROM(X) → A(X)
SEL(0, cons(X, Y)) → A(X)
INDX(cons(X, Y), Z) → A(X)
A(indxInact(x1, x2)) → INDX(x1, x2)
DBLS(cons(X, Y)) → A(Y)
DBL(s(X)) → A(X)
A(fromInact(x1)) → FROM(x1)
INDX(cons(X, Y), Z) → A(Y)
SEL(s(X), cons(Y, Z)) → A(Z)
A(selInact(x1, x2)) → SEL(x1, x2)
A(dblInact(x1)) → DBL(x1)
SEL(s(X), cons(Y, Z)) → A(X)
INDX(cons(X, Y), Z) → A(Z)
A(dblsInact(x1)) → DBLS(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
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(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(sInact(x0)), cons(y1, y2)) → SEL(s(x0), a(y2))
SEL(s(x0), cons(y1, y2)) → SEL(x0, a(y2))
SEL(s(indxInact(x0, x1)), cons(y1, y2)) → SEL(indx(x0, x1), a(y2))
SEL(s(dblInact(x0)), cons(y1, y2)) → SEL(dbl(x0), a(y2))
SEL(s(fromInact(x0)), cons(y1, y2)) → SEL(from(x0), a(y2))
SEL(s(dblsInact(x0)), cons(y1, y2)) → SEL(dbls(x0), a(y2))



↳ CSR
  ↳ Zantema-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
  ↳ Incomplete Giesl Middeldorp-Transformation
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

DBLS(cons(X, Y)) → A(X)
SEL(s(selInact(x0, x1)), cons(y1, y2)) → SEL(sel(x0, x1), a(y2))
SEL(s(x0), cons(y1, y2)) → SEL(x0, a(y2))
FROM(X) → A(X)
SEL(0, cons(X, Y)) → A(X)
INDX(cons(X, Y), Z) → A(X)
SEL(s(dblsInact(x0)), cons(y1, y2)) → SEL(dbls(x0), a(y2))
SEL(s(fromInact(x0)), cons(y1, y2)) → SEL(from(x0), a(y2))
DBLS(cons(X, Y)) → A(Y)
A(indxInact(x1, x2)) → INDX(x1, x2)
DBL(s(X)) → A(X)
INDX(cons(X, Y), Z) → A(Y)
A(fromInact(x1)) → FROM(x1)
SEL(s(X), cons(Y, Z)) → A(Z)
SEL(s(sInact(x0)), cons(y1, y2)) → SEL(s(x0), a(y2))
A(selInact(x1, x2)) → SEL(x1, x2)
A(dblInact(x1)) → DBL(x1)
SEL(s(X), cons(Y, Z)) → A(X)
INDX(cons(X, Y), Z) → A(Z)
SEL(s(dblInact(x0)), cons(y1, y2)) → SEL(dbl(x0), a(y2))
SEL(s(indxInact(x0, x1)), cons(y1, y2)) → SEL(indx(x0, x1), a(y2))
A(dblsInact(x1)) → DBLS(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
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(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, fromInact(x0))) → SEL(sel(y0, y1), from(x0))
SEL(s(selInact(y0, y1)), cons(y2, sInact(x0))) → SEL(sel(y0, y1), s(x0))
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(y0, y1), dbls(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(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(y0, y1), dbl(x0))



↳ CSR
  ↳ Zantema-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
  ↳ Incomplete Giesl Middeldorp-Transformation
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

DBLS(cons(X, Y)) → A(X)
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(0, cons(X, Y)) → A(X)
SEL(s(dblsInact(x0)), cons(y1, y2)) → SEL(dbls(x0), a(y2))
SEL(s(fromInact(x0)), cons(y1, y2)) → SEL(from(x0), a(y2))
INDX(cons(X, Y), Z) → A(Y)
A(fromInact(x1)) → FROM(x1)
A(selInact(x1, x2)) → SEL(x1, x2)
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
A(dblsInact(x1)) → DBLS(x1)
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
FROM(X) → A(X)
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(y0, y1), dbl(x0))
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))
INDX(cons(X, Y), Z) → A(X)
DBLS(cons(X, Y)) → A(Y)
A(indxInact(x1, x2)) → INDX(x1, x2)
DBL(s(X)) → A(X)
SEL(s(X), cons(Y, Z)) → A(Z)
SEL(s(sInact(x0)), cons(y1, y2)) → SEL(s(x0), a(y2))
SEL(s(selInact(y0, y1)), cons(y2, sInact(x0))) → SEL(sel(y0, y1), s(x0))
A(dblInact(x1)) → DBL(x1)
SEL(s(X), cons(Y, Z)) → A(X)
INDX(cons(X, Y), Z) → A(Z)
SEL(s(dblInact(x0)), cons(y1, y2)) → SEL(dbl(x0), a(y2))
SEL(s(indxInact(x0, x1)), cons(y1, y2)) → SEL(indx(x0, x1), 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
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(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, sInact(x0))) → SEL(s(y0), s(x0))
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(x0, x1))
SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(x0))
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)



↳ CSR
  ↳ Zantema-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
  ↳ Incomplete Giesl Middeldorp-Transformation
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

DBLS(cons(X, Y)) → A(X)
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(0, cons(X, Y)) → A(X)
SEL(s(fromInact(x0)), cons(y1, y2)) → SEL(from(x0), a(y2))
SEL(s(dblsInact(x0)), cons(y1, y2)) → SEL(dbls(x0), a(y2))
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)
A(fromInact(x1)) → FROM(x1)
INDX(cons(X, Y), Z) → A(Y)
A(selInact(x1, x2)) → SEL(x1, x2)
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(x0, x1))
A(dblsInact(x1)) → DBLS(x1)
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
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))
FROM(X) → A(X)
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(y0, y1), dbl(x0))
INDX(cons(X, Y), Z) → A(X)
SEL(s(sInact(y0)), cons(y1, sInact(x0))) → SEL(s(y0), s(x0))
SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(x0))
A(indxInact(x1, x2)) → INDX(x1, x2)
DBLS(cons(X, Y)) → A(Y)
DBL(s(X)) → A(X)
SEL(s(X), cons(Y, Z)) → A(Z)
SEL(s(selInact(y0, y1)), cons(y2, sInact(x0))) → SEL(sel(y0, y1), s(x0))
A(dblInact(x1)) → DBL(x1)
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(x0, x1))
SEL(s(X), cons(Y, Z)) → A(X)
INDX(cons(X, Y), Z) → A(Z)
SEL(s(indxInact(x0, x1)), cons(y1, y2)) → SEL(indx(x0, x1), 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
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(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, sInact(x0))) → SEL(y0, s(x0))
SEL(s(y0), cons(y1, dblInact(x0))) → SEL(y0, dbl(x0))
SEL(s(y0), cons(y1, dblsInact(x0))) → SEL(y0, dbls(x0))
SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(y0), cons(y1, fromInact(x0))) → SEL(y0, from(x0))
SEL(s(y0), cons(y1, selInact(x0, x1))) → SEL(y0, sel(x0, x1))



↳ CSR
  ↳ Zantema-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
  ↳ Incomplete Giesl Middeldorp-Transformation
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

DBLS(cons(X, Y)) → A(X)
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(y0, y1), from(x0))
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(y0, y1), dbls(x0))
SEL(0, cons(X, Y)) → A(X)
SEL(s(dblsInact(x0)), cons(y1, y2)) → SEL(dbls(x0), a(y2))
SEL(s(fromInact(x0)), cons(y1, y2)) → SEL(from(x0), a(y2))
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)
SEL(s(y0), cons(y1, indxInact(x0, x1))) → SEL(y0, indx(x0, x1))
INDX(cons(X, Y), Z) → A(Y)
A(fromInact(x1)) → FROM(x1)
A(selInact(x1, x2)) → SEL(x1, x2)
SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
A(dblsInact(x1)) → DBLS(x1)
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(x0, x1))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
SEL(s(y0), cons(y1, sInact(x0))) → SEL(y0, s(x0))
SEL(s(y0), cons(y1, dblsInact(x0))) → SEL(y0, dbls(x0))
SEL(s(sInact(y0)), cons(y1, dblInact(x0))) → SEL(s(y0), dbl(x0))
FROM(X) → A(X)
SEL(s(sInact(y0)), cons(y1, fromInact(x0))) → SEL(s(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, selInact(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))
INDX(cons(X, Y), Z) → A(X)
SEL(s(sInact(y0)), cons(y1, sInact(x0))) → SEL(s(y0), s(x0))
SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(x0))
DBLS(cons(X, Y)) → A(Y)
A(indxInact(x1, x2)) → INDX(x1, x2)
DBL(s(X)) → A(X)
SEL(s(X), cons(Y, Z)) → A(Z)
SEL(s(y0), cons(y1, dblInact(x0))) → SEL(y0, dbl(x0))
SEL(s(selInact(y0, y1)), cons(y2, sInact(x0))) → SEL(sel(y0, y1), s(x0))
A(dblInact(x1)) → DBL(x1)
SEL(s(X), cons(Y, Z)) → A(X)
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(x0, x1))
INDX(cons(X, Y), Z) → A(Z)
SEL(s(dblInact(x0)), cons(y1, y2)) → SEL(dbl(x0), a(y2))
SEL(s(indxInact(x0, x1)), cons(y1, y2)) → SEL(indx(x0, x1), a(y2))
SEL(s(y0), cons(y1, fromInact(x0))) → SEL(y0, from(x0))
SEL(s(y0), cons(y1, selInact(x0, x1))) → SEL(y0, sel(x0, 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
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(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(nil, x0)), cons(y2, y3)) → SEL(nil, a(y3))
SEL(s(indxInact(x0, x1)), cons(y2, y3)) → SEL(indxInact(x0, x1), a(y3))
SEL(s(indxInact(cons(x0, x1), x2)), cons(y2, y3)) → SEL(cons(selInact(a(x0), a(x2)), indxInact(a(x1), a(x2))), a(y3))



↳ CSR
  ↳ Zantema-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
  ↳ Incomplete Giesl Middeldorp-Transformation
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

DBLS(cons(X, Y)) → A(X)
SEL(s(indxInact(nil, x0)), cons(y2, y3)) → SEL(nil, a(y3))
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(y0, y1), from(x0))
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(y0, y1), dbls(x0))
SEL(0, cons(X, Y)) → A(X)
SEL(s(fromInact(x0)), cons(y1, y2)) → SEL(from(x0), a(y2))
SEL(s(dblsInact(x0)), cons(y1, y2)) → SEL(dbls(x0), a(y2))
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)
A(fromInact(x1)) → FROM(x1)
INDX(cons(X, Y), Z) → A(Y)
SEL(s(y0), cons(y1, indxInact(x0, x1))) → SEL(y0, indx(x0, x1))
SEL(s(indxInact(x0, x1)), cons(y2, y3)) → SEL(indxInact(x0, x1), a(y3))
A(selInact(x1, x2)) → SEL(x1, x2)
SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(x0, x1))
A(dblsInact(x1)) → DBLS(x1)
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
SEL(s(y0), cons(y1, sInact(x0))) → SEL(y0, s(x0))
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(y0), cons(y1, dblsInact(x0))) → SEL(y0, dbls(x0))
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))
FROM(X) → A(X)
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(y0, y1), dbl(x0))
INDX(cons(X, Y), Z) → A(X)
SEL(s(sInact(y0)), cons(y1, sInact(x0))) → SEL(s(y0), s(x0))
SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(x0))
A(indxInact(x1, x2)) → INDX(x1, x2)
DBLS(cons(X, Y)) → A(Y)
DBL(s(X)) → A(X)
SEL(s(X), cons(Y, Z)) → A(Z)
SEL(s(selInact(y0, y1)), cons(y2, sInact(x0))) → SEL(sel(y0, y1), s(x0))
SEL(s(y0), cons(y1, dblInact(x0))) → SEL(y0, dbl(x0))
A(dblInact(x1)) → DBL(x1)
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(x0, x1))
SEL(s(X), cons(Y, Z)) → A(X)
INDX(cons(X, Y), Z) → A(Z)
SEL(s(dblInact(x0)), cons(y1, y2)) → SEL(dbl(x0), a(y2))
SEL(s(y0), cons(y1, fromInact(x0))) → SEL(y0, from(x0))
SEL(s(y0), cons(y1, selInact(x0, x1))) → SEL(y0, sel(x0, 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
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(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
  ↳ Zantema-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
  ↳ Incomplete Giesl Middeldorp-Transformation
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

DBLS(cons(X, Y)) → A(X)
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(y0, y1), from(x0))
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(y0, y1), dbls(x0))
SEL(0, cons(X, Y)) → A(X)
SEL(s(fromInact(x0)), cons(y1, y2)) → SEL(from(x0), a(y2))
SEL(s(dblsInact(x0)), cons(y1, y2)) → SEL(dbls(x0), a(y2))
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)
SEL(s(y0), cons(y1, indxInact(x0, x1))) → SEL(y0, indx(x0, x1))
INDX(cons(X, Y), Z) → A(Y)
A(fromInact(x1)) → FROM(x1)
A(selInact(x1, x2)) → SEL(x1, x2)
SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(x0, x1))
A(dblsInact(x1)) → DBLS(x1)
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
SEL(s(y0), cons(y1, sInact(x0))) → SEL(y0, s(x0))
SEL(s(y0), cons(y1, dblsInact(x0))) → SEL(y0, dbls(x0))
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))
FROM(X) → A(X)
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(y0, y1), dbl(x0))
INDX(cons(X, Y), Z) → A(X)
SEL(s(sInact(y0)), cons(y1, sInact(x0))) → SEL(s(y0), s(x0))
SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(x0))
DBLS(cons(X, Y)) → A(Y)
A(indxInact(x1, x2)) → INDX(x1, x2)
DBL(s(X)) → A(X)
SEL(s(X), cons(Y, Z)) → A(Z)
SEL(s(y0), cons(y1, dblInact(x0))) → SEL(y0, dbl(x0))
SEL(s(selInact(y0, y1)), cons(y2, sInact(x0))) → SEL(sel(y0, y1), s(x0))
A(dblInact(x1)) → DBL(x1)
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(x0, x1))
SEL(s(X), cons(Y, Z)) → A(X)
INDX(cons(X, Y), Z) → A(Z)
SEL(s(dblInact(x0)), cons(y1, y2)) → SEL(dbl(x0), a(y2))
SEL(s(y0), cons(y1, fromInact(x0))) → SEL(y0, from(x0))
SEL(s(y0), cons(y1, selInact(x0, x1))) → SEL(y0, sel(x0, 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
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(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, x0)) → SEL(dbl(y0), x0)
SEL(s(dblInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbl(y0), indx(x0, x1))
SEL(s(dblInact(y0)), cons(y1, fromInact(x0))) → SEL(dbl(y0), from(x0))
SEL(s(dblInact(y0)), cons(y1, sInact(x0))) → SEL(dbl(y0), s(x0))
SEL(s(dblInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbl(y0), sel(x0, x1))
SEL(s(dblInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbl(y0), dbls(x0))
SEL(s(dblInact(y0)), cons(y1, dblInact(x0))) → SEL(dbl(y0), dbl(x0))



↳ CSR
  ↳ Zantema-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ Narrowing
  ↳ Incomplete Giesl Middeldorp-Transformation
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

DBLS(cons(X, Y)) → A(X)
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(y0, y1), from(x0))
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(y0, y1), dbls(x0))
SEL(0, cons(X, Y)) → A(X)
SEL(s(dblsInact(x0)), cons(y1, y2)) → SEL(dbls(x0), a(y2))
SEL(s(fromInact(x0)), cons(y1, y2)) → SEL(from(x0), a(y2))
SEL(s(dblInact(y0)), cons(y1, dblInact(x0))) → SEL(dbl(y0), dbl(x0))
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)
SEL(s(dblInact(y0)), cons(y1, x0)) → SEL(dbl(y0), x0)
A(fromInact(x1)) → FROM(x1)
INDX(cons(X, Y), Z) → A(Y)
SEL(s(y0), cons(y1, indxInact(x0, x1))) → SEL(y0, indx(x0, x1))
A(selInact(x1, x2)) → SEL(x1, x2)
SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(dblInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbl(y0), sel(x0, x1))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
A(dblsInact(x1)) → DBLS(x1)
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(x0, x1))
SEL(s(dblInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbl(y0), dbls(x0))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
SEL(s(y0), cons(y1, sInact(x0))) → SEL(y0, s(x0))
SEL(s(dblInact(y0)), cons(y1, fromInact(x0))) → SEL(dbl(y0), from(x0))
SEL(s(y0), cons(y1, dblsInact(x0))) → SEL(y0, 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))
FROM(X) → A(X)
SEL(s(sInact(y0)), cons(y1, fromInact(x0))) → SEL(s(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, selInact(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))
INDX(cons(X, Y), Z) → A(X)
SEL(s(sInact(y0)), cons(y1, sInact(x0))) → SEL(s(y0), s(x0))
SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(x0))
A(indxInact(x1, x2)) → INDX(x1, x2)
DBLS(cons(X, Y)) → A(Y)
DBL(s(X)) → A(X)
SEL(s(dblInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbl(y0), indx(x0, x1))
SEL(s(X), cons(Y, Z)) → A(Z)
SEL(s(selInact(y0, y1)), cons(y2, sInact(x0))) → SEL(sel(y0, y1), s(x0))
SEL(s(y0), cons(y1, dblInact(x0))) → SEL(y0, dbl(x0))
A(dblInact(x1)) → DBL(x1)
SEL(s(X), cons(Y, Z)) → A(X)
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(x0, x1))
INDX(cons(X, Y), Z) → A(Z)
SEL(s(y0), cons(y1, fromInact(x0))) → SEL(y0, from(x0))
SEL(s(y0), cons(y1, selInact(x0, x1))) → SEL(y0, sel(x0, 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
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(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(cons(a(x0), fromInact(sInact(a(x0)))), a(y2))
SEL(s(fromInact(x0)), cons(y1, y2)) → SEL(fromInact(x0), a(y2))



