Termination w.r.t. Q proof of /tmp/tmpFCYnkY/Ex3_2_Luc97

Termination w.r.t. Q of the following Term Rewriting System could not be shown:

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

a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(nil) → nil
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__indx(nil, X) → nil
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__from(X) → cons(X, from(s(X)))
mark(dbl(X)) → a__dbl(mark(X))
mark(dbls(X)) → a__dbls(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__from(X) → from(X)

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(nil) → nil
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__indx(nil, X) → nil
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__from(X) → cons(X, from(s(X)))
mark(dbl(X)) → a__dbl(mark(X))
mark(dbls(X)) → a__dbls(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__from(X) → from(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(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
MARK(indx(X1, X2)) → MARK(X1)
A__SEL(s(X), cons(Y, Z)) → A__SEL(mark(X), mark(Z))
MARK(dbl(X)) → MARK(X)
MARK(dbls(X)) → MARK(X)
A__SEL(0, cons(X, Y)) → MARK(X)
MARK(indx(X1, X2)) → A__INDX(mark(X1), X2)
MARK(dbls(X)) → A__DBLS(mark(X))
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(from(X)) → A__FROM(X)
MARK(dbl(X)) → A__DBL(mark(X))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(nil) → nil
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__indx(nil, X) → nil
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__from(X) → cons(X, from(s(X)))
mark(dbl(X)) → a__dbl(mark(X))
mark(dbls(X)) → a__dbls(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__from(X) → from(X)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

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

MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
MARK(indx(X1, X2)) → MARK(X1)
A__SEL(s(X), cons(Y, Z)) → A__SEL(mark(X), mark(Z))
MARK(dbl(X)) → MARK(X)
MARK(dbls(X)) → MARK(X)
A__SEL(0, cons(X, Y)) → MARK(X)
MARK(indx(X1, X2)) → A__INDX(mark(X1), X2)
MARK(dbls(X)) → A__DBLS(mark(X))
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(from(X)) → A__FROM(X)
MARK(dbl(X)) → A__DBL(mark(X))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(nil) → nil
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__indx(nil, X) → nil
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__from(X) → cons(X, from(s(X)))
mark(dbl(X)) → a__dbl(mark(X))
mark(dbls(X)) → a__dbls(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__from(X) → from(X)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

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

MARK(sel(X1, X2)) → MARK(X1)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
MARK(indx(X1, X2)) → MARK(X1)
A__SEL(s(X), cons(Y, Z)) → A__SEL(mark(X), mark(Z))
MARK(dbl(X)) → MARK(X)
MARK(dbls(X)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X2)
A__SEL(0, cons(X, Y)) → MARK(X)

The TRS R consists of the following rules:

a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(nil) → nil
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__indx(nil, X) → nil
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__from(X) → cons(X, from(s(X)))
mark(dbl(X)) → a__dbl(mark(X))
mark(dbls(X)) → a__dbls(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__from(X) → from(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.

A__SEL(0, cons(X, Y)) → MARK(X)

The remaining pairs can at least be oriented weakly.

MARK(sel(X1, X2)) → MARK(X1)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
MARK(indx(X1, X2)) → MARK(X1)
A__SEL(s(X), cons(Y, Z)) → A__SEL(mark(X), mark(Z))
MARK(dbl(X)) → MARK(X)
MARK(dbls(X)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X2)

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

POL(0) = 1
POL(A__SEL(x₁, x₂)) = x₁ + x₂
POL(MARK(x₁)) = x₁
POL(a__dbl(x₁)) = x₁
POL(a__dbls(x₁)) = x₁
POL(a__from(x₁)) = x₁
POL(a__indx(x₁, x₂)) = x₁ + x₂
POL(a__sel(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = max(x₁, x₂)
POL(dbl(x₁)) = x₁
POL(dbls(x₁)) = x₁
POL(from(x₁)) = x₁
POL(indx(x₁, x₂)) = x₁ + x₂
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(sel(x₁, x₂)) = x₁ + x₂

The following usable rules [17] were oriented:

a__from(X) → from(X)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__dbls(nil) → nil
mark(dbl(X)) → a__dbl(mark(X))
a__from(X) → cons(X, from(s(X)))
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__indx(nil, X) → nil
mark(from(X)) → a__from(X)
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(dbls(X)) → a__dbls(mark(X))
mark(cons(X1, X2)) → cons(X1, X2)
mark(nil) → nil
mark(s(X)) → s(X)
mark(0) → 0

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

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

MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(indx(X1, X2)) → MARK(X1)
A__SEL(s(X), cons(Y, Z)) → A__SEL(mark(X), mark(Z))
MARK(dbl(X)) → MARK(X)
MARK(dbls(X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X2)
A__SEL(s(X), cons(Y, Z)) → MARK(X)

The TRS R consists of the following rules:

a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(nil) → nil
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__indx(nil, X) → nil
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__from(X) → cons(X, from(s(X)))
mark(dbl(X)) → a__dbl(mark(X))
mark(dbls(X)) → a__dbls(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__from(X) → from(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.