↳ CSR
  ↳ Zantema-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ DependencyGraphProof
  ↳ Incomplete Giesl Middeldorp-Transformation
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

DBLS(cons(X, Y)) → A(X)
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(y0, y1), from(x0))
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(y0, y1), dbls(x0))
SEL(0, cons(X, Y)) → A(X)
SEL(s(dblsInact(x0)), cons(y1, y2)) → SEL(dbls(x0), a(y2))
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)
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, indxInact(x0, x1))) → SEL(y0, indx(x0, x1))
INDX(cons(X, Y), Z) → A(Y)
A(fromInact(x1)) → FROM(x1)
SEL(s(fromInact(x0)), cons(y1, y2)) → SEL(fromInact(x0), a(y2))
A(selInact(x1, x2)) → SEL(x1, x2)
SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(dblInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbl(y0), sel(x0, x1))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(x0, x1))
A(dblsInact(x1)) → DBLS(x1)
SEL(s(dblInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbl(y0), dbls(x0))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
SEL(s(y0), cons(y1, sInact(x0))) → SEL(y0, s(x0))
SEL(s(dblInact(y0)), cons(y1, fromInact(x0))) → SEL(dbl(y0), from(x0))
SEL(s(y0), cons(y1, dblsInact(x0))) → SEL(y0, 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(fromInact(x0)), cons(y1, y2)) → SEL(cons(a(x0), fromInact(sInact(a(x0)))), a(y2))
SEL(s(sInact(y0)), cons(y1, fromInact(x0))) → SEL(s(y0), from(x0))
FROM(X) → A(X)
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(y0, y1), dbl(x0))
INDX(cons(X, Y), Z) → A(X)
SEL(s(sInact(y0)), cons(y1, sInact(x0))) → SEL(s(y0), s(x0))
SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(x0))
DBLS(cons(X, Y)) → A(Y)
A(indxInact(x1, x2)) → INDX(x1, x2)
DBL(s(X)) → A(X)
SEL(s(dblInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbl(y0), indx(x0, x1))
SEL(s(X), cons(Y, Z)) → A(Z)
SEL(s(y0), cons(y1, dblInact(x0))) → SEL(y0, dbl(x0))
SEL(s(selInact(y0, y1)), cons(y2, sInact(x0))) → SEL(sel(y0, y1), s(x0))
A(dblInact(x1)) → DBL(x1)
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(x0, x1))
SEL(s(X), cons(Y, Z)) → A(X)
INDX(cons(X, Y), Z) → A(Z)
SEL(s(y0), cons(y1, fromInact(x0))) → SEL(y0, from(x0))
SEL(s(y0), cons(y1, selInact(x0, x1))) → SEL(y0, sel(x0, 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
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(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
  ↳ Zantema-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ DependencyGraphProof
                                                ↳ QDP
                                                  ↳ Narrowing
  ↳ Incomplete Giesl Middeldorp-Transformation
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

DBLS(cons(X, Y)) → A(X)
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(y0, y1), from(x0))
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(y0, y1), dbls(x0))
SEL(0, cons(X, Y)) → A(X)
SEL(s(dblsInact(x0)), cons(y1, y2)) → SEL(dbls(x0), a(y2))
SEL(s(dblInact(y0)), cons(y1, dblInact(x0))) → SEL(dbl(y0), dbl(x0))
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)
SEL(s(dblInact(y0)), cons(y1, x0)) → SEL(dbl(y0), x0)
SEL(s(y0), cons(y1, indxInact(x0, x1))) → SEL(y0, indx(x0, x1))
INDX(cons(X, Y), Z) → A(Y)
A(fromInact(x1)) → FROM(x1)
A(selInact(x1, x2)) → SEL(x1, x2)
SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(dblInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbl(y0), sel(x0, x1))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(x0, x1))
A(dblsInact(x1)) → DBLS(x1)
SEL(s(dblInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbl(y0), dbls(x0))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
SEL(s(y0), cons(y1, sInact(x0))) → SEL(y0, s(x0))
SEL(s(dblInact(y0)), cons(y1, fromInact(x0))) → SEL(dbl(y0), from(x0))
SEL(s(y0), cons(y1, dblsInact(x0))) → SEL(y0, 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))
FROM(X) → A(X)
SEL(s(sInact(y0)), cons(y1, fromInact(x0))) → SEL(s(y0), from(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, dblInact(x0))) → SEL(sel(y0, y1), dbl(x0))
INDX(cons(X, Y), Z) → A(X)
SEL(s(sInact(y0)), cons(y1, sInact(x0))) → SEL(s(y0), s(x0))
SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(x0))
DBLS(cons(X, Y)) → A(Y)
A(indxInact(x1, x2)) → INDX(x1, x2)
DBL(s(X)) → A(X)
SEL(s(dblInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbl(y0), indx(x0, x1))
SEL(s(X), cons(Y, Z)) → A(Z)
SEL(s(y0), cons(y1, dblInact(x0))) → SEL(y0, dbl(x0))
SEL(s(selInact(y0, y1)), cons(y2, sInact(x0))) → SEL(sel(y0, y1), s(x0))
A(dblInact(x1)) → DBL(x1)
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(x0, x1))
SEL(s(X), cons(Y, Z)) → A(X)
INDX(cons(X, Y), Z) → A(Z)
SEL(s(y0), cons(y1, fromInact(x0))) → SEL(y0, from(x0))
SEL(s(y0), cons(y1, selInact(x0, x1))) → SEL(y0, sel(x0, 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
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(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(nil)), cons(y1, y2)) → SEL(nil, a(y2))
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))



↳ CSR
  ↳ Zantema-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ DependencyGraphProof
                                                ↳ QDP
                                                  ↳ Narrowing
                                                    ↳ QDP
                                                      ↳ DependencyGraphProof
  ↳ Incomplete Giesl Middeldorp-Transformation
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

DBLS(cons(X, Y)) → A(X)
SEL(s(dblsInact(nil)), cons(y1, y2)) → SEL(nil, a(y2))
SEL(s(dblsInact(x0)), cons(y1, y2)) → SEL(dblsInact(x0), a(y2))
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(y0, y1), from(x0))
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(y0, y1), dbls(x0))
SEL(0, cons(X, Y)) → A(X)
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)
SEL(s(dblInact(y0)), cons(y1, dblInact(x0))) → SEL(dbl(y0), dbl(x0))
SEL(s(dblInact(y0)), cons(y1, x0)) → SEL(dbl(y0), x0)
A(fromInact(x1)) → FROM(x1)
INDX(cons(X, Y), Z) → A(Y)
SEL(s(y0), cons(y1, indxInact(x0, x1))) → SEL(y0, indx(x0, x1))
A(selInact(x1, x2)) → SEL(x1, x2)
SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(dblInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbl(y0), sel(x0, x1))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
A(dblsInact(x1)) → DBLS(x1)
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(x0, x1))
SEL(s(dblInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbl(y0), dbls(x0))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
SEL(s(y0), cons(y1, sInact(x0))) → SEL(y0, s(x0))
SEL(s(dblInact(y0)), cons(y1, fromInact(x0))) → SEL(dbl(y0), from(x0))
SEL(s(y0), cons(y1, dblsInact(x0))) → SEL(y0, 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, fromInact(x0))) → SEL(s(y0), from(x0))
FROM(X) → A(X)
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(y0, y1), dbl(x0))
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))
INDX(cons(X, Y), Z) → A(X)
SEL(s(sInact(y0)), cons(y1, sInact(x0))) → SEL(s(y0), s(x0))
SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(x0))
A(indxInact(x1, x2)) → INDX(x1, x2)
DBLS(cons(X, Y)) → A(Y)
DBL(s(X)) → A(X)
SEL(s(dblInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbl(y0), indx(x0, x1))
SEL(s(X), cons(Y, Z)) → A(Z)
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, sInact(x0))) → SEL(sel(y0, y1), s(x0))
SEL(s(y0), cons(y1, dblInact(x0))) → SEL(y0, dbl(x0))
A(dblInact(x1)) → DBL(x1)
SEL(s(X), cons(Y, Z)) → A(X)
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(x0, x1))
INDX(cons(X, Y), Z) → A(Z)
SEL(s(y0), cons(y1, fromInact(x0))) → SEL(y0, from(x0))
SEL(s(y0), cons(y1, selInact(x0, x1))) → SEL(y0, sel(x0, 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
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(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
  ↳ Zantema-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ DependencyGraphProof
                                                ↳ QDP
                                                  ↳ Narrowing
                                                    ↳ QDP
                                                      ↳ DependencyGraphProof
                                                        ↳ QDP
                                                          ↳ Narrowing
  ↳ Incomplete Giesl Middeldorp-Transformation
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

DBLS(cons(X, Y)) → A(X)
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(y0, y1), from(x0))
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(y0, y1), dbls(x0))
SEL(0, cons(X, Y)) → A(X)
SEL(s(dblInact(y0)), cons(y1, dblInact(x0))) → SEL(dbl(y0), dbl(x0))
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)
SEL(s(dblInact(y0)), cons(y1, x0)) → SEL(dbl(y0), x0)
SEL(s(y0), cons(y1, indxInact(x0, x1))) → SEL(y0, indx(x0, x1))
INDX(cons(X, Y), Z) → A(Y)
A(fromInact(x1)) → FROM(x1)
A(selInact(x1, x2)) → SEL(x1, x2)
SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(dblInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbl(y0), sel(x0, x1))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(x0, x1))
A(dblsInact(x1)) → DBLS(x1)
SEL(s(dblInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbl(y0), dbls(x0))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
SEL(s(y0), cons(y1, sInact(x0))) → SEL(y0, s(x0))
SEL(s(dblInact(y0)), cons(y1, fromInact(x0))) → SEL(dbl(y0), from(x0))
SEL(s(y0), cons(y1, dblsInact(x0))) → SEL(y0, 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))
FROM(X) → A(X)
SEL(s(sInact(y0)), cons(y1, fromInact(x0))) → SEL(s(y0), from(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, dblInact(x0))) → SEL(sel(y0, y1), dbl(x0))
INDX(cons(X, Y), Z) → A(X)
SEL(s(sInact(y0)), cons(y1, sInact(x0))) → SEL(s(y0), s(x0))
SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(x0))
DBLS(cons(X, Y)) → A(Y)
A(indxInact(x1, x2)) → INDX(x1, x2)
DBL(s(X)) → A(X)
SEL(s(dblInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbl(y0), indx(x0, x1))
SEL(s(X), cons(Y, Z)) → A(Z)
SEL(s(y0), cons(y1, dblInact(x0))) → SEL(y0, dbl(x0))
SEL(s(selInact(y0, y1)), cons(y2, sInact(x0))) → SEL(sel(y0, y1), s(x0))
A(dblInact(x1)) → DBL(x1)
SEL(s(X), cons(Y, Z)) → A(X)
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(x0, x1))
INDX(cons(X, Y), Z) → A(Z)
SEL(s(y0), cons(y1, fromInact(x0))) → SEL(y0, from(x0))
SEL(s(y0), cons(y1, selInact(x0, x1))) → SEL(y0, sel(x0, 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
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(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
  ↳ Zantema-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ DependencyGraphProof
                                                ↳ QDP
                                                  ↳ Narrowing
                                                    ↳ QDP
                                                      ↳ DependencyGraphProof
                                                        ↳ QDP
                                                          ↳ Narrowing
                                                            ↳ QDP
                                                              ↳ DependencyGraphProof
  ↳ Incomplete Giesl Middeldorp-Transformation
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

DBLS(cons(X, Y)) → A(X)
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(y0, y1), from(x0))
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(y0, y1), dbls(x0))
SEL(0, cons(X, Y)) → A(X)
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)
SEL(s(dblInact(y0)), cons(y1, dblInact(x0))) → SEL(dbl(y0), dbl(x0))
SEL(s(dblInact(y0)), cons(y1, x0)) → SEL(dbl(y0), x0)
A(fromInact(x1)) → FROM(x1)
INDX(cons(X, Y), Z) → A(Y)
SEL(s(y0), cons(y1, indxInact(x0, x1))) → SEL(y0, indx(x0, x1))
A(selInact(x1, x2)) → SEL(x1, x2)
SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(dblInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbl(y0), sel(x0, x1))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
A(dblsInact(x1)) → DBLS(x1)
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(x0, x1))
SEL(s(dblInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbl(y0), dbls(x0))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
SEL(s(y0), cons(y1, sInact(x0))) → SEL(y0, s(x0))
SEL(s(dblInact(y0)), cons(y1, fromInact(x0))) → SEL(dbl(y0), from(x0))
SEL(s(y0), cons(y1, dblsInact(x0))) → SEL(y0, 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, fromInact(x0))) → SEL(s(y0), from(x0))
FROM(X) → A(X)
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(y0, y1), dbl(x0))
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))
INDX(cons(X, Y), Z) → A(X)
SEL(s(sInact(y0)), cons(y1, sInact(x0))) → SEL(s(y0), s(x0))
SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(x0))
A(indxInact(x1, x2)) → INDX(x1, x2)
DBLS(cons(X, Y)) → A(Y)
SEL(s(selInact(y0, y1)), cons(y2, sInact(x0))) → SEL(sel(y0, y1), sInact(x0))
DBL(s(X)) → A(X)
SEL(s(dblInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbl(y0), indx(x0, x1))
SEL(s(X), cons(Y, Z)) → A(Z)
SEL(s(y0), cons(y1, dblInact(x0))) → SEL(y0, dbl(x0))
A(dblInact(x1)) → DBL(x1)
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(x0, x1))
SEL(s(X), cons(Y, Z)) → A(X)
INDX(cons(X, Y), Z) → A(Z)
SEL(s(y0), cons(y1, fromInact(x0))) → SEL(y0, from(x0))
SEL(s(y0), cons(y1, selInact(x0, x1))) → SEL(y0, sel(x0, 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
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(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
  ↳ Zantema-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ DependencyGraphProof
                                                ↳ QDP
                                                  ↳ Narrowing
                                                    ↳ QDP
                                                      ↳ DependencyGraphProof
                                                        ↳ QDP
                                                          ↳ Narrowing
                                                            ↳ QDP
                                                              ↳ DependencyGraphProof
                                                                ↳ QDP
                                                                  ↳ Narrowing
  ↳ Incomplete Giesl Middeldorp-Transformation
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

DBLS(cons(X, Y)) → A(X)
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(y0, y1), from(x0))
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(y0, y1), dbls(x0))
SEL(0, cons(X, Y)) → A(X)
SEL(s(dblInact(y0)), cons(y1, dblInact(x0))) → SEL(dbl(y0), dbl(x0))
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)
SEL(s(dblInact(y0)), cons(y1, x0)) → SEL(dbl(y0), x0)
SEL(s(y0), cons(y1, indxInact(x0, x1))) → SEL(y0, indx(x0, x1))
INDX(cons(X, Y), Z) → A(Y)
A(fromInact(x1)) → FROM(x1)
A(selInact(x1, x2)) → SEL(x1, x2)
SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(dblInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbl(y0), sel(x0, x1))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(x0, x1))
A(dblsInact(x1)) → DBLS(x1)
SEL(s(dblInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbl(y0), dbls(x0))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
SEL(s(y0), cons(y1, sInact(x0))) → SEL(y0, s(x0))
SEL(s(dblInact(y0)), cons(y1, fromInact(x0))) → SEL(dbl(y0), from(x0))
SEL(s(y0), cons(y1, dblsInact(x0))) → SEL(y0, 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))
FROM(X) → A(X)
SEL(s(sInact(y0)), cons(y1, fromInact(x0))) → SEL(s(y0), from(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, dblInact(x0))) → SEL(sel(y0, y1), dbl(x0))
INDX(cons(X, Y), Z) → A(X)
SEL(s(sInact(y0)), cons(y1, sInact(x0))) → SEL(s(y0), s(x0))
SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(x0))
DBLS(cons(X, Y)) → A(Y)
A(indxInact(x1, x2)) → INDX(x1, x2)
DBL(s(X)) → A(X)
SEL(s(dblInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbl(y0), indx(x0, x1))
SEL(s(X), cons(Y, Z)) → A(Z)
SEL(s(y0), cons(y1, dblInact(x0))) → SEL(y0, dbl(x0))
A(dblInact(x1)) → DBL(x1)
SEL(s(X), cons(Y, Z)) → A(X)
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(x0, x1))
INDX(cons(X, Y), Z) → A(Z)
SEL(s(y0), cons(y1, fromInact(x0))) → SEL(y0, from(x0))
SEL(s(y0), cons(y1, selInact(x0, x1))) → SEL(y0, sel(x0, 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
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(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
  ↳ Zantema-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ DependencyGraphProof
                                                ↳ QDP
                                                  ↳ Narrowing
                                                    ↳ QDP
                                                      ↳ DependencyGraphProof
                                                        ↳ QDP
                                                          ↳ Narrowing
                                                            ↳ QDP
                                                              ↳ DependencyGraphProof
                                                                ↳ QDP
                                                                  ↳ Narrowing
                                                                    ↳ QDP
                                                                      ↳ DependencyGraphProof
  ↳ Incomplete Giesl Middeldorp-Transformation
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