MARK(dbls(X)) → MARK(X)

The remaining pairs can at least be oriented weakly.

MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(indx(X1, X2)) → MARK(X1)
A__SEL(s(X), cons(Y, Z)) → A__SEL(mark(X), mark(Z))
MARK(dbl(X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X2)
A__SEL(s(X), cons(Y, Z)) → MARK(X)

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

POL(0) = 0
POL(A__SEL(x₁, x₂)) = x₁ + x₂
POL(MARK(x₁)) = x₁
POL(a__dbl(x₁)) = x₁
POL(a__dbls(x₁)) = 1 + x₁
POL(a__from(x₁)) = x₁
POL(a__indx(x₁, x₂)) = x₁ + x₂
POL(a__sel(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = max(x₁, x₂)
POL(dbl(x₁)) = x₁
POL(dbls(x₁)) = 1 + x₁
POL(from(x₁)) = x₁
POL(indx(x₁, x₂)) = x₁ + x₂
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(sel(x₁, x₂)) = x₁ + x₂

The following usable rules [17] were oriented:

a__from(X) → from(X)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__dbls(nil) → nil
mark(dbl(X)) → a__dbl(mark(X))
a__from(X) → cons(X, from(s(X)))
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__indx(nil, X) → nil
mark(from(X)) → a__from(X)
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(dbls(X)) → a__dbls(mark(X))
mark(cons(X1, X2)) → cons(X1, X2)
mark(nil) → nil
mark(s(X)) → s(X)
mark(0) → 0

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ QDPOrderProof

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

MARK(sel(X1, X2)) → MARK(X1)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
MARK(indx(X1, X2)) → MARK(X1)
A__SEL(s(X), cons(Y, Z)) → A__SEL(mark(X), mark(Z))
MARK(dbl(X)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(nil) → nil
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__indx(nil, X) → nil
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__from(X) → cons(X, from(s(X)))
mark(dbl(X)) → a__dbl(mark(X))
mark(dbls(X)) → a__dbls(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__from(X) → from(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.

MARK(indx(X1, X2)) → MARK(X1)

The remaining pairs can at least be oriented weakly.

MARK(sel(X1, X2)) → MARK(X1)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
A__SEL(s(X), cons(Y, Z)) → A__SEL(mark(X), mark(Z))
MARK(dbl(X)) → MARK(X)
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X2)

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

POL(0) = 0
POL(A__SEL(x₁, x₂)) = x₁ + x₂
POL(MARK(x₁)) = x₁
POL(a__dbl(x₁)) = x₁
POL(a__dbls(x₁)) = x₁
POL(a__from(x₁)) = x₁
POL(a__indx(x₁, x₂)) = 1 + x₁ + x₂
POL(a__sel(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = max(x₁, x₂)
POL(dbl(x₁)) = x₁
POL(dbls(x₁)) = x₁
POL(from(x₁)) = x₁
POL(indx(x₁, x₂)) = 1 + x₁ + x₂
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(sel(x₁, x₂)) = x₁ + x₂

The following usable rules [17] were oriented:

a__from(X) → from(X)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__dbls(nil) → nil
mark(dbl(X)) → a__dbl(mark(X))
a__from(X) → cons(X, from(s(X)))
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__indx(nil, X) → nil
mark(from(X)) → a__from(X)
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(dbls(X)) → a__dbls(mark(X))
mark(cons(X1, X2)) → cons(X1, X2)
mark(nil) → nil
mark(s(X)) → s(X)
mark(0) → 0

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

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

MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
A__SEL(s(X), cons(Y, Z)) → A__SEL(mark(X), mark(Z))
MARK(dbl(X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X2)
A__SEL(s(X), cons(Y, Z)) → MARK(X)

The TRS R consists of the following rules:

a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(nil) → nil
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__indx(nil, X) → nil
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__from(X) → cons(X, from(s(X)))
mark(dbl(X)) → a__dbl(mark(X))
mark(dbls(X)) → a__dbls(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__from(X) → from(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.

MARK(dbl(X)) → MARK(X)

The remaining pairs can at least be oriented weakly.

MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
A__SEL(s(X), cons(Y, Z)) → A__SEL(mark(X), mark(Z))
MARK(sel(X1, X2)) → MARK(X2)
A__SEL(s(X), cons(Y, Z)) → MARK(X)

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

POL(0) = 0
POL(A__SEL(x₁, x₂)) = x₁ + x₂
POL(MARK(x₁)) = x₁
POL(a__dbl(x₁)) = 1 + x₁
POL(a__dbls(x₁)) = 1 + x₁
POL(a__from(x₁)) = x₁
POL(a__indx(x₁, x₂)) = x₁ + x₂
POL(a__sel(x₁, x₂)) = x₁ + x₂
POL(cons(x₁, x₂)) = max(x₁, x₂)
POL(dbl(x₁)) = 1 + x₁
POL(dbls(x₁)) = 1 + x₁
POL(from(x₁)) = x₁
POL(indx(x₁, x₂)) = x₁ + x₂
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(sel(x₁, x₂)) = x₁ + x₂

The following usable rules [17] were oriented:

a__from(X) → from(X)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__dbls(nil) → nil
mark(dbl(X)) → a__dbl(mark(X))
a__from(X) → cons(X, from(s(X)))
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__indx(nil, X) → nil
mark(from(X)) → a__from(X)
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(dbls(X)) → a__dbls(mark(X))
mark(cons(X1, X2)) → cons(X1, X2)
mark(nil) → nil
mark(s(X)) → s(X)
mark(0) → 0

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ QDPOrderProof

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

MARK(sel(X1, X2)) → MARK(X1)
A__SEL(s(X), cons(Y, Z)) → MARK(Z)
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
A__SEL(s(X), cons(Y, Z)) → A__SEL(mark(X), mark(Z))
A__SEL(s(X), cons(Y, Z)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(nil) → nil
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__indx(nil, X) → nil
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__from(X) → cons(X, from(s(X)))
mark(dbl(X)) → a__dbl(mark(X))
mark(dbls(X)) → a__dbls(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__from(X) → from(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.

MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(X1, X2)) → A__SEL(mark(X1), mark(X2))
MARK(sel(X1, X2)) → MARK(X2)

The remaining pairs can at least be oriented weakly.

A__SEL(s(X), cons(Y, Z)) → MARK(Z)
A__SEL(s(X), cons(Y, Z)) → A__SEL(mark(X), mark(Z))
A__SEL(s(X), cons(Y, Z)) → MARK(X)

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

POL(0) = 0
POL(A__SEL(x₁, x₂)) = x₁ + x₂
POL(MARK(x₁)) = x₁
POL(a__dbl(x₁)) = 0
POL(a__dbls(x₁)) = 0
POL(a__from(x₁)) = x₁
POL(a__indx(x₁, x₂)) = 1 + x₁ + x₂
POL(a__sel(x₁, x₂)) = 1 + x₁ + x₂
POL(cons(x₁, x₂)) = max(x₁, x₂)
POL(dbl(x₁)) = 0
POL(dbls(x₁)) = 0
POL(from(x₁)) = x₁
POL(indx(x₁, x₂)) = 1 + x₁ + x₂
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(s(x₁)) = x₁
POL(sel(x₁, x₂)) = 1 + x₁ + x₂

The following usable rules [17] were oriented:

a__from(X) → from(X)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__dbls(nil) → nil
mark(dbl(X)) → a__dbl(mark(X))
a__from(X) → cons(X, from(s(X)))
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__indx(nil, X) → nil
mark(from(X)) → a__from(X)
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(dbls(X)) → a__dbls(mark(X))
mark(cons(X1, X2)) → cons(X1, X2)
mark(nil) → nil
mark(s(X)) → s(X)
mark(0) → 0

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

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

A__SEL(s(X), cons(Y, Z)) → MARK(Z)
A__SEL(s(X), cons(Y, Z)) → A__SEL(mark(X), mark(Z))
A__SEL(s(X), cons(Y, Z)) → MARK(X)

The TRS R consists of the following rules:

a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(nil) → nil
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__indx(nil, X) → nil
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__from(X) → cons(X, from(s(X)))
mark(dbl(X)) → a__dbl(mark(X))
mark(dbls(X)) → a__dbls(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__from(X) → from(X)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

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

A__SEL(s(X), cons(Y, Z)) → A__SEL(mark(X), mark(Z))

The TRS R consists of the following rules:

a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(nil) → nil
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__indx(nil, X) → nil
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__from(X) → cons(X, from(s(X)))
mark(dbl(X)) → a__dbl(mark(X))
mark(dbls(X)) → a__dbls(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__from(X) → from(X)