DBLS(cons(X, Y)) → A(X)
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(y0, y1), from(x0))
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(y0, y1), dbls(x0))
SEL(0, cons(X, Y)) → A(X)
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)
SEL(s(dblInact(y0)), cons(y1, dblInact(x0))) → SEL(dbl(y0), dbl(x0))
SEL(s(dblInact(y0)), cons(y1, x0)) → SEL(dbl(y0), x0)
A(fromInact(x1)) → FROM(x1)
INDX(cons(X, Y), Z) → A(Y)
SEL(s(y0), cons(y1, indxInact(x0, x1))) → SEL(y0, indx(x0, x1))
A(selInact(x1, x2)) → SEL(x1, x2)
SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(dblInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbl(y0), sel(x0, x1))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
A(dblsInact(x1)) → DBLS(x1)
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(x0, x1))
SEL(s(dblInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbl(y0), dbls(x0))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
SEL(s(y0), cons(y1, sInact(x0))) → SEL(y0, s(x0))
SEL(s(dblInact(y0)), cons(y1, fromInact(x0))) → SEL(dbl(y0), from(x0))
SEL(s(y0), cons(y1, dblsInact(x0))) → SEL(y0, dbls(x0))
SEL(s(sInact(y0)), cons(y1, sInact(x0))) → SEL(s(y0), sInact(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, fromInact(x0))) → SEL(s(y0), from(x0))
FROM(X) → A(X)
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(y0, y1), dbl(x0))
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))
INDX(cons(X, Y), Z) → A(X)
SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(x0))
A(indxInact(x1, x2)) → INDX(x1, x2)
DBLS(cons(X, Y)) → A(Y)
DBL(s(X)) → A(X)
SEL(s(dblInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbl(y0), indx(x0, x1))
SEL(s(X), cons(Y, Z)) → A(Z)
SEL(s(y0), cons(y1, dblInact(x0))) → SEL(y0, dbl(x0))
A(dblInact(x1)) → DBL(x1)
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(x0, x1))
SEL(s(X), cons(Y, Z)) → A(X)
INDX(cons(X, Y), Z) → A(Z)
SEL(s(y0), cons(y1, fromInact(x0))) → SEL(y0, from(x0))
SEL(s(y0), cons(y1, selInact(x0, x1))) → SEL(y0, sel(x0, 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
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(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
  ↳ Zantema-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ DependencyGraphProof
                                                ↳ QDP
                                                  ↳ Narrowing
                                                    ↳ QDP
                                                      ↳ DependencyGraphProof
                                                        ↳ QDP
                                                          ↳ Narrowing
                                                            ↳ QDP
                                                              ↳ DependencyGraphProof
                                                                ↳ QDP
                                                                  ↳ Narrowing
                                                                    ↳ QDP
                                                                      ↳ DependencyGraphProof
                                                                        ↳ QDP
                                                                          ↳ Narrowing
  ↳ Incomplete Giesl Middeldorp-Transformation
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

DBLS(cons(X, Y)) → A(X)
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(y0, y1), from(x0))
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(y0, y1), dbls(x0))
SEL(0, cons(X, Y)) → A(X)
SEL(s(dblInact(y0)), cons(y1, dblInact(x0))) → SEL(dbl(y0), dbl(x0))
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)
SEL(s(dblInact(y0)), cons(y1, x0)) → SEL(dbl(y0), x0)
SEL(s(y0), cons(y1, indxInact(x0, x1))) → SEL(y0, indx(x0, x1))
INDX(cons(X, Y), Z) → A(Y)
A(fromInact(x1)) → FROM(x1)
A(selInact(x1, x2)) → SEL(x1, x2)
SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(dblInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbl(y0), sel(x0, x1))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
A(dblsInact(x1)) → DBLS(x1)
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(x0, x1))
SEL(s(dblInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbl(y0), dbls(x0))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
SEL(s(y0), cons(y1, sInact(x0))) → SEL(y0, s(x0))
SEL(s(dblInact(y0)), cons(y1, fromInact(x0))) → SEL(dbl(y0), from(x0))
SEL(s(y0), cons(y1, dblsInact(x0))) → SEL(y0, 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))
FROM(X) → A(X)
SEL(s(sInact(y0)), cons(y1, fromInact(x0))) → SEL(s(y0), from(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, dblInact(x0))) → SEL(sel(y0, y1), dbl(x0))
INDX(cons(X, Y), Z) → A(X)
SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(x0))
DBLS(cons(X, Y)) → A(Y)
A(indxInact(x1, x2)) → INDX(x1, x2)
DBL(s(X)) → A(X)
SEL(s(dblInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbl(y0), indx(x0, x1))
SEL(s(X), cons(Y, Z)) → A(Z)
SEL(s(y0), cons(y1, dblInact(x0))) → SEL(y0, dbl(x0))
A(dblInact(x1)) → DBL(x1)
SEL(s(X), cons(Y, Z)) → A(X)
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(x0, x1))
INDX(cons(X, Y), Z) → A(Z)
SEL(s(y0), cons(y1, fromInact(x0))) → SEL(y0, from(x0))
SEL(s(y0), cons(y1, selInact(x0, x1))) → SEL(y0, sel(x0, 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
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(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
  ↳ Zantema-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ DependencyGraphProof
                                                ↳ QDP
                                                  ↳ Narrowing
                                                    ↳ QDP
                                                      ↳ DependencyGraphProof
                                                        ↳ QDP
                                                          ↳ Narrowing
                                                            ↳ QDP
                                                              ↳ DependencyGraphProof
                                                                ↳ QDP
                                                                  ↳ Narrowing
                                                                    ↳ QDP
                                                                      ↳ DependencyGraphProof
                                                                        ↳ QDP
                                                                          ↳ Narrowing
                                                                            ↳ QDP
                                                                              ↳ DependencyGraphProof
  ↳ Incomplete Giesl Middeldorp-Transformation
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

DBLS(cons(X, Y)) → A(X)
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(y0, y1), from(x0))
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(y0, y1), dbls(x0))
SEL(0, cons(X, Y)) → A(X)
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)
SEL(s(dblInact(y0)), cons(y1, dblInact(x0))) → SEL(dbl(y0), dbl(x0))
SEL(s(dblInact(y0)), cons(y1, x0)) → SEL(dbl(y0), x0)
A(fromInact(x1)) → FROM(x1)
INDX(cons(X, Y), Z) → A(Y)
SEL(s(y0), cons(y1, indxInact(x0, x1))) → SEL(y0, indx(x0, x1))
A(selInact(x1, x2)) → SEL(x1, x2)
SEL(s(y0), cons(y1, sInact(x0))) → SEL(y0, sInact(x0))
SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(dblInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbl(y0), sel(x0, x1))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(x0, x1))
A(dblsInact(x1)) → DBLS(x1)
SEL(s(dblInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbl(y0), dbls(x0))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
SEL(s(dblInact(y0)), cons(y1, fromInact(x0))) → SEL(dbl(y0), from(x0))
SEL(s(y0), cons(y1, dblsInact(x0))) → SEL(y0, 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, fromInact(x0))) → SEL(s(y0), from(x0))
FROM(X) → A(X)
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(y0, y1), dbl(x0))
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))
INDX(cons(X, Y), Z) → A(X)
SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(x0))
A(indxInact(x1, x2)) → INDX(x1, x2)
DBLS(cons(X, Y)) → A(Y)
DBL(s(X)) → A(X)
SEL(s(dblInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbl(y0), indx(x0, x1))
SEL(s(X), cons(Y, Z)) → A(Z)
SEL(s(y0), cons(y1, dblInact(x0))) → SEL(y0, dbl(x0))
A(dblInact(x1)) → DBL(x1)
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(x0, x1))
SEL(s(X), cons(Y, Z)) → A(X)
INDX(cons(X, Y), Z) → A(Z)
SEL(s(y0), cons(y1, fromInact(x0))) → SEL(y0, from(x0))
SEL(s(y0), cons(y1, selInact(x0, x1))) → SEL(y0, sel(x0, 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
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(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
  ↳ Zantema-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ DependencyGraphProof
                                                ↳ QDP
                                                  ↳ Narrowing
                                                    ↳ QDP
                                                      ↳ DependencyGraphProof
                                                        ↳ QDP
                                                          ↳ Narrowing
                                                            ↳ QDP
                                                              ↳ DependencyGraphProof
                                                                ↳ QDP
                                                                  ↳ Narrowing
                                                                    ↳ QDP
                                                                      ↳ DependencyGraphProof
                                                                        ↳ QDP
                                                                          ↳ Narrowing
                                                                            ↳ QDP
                                                                              ↳ DependencyGraphProof
                                                                                ↳ QDP
                                                                                  ↳ Narrowing
  ↳ Incomplete Giesl Middeldorp-Transformation
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

DBLS(cons(X, Y)) → A(X)
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(y0, y1), from(x0))
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(y0, y1), dbls(x0))
SEL(0, cons(X, Y)) → A(X)
SEL(s(dblInact(y0)), cons(y1, dblInact(x0))) → SEL(dbl(y0), dbl(x0))
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)
SEL(s(dblInact(y0)), cons(y1, x0)) → SEL(dbl(y0), x0)
SEL(s(y0), cons(y1, indxInact(x0, x1))) → SEL(y0, indx(x0, x1))
INDX(cons(X, Y), Z) → A(Y)
A(fromInact(x1)) → FROM(x1)
A(selInact(x1, x2)) → SEL(x1, x2)
SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(dblInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbl(y0), sel(x0, x1))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
A(dblsInact(x1)) → DBLS(x1)
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(x0, x1))
SEL(s(dblInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbl(y0), dbls(x0))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
SEL(s(dblInact(y0)), cons(y1, fromInact(x0))) → SEL(dbl(y0), from(x0))
SEL(s(y0), cons(y1, dblsInact(x0))) → SEL(y0, 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))
FROM(X) → A(X)
SEL(s(sInact(y0)), cons(y1, fromInact(x0))) → SEL(s(y0), from(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, dblInact(x0))) → SEL(sel(y0, y1), dbl(x0))
INDX(cons(X, Y), Z) → A(X)
SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(x0))
DBLS(cons(X, Y)) → A(Y)
A(indxInact(x1, x2)) → INDX(x1, x2)
DBL(s(X)) → A(X)
SEL(s(dblInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbl(y0), indx(x0, x1))
SEL(s(X), cons(Y, Z)) → A(Z)
SEL(s(y0), cons(y1, dblInact(x0))) → SEL(y0, dbl(x0))
A(dblInact(x1)) → DBL(x1)
SEL(s(X), cons(Y, Z)) → A(X)
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(x0, x1))
INDX(cons(X, Y), Z) → A(Z)
SEL(s(y0), cons(y1, fromInact(x0))) → SEL(y0, from(x0))
SEL(s(y0), cons(y1, selInact(x0, x1))) → SEL(y0, sel(x0, 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
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(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
  ↳ Zantema-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ 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
  ↳ Incomplete Giesl Middeldorp-Transformation
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

DBLS(cons(X, Y)) → A(X)
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(y0, y1), from(x0))
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(y0, y1), dbls(x0))
SEL(0, cons(X, Y)) → A(X)
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)
SEL(s(dblInact(y0)), cons(y1, dblInact(x0))) → SEL(dbl(y0), dbl(x0))
SEL(s(dblInact(y0)), cons(y1, x0)) → SEL(dbl(y0), x0)
A(fromInact(x1)) → FROM(x1)
INDX(cons(X, Y), Z) → A(Y)
SEL(s(y0), cons(y1, indxInact(x0, x1))) → SEL(y0, indx(x0, x1))
A(selInact(x1, x2)) → SEL(x1, x2)
SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(dblInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbl(y0), sel(x0, x1))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(x0, x1))
A(dblsInact(x1)) → DBLS(x1)
SEL(s(dblInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbl(y0), dbls(x0))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
SEL(s(dblInact(y0)), cons(y1, fromInact(x0))) → SEL(dbl(y0), from(x0))
SEL(s(y0), cons(y1, dblsInact(x0))) → SEL(y0, dbls(x0))
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))
FROM(X) → A(X)
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(y0, y1), dbl(x0))
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(y0, y1), sel(x0, x1))
INDX(cons(X, Y), Z) → A(X)
SEL(s(dblInact(y0)), cons(y1, sInact(x0))) → SEL(dbl(y0), sInact(x0))
SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(x0))
A(indxInact(x1, x2)) → INDX(x1, x2)
DBLS(cons(X, Y)) → A(Y)
DBL(s(X)) → A(X)
SEL(s(dblInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbl(y0), indx(x0, x1))
SEL(s(X), cons(Y, Z)) → A(Z)
SEL(s(y0), cons(y1, dblInact(x0))) → SEL(y0, dbl(x0))
A(dblInact(x1)) → DBL(x1)
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(x0, x1))
SEL(s(X), cons(Y, Z)) → A(X)
INDX(cons(X, Y), Z) → A(Z)
SEL(s(y0), cons(y1, fromInact(x0))) → SEL(y0, from(x0))
SEL(s(y0), cons(y1, selInact(x0, x1))) → SEL(y0, sel(x0, 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
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(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
  ↳ Zantema-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ 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
  ↳ Incomplete Giesl Middeldorp-Transformation
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

DBLS(cons(X, Y)) → A(X)
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(y0, y1), from(x0))
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(y0, y1), dbls(x0))
SEL(0, cons(X, Y)) → A(X)
SEL(s(dblInact(y0)), cons(y1, dblInact(x0))) → SEL(dbl(y0), dbl(x0))
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)
SEL(s(dblInact(y0)), cons(y1, x0)) → SEL(dbl(y0), x0)
SEL(s(y0), cons(y1, indxInact(x0, x1))) → SEL(y0, indx(x0, x1))
INDX(cons(X, Y), Z) → A(Y)
A(fromInact(x1)) → FROM(x1)
A(selInact(x1, x2)) → SEL(x1, x2)
SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(dblInact(y0)), cons(y1, selInact(x0, x1))) → SEL(dbl(y0), sel(x0, x1))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(y0, y1), indx(x0, x1))
A(dblsInact(x1)) → DBLS(x1)
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(x0, x1))
SEL(s(dblInact(y0)), cons(y1, dblsInact(x0))) → SEL(dbl(y0), dbls(x0))
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(y0, y1), x0)
SEL(s(dblInact(y0)), cons(y1, fromInact(x0))) → SEL(dbl(y0), from(x0))
SEL(s(y0), cons(y1, dblsInact(x0))) → SEL(y0, dbls(x0))
SEL(s(sInact(y0)), cons(y1, dblInact(x0))) → SEL(s(y0), dbl(x0))
FROM(X) → A(X)
SEL(s(sInact(y0)), cons(y1, fromInact(x0))) → SEL(s(y0), from(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, dblInact(x0))) → SEL(sel(y0, y1), dbl(x0))
INDX(cons(X, Y), Z) → A(X)
SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(x0))
DBLS(cons(X, Y)) → A(Y)
A(indxInact(x1, x2)) → INDX(x1, x2)
DBL(s(X)) → A(X)
SEL(s(dblInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(dbl(y0), indx(x0, x1))
SEL(s(X), cons(Y, Z)) → A(Z)
SEL(s(y0), cons(y1, dblInact(x0))) → SEL(y0, dbl(x0))
A(dblInact(x1)) → DBL(x1)
SEL(s(X), cons(Y, Z)) → A(X)
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(x0, x1))
INDX(cons(X, Y), Z) → A(Z)
SEL(s(y0), cons(y1, fromInact(x0))) → SEL(y0, from(x0))
SEL(s(y0), cons(y1, selInact(x0, x1))) → SEL(y0, sel(x0, 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
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(x1, x2)
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(x1, x2)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(x1)
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(x1)
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)

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

↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

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

Q is empty.

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

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

The TRS R consists of the following rules:

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

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

↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

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

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 4 less nodes.

↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

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

The TRS R consists of the following rules:

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

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

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



↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

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

The TRS R consists of the following rules:

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

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

↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

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

The TRS R consists of the following rules:

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

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

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



↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

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

The TRS R consists of the following rules:

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

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

↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

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

The TRS R consists of the following rules:

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

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

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



↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

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

The TRS R consists of the following rules:

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

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

↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

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

The TRS R consists of the following rules:

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

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

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



↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

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

The TRS R consists of the following rules:

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

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

↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

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

The TRS R consists of the following rules:

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

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

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



↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

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

The TRS R consists of the following rules:

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

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

↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

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

The TRS R consists of the following rules:

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

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

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



↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

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

The TRS R consists of the following rules:

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

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

↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

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

The TRS R consists of the following rules:

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

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

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



↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

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

The TRS R consists of the following rules:

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

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

↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

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

The TRS R consists of the following rules:

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

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

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



↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

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

The TRS R consists of the following rules:

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

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

↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

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

The TRS R consists of the following rules:

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

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

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



↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

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

The TRS R consists of the following rules:

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

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

↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

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

The TRS R consists of the following rules:

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

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

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



↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

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

The TRS R consists of the following rules:

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

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

↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

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

The TRS R consists of the following rules:

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

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

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



↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

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

The TRS R consists of the following rules:

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

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

↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

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

The TRS R consists of the following rules:

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

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

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



↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

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

The TRS R consists of the following rules:

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

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

↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

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

The TRS R consists of the following rules:

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

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

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



↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

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

The TRS R consists of the following rules:

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

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

↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

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

The TRS R consists of the following rules:

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

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

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



↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ DependencyGraphProof
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

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

The TRS R consists of the following rules:

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

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

↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ DependencyGraphProof
                                                                                                                            ↳ QDP
                                                                                                                              ↳ Narrowing
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

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

The TRS R consists of the following rules:

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

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

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



↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ DependencyGraphProof
                                                                                                                            ↳ QDP
                                                                                                                              ↳ Narrowing
                                                                                                                                ↳ QDP
                                                                                                                                  ↳ DependencyGraphProof
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

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

The TRS R consists of the following rules:

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

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

↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ DependencyGraphProof
                                                                                                                            ↳ QDP
                                                                                                                              ↳ Narrowing
                                                                                                                                ↳ QDP
                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                    ↳ QDP
                                                                                                                                      ↳ Narrowing
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

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

The TRS R consists of the following rules:

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

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

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



↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ DependencyGraphProof
                                                                                                                            ↳ QDP
                                                                                                                              ↳ Narrowing
                                                                                                                                ↳ QDP
                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                    ↳ QDP
                                                                                                                                      ↳ Narrowing
                                                                                                                                        ↳ QDP
                                                                                                                                          ↳ DependencyGraphProof
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

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

The TRS R consists of the following rules:

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

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

↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ DependencyGraphProof
                                                                                                                            ↳ QDP
                                                                                                                              ↳ Narrowing
                                                                                                                                ↳ QDP
                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                    ↳ QDP
                                                                                                                                      ↳ Narrowing
                                                                                                                                        ↳ QDP
                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                            ↳ QDP
                                                                                                                                              ↳ QDPOrderProof
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

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

The TRS R consists of the following rules:

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

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


The following pairs can be oriented strictly and are deleted.


SELACTIVE(s(dbls(y0)), cons(y1, dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
MARK(sel(dbl(y0), dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(s(0), cons(y0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
SELACTIVE(s(indx(y0, y1)), cons(y2, dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(s(y0), dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel(0, dbl(x0))) → SELACTIVE(0, dblActive(mark(x0)))
MARK(sel(sel(y0, y1), dbl(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblActive(mark(x0)))
MARK(sel(indx(y0, y1), dbl(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblActive(mark(x0)))
SELACTIVE(s(dbl(y0)), cons(y1, dbl(x0))) → SELACTIVE(dblActive(mark(y0)), dblActive(mark(x0)))
SELACTIVE(s(s(y0)), cons(y1, dbl(x0))) → SELACTIVE(s(y0), dblActive(mark(x0)))
MARK(sel(dbls(y0), dbl(x0))) → SELACTIVE(dblsActive(mark(y0)), dblActive(mark(x0)))
The remaining pairs can at least be oriented weakly.

MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
SELACTIVE(s(dbl(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbl(y0), indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(s(y0), sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
SELACTIVE(s(indx(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
SELACTIVE(s(dbls(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(s(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
SELACTIVE(s(s(y0)), cons(y1, dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
SELACTIVE(s(dbls(y0)), cons(y1, from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(dbl(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(0), cons(y0, from(x0))) → SELACTIVE(0, fromActive(x0))
SELACTIVE(s(dbls(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(dbls(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
SELACTIVE(s(s(y0)), cons(y1, from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SELACTIVE(s(dbl(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(0), cons(y0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
SELACTIVE(s(dbl(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(indx(y0, y1)), cons(y2, from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(sel(y0, y1), dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(sel(dbls(y0), sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(0), cons(y0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(indx(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel(dbls(y0), indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(indx(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(s(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
MARK(sel(sel(y0, y1), indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
SELACTIVE(s(0), cons(y0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(x1, x2)) → MARK(x1)
SELACTIVE(s(dbls(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(dbl(y0), sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(sel(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
SELACTIVE(0, cons(X, Y)) → MARK(X)
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SELACTIVE(s(s(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
SELACTIVE(s(sel(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
MARK(dbl(x1)) → MARK(x1)
SELACTIVE(s(sel(y0, y1)), cons(y2, from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
SELACTIVE(s(indx(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(x1, x2)) → MARK(x2)
SELACTIVE(s(dbl(y0)), cons(y1, from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(0, from(x0))) → SELACTIVE(0, fromActive(x0))
MARK(sel(s(y0), indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
SELACTIVE(s(0), cons(y0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
Used ordering: Polynomial interpretation [25]:

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

The following usable rules [17] were oriented:

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



↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ DependencyGraphProof
                                                                                                                            ↳ QDP
                                                                                                                              ↳ Narrowing
                                                                                                                                ↳ QDP
                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                    ↳ QDP
                                                                                                                                      ↳ Narrowing
                                                                                                                                        ↳ QDP
                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                            ↳ QDP
                                                                                                                                              ↳ QDPOrderProof
                                                                                                                                                ↳ QDP
                                                                                                                                                  ↳ QDPOrderProof
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

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

The TRS R consists of the following rules:

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

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


The following pairs can be oriented strictly and are deleted.


SELACTIVE(s(indx(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
SELACTIVE(s(dbls(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(dbls(y0)), cons(y1, from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
SELACTIVE(s(dbls(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(dbls(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), cons(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(indx(y0, y1), from(x0))) → SELACTIVE(indxActive(mark(y0), y1), fromActive(x0))
MARK(sel(dbls(y0), sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
SELACTIVE(s(indx(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(indxActive(mark(y0), y1), dblsActive(mark(x0)))
MARK(sel(dbls(y0), indx(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(dbls(y0), dbls(x0))) → SELACTIVE(dblsActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(indx(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
MARK(sel(dbls(y0), from(x0))) → SELACTIVE(dblsActive(mark(y0)), fromActive(x0))
SELACTIVE(s(dbls(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblsActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), sel(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), selActive(mark(x0), mark(x1)))
MARK(sel(indx(y0, y1), indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
SELACTIVE(s(indx(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), indxActive(mark(x0), x1))
MARK(sel(indx(y0, y1), cons(x0, x1))) → SELACTIVE(indxActive(mark(y0), y1), cons(x0, x1))
The remaining pairs can at least be oriented weakly.

MARK(sel(0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
SELACTIVE(s(dbl(y0)), cons(y1, dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
MARK(sel(dbl(y0), indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(s(y0), sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
SELACTIVE(s(s(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
SELACTIVE(s(s(y0)), cons(y1, dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(dbl(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(dblActive(mark(y0)), indxActive(mark(x0), x1))
MARK(sel(sel(y0, y1), sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(0), cons(y0, from(x0))) → SELACTIVE(0, fromActive(x0))
MARK(sel(sel(y0, y1), from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
SELACTIVE(s(dbl(y0)), cons(y1, cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(s(y0)), cons(y1, from(x0))) → SELACTIVE(s(y0), fromActive(x0))
SELACTIVE(s(0), cons(y0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
SELACTIVE(s(dbl(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
MARK(sel(dbl(y0), from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(sel(y0, y1), dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
SELACTIVE(s(0), cons(y0, indx(x0, x1))) → SELACTIVE(0, indxActive(mark(x0), x1))
SELACTIVE(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), selActive(mark(x0), mark(x1)))
MARK(sel(sel(y0, y1), cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(dbl(y0), cons(x0, x1))) → SELACTIVE(dblActive(mark(y0)), cons(x0, x1))
SELACTIVE(s(s(y0)), cons(y1, indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(indx(x1, x2)) → MARK(x1)
MARK(sel(sel(y0, y1), indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
MARK(sel(0, sel(x0, x1))) → SELACTIVE(0, selActive(mark(x0), mark(x1)))
SELACTIVE(s(0), cons(y0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
SELACTIVE(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), cons(x0, x1))
MARK(sel(x1, x2)) → MARK(x1)
MARK(sel(dbl(y0), sel(x0, x1))) → SELACTIVE(dblActive(mark(y0)), selActive(mark(x0), mark(x1)))
SELACTIVE(s(sel(y0, y1)), cons(y2, indx(x0, x1))) → SELACTIVE(selActive(mark(y0), mark(y1)), indxActive(mark(x0), x1))
SELACTIVE(0, cons(X, Y)) → MARK(X)
SELACTIVE(s(sel(y0, y1)), cons(y2, from(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), fromActive(x0))
MARK(dbl(x1)) → MARK(x1)
SELACTIVE(s(sel(y0, y1)), cons(y2, dbls(x0))) → SELACTIVE(selActive(mark(y0), mark(y1)), dblsActive(mark(x0)))
SELACTIVE(s(s(y0)), cons(y1, sel(x0, x1))) → SELACTIVE(s(y0), selActive(mark(x0), mark(x1)))
MARK(sel(s(y0), from(x0))) → SELACTIVE(s(y0), fromActive(x0))
MARK(sel(s(y0), cons(x0, x1))) → SELACTIVE(s(y0), cons(x0, x1))
MARK(sel(0, dbls(x0))) → SELACTIVE(0, dblsActive(mark(x0)))
MARK(dbls(x1)) → MARK(x1)
MARK(sel(x1, x2)) → MARK(x2)
SELACTIVE(s(dbl(y0)), cons(y1, from(x0))) → SELACTIVE(dblActive(mark(y0)), fromActive(x0))
MARK(sel(0, from(x0))) → SELACTIVE(0, fromActive(x0))
MARK(sel(s(y0), indx(x0, x1))) → SELACTIVE(s(y0), indxActive(mark(x0), x1))
MARK(sel(dbl(y0), dbls(x0))) → SELACTIVE(dblActive(mark(y0)), dblsActive(mark(x0)))
SELACTIVE(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(s(y0), dbls(x0))) → SELACTIVE(s(y0), dblsActive(mark(x0)))
SELACTIVE(s(0), cons(y0, cons(x0, x1))) → SELACTIVE(0, cons(x0, x1))
Used ordering: Polynomial interpretation [25]:

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

The following usable rules [17] were oriented:

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



↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ DependencyGraphProof
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ DependencyGraphProof
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ DependencyGraphProof
                                    ↳ QDP
                                      ↳ Narrowing
                                        ↳ QDP
                                          ↳ DependencyGraphProof
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ DependencyGraphProof
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ DependencyGraphProof
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ DependencyGraphProof
                                                                    ↳ QDP
                                                                      ↳ Narrowing
                                                                        ↳ QDP
                                                                          ↳ DependencyGraphProof
                                                                            ↳ QDP
                                                                              ↳ Narrowing
                                                                                ↳ QDP
                                                                                  ↳ DependencyGraphProof
                                                                                    ↳ QDP
                                                                                      ↳ Narrowing
                                                                                        ↳ QDP
                                                                                          ↳ DependencyGraphProof
                                                                                            ↳ QDP
                                                                                              ↳ Narrowing
                                                                                                ↳ QDP
                                                                                                  ↳ DependencyGraphProof
                                                                                                    ↳ QDP
                                                                                                      ↳ Narrowing
                                                                                                        ↳ QDP
                                                                                                          ↳ DependencyGraphProof
                                                                                                            ↳ QDP
                                                                                                              ↳ Narrowing
                                                                                                                ↳ QDP
                                                                                                                  ↳ DependencyGraphProof
                                                                                                                    ↳ QDP
                                                                                                                      ↳ Narrowing
                                                                                                                        ↳ QDP
                                                                                                                          ↳ DependencyGraphProof
                                                                                                                            ↳ QDP
                                                                                                                              ↳ Narrowing
                                                                                                                                ↳ QDP
                                                                                                                                  ↳ DependencyGraphProof
                                                                                                                                    ↳ QDP
                                                                                                                                      ↳ Narrowing
                                                                                                                                        ↳ QDP
                                                                                                                                          ↳ DependencyGraphProof
                                                                                                                                            ↳ QDP
                                                                                                                                              ↳ QDPOrderProof
                                                                                                                                                ↳ QDP
                                                                                                                                                  ↳ QDPOrderProof
                                                                                                                                                    ↳ QDP
  ↳ Improved Ferreira Ribeiro-Transformation
  ↳ Complete Giesl Middeldorp-Transformation

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

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

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We applied the Improved Ferreira Ribeiro [5,11] to transform the context-sensitive TRS to an usual TRS.

↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
  ↳ Improved Ferreira Ribeiro-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
  ↳ Complete Giesl Middeldorp-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
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(a(x1), a(x2))
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(a(x1))
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(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:

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

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

↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
  ↳ Improved Ferreira Ribeiro-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
  ↳ Complete Giesl Middeldorp-Transformation

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

DBLS(cons(X, Y)) → A(X)
SEL(s(X), cons(Y, Z)) → SEL(a(X), a(Z))
A(selInact(x1, x2)) → A(x2)
A(selInact(x1, x2)) → A(x1)
A(sInact(x1)) → S(x1)
A(dblInact(x1)) → DBL(a(x1))
FROM(X) → A(X)
SEL(0, cons(X, Y)) → A(X)
A(dblsInact(x1)) → DBLS(a(x1))
INDX(cons(X, Y), Z) → A(X)
A(indxInact(x1, x2)) → A(x1)
DBLS(cons(X, Y)) → A(Y)
DBL(s(X)) → A(X)
INDX(cons(X, Y), Z) → A(Y)
A(fromInact(x1)) → FROM(x1)
SEL(s(X), cons(Y, Z)) → A(Z)
A(indxInact(x1, x2)) → INDX(a(x1), x2)
A(dblInact(x1)) → A(x1)
A(selInact(x1, x2)) → SEL(a(x1), a(x2))
SEL(s(X), cons(Y, Z)) → A(X)
INDX(cons(X, Y), Z) → A(Z)
A(dblsInact(x1)) → A(x1)
DBL(s(X)) → S(sInact(dblInact(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
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(a(x1), a(x2))
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(a(x1))
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(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
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
  ↳ Improved Ferreira Ribeiro-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
  ↳ Complete Giesl Middeldorp-Transformation

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

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

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

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



↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
  ↳ Improved Ferreira Ribeiro-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
  ↳ Complete Giesl Middeldorp-Transformation

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

SEL(s(X), cons(Y, Z)) → SEL(a(X), a(Z))
DBLS(cons(X, Y)) → A(X)
A(selInact(x1, x2)) → A(x2)
A(selInact(x1, x2)) → A(x1)
A(selInact(sInact(x0), y1)) → SEL(s(x0), a(y1))
SEL(0, cons(X, Y)) → A(X)
A(selInact(x0, y1)) → SEL(x0, a(y1))
A(selInact(fromInact(x0), y1)) → SEL(from(x0), a(y1))
INDX(cons(X, Y), Z) → A(Y)
A(fromInact(x1)) → FROM(x1)
A(dblInact(x1)) → A(x1)
A(selInact(indxInact(x0, x1), y1)) → SEL(indx(a(x0), x1), a(y1))
A(dblsInact(x1)) → A(x1)
A(dblInact(x1)) → DBL(a(x1))
FROM(X) → A(X)
INDX(cons(X, Y), Z) → A(X)
A(dblsInact(x1)) → DBLS(a(x1))
DBLS(cons(X, Y)) → A(Y)
A(indxInact(x1, x2)) → A(x1)
DBL(s(X)) → A(X)
SEL(s(X), cons(Y, Z)) → A(Z)
A(indxInact(x1, x2)) → INDX(a(x1), x2)
A(selInact(dblInact(x0), y1)) → SEL(dbl(a(x0)), a(y1))
A(selInact(dblsInact(x0), y1)) → SEL(dbls(a(x0)), a(y1))
SEL(s(X), cons(Y, Z)) → A(X)
A(selInact(selInact(x0, x1), y1)) → SEL(sel(a(x0), a(x1)), a(y1))
INDX(cons(X, Y), Z) → 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
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(a(x1), a(x2))
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(a(x1))
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)

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

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



↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
  ↳ Improved Ferreira Ribeiro-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
  ↳ Complete Giesl Middeldorp-Transformation

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

DBLS(cons(X, Y)) → A(X)
SEL(s(X), cons(Y, Z)) → SEL(a(X), a(Z))
A(selInact(x1, x2)) → A(x2)
A(selInact(x1, x2)) → A(x1)
A(selInact(sInact(x0), y1)) → SEL(s(x0), a(y1))
A(indxInact(dblsInact(x0), y1)) → INDX(dbls(a(x0)), y1)
SEL(0, cons(X, Y)) → A(X)
A(indxInact(fromInact(x0), y1)) → INDX(from(x0), y1)
A(selInact(x0, y1)) → SEL(x0, a(y1))
A(selInact(fromInact(x0), y1)) → SEL(from(x0), a(y1))
A(fromInact(x1)) → FROM(x1)
INDX(cons(X, Y), Z) → A(Y)
A(dblInact(x1)) → A(x1)
A(selInact(indxInact(x0, x1), y1)) → SEL(indx(a(x0), x1), a(y1))
A(indxInact(indxInact(x0, x1), y1)) → INDX(indx(a(x0), x1), y1)
A(dblsInact(x1)) → A(x1)
A(indxInact(selInact(x0, x1), y1)) → INDX(sel(a(x0), a(x1)), y1)
A(dblInact(x1)) → DBL(a(x1))
FROM(X) → A(X)
A(indxInact(dblInact(x0), y1)) → INDX(dbl(a(x0)), y1)
A(dblsInact(x1)) → DBLS(a(x1))
INDX(cons(X, Y), Z) → A(X)
A(indxInact(x0, y1)) → INDX(x0, y1)
A(indxInact(x1, x2)) → A(x1)
DBLS(cons(X, Y)) → A(Y)
DBL(s(X)) → A(X)
SEL(s(X), cons(Y, Z)) → A(Z)
A(selInact(dblsInact(x0), y1)) → SEL(dbls(a(x0)), a(y1))
A(selInact(dblInact(x0), y1)) → SEL(dbl(a(x0)), a(y1))
SEL(s(X), cons(Y, Z)) → A(X)
A(indxInact(sInact(x0), y1)) → INDX(s(x0), y1)
INDX(cons(X, Y), Z) → A(Z)
A(selInact(selInact(x0, x1), y1)) → SEL(sel(a(x0), a(x1)), a(y1))