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

A__SEL(s(sel(x0, x1)), cons(y1, y2)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y2))
A__SEL(s(0), cons(y1, y2)) → A__SEL(0, mark(y2))
A__SEL(s(from(x0)), cons(y1, y2)) → A__SEL(a__from(x0), mark(y2))
A__SEL(s(dbl(x0)), cons(y1, y2)) → A__SEL(a__dbl(mark(x0)), mark(y2))
A__SEL(s(nil), cons(y1, y2)) → A__SEL(nil, mark(y2))
A__SEL(s(indx(x0, x1)), cons(y1, y2)) → A__SEL(a__indx(mark(x0), x1), mark(y2))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(x0), mark(y2))
A__SEL(s(cons(x0, x1)), cons(y1, y2)) → A__SEL(cons(x0, x1), mark(y2))
A__SEL(s(dbls(x0)), cons(y1, y2)) → A__SEL(a__dbls(mark(x0)), mark(y2))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

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

A__SEL(s(0), cons(y1, y2)) → A__SEL(0, mark(y2))
A__SEL(s(sel(x0, x1)), cons(y1, y2)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y2))
A__SEL(s(from(x0)), cons(y1, y2)) → A__SEL(a__from(x0), mark(y2))
A__SEL(s(dbl(x0)), cons(y1, y2)) → A__SEL(a__dbl(mark(x0)), mark(y2))
A__SEL(s(indx(x0, x1)), cons(y1, y2)) → A__SEL(a__indx(mark(x0), x1), mark(y2))
A__SEL(s(nil), cons(y1, y2)) → A__SEL(nil, mark(y2))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(x0), mark(y2))
A__SEL(s(cons(x0, x1)), cons(y1, y2)) → A__SEL(cons(x0, x1), mark(y2))
A__SEL(s(dbls(x0)), cons(y1, y2)) → A__SEL(a__dbls(mark(x0)), mark(y2))

The TRS R consists of the following rules:

a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(nil) → nil
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__indx(nil, X) → nil
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__from(X) → cons(X, from(s(X)))
mark(dbl(X)) → a__dbl(mark(X))
mark(dbls(X)) → a__dbls(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__from(X) → from(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.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

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

A__SEL(s(sel(x0, x1)), cons(y1, y2)) → A__SEL(a__sel(mark(x0), mark(x1)), mark(y2))
A__SEL(s(from(x0)), cons(y1, y2)) → A__SEL(a__from(x0), mark(y2))
A__SEL(s(dbl(x0)), cons(y1, y2)) → A__SEL(a__dbl(mark(x0)), mark(y2))
A__SEL(s(indx(x0, x1)), cons(y1, y2)) → A__SEL(a__indx(mark(x0), x1), mark(y2))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(x0), mark(y2))
A__SEL(s(dbls(x0)), cons(y1, y2)) → A__SEL(a__dbls(mark(x0)), mark(y2))

The TRS R consists of the following rules:

a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(nil) → nil
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__indx(nil, X) → nil
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__from(X) → cons(X, from(s(X)))
mark(dbl(X)) → a__dbl(mark(X))
mark(dbls(X)) → a__dbls(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__from(X) → from(X)

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

A__SEL(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(x0))
A__SEL(s(sel(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbl(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, 0)) → A__SEL(a__sel(mark(y0), mark(y1)), 0)
A__SEL(s(sel(y0, y1)), cons(y2, nil)) → A__SEL(a__sel(mark(y0), mark(y1)), nil)
A__SEL(s(sel(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, s(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), s(x0))
A__SEL(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(x0, x1))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

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

A__SEL(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbl(x0)), cons(y1, y2)) → A__SEL(a__dbl(mark(x0)), mark(y2))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(x0))
A__SEL(s(sel(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbl(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, 0)) → A__SEL(a__sel(mark(y0), mark(y1)), 0)
A__SEL(s(sel(y0, y1)), cons(y2, nil)) → A__SEL(a__sel(mark(y0), mark(y1)), nil)
A__SEL(s(sel(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(from(x0)), cons(y1, y2)) → A__SEL(a__from(x0), mark(y2))
A__SEL(s(sel(y0, y1)), cons(y2, s(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), s(x0))
A__SEL(s(indx(x0, x1)), cons(y1, y2)) → A__SEL(a__indx(mark(x0), x1), mark(y2))
A__SEL(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(x0, x1))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(x0), mark(y2))
A__SEL(s(dbls(x0)), cons(y1, y2)) → A__SEL(a__dbls(mark(x0)), mark(y2))