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
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(a(x1), a(x2))
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(a(x1))
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)

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

A(dblInact(dblsInact(x0))) → DBL(dbls(a(x0)))
A(dblInact(dblInact(x0))) → DBL(dbl(a(x0)))
A(dblInact(indxInact(x0, x1))) → DBL(indx(a(x0), x1))
A(dblInact(sInact(x0))) → DBL(s(x0))
A(dblInact(x0)) → DBL(x0)
A(dblInact(fromInact(x0))) → DBL(from(x0))
A(dblInact(selInact(x0, x1))) → DBL(sel(a(x0), a(x1)))



↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
  ↳ Improved Ferreira Ribeiro-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
  ↳ Complete Giesl Middeldorp-Transformation

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

SEL(s(X), cons(Y, Z)) → SEL(a(X), a(Z))
DBLS(cons(X, Y)) → A(X)
A(selInact(x1, x2)) → A(x2)
A(selInact(x1, x2)) → A(x1)
A(selInact(sInact(x0), y1)) → SEL(s(x0), a(y1))
SEL(0, cons(X, Y)) → A(X)
A(indxInact(dblsInact(x0), y1)) → INDX(dbls(a(x0)), y1)
A(selInact(x0, y1)) → SEL(x0, a(y1))
A(indxInact(fromInact(x0), y1)) → INDX(from(x0), y1)
A(dblInact(dblsInact(x0))) → DBL(dbls(a(x0)))
A(dblInact(dblInact(x0))) → DBL(dbl(a(x0)))
A(selInact(fromInact(x0), y1)) → SEL(from(x0), a(y1))
A(dblInact(indxInact(x0, x1))) → DBL(indx(a(x0), x1))
INDX(cons(X, Y), Z) → A(Y)
A(fromInact(x1)) → FROM(x1)
A(dblInact(sInact(x0))) → DBL(s(x0))
A(dblInact(x1)) → A(x1)
A(selInact(indxInact(x0, x1), y1)) → SEL(indx(a(x0), x1), a(y1))
A(indxInact(indxInact(x0, x1), y1)) → INDX(indx(a(x0), x1), y1)
A(dblsInact(x1)) → A(x1)
A(indxInact(selInact(x0, x1), y1)) → INDX(sel(a(x0), a(x1)), y1)
FROM(X) → A(X)
INDX(cons(X, Y), Z) → A(X)
A(dblsInact(x1)) → DBLS(a(x1))
A(indxInact(dblInact(x0), y1)) → INDX(dbl(a(x0)), y1)
DBLS(cons(X, Y)) → A(Y)
A(indxInact(x1, x2)) → A(x1)
A(indxInact(x0, y1)) → INDX(x0, y1)
DBL(s(X)) → A(X)
SEL(s(X), cons(Y, Z)) → A(Z)
A(selInact(dblInact(x0), y1)) → SEL(dbl(a(x0)), a(y1))
A(selInact(dblsInact(x0), y1)) → SEL(dbls(a(x0)), a(y1))
A(dblInact(x0)) → DBL(x0)
SEL(s(X), cons(Y, Z)) → A(X)
A(indxInact(sInact(x0), y1)) → INDX(s(x0), y1)
A(selInact(selInact(x0, x1), y1)) → SEL(sel(a(x0), a(x1)), a(y1))
INDX(cons(X, Y), Z) → A(Z)
A(dblInact(selInact(x0, x1))) → DBL(sel(a(x0), a(x1)))
A(dblInact(fromInact(x0))) → DBL(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
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(a(x1), a(x2))
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(a(x1))
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)

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

A(dblsInact(dblsInact(x0))) → DBLS(dbls(a(x0)))
A(dblsInact(selInact(x0, x1))) → DBLS(sel(a(x0), a(x1)))
A(dblsInact(fromInact(x0))) → DBLS(from(x0))
A(dblsInact(dblInact(x0))) → DBLS(dbl(a(x0)))
A(dblsInact(x0)) → DBLS(x0)
A(dblsInact(indxInact(x0, x1))) → DBLS(indx(a(x0), x1))
A(dblsInact(sInact(x0))) → DBLS(s(x0))



↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
  ↳ Improved Ferreira Ribeiro-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
  ↳ Complete Giesl Middeldorp-Transformation

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

DBLS(cons(X, Y)) → A(X)
SEL(s(X), cons(Y, Z)) → SEL(a(X), a(Z))
A(selInact(x1, x2)) → A(x2)
A(selInact(x1, x2)) → A(x1)
A(selInact(sInact(x0), y1)) → SEL(s(x0), a(y1))
A(dblsInact(dblInact(x0))) → DBLS(dbl(a(x0)))
A(indxInact(dblsInact(x0), y1)) → INDX(dbls(a(x0)), y1)
SEL(0, cons(X, Y)) → A(X)
A(indxInact(fromInact(x0), y1)) → INDX(from(x0), y1)
A(selInact(x0, y1)) → SEL(x0, a(y1))
A(dblsInact(indxInact(x0, x1))) → DBLS(indx(a(x0), x1))
A(dblInact(dblsInact(x0))) → DBL(dbls(a(x0)))
A(selInact(fromInact(x0), y1)) → SEL(from(x0), a(y1))
A(dblInact(dblInact(x0))) → DBL(dbl(a(x0)))
A(fromInact(x1)) → FROM(x1)
INDX(cons(X, Y), Z) → A(Y)
A(dblInact(indxInact(x0, x1))) → DBL(indx(a(x0), x1))
A(dblInact(sInact(x0))) → DBL(s(x0))
A(dblInact(x1)) → A(x1)
A(selInact(indxInact(x0, x1), y1)) → SEL(indx(a(x0), x1), a(y1))
A(indxInact(indxInact(x0, x1), y1)) → INDX(indx(a(x0), x1), y1)
A(dblsInact(x1)) → A(x1)
A(dblsInact(x0)) → DBLS(x0)
A(dblsInact(dblsInact(x0))) → DBLS(dbls(a(x0)))
A(dblsInact(fromInact(x0))) → DBLS(from(x0))
A(indxInact(selInact(x0, x1), y1)) → INDX(sel(a(x0), a(x1)), y1)
FROM(X) → A(X)
A(indxInact(dblInact(x0), y1)) → INDX(dbl(a(x0)), y1)
INDX(cons(X, Y), Z) → A(X)
A(dblsInact(sInact(x0))) → DBLS(s(x0))
A(indxInact(x0, y1)) → INDX(x0, y1)
A(indxInact(x1, x2)) → A(x1)
DBLS(cons(X, Y)) → A(Y)
DBL(s(X)) → A(X)
A(dblsInact(selInact(x0, x1))) → DBLS(sel(a(x0), a(x1)))
SEL(s(X), cons(Y, Z)) → A(Z)
A(selInact(dblsInact(x0), y1)) → SEL(dbls(a(x0)), a(y1))
A(selInact(dblInact(x0), y1)) → SEL(dbl(a(x0)), a(y1))
A(dblInact(x0)) → DBL(x0)
SEL(s(X), cons(Y, Z)) → A(X)
A(indxInact(sInact(x0), y1)) → INDX(s(x0), y1)
INDX(cons(X, Y), Z) → A(Z)
A(selInact(selInact(x0, x1), y1)) → SEL(sel(a(x0), a(x1)), a(y1))
A(dblInact(fromInact(x0))) → DBL(from(x0))
A(dblInact(selInact(x0, x1))) → DBL(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
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(a(x1), a(x2))
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(a(x1))
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(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(sInact(x0)), cons(y1, y2)) → SEL(s(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))
SEL(s(fromInact(x0)), cons(y1, y2)) → SEL(from(x0), a(y2))
SEL(s(dblsInact(x0)), cons(y1, y2)) → SEL(dbls(a(x0)), a(y2))
SEL(s(dblInact(x0)), cons(y1, y2)) → SEL(dbl(a(x0)), a(y2))



↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
  ↳ Improved Ferreira Ribeiro-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ Narrowing
  ↳ Complete Giesl Middeldorp-Transformation

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

DBLS(cons(X, Y)) → A(X)
A(selInact(x1, x2)) → A(x1)
SEL(s(x0), cons(y1, y2)) → SEL(x0, a(y2))
A(selInact(sInact(x0), y1)) → SEL(s(x0), a(y1))
A(dblsInact(dblInact(x0))) → DBLS(dbl(a(x0)))
A(indxInact(dblsInact(x0), y1)) → INDX(dbls(a(x0)), y1)
SEL(s(fromInact(x0)), cons(y1, y2)) → SEL(from(x0), a(y2))
A(dblsInact(indxInact(x0, x1))) → DBLS(indx(a(x0), x1))
INDX(cons(X, Y), Z) → A(Y)
A(dblInact(sInact(x0))) → DBL(s(x0))
A(selInact(indxInact(x0, x1), y1)) → SEL(indx(a(x0), x1), a(y1))
A(dblsInact(x1)) → A(x1)
A(dblsInact(dblsInact(x0))) → DBLS(dbls(a(x0)))
A(dblsInact(fromInact(x0))) → DBLS(from(x0))
A(indxInact(selInact(x0, x1), y1)) → INDX(sel(a(x0), a(x1)), y1)
SEL(s(indxInact(x0, x1)), cons(y1, y2)) → SEL(indx(a(x0), x1), a(y2))
INDX(cons(X, Y), Z) → A(X)
SEL(s(dblInact(x0)), cons(y1, y2)) → SEL(dbl(a(x0)), a(y2))
A(dblsInact(sInact(x0))) → DBLS(s(x0))
DBL(s(X)) → A(X)
A(dblInact(fromInact(x0))) → DBL(from(x0))
SEL(s(selInact(x0, x1)), cons(y1, y2)) → SEL(sel(a(x0), a(x1)), a(y2))
A(selInact(x1, x2)) → A(x2)
SEL(0, cons(X, Y)) → A(X)
A(indxInact(fromInact(x0), y1)) → INDX(from(x0), y1)
A(selInact(x0, y1)) → SEL(x0, a(y1))
A(dblInact(dblsInact(x0))) → DBL(dbls(a(x0)))
A(selInact(fromInact(x0), y1)) → SEL(from(x0), a(y1))
A(dblInact(dblInact(x0))) → DBL(dbl(a(x0)))
A(fromInact(x1)) → FROM(x1)
A(dblInact(indxInact(x0, x1))) → DBL(indx(a(x0), x1))
A(dblInact(x1)) → A(x1)
A(indxInact(indxInact(x0, x1), y1)) → INDX(indx(a(x0), x1), y1)
SEL(s(dblsInact(x0)), cons(y1, y2)) → SEL(dbls(a(x0)), a(y2))
A(dblsInact(x0)) → DBLS(x0)
FROM(X) → A(X)
A(indxInact(dblInact(x0), y1)) → INDX(dbl(a(x0)), y1)
A(indxInact(x0, y1)) → INDX(x0, y1)
A(indxInact(x1, x2)) → A(x1)
DBLS(cons(X, Y)) → A(Y)
A(dblsInact(selInact(x0, x1))) → DBLS(sel(a(x0), a(x1)))
SEL(s(X), cons(Y, Z)) → A(Z)
SEL(s(sInact(x0)), cons(y1, y2)) → SEL(s(x0), a(y2))
A(selInact(dblsInact(x0), y1)) → SEL(dbls(a(x0)), a(y1))
A(selInact(dblInact(x0), y1)) → SEL(dbl(a(x0)), a(y1))
A(dblInact(x0)) → DBL(x0)
SEL(s(X), cons(Y, Z)) → A(X)
A(indxInact(sInact(x0), y1)) → INDX(s(x0), y1)
INDX(cons(X, Y), Z) → A(Z)
A(selInact(selInact(x0, x1), y1)) → SEL(sel(a(x0), a(x1)), a(y1))
A(dblInact(selInact(x0, x1))) → DBL(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
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(a(x1), a(x2))
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(a(x1))
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)

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

A(selInact(fromInact(x0), y1)) → SEL(fromInact(x0), a(y1))
A(selInact(fromInact(x0), y1)) → SEL(cons(a(x0), fromInact(sInact(a(x0)))), a(y1))



↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
  ↳ Improved Ferreira Ribeiro-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ DependencyGraphProof
  ↳ Complete Giesl Middeldorp-Transformation

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

DBLS(cons(X, Y)) → A(X)
A(selInact(x1, x2)) → A(x1)
SEL(s(x0), cons(y1, y2)) → SEL(x0, a(y2))
A(selInact(sInact(x0), y1)) → SEL(s(x0), a(y1))
A(indxInact(dblsInact(x0), y1)) → INDX(dbls(a(x0)), y1)
A(dblsInact(dblInact(x0))) → DBLS(dbl(a(x0)))
SEL(s(fromInact(x0)), cons(y1, y2)) → SEL(from(x0), a(y2))
A(dblsInact(indxInact(x0, x1))) → DBLS(indx(a(x0), x1))
INDX(cons(X, Y), Z) → A(Y)
A(dblInact(sInact(x0))) → DBL(s(x0))
A(selInact(indxInact(x0, x1), y1)) → SEL(indx(a(x0), x1), a(y1))
A(dblsInact(x1)) → A(x1)
A(dblsInact(dblsInact(x0))) → DBLS(dbls(a(x0)))
A(dblsInact(fromInact(x0))) → DBLS(from(x0))
A(indxInact(selInact(x0, x1), y1)) → INDX(sel(a(x0), a(x1)), y1)
SEL(s(indxInact(x0, x1)), cons(y1, y2)) → SEL(indx(a(x0), x1), a(y2))
INDX(cons(X, Y), Z) → A(X)
SEL(s(dblInact(x0)), cons(y1, y2)) → SEL(dbl(a(x0)), a(y2))
A(dblsInact(sInact(x0))) → DBLS(s(x0))
DBL(s(X)) → A(X)
A(selInact(fromInact(x0), y1)) → SEL(cons(a(x0), fromInact(sInact(a(x0)))), a(y1))
A(dblInact(fromInact(x0))) → DBL(from(x0))
SEL(s(selInact(x0, x1)), cons(y1, y2)) → SEL(sel(a(x0), a(x1)), a(y2))
A(selInact(x1, x2)) → A(x2)
SEL(0, cons(X, Y)) → A(X)
A(selInact(x0, y1)) → SEL(x0, a(y1))
A(indxInact(fromInact(x0), y1)) → INDX(from(x0), y1)
A(dblInact(dblsInact(x0))) → DBL(dbls(a(x0)))
A(dblInact(dblInact(x0))) → DBL(dbl(a(x0)))
A(dblInact(indxInact(x0, x1))) → DBL(indx(a(x0), x1))
A(fromInact(x1)) → FROM(x1)
A(dblInact(x1)) → A(x1)
A(indxInact(indxInact(x0, x1), y1)) → INDX(indx(a(x0), x1), y1)
A(dblsInact(x0)) → DBLS(x0)
SEL(s(dblsInact(x0)), cons(y1, y2)) → SEL(dbls(a(x0)), a(y2))
FROM(X) → A(X)
A(indxInact(dblInact(x0), y1)) → INDX(dbl(a(x0)), y1)
DBLS(cons(X, Y)) → A(Y)
A(indxInact(x1, x2)) → A(x1)
A(indxInact(x0, y1)) → INDX(x0, y1)
SEL(s(X), cons(Y, Z)) → A(Z)
A(dblsInact(selInact(x0, x1))) → DBLS(sel(a(x0), a(x1)))
SEL(s(sInact(x0)), cons(y1, y2)) → SEL(s(x0), a(y2))
A(selInact(dblInact(x0), y1)) → SEL(dbl(a(x0)), a(y1))
A(selInact(dblsInact(x0), y1)) → SEL(dbls(a(x0)), a(y1))
A(dblInact(x0)) → DBL(x0)
SEL(s(X), cons(Y, Z)) → A(X)
A(indxInact(sInact(x0), y1)) → INDX(s(x0), y1)
A(selInact(fromInact(x0), y1)) → SEL(fromInact(x0), a(y1))
A(selInact(selInact(x0, x1), y1)) → SEL(sel(a(x0), a(x1)), a(y1))
INDX(cons(X, Y), Z) → A(Z)
A(dblInact(selInact(x0, x1))) → DBL(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
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(a(x1), a(x2))
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(a(x1))
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(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
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
  ↳ Improved Ferreira Ribeiro-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ DependencyGraphProof
                                        ↳ QDP
                                          ↳ Narrowing
  ↳ Complete Giesl Middeldorp-Transformation

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

DBLS(cons(X, Y)) → A(X)
A(selInact(x1, x2)) → A(x1)
SEL(s(x0), cons(y1, y2)) → SEL(x0, a(y2))
A(selInact(sInact(x0), y1)) → SEL(s(x0), a(y1))
A(indxInact(dblsInact(x0), y1)) → INDX(dbls(a(x0)), y1)
A(dblsInact(dblInact(x0))) → DBLS(dbl(a(x0)))
SEL(s(fromInact(x0)), cons(y1, y2)) → SEL(from(x0), a(y2))
A(dblsInact(indxInact(x0, x1))) → DBLS(indx(a(x0), x1))
INDX(cons(X, Y), Z) → A(Y)
A(dblInact(sInact(x0))) → DBL(s(x0))
A(selInact(indxInact(x0, x1), y1)) → SEL(indx(a(x0), x1), a(y1))
A(dblsInact(x1)) → A(x1)
A(dblsInact(dblsInact(x0))) → DBLS(dbls(a(x0)))
A(dblsInact(fromInact(x0))) → DBLS(from(x0))
A(indxInact(selInact(x0, x1), y1)) → INDX(sel(a(x0), a(x1)), y1)
SEL(s(indxInact(x0, x1)), cons(y1, y2)) → SEL(indx(a(x0), x1), a(y2))
INDX(cons(X, Y), Z) → A(X)
SEL(s(dblInact(x0)), cons(y1, y2)) → SEL(dbl(a(x0)), a(y2))
A(dblsInact(sInact(x0))) → DBLS(s(x0))
DBL(s(X)) → A(X)
A(dblInact(fromInact(x0))) → DBL(from(x0))
SEL(s(selInact(x0, x1)), cons(y1, y2)) → SEL(sel(a(x0), a(x1)), a(y2))
A(selInact(x1, x2)) → A(x2)
SEL(0, cons(X, Y)) → A(X)
A(selInact(x0, y1)) → SEL(x0, a(y1))
A(indxInact(fromInact(x0), y1)) → INDX(from(x0), y1)
A(dblInact(dblsInact(x0))) → DBL(dbls(a(x0)))
A(dblInact(dblInact(x0))) → DBL(dbl(a(x0)))
A(fromInact(x1)) → FROM(x1)
A(dblInact(indxInact(x0, x1))) → DBL(indx(a(x0), x1))
A(dblInact(x1)) → A(x1)
A(indxInact(indxInact(x0, x1), y1)) → INDX(indx(a(x0), x1), y1)
SEL(s(dblsInact(x0)), cons(y1, y2)) → SEL(dbls(a(x0)), a(y2))
A(dblsInact(x0)) → DBLS(x0)
FROM(X) → A(X)
A(indxInact(dblInact(x0), y1)) → INDX(dbl(a(x0)), y1)
A(indxInact(x1, x2)) → A(x1)
A(indxInact(x0, y1)) → INDX(x0, y1)
DBLS(cons(X, Y)) → A(Y)
SEL(s(X), cons(Y, Z)) → A(Z)
A(dblsInact(selInact(x0, x1))) → DBLS(sel(a(x0), a(x1)))
SEL(s(sInact(x0)), cons(y1, y2)) → SEL(s(x0), a(y2))
A(selInact(dblInact(x0), y1)) → SEL(dbl(a(x0)), a(y1))
A(selInact(dblsInact(x0), y1)) → SEL(dbls(a(x0)), a(y1))
A(dblInact(x0)) → DBL(x0)
SEL(s(X), cons(Y, Z)) → A(X)
A(indxInact(sInact(x0), y1)) → INDX(s(x0), y1)
A(selInact(selInact(x0, x1), y1)) → SEL(sel(a(x0), a(x1)), a(y1))
INDX(cons(X, Y), Z) → A(Z)
A(dblInact(selInact(x0, x1))) → DBL(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
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(a(x1), a(x2))
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(a(x1))
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)

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

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



↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
  ↳ Improved Ferreira Ribeiro-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ DependencyGraphProof
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ Narrowing
  ↳ Complete Giesl Middeldorp-Transformation

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

DBLS(cons(X, Y)) → A(X)
A(selInact(x1, x2)) → A(x1)
SEL(s(x0), cons(y1, y2)) → SEL(x0, a(y2))
A(selInact(sInact(x0), y1)) → SEL(s(x0), a(y1))
A(dblsInact(dblInact(x0))) → DBLS(dbl(a(x0)))
A(indxInact(dblsInact(x0), y1)) → INDX(dbls(a(x0)), y1)
SEL(s(fromInact(x0)), cons(y1, y2)) → SEL(from(x0), a(y2))
A(dblsInact(indxInact(x0, x1))) → DBLS(indx(a(x0), x1))
INDX(cons(X, Y), Z) → A(Y)
A(dblInact(sInact(x0))) → DBL(s(x0))
A(selInact(indxInact(x0, x1), y1)) → SEL(indx(a(x0), x1), a(y1))
A(selInact(dblInact(y0), selInact(x0, x1))) → SEL(dbl(a(y0)), sel(a(x0), a(x1)))
A(dblsInact(x1)) → A(x1)
A(selInact(dblInact(y0), sInact(x0))) → SEL(dbl(a(y0)), s(x0))
A(selInact(dblInact(y0), dblInact(x0))) → SEL(dbl(a(y0)), dbl(a(x0)))
A(dblsInact(dblsInact(x0))) → DBLS(dbls(a(x0)))
A(dblsInact(fromInact(x0))) → DBLS(from(x0))
A(indxInact(selInact(x0, x1), y1)) → INDX(sel(a(x0), a(x1)), y1)
SEL(s(indxInact(x0, x1)), cons(y1, y2)) → SEL(indx(a(x0), x1), a(y2))
INDX(cons(X, Y), Z) → A(X)
SEL(s(dblInact(x0)), cons(y1, y2)) → SEL(dbl(a(x0)), a(y2))
A(dblsInact(sInact(x0))) → DBLS(s(x0))
A(selInact(dblInact(y0), dblsInact(x0))) → SEL(dbl(a(y0)), dbls(a(x0)))
DBL(s(X)) → A(X)
A(dblInact(fromInact(x0))) → DBL(from(x0))
SEL(s(selInact(x0, x1)), cons(y1, y2)) → SEL(sel(a(x0), a(x1)), a(y2))
A(selInact(x1, x2)) → A(x2)
SEL(0, cons(X, Y)) → A(X)
A(selInact(dblInact(y0), fromInact(x0))) → SEL(dbl(a(y0)), from(x0))
A(indxInact(fromInact(x0), y1)) → INDX(from(x0), y1)
A(selInact(x0, y1)) → SEL(x0, a(y1))
A(dblInact(dblsInact(x0))) → DBL(dbls(a(x0)))
A(dblInact(dblInact(x0))) → DBL(dbl(a(x0)))
A(dblInact(indxInact(x0, x1))) → DBL(indx(a(x0), x1))
A(fromInact(x1)) → FROM(x1)
A(dblInact(x1)) → A(x1)
A(indxInact(indxInact(x0, x1), y1)) → INDX(indx(a(x0), x1), y1)
A(dblsInact(x0)) → DBLS(x0)
SEL(s(dblsInact(x0)), cons(y1, y2)) → SEL(dbls(a(x0)), a(y2))
A(selInact(dblInact(y0), indxInact(x0, x1))) → SEL(dbl(a(y0)), indx(a(x0), x1))
A(selInact(dblInact(y0), x0)) → SEL(dbl(a(y0)), x0)
FROM(X) → A(X)
A(indxInact(dblInact(x0), y1)) → INDX(dbl(a(x0)), y1)
DBLS(cons(X, Y)) → A(Y)
A(indxInact(x0, y1)) → INDX(x0, y1)
A(indxInact(x1, x2)) → A(x1)
A(dblsInact(selInact(x0, x1))) → DBLS(sel(a(x0), a(x1)))
SEL(s(X), cons(Y, Z)) → A(Z)
SEL(s(sInact(x0)), cons(y1, y2)) → SEL(s(x0), a(y2))
A(selInact(dblsInact(x0), y1)) → SEL(dbls(a(x0)), a(y1))
A(dblInact(x0)) → DBL(x0)
SEL(s(X), cons(Y, Z)) → A(X)
A(indxInact(sInact(x0), y1)) → INDX(s(x0), y1)
INDX(cons(X, Y), Z) → A(Z)
A(selInact(selInact(x0, x1), y1)) → SEL(sel(a(x0), a(x1)), a(y1))
A(dblInact(selInact(x0, x1))) → DBL(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
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(a(x1), a(x2))
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(a(x1))
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)

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

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



↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
  ↳ Improved Ferreira Ribeiro-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ DependencyGraphProof
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ Narrowing
  ↳ Complete Giesl Middeldorp-Transformation

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

DBLS(cons(X, Y)) → A(X)
A(selInact(x1, x2)) → A(x1)
SEL(s(x0), cons(y1, y2)) → SEL(x0, a(y2))
A(selInact(sInact(x0), y1)) → SEL(s(x0), a(y1))
A(indxInact(dblsInact(x0), y1)) → INDX(dbls(a(x0)), y1)
A(dblsInact(dblInact(x0))) → DBLS(dbl(a(x0)))
SEL(s(fromInact(x0)), cons(y1, y2)) → SEL(from(x0), a(y2))
A(dblsInact(indxInact(x0, x1))) → DBLS(indx(a(x0), x1))
INDX(cons(X, Y), Z) → A(Y)
A(dblInact(sInact(x0))) → DBL(s(x0))
A(selInact(indxInact(x0, x1), y1)) → SEL(indx(a(x0), x1), a(y1))
A(selInact(dblInact(y0), selInact(x0, x1))) → SEL(dbl(a(y0)), sel(a(x0), a(x1)))
A(selInact(dblsInact(y0), selInact(x0, x1))) → SEL(dbls(a(y0)), sel(a(x0), a(x1)))
A(dblsInact(x1)) → A(x1)
A(selInact(dblInact(y0), sInact(x0))) → SEL(dbl(a(y0)), s(x0))
A(selInact(dblInact(y0), dblInact(x0))) → SEL(dbl(a(y0)), dbl(a(x0)))
A(dblsInact(dblsInact(x0))) → DBLS(dbls(a(x0)))
A(dblsInact(fromInact(x0))) → DBLS(from(x0))
A(indxInact(selInact(x0, x1), y1)) → INDX(sel(a(x0), a(x1)), y1)
SEL(s(indxInact(x0, x1)), cons(y1, y2)) → SEL(indx(a(x0), x1), a(y2))
INDX(cons(X, Y), Z) → A(X)
SEL(s(dblInact(x0)), cons(y1, y2)) → SEL(dbl(a(x0)), a(y2))
A(dblsInact(sInact(x0))) → DBLS(s(x0))
A(selInact(dblsInact(y0), dblInact(x0))) → SEL(dbls(a(y0)), dbl(a(x0)))
A(selInact(dblInact(y0), dblsInact(x0))) → SEL(dbl(a(y0)), dbls(a(x0)))
DBL(s(X)) → A(X)
A(dblInact(fromInact(x0))) → DBL(from(x0))
SEL(s(selInact(x0, x1)), cons(y1, y2)) → SEL(sel(a(x0), a(x1)), a(y2))
A(selInact(x1, x2)) → A(x2)
SEL(0, cons(X, Y)) → A(X)
A(selInact(dblsInact(y0), fromInact(x0))) → SEL(dbls(a(y0)), from(x0))
A(selInact(x0, y1)) → SEL(x0, a(y1))
A(indxInact(fromInact(x0), y1)) → INDX(from(x0), y1)
A(selInact(dblInact(y0), fromInact(x0))) → SEL(dbl(a(y0)), from(x0))
A(selInact(dblsInact(y0), x0)) → SEL(dbls(a(y0)), x0)
A(dblInact(dblsInact(x0))) → DBL(dbls(a(x0)))
A(dblInact(dblInact(x0))) → DBL(dbl(a(x0)))
A(fromInact(x1)) → FROM(x1)
A(dblInact(indxInact(x0, x1))) → DBL(indx(a(x0), x1))
A(selInact(dblsInact(y0), sInact(x0))) → SEL(dbls(a(y0)), s(x0))
A(dblInact(x1)) → A(x1)
A(indxInact(indxInact(x0, x1), y1)) → INDX(indx(a(x0), x1), y1)
SEL(s(dblsInact(x0)), cons(y1, y2)) → SEL(dbls(a(x0)), a(y2))
A(dblsInact(x0)) → DBLS(x0)
A(selInact(dblInact(y0), indxInact(x0, x1))) → SEL(dbl(a(y0)), indx(a(x0), x1))
A(selInact(dblInact(y0), x0)) → SEL(dbl(a(y0)), x0)
A(selInact(dblsInact(y0), indxInact(x0, x1))) → SEL(dbls(a(y0)), indx(a(x0), x1))
FROM(X) → A(X)
A(indxInact(dblInact(x0), y1)) → INDX(dbl(a(x0)), y1)
A(indxInact(x1, x2)) → A(x1)
A(indxInact(x0, y1)) → INDX(x0, y1)
DBLS(cons(X, Y)) → A(Y)
A(selInact(dblsInact(y0), dblsInact(x0))) → SEL(dbls(a(y0)), dbls(a(x0)))
SEL(s(X), cons(Y, Z)) → A(Z)
A(dblsInact(selInact(x0, x1))) → DBLS(sel(a(x0), a(x1)))
SEL(s(sInact(x0)), cons(y1, y2)) → SEL(s(x0), a(y2))
A(dblInact(x0)) → DBL(x0)
SEL(s(X), cons(Y, Z)) → A(X)
A(indxInact(sInact(x0), y1)) → INDX(s(x0), y1)
A(selInact(selInact(x0, x1), y1)) → SEL(sel(a(x0), a(x1)), a(y1))
INDX(cons(X, Y), Z) → A(Z)
A(dblInact(selInact(x0, x1))) → DBL(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
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(a(x1), a(x2))
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(a(x1))
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)

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

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



↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
  ↳ Improved Ferreira Ribeiro-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ DependencyGraphProof
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ Narrowing
                                                    ↳ QDP
                                                      ↳ Narrowing
  ↳ Complete Giesl Middeldorp-Transformation

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

DBLS(cons(X, Y)) → A(X)
A(selInact(x1, x2)) → A(x1)
A(selInact(sInact(y0), dblInact(x0))) → SEL(s(y0), dbl(a(x0)))
SEL(s(x0), cons(y1, y2)) → SEL(x0, a(y2))
A(dblsInact(dblInact(x0))) → DBLS(dbl(a(x0)))
A(indxInact(dblsInact(x0), y1)) → INDX(dbls(a(x0)), y1)
SEL(s(fromInact(x0)), cons(y1, y2)) → SEL(from(x0), a(y2))
A(dblsInact(indxInact(x0, x1))) → DBLS(indx(a(x0), x1))
INDX(cons(X, Y), Z) → A(Y)
A(dblInact(sInact(x0))) → DBL(s(x0))
A(selInact(sInact(y0), x0)) → SEL(s(y0), x0)
A(selInact(indxInact(x0, x1), y1)) → SEL(indx(a(x0), x1), a(y1))
A(selInact(dblInact(y0), selInact(x0, x1))) → SEL(dbl(a(y0)), sel(a(x0), a(x1)))
A(dblsInact(x1)) → A(x1)
A(selInact(dblsInact(y0), selInact(x0, x1))) → SEL(dbls(a(y0)), sel(a(x0), a(x1)))
A(selInact(dblInact(y0), sInact(x0))) → SEL(dbl(a(y0)), s(x0))
A(selInact(dblInact(y0), dblInact(x0))) → SEL(dbl(a(y0)), dbl(a(x0)))
A(dblsInact(dblsInact(x0))) → DBLS(dbls(a(x0)))
A(selInact(sInact(y0), fromInact(x0))) → SEL(s(y0), from(x0))
A(dblsInact(fromInact(x0))) → DBLS(from(x0))
A(indxInact(selInact(x0, x1), y1)) → INDX(sel(a(x0), a(x1)), y1)
SEL(s(indxInact(x0, x1)), cons(y1, y2)) → SEL(indx(a(x0), x1), a(y2))
INDX(cons(X, Y), Z) → A(X)
SEL(s(dblInact(x0)), cons(y1, y2)) → SEL(dbl(a(x0)), a(y2))
A(dblsInact(sInact(x0))) → DBLS(s(x0))
A(selInact(dblInact(y0), dblsInact(x0))) → SEL(dbl(a(y0)), dbls(a(x0)))
A(selInact(dblsInact(y0), dblInact(x0))) → SEL(dbls(a(y0)), dbl(a(x0)))
DBL(s(X)) → A(X)
A(dblInact(fromInact(x0))) → DBL(from(x0))
A(selInact(sInact(y0), dblsInact(x0))) → SEL(s(y0), dbls(a(x0)))
SEL(s(selInact(x0, x1)), cons(y1, y2)) → SEL(sel(a(x0), a(x1)), a(y2))
A(selInact(x1, x2)) → A(x2)
SEL(0, cons(X, Y)) → A(X)
A(selInact(dblsInact(y0), fromInact(x0))) → SEL(dbls(a(y0)), from(x0))
A(selInact(dblInact(y0), fromInact(x0))) → SEL(dbl(a(y0)), from(x0))
A(indxInact(fromInact(x0), y1)) → INDX(from(x0), y1)
A(selInact(x0, y1)) → SEL(x0, a(y1))
A(selInact(dblsInact(y0), x0)) → SEL(dbls(a(y0)), x0)
A(dblInact(dblsInact(x0))) → DBL(dbls(a(x0)))
A(dblInact(dblInact(x0))) → DBL(dbl(a(x0)))
A(dblInact(indxInact(x0, x1))) → DBL(indx(a(x0), x1))
A(fromInact(x1)) → FROM(x1)
A(selInact(dblsInact(y0), sInact(x0))) → SEL(dbls(a(y0)), s(x0))
A(dblInact(x1)) → A(x1)
A(indxInact(indxInact(x0, x1), y1)) → INDX(indx(a(x0), x1), y1)
A(dblsInact(x0)) → DBLS(x0)
SEL(s(dblsInact(x0)), cons(y1, y2)) → SEL(dbls(a(x0)), a(y2))
A(selInact(sInact(y0), sInact(x0))) → SEL(s(y0), s(x0))
A(selInact(sInact(y0), indxInact(x0, x1))) → SEL(s(y0), indx(a(x0), x1))
A(selInact(dblInact(y0), indxInact(x0, x1))) → SEL(dbl(a(y0)), indx(a(x0), x1))
A(selInact(sInact(y0), selInact(x0, x1))) → SEL(s(y0), sel(a(x0), a(x1)))
A(selInact(dblInact(y0), x0)) → SEL(dbl(a(y0)), x0)
FROM(X) → A(X)
A(selInact(dblsInact(y0), indxInact(x0, x1))) → SEL(dbls(a(y0)), indx(a(x0), x1))
A(indxInact(dblInact(x0), y1)) → INDX(dbl(a(x0)), y1)
DBLS(cons(X, Y)) → A(Y)
A(indxInact(x0, y1)) → INDX(x0, y1)
A(indxInact(x1, x2)) → A(x1)
A(selInact(dblsInact(y0), dblsInact(x0))) → SEL(dbls(a(y0)), dbls(a(x0)))
A(dblsInact(selInact(x0, x1))) → DBLS(sel(a(x0), a(x1)))
SEL(s(X), cons(Y, Z)) → A(Z)
SEL(s(sInact(x0)), cons(y1, y2)) → SEL(s(x0), a(y2))
A(dblInact(x0)) → DBL(x0)
SEL(s(X), cons(Y, Z)) → A(X)
A(indxInact(sInact(x0), y1)) → INDX(s(x0), y1)
INDX(cons(X, Y), Z) → A(Z)
A(selInact(selInact(x0, x1), y1)) → SEL(sel(a(x0), a(x1)), a(y1))
A(dblInact(selInact(x0, x1))) → DBL(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
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(a(x1), a(x2))
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(a(x1))
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(x1)

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

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



↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
  ↳ Improved Ferreira Ribeiro-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ DependencyGraphProof
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ Narrowing
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ Narrowing
  ↳ Complete Giesl Middeldorp-Transformation

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

A(selInact(indxInact(y0, y1), selInact(x0, x1))) → SEL(indx(a(y0), y1), sel(a(x0), a(x1)))
DBLS(cons(X, Y)) → A(X)
A(selInact(indxInact(y0, y1), fromInact(x0))) → SEL(indx(a(y0), y1), from(x0))
A(selInact(x1, x2)) → A(x1)
SEL(s(x0), cons(y1, y2)) → SEL(x0, a(y2))
A(selInact(sInact(y0), dblInact(x0))) → SEL(s(y0), dbl(a(x0)))
A(indxInact(dblsInact(x0), y1)) → INDX(dbls(a(x0)), y1)
A(dblsInact(dblInact(x0))) → DBLS(dbl(a(x0)))
A(selInact(indxInact(y0, y1), sInact(x0))) → SEL(indx(a(y0), y1), s(x0))
SEL(s(fromInact(x0)), cons(y1, y2)) → SEL(from(x0), a(y2))
A(dblsInact(indxInact(x0, x1))) → DBLS(indx(a(x0), x1))
INDX(cons(X, Y), Z) → A(Y)
A(dblInact(sInact(x0))) → DBL(s(x0))
A(selInact(sInact(y0), x0)) → SEL(s(y0), x0)
A(selInact(dblInact(y0), selInact(x0, x1))) → SEL(dbl(a(y0)), sel(a(x0), a(x1)))
A(selInact(dblsInact(y0), selInact(x0, x1))) → SEL(dbls(a(y0)), sel(a(x0), a(x1)))
A(dblsInact(x1)) → A(x1)
A(selInact(dblInact(y0), sInact(x0))) → SEL(dbl(a(y0)), s(x0))
A(selInact(dblInact(y0), dblInact(x0))) → SEL(dbl(a(y0)), dbl(a(x0)))
A(dblsInact(dblsInact(x0))) → DBLS(dbls(a(x0)))
A(selInact(sInact(y0), fromInact(x0))) → SEL(s(y0), from(x0))
A(selInact(indxInact(y0, y1), x0)) → SEL(indx(a(y0), y1), x0)
A(dblsInact(fromInact(x0))) → DBLS(from(x0))
A(indxInact(selInact(x0, x1), y1)) → INDX(sel(a(x0), a(x1)), y1)
SEL(s(indxInact(x0, x1)), cons(y1, y2)) → SEL(indx(a(x0), x1), a(y2))
INDX(cons(X, Y), Z) → A(X)
SEL(s(dblInact(x0)), cons(y1, y2)) → SEL(dbl(a(x0)), a(y2))
A(dblsInact(sInact(x0))) → DBLS(s(x0))
A(selInact(dblsInact(y0), dblInact(x0))) → SEL(dbls(a(y0)), dbl(a(x0)))
A(selInact(dblInact(y0), dblsInact(x0))) → SEL(dbl(a(y0)), dbls(a(x0)))
DBL(s(X)) → A(X)
A(dblInact(fromInact(x0))) → DBL(from(x0))
A(selInact(sInact(y0), dblsInact(x0))) → SEL(s(y0), dbls(a(x0)))
SEL(s(selInact(x0, x1)), cons(y1, y2)) → SEL(sel(a(x0), a(x1)), a(y2))
A(selInact(x1, x2)) → A(x2)
SEL(0, cons(X, Y)) → A(X)
A(selInact(dblsInact(y0), fromInact(x0))) → SEL(dbls(a(y0)), from(x0))
A(selInact(indxInact(y0, y1), indxInact(x0, x1))) → SEL(indx(a(y0), y1), indx(a(x0), x1))
A(selInact(x0, y1)) → SEL(x0, a(y1))
A(indxInact(fromInact(x0), y1)) → INDX(from(x0), y1)
A(selInact(dblInact(y0), fromInact(x0))) → SEL(dbl(a(y0)), from(x0))
A(selInact(dblsInact(y0), x0)) → SEL(dbls(a(y0)), x0)
A(selInact(indxInact(y0, y1), dblsInact(x0))) → SEL(indx(a(y0), y1), dbls(a(x0)))
A(dblInact(dblsInact(x0))) → DBL(dbls(a(x0)))
A(dblInact(dblInact(x0))) → DBL(dbl(a(x0)))
A(fromInact(x1)) → FROM(x1)
A(dblInact(indxInact(x0, x1))) → DBL(indx(a(x0), x1))
A(selInact(dblsInact(y0), sInact(x0))) → SEL(dbls(a(y0)), s(x0))
A(dblInact(x1)) → A(x1)
A(indxInact(indxInact(x0, x1), y1)) → INDX(indx(a(x0), x1), y1)
SEL(s(dblsInact(x0)), cons(y1, y2)) → SEL(dbls(a(x0)), a(y2))
A(dblsInact(x0)) → DBLS(x0)
A(selInact(sInact(y0), sInact(x0))) → SEL(s(y0), s(x0))
A(selInact(dblInact(y0), indxInact(x0, x1))) → SEL(dbl(a(y0)), indx(a(x0), x1))
A(selInact(sInact(y0), indxInact(x0, x1))) → SEL(s(y0), indx(a(x0), x1))
A(selInact(sInact(y0), selInact(x0, x1))) → SEL(s(y0), sel(a(x0), a(x1)))
A(selInact(dblInact(y0), x0)) → SEL(dbl(a(y0)), x0)
A(selInact(dblsInact(y0), indxInact(x0, x1))) → SEL(dbls(a(y0)), indx(a(x0), x1))
FROM(X) → A(X)
A(indxInact(dblInact(x0), y1)) → INDX(dbl(a(x0)), y1)
A(indxInact(x1, x2)) → A(x1)
A(indxInact(x0, y1)) → INDX(x0, y1)
DBLS(cons(X, Y)) → A(Y)
A(selInact(dblsInact(y0), dblsInact(x0))) → SEL(dbls(a(y0)), dbls(a(x0)))
SEL(s(X), cons(Y, Z)) → A(Z)
A(dblsInact(selInact(x0, x1))) → DBLS(sel(a(x0), a(x1)))
SEL(s(sInact(x0)), cons(y1, y2)) → SEL(s(x0), a(y2))
A(dblInact(x0)) → DBL(x0)
SEL(s(X), cons(Y, Z)) → A(X)
A(indxInact(sInact(x0), y1)) → INDX(s(x0), y1)
A(selInact(selInact(x0, x1), y1)) → SEL(sel(a(x0), a(x1)), a(y1))
INDX(cons(X, Y), Z) → A(Z)
A(dblInact(selInact(x0, x1))) → DBL(sel(a(x0), a(x1)))
A(selInact(indxInact(y0, y1), dblInact(x0))) → SEL(indx(a(y0), y1), 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
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(a(x1), a(x2))
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(a(x1))
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(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, dblInact(x0))) → SEL(sel(a(y0), a(y1)), dbl(a(x0)))
SEL(s(selInact(y0, y1)), cons(y2, sInact(x0))) → SEL(sel(a(y0), a(y1)), s(x0))
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(a(y0), a(y1)), sel(a(x0), a(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, dblsInact(x0))) → SEL(sel(a(y0), a(y1)), dbls(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, x0)) → SEL(sel(a(y0), a(y1)), x0)



↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
  ↳ Improved Ferreira Ribeiro-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ DependencyGraphProof
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ Narrowing
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ Narrowing
                                                            ↳ QDP
                                                              ↳ Narrowing
  ↳ Complete Giesl Middeldorp-Transformation

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

DBLS(cons(X, Y)) → A(X)
A(selInact(indxInact(y0, y1), selInact(x0, x1))) → SEL(indx(a(y0), y1), sel(a(x0), a(x1)))
A(selInact(x1, x2)) → A(x1)
A(selInact(indxInact(y0, y1), fromInact(x0))) → SEL(indx(a(y0), y1), from(x0))
A(selInact(sInact(y0), dblInact(x0))) → SEL(s(y0), dbl(a(x0)))
SEL(s(x0), cons(y1, y2)) → SEL(x0, a(y2))
A(dblsInact(dblInact(x0))) → DBLS(dbl(a(x0)))
A(indxInact(dblsInact(x0), y1)) → INDX(dbls(a(x0)), y1)
A(selInact(indxInact(y0, y1), sInact(x0))) → SEL(indx(a(y0), y1), s(x0))
SEL(s(fromInact(x0)), cons(y1, y2)) → SEL(from(x0), a(y2))
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(a(y0), a(y1)), sel(a(x0), a(x1)))
A(dblsInact(indxInact(x0, x1))) → DBLS(indx(a(x0), x1))
INDX(cons(X, Y), Z) → A(Y)
A(dblInact(sInact(x0))) → DBL(s(x0))
A(selInact(sInact(y0), x0)) → SEL(s(y0), x0)
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(a(y0), a(y1)), x0)
A(selInact(dblInact(y0), selInact(x0, x1))) → SEL(dbl(a(y0)), sel(a(x0), a(x1)))
A(dblsInact(x1)) → A(x1)
A(selInact(dblsInact(y0), selInact(x0, x1))) → SEL(dbls(a(y0)), sel(a(x0), a(x1)))
A(selInact(dblInact(y0), sInact(x0))) → SEL(dbl(a(y0)), s(x0))
A(selInact(dblInact(y0), dblInact(x0))) → SEL(dbl(a(y0)), dbl(a(x0)))
A(dblsInact(dblsInact(x0))) → DBLS(dbls(a(x0)))
A(selInact(sInact(y0), fromInact(x0))) → SEL(s(y0), from(x0))
A(selInact(indxInact(y0, y1), x0)) → SEL(indx(a(y0), y1), x0)
A(dblsInact(fromInact(x0))) → DBLS(from(x0))
A(indxInact(selInact(x0, x1), y1)) → INDX(sel(a(x0), a(x1)), y1)
SEL(s(indxInact(x0, x1)), cons(y1, y2)) → SEL(indx(a(x0), x1), a(y2))
INDX(cons(X, Y), Z) → A(X)
SEL(s(dblInact(x0)), cons(y1, y2)) → SEL(dbl(a(x0)), a(y2))
A(dblsInact(sInact(x0))) → DBLS(s(x0))
A(selInact(dblInact(y0), dblsInact(x0))) → SEL(dbl(a(y0)), dbls(a(x0)))
A(selInact(dblsInact(y0), dblInact(x0))) → SEL(dbls(a(y0)), dbl(a(x0)))
DBL(s(X)) → A(X)
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(a(y0), a(y1)), from(x0))
A(dblInact(fromInact(x0))) → DBL(from(x0))
A(selInact(sInact(y0), dblsInact(x0))) → SEL(s(y0), dbls(a(x0)))
A(selInact(x1, x2)) → A(x2)
SEL(0, cons(X, Y)) → A(X)
A(selInact(dblsInact(y0), fromInact(x0))) → SEL(dbls(a(y0)), from(x0))
A(selInact(indxInact(y0, y1), indxInact(x0, x1))) → SEL(indx(a(y0), y1), indx(a(x0), x1))
A(selInact(dblInact(y0), fromInact(x0))) → SEL(dbl(a(y0)), from(x0))
A(indxInact(fromInact(x0), y1)) → INDX(from(x0), y1)
A(selInact(x0, y1)) → SEL(x0, a(y1))
A(selInact(dblsInact(y0), x0)) → SEL(dbls(a(y0)), x0)
A(dblInact(dblsInact(x0))) → DBL(dbls(a(x0)))
A(selInact(indxInact(y0, y1), dblsInact(x0))) → SEL(indx(a(y0), y1), dbls(a(x0)))
A(dblInact(dblInact(x0))) → DBL(dbl(a(x0)))
A(dblInact(indxInact(x0, x1))) → DBL(indx(a(x0), x1))
A(fromInact(x1)) → FROM(x1)
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(a(y0), a(y1)), dbls(a(x0)))
A(selInact(dblsInact(y0), sInact(x0))) → SEL(dbls(a(y0)), s(x0))
A(dblInact(x1)) → A(x1)
A(indxInact(indxInact(x0, x1), y1)) → INDX(indx(a(x0), x1), y1)
A(dblsInact(x0)) → DBLS(x0)
SEL(s(dblsInact(x0)), cons(y1, y2)) → SEL(dbls(a(x0)), a(y2))
A(selInact(sInact(y0), sInact(x0))) → SEL(s(y0), s(x0))
A(selInact(sInact(y0), indxInact(x0, x1))) → SEL(s(y0), indx(a(x0), x1))
A(selInact(dblInact(y0), indxInact(x0, x1))) → SEL(dbl(a(y0)), indx(a(x0), x1))
A(selInact(sInact(y0), selInact(x0, x1))) → SEL(s(y0), sel(a(x0), a(x1)))
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(a(y0), a(y1)), dbl(a(x0)))
A(selInact(dblInact(y0), x0)) → SEL(dbl(a(y0)), x0)
SEL(s(selInact(y0, y1)), cons(y2, sInact(x0))) → SEL(sel(a(y0), a(y1)), s(x0))
FROM(X) → A(X)
A(selInact(dblsInact(y0), indxInact(x0, x1))) → SEL(dbls(a(y0)), indx(a(x0), x1))
A(indxInact(dblInact(x0), y1)) → INDX(dbl(a(x0)), y1)
DBLS(cons(X, Y)) → A(Y)
A(indxInact(x0, y1)) → INDX(x0, y1)
A(indxInact(x1, x2)) → A(x1)
A(selInact(dblsInact(y0), dblsInact(x0))) → SEL(dbls(a(y0)), dbls(a(x0)))
SEL(s(selInact(y0, y1)), cons(y2, indxInact(x0, x1))) → SEL(sel(a(y0), a(y1)), indx(a(x0), x1))
A(dblsInact(selInact(x0, x1))) → DBLS(sel(a(x0), a(x1)))
SEL(s(X), cons(Y, Z)) → A(Z)
SEL(s(sInact(x0)), cons(y1, y2)) → SEL(s(x0), a(y2))
A(dblInact(x0)) → DBL(x0)
SEL(s(X), cons(Y, Z)) → A(X)
A(indxInact(sInact(x0), y1)) → INDX(s(x0), y1)
INDX(cons(X, Y), Z) → A(Z)
A(selInact(selInact(x0, x1), y1)) → SEL(sel(a(x0), a(x1)), a(y1))
A(dblInact(selInact(x0, x1))) → DBL(sel(a(x0), a(x1)))
A(selInact(indxInact(y0, y1), dblInact(x0))) → SEL(indx(a(y0), y1), 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
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(a(x1), a(x2))
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(a(x1))
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(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, dblsInact(x0))) → SEL(s(y0), dbls(a(x0)))
SEL(s(sInact(y0)), cons(y1, dblInact(x0))) → SEL(s(y0), dbl(a(x0)))
SEL(s(sInact(y0)), cons(y1, fromInact(x0))) → SEL(s(y0), from(x0))
SEL(s(sInact(y0)), cons(y1, sInact(x0))) → SEL(s(y0), s(x0))
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(a(x0), a(x1)))
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(a(x0), x1))
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)



↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
  ↳ Improved Ferreira Ribeiro-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ DependencyGraphProof
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ Narrowing
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ Narrowing
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ Narrowing
  ↳ Complete Giesl Middeldorp-Transformation

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

A(selInact(indxInact(y0, y1), selInact(x0, x1))) → SEL(indx(a(y0), y1), sel(a(x0), a(x1)))
DBLS(cons(X, Y)) → A(X)
A(selInact(indxInact(y0, y1), fromInact(x0))) → SEL(indx(a(y0), y1), from(x0))
A(selInact(x1, x2)) → A(x1)
SEL(s(x0), cons(y1, y2)) → SEL(x0, a(y2))
A(selInact(sInact(y0), dblInact(x0))) → SEL(s(y0), dbl(a(x0)))
A(indxInact(dblsInact(x0), y1)) → INDX(dbls(a(x0)), y1)
A(dblsInact(dblInact(x0))) → DBLS(dbl(a(x0)))
A(selInact(indxInact(y0, y1), sInact(x0))) → SEL(indx(a(y0), y1), s(x0))
SEL(s(fromInact(x0)), cons(y1, y2)) → SEL(from(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(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)
A(dblsInact(indxInact(x0, x1))) → DBLS(indx(a(x0), x1))
INDX(cons(X, Y), Z) → A(Y)
A(dblInact(sInact(x0))) → DBL(s(x0))
A(selInact(sInact(y0), x0)) → SEL(s(y0), x0)
A(selInact(dblInact(y0), 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)
A(selInact(dblsInact(y0), selInact(x0, x1))) → SEL(dbls(a(y0)), sel(a(x0), a(x1)))
A(dblsInact(x1)) → A(x1)
A(selInact(dblInact(y0), sInact(x0))) → SEL(dbl(a(y0)), s(x0))
A(selInact(dblInact(y0), dblInact(x0))) → SEL(dbl(a(y0)), dbl(a(x0)))
A(dblsInact(dblsInact(x0))) → DBLS(dbls(a(x0)))
A(selInact(sInact(y0), fromInact(x0))) → SEL(s(y0), from(x0))
A(selInact(indxInact(y0, y1), x0)) → SEL(indx(a(y0), y1), x0)
A(dblsInact(fromInact(x0))) → DBLS(from(x0))
A(indxInact(selInact(x0, x1), y1)) → INDX(sel(a(x0), a(x1)), y1)
SEL(s(indxInact(x0, x1)), cons(y1, y2)) → SEL(indx(a(x0), x1), a(y2))
INDX(cons(X, Y), Z) → A(X)
SEL(s(sInact(y0)), cons(y1, sInact(x0))) → SEL(s(y0), s(x0))
SEL(s(dblInact(x0)), cons(y1, y2)) → SEL(dbl(a(x0)), a(y2))
A(dblsInact(sInact(x0))) → DBLS(s(x0))
A(selInact(dblsInact(y0), dblInact(x0))) → SEL(dbls(a(y0)), dbl(a(x0)))
A(selInact(dblInact(y0), dblsInact(x0))) → SEL(dbl(a(y0)), dbls(a(x0)))
DBL(s(X)) → A(X)
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(a(y0), a(y1)), from(x0))
A(dblInact(fromInact(x0))) → DBL(from(x0))
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(a(x0), x1))
A(selInact(sInact(y0), dblsInact(x0))) → SEL(s(y0), dbls(a(x0)))
A(selInact(x1, x2)) → A(x2)
SEL(s(sInact(y0)), cons(y1, dblInact(x0))) → SEL(s(y0), dbl(a(x0)))
SEL(0, cons(X, Y)) → A(X)
A(selInact(dblsInact(y0), fromInact(x0))) → SEL(dbls(a(y0)), from(x0))
A(selInact(indxInact(y0, y1), indxInact(x0, x1))) → SEL(indx(a(y0), y1), indx(a(x0), x1))
A(selInact(x0, y1)) → SEL(x0, a(y1))
A(indxInact(fromInact(x0), y1)) → INDX(from(x0), y1)
A(selInact(dblInact(y0), fromInact(x0))) → SEL(dbl(a(y0)), from(x0))
A(selInact(dblsInact(y0), x0)) → SEL(dbls(a(y0)), x0)
A(selInact(indxInact(y0, y1), dblsInact(x0))) → SEL(indx(a(y0), y1), dbls(a(x0)))
A(dblInact(dblsInact(x0))) → DBL(dbls(a(x0)))
A(dblInact(dblInact(x0))) → DBL(dbl(a(x0)))
A(fromInact(x1)) → FROM(x1)
A(dblInact(indxInact(x0, x1))) → DBL(indx(a(x0), x1))
A(selInact(dblsInact(y0), sInact(x0))) → SEL(dbls(a(y0)), s(x0))
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(a(y0), a(y1)), dbls(a(x0)))
A(dblInact(x1)) → A(x1)
A(indxInact(indxInact(x0, x1), y1)) → INDX(indx(a(x0), x1), y1)
SEL(s(dblsInact(x0)), cons(y1, y2)) → SEL(dbls(a(x0)), a(y2))
A(dblsInact(x0)) → DBLS(x0)
A(selInact(sInact(y0), sInact(x0))) → SEL(s(y0), s(x0))
SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(a(x0)))
A(selInact(dblInact(y0), indxInact(x0, x1))) → SEL(dbl(a(y0)), indx(a(x0), x1))
A(selInact(sInact(y0), indxInact(x0, x1))) → SEL(s(y0), indx(a(x0), x1))
A(selInact(sInact(y0), selInact(x0, x1))) → SEL(s(y0), sel(a(x0), a(x1)))
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(a(y0), a(y1)), dbl(a(x0)))
A(selInact(dblInact(y0), x0)) → SEL(dbl(a(y0)), x0)
SEL(s(sInact(y0)), cons(y1, fromInact(x0))) → SEL(s(y0), from(x0))
A(selInact(dblsInact(y0), indxInact(x0, x1))) → SEL(dbls(a(y0)), indx(a(x0), x1))
FROM(X) → A(X)
SEL(s(selInact(y0, y1)), cons(y2, sInact(x0))) → SEL(sel(a(y0), a(y1)), s(x0))
A(indxInact(dblInact(x0), y1)) → INDX(dbl(a(x0)), y1)
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y0), sel(a(x0), a(x1)))
A(indxInact(x1, x2)) → A(x1)
A(indxInact(x0, y1)) → INDX(x0, y1)
DBLS(cons(X, Y)) → A(Y)
A(selInact(dblsInact(y0), dblsInact(x0))) → SEL(dbls(a(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(X), cons(Y, Z)) → A(Z)
A(dblsInact(selInact(x0, x1))) → DBLS(sel(a(x0), a(x1)))
A(dblInact(x0)) → DBL(x0)
SEL(s(X), cons(Y, Z)) → A(X)
A(indxInact(sInact(x0), y1)) → INDX(s(x0), y1)
A(selInact(selInact(x0, x1), y1)) → SEL(sel(a(x0), a(x1)), a(y1))
INDX(cons(X, Y), Z) → A(Z)
A(dblInact(selInact(x0, x1))) → DBL(sel(a(x0), a(x1)))
A(selInact(indxInact(y0, y1), dblInact(x0))) → SEL(indx(a(y0), y1), 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
from(x1) → fromInact(x1)
a(fromInact(x1)) → from(x1)
sel(x1, x2) → selInact(x1, x2)
a(selInact(x1, x2)) → sel(a(x1), a(x2))
indx(x1, x2) → indxInact(x1, x2)
a(indxInact(x1, x2)) → indx(a(x1), x2)
dbl(x1) → dblInact(x1)
a(dblInact(x1)) → dbl(a(x1))
dbls(x1) → dblsInact(x1)
a(dblsInact(x1)) → dbls(a(x1))
s(x1) → sInact(x1)
a(sInact(x1)) → s(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, sInact(x0))) → SEL(y0, s(x0))
SEL(s(y0), cons(y1, x0)) → SEL(y0, x0)
SEL(s(y0), cons(y1, selInact(x0, x1))) → SEL(y0, sel(a(x0), a(x1)))
SEL(s(y0), cons(y1, indxInact(x0, x1))) → SEL(y0, indx(a(x0), x1))
SEL(s(y0), cons(y1, dblInact(x0))) → SEL(y0, dbl(a(x0)))
SEL(s(y0), cons(y1, fromInact(x0))) → SEL(y0, from(x0))



↳ CSR
  ↳ Zantema-Transformation
  ↳ Incomplete Giesl Middeldorp-Transformation
  ↳ Improved Ferreira Ribeiro-Transformation
    ↳ QTRS
      ↳ DependencyPairsProof
        ↳ QDP
          ↳ DependencyGraphProof
            ↳ QDP
              ↳ Narrowing
                ↳ QDP
                  ↳ Narrowing
                    ↳ QDP
                      ↳ Narrowing
                        ↳ QDP
                          ↳ Narrowing
                            ↳ QDP
                              ↳ Narrowing
                                ↳ QDP
                                  ↳ Narrowing
                                    ↳ QDP
                                      ↳ DependencyGraphProof
                                        ↳ QDP
                                          ↳ Narrowing
                                            ↳ QDP
                                              ↳ Narrowing
                                                ↳ QDP
                                                  ↳ Narrowing
                                                    ↳ QDP
                                                      ↳ Narrowing
                                                        ↳ QDP
                                                          ↳ Narrowing
                                                            ↳ QDP
                                                              ↳ Narrowing
                                                                ↳ QDP
                                                                  ↳ Narrowing
                                                                    ↳ QDP
                                                                      ↳ Narrowing
  ↳ Complete Giesl Middeldorp-Transformation

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

DBLS(cons(X, Y)) → A(X)
A(selInact(indxInact(y0, y1), selInact(x0, x1))) → SEL(indx(a(y0), y1), sel(a(x0), a(x1)))
A(selInact(x1, x2)) → A(x1)
A(selInact(indxInact(y0, y1), fromInact(x0))) → SEL(indx(a(y0), y1), from(x0))
A(selInact(sInact(y0), dblInact(x0))) → SEL(s(y0), dbl(a(x0)))
A(dblsInact(dblInact(x0))) → DBLS(dbl(a(x0)))
A(indxInact(dblsInact(x0), y1)) → INDX(dbls(a(x0)), y1)
A(selInact(indxInact(y0, y1), sInact(x0))) → SEL(indx(a(y0), y1), s(x0))
SEL(s(fromInact(x0)), cons(y1, y2)) → SEL(from(x0), a(y2))
SEL(s(selInact(y0, y1)), cons(y2, selInact(x0, x1))) → SEL(sel(a(y0), a(y1)), sel(a(x0), a(x1)))
A(dblsInact(indxInact(x0, x1))) → DBLS(indx(a(x0), x1))
SEL(s(sInact(y0)), cons(y1, x0)) → SEL(s(y0), x0)
INDX(cons(X, Y), Z) → A(Y)
A(dblInact(sInact(x0))) → DBL(s(x0))
A(selInact(sInact(y0), x0)) → SEL(s(y0), x0)
SEL(s(selInact(y0, y1)), cons(y2, x0)) → SEL(sel(a(y0), a(y1)), x0)
A(selInact(dblInact(y0), selInact(x0, x1))) → SEL(dbl(a(y0)), sel(a(x0), a(x1)))
A(dblsInact(x1)) → A(x1)
A(selInact(dblsInact(y0), selInact(x0, x1))) → SEL(dbls(a(y0)), sel(a(x0), a(x1)))
A(selInact(dblInact(y0), sInact(x0))) → SEL(dbl(a(y0)), s(x0))
A(selInact(dblInact(y0), dblInact(x0))) → SEL(dbl(a(y0)), dbl(a(x0)))
SEL(s(y0), cons(y1, dblsInact(x0))) → SEL(y0, dbls(a(x0)))
A(dblsInact(dblsInact(x0))) → DBLS(dbls(a(x0)))
A(selInact(sInact(y0), fromInact(x0))) → SEL(s(y0), from(x0))
A(selInact(indxInact(y0, y1), x0)) → SEL(indx(a(y0), y1), x0)
A(dblsInact(fromInact(x0))) → DBLS(from(x0))
A(indxInact(selInact(x0, x1), y1)) → INDX(sel(a(x0), a(x1)), y1)
SEL(s(indxInact(x0, x1)), cons(y1, y2)) → SEL(indx(a(x0), x1), a(y2))
INDX(cons(X, Y), Z) → A(X)
SEL(s(sInact(y0)), cons(y1, sInact(x0))) → SEL(s(y0), s(x0))
SEL(s(dblInact(x0)), cons(y1, y2)) → SEL(dbl(a(x0)), a(y2))
A(dblsInact(sInact(x0))) → DBLS(s(x0))
A(selInact(dblInact(y0), dblsInact(x0))) → SEL(dbl(a(y0)), dbls(a(x0)))
A(selInact(dblsInact(y0), dblInact(x0))) → SEL(dbls(a(y0)), dbl(a(x0)))
DBL(s(X)) → A(X)
SEL(s(selInact(y0, y1)), cons(y2, fromInact(x0))) → SEL(sel(a(y0), a(y1)), from(x0))
A(dblInact(fromInact(x0))) → DBL(from(x0))
SEL(s(sInact(y0)), cons(y1, indxInact(x0, x1))) → SEL(s(y0), indx(a(x0), x1))
A(selInact(sInact(y0), dblsInact(x0))) → SEL(s(y0), dbls(a(x0)))
A(selInact(x1, x2)) → A(x2)
SEL(s(sInact(y0)), cons(y1, dblInact(x0))) → SEL(s(y0), dbl(a(x0)))
SEL(0, cons(X, Y)) → A(X)
A(selInact(dblsInact(y0), fromInact(x0))) → SEL(dbls(a(y0)), from(x0))
A(selInact(indxInact(y0, y1), indxInact(x0, x1))) → SEL(indx(a(y0), y1), indx(a(x0), x1))
A(selInact(dblInact(y0), fromInact(x0))) → SEL(dbl(a(y0)), from(x0))
A(indxInact(fromInact(x0), y1)) → INDX(from(x0), y1)
A(selInact(x0, y1)) → SEL(x0, a(y1))
A(selInact(dblsInact(y0), x0)) → SEL(dbls(a(y0)), x0)
A(dblInact(dblsInact(x0))) → DBL(dbls(a(x0)))
A(selInact(indxInact(y0, y1), dblsInact(x0))) → SEL(indx(a(y0), y1), dbls(a(x0)))
A(dblInact(dblInact(x0))) → DBL(dbl(a(x0)))
A(dblInact(indxInact(x0, x1))) → DBL(indx(a(x0), x1))
A(fromInact(x1)) → FROM(x1)
SEL(s(selInact(y0, y1)), cons(y2, dblsInact(x0))) → SEL(sel(a(y0), a(y1)), dbls(a(x0)))
A(selInact(dblsInact(y0), 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(y0), cons(y1, x0)) → SEL(y0, x0)
A(dblInact(x1)) → A(x1)
SEL(s(y0), cons(y1, indxInact(x0, x1))) → SEL(y0, indx(a(x0), x1))
A(indxInact(indxInact(x0, x1), y1)) → INDX(indx(a(x0), x1), y1)
A(dblsInact(x0)) → DBLS(x0)
SEL(s(dblsInact(x0)), cons(y1, y2)) → SEL(dbls(a(x0)), a(y2))
A(selInact(sInact(y0), sInact(x0))) → SEL(s(y0), s(x0))
A(selInact(sInact(y0), indxInact(x0, x1))) → SEL(s(y0), indx(a(x0), x1))
A(selInact(dblInact(y0), indxInact(x0, x1))) → SEL(dbl(a(y0)), indx(a(x0), x1))
SEL(s(sInact(y0)), cons(y1, dblsInact(x0))) → SEL(s(y0), dbls(a(x0)))
A(selInact(sInact(y0), selInact(x0, x1))) → SEL(s(y0), sel(a(x0), a(x1)))
SEL(s(y0), cons(y1, sInact(x0))) → SEL(y0, s(x0))
SEL(s(selInact(y0, y1)), cons(y2, dblInact(x0))) → SEL(sel(a(y0), a(y1)), dbl(a(x0)))
A(selInact(dblInact(y0), x0)) → SEL(dbl(a(y0)), x0)
SEL(s(selInact(y0, y1)), cons(y2, sInact(x0))) → SEL(sel(a(y0), a(y1)), s(x0))
FROM(X) → A(X)
A(selInact(dblsInact(y0), indxInact(x0, x1))) → SEL(dbls(a(y0)), indx(a(x0), x1))
SEL(s(sInact(y0)), cons(y1, fromInact(x0))) → SEL(s(y0), from(x0))
A(indxInact(dblInact(x0), y1)) → INDX(dbl(a(x0)), y1)
SEL(s(y0), cons(y1, dblInact(x0))) → SEL(y0, dbl(a(x0)))
SEL(s(sInact(y0)), cons(y1, selInact(x0, x1))) → SEL(s(y