The TRS R consists of the following rules:

a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(nil) → nil
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__indx(nil, X) → nil
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__from(X) → cons(X, from(s(X)))
mark(dbl(X)) → a__dbl(mark(X))
mark(dbls(X)) → a__dbls(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__from(X) → from(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.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

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

A__SEL(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(from(x0)), cons(y1, y2)) → A__SEL(a__from(x0), mark(y2))
A__SEL(s(dbl(x0)), cons(y1, y2)) → A__SEL(a__dbl(mark(x0)), mark(y2))
A__SEL(s(indx(x0, x1)), cons(y1, y2)) → A__SEL(a__indx(mark(x0), x1), mark(y2))
A__SEL(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(x0, x1))
A__SEL(s(sel(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbl(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(x0))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(x0), mark(y2))
A__SEL(s(dbls(x0)), cons(y1, y2)) → A__SEL(a__dbls(mark(x0)), mark(y2))

The TRS R consists of the following rules:

a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(nil) → nil
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__indx(nil, X) → nil
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__from(X) → cons(X, from(s(X)))
mark(dbl(X)) → a__dbl(mark(X))
mark(dbls(X)) → a__dbls(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__from(X) → from(X)

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

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

A__SEL(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(from(x0)), cons(y1, y2)) → A__SEL(cons(x0, from(s(x0))), mark(y2))
A__SEL(s(dbl(x0)), cons(y1, y2)) → A__SEL(a__dbl(mark(x0)), mark(y2))
A__SEL(s(from(x0)), cons(y1, y2)) → A__SEL(from(x0), mark(y2))
A__SEL(s(sel(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbl(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(x0))
A__SEL(s(sel(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(indx(x0, x1)), cons(y1, y2)) → A__SEL(a__indx(mark(x0), x1), mark(y2))
A__SEL(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(x0, x1))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(x0), mark(y2))
A__SEL(s(dbls(x0)), cons(y1, y2)) → A__SEL(a__dbls(mark(x0)), mark(y2))

The TRS R consists of the following rules:

a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(nil) → nil
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__indx(nil, X) → nil
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__from(X) → cons(X, from(s(X)))
mark(dbl(X)) → a__dbl(mark(X))
mark(dbls(X)) → a__dbls(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__from(X) → from(X)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

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

A__SEL(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(dbl(x0)), cons(y1, y2)) → A__SEL(a__dbl(mark(x0)), mark(y2))
A__SEL(s(indx(x0, x1)), cons(y1, y2)) → A__SEL(a__indx(mark(x0), x1), mark(y2))
A__SEL(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(x0, x1))
A__SEL(s(sel(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbl(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(x0))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(x0), mark(y2))
A__SEL(s(dbls(x0)), cons(y1, y2)) → A__SEL(a__dbls(mark(x0)), mark(y2))

The TRS R consists of the following rules:

a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(nil) → nil
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__indx(nil, X) → nil
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__from(X) → cons(X, from(s(X)))
mark(dbl(X)) → a__dbl(mark(X))
mark(dbls(X)) → a__dbls(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__from(X) → from(X)

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

A__SEL(s(dbl(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbl(mark(y0)), cons(x0, x1))
A__SEL(s(dbl(y0)), cons(y1, nil)) → A__SEL(a__dbl(mark(y0)), nil)
A__SEL(s(dbl(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(dbl(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbl(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(dbl(y0)), cons(y1, s(x0))) → A__SEL(a__dbl(mark(y0)), s(x0))
A__SEL(s(dbl(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbl(mark(y0)), a__dbls(mark(x0)))
A__SEL(s(dbl(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbl(y0)), cons(y1, 0)) → A__SEL(a__dbl(mark(y0)), 0)
A__SEL(s(dbl(y0)), cons(y1, from(x0))) → A__SEL(a__dbl(mark(y0)), a__from(x0))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

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

A__SEL(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbl(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbl(mark(y0)), a__dbls(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbl(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(x0))
A__SEL(s(dbl(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbl(y0)), cons(y1, from(x0))) → A__SEL(a__dbl(mark(y0)), a__from(x0))
A__SEL(s(dbl(y0)), cons(y1, 0)) → A__SEL(a__dbl(mark(y0)), 0)
A__SEL(s(dbl(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbl(mark(y0)), cons(x0, x1))
A__SEL(s(dbl(y0)), cons(y1, nil)) → A__SEL(a__dbl(mark(y0)), nil)
A__SEL(s(dbl(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(dbl(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbl(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(dbl(y0)), cons(y1, s(x0))) → A__SEL(a__dbl(mark(y0)), s(x0))
A__SEL(s(indx(x0, x1)), cons(y1, y2)) → A__SEL(a__indx(mark(x0), x1), mark(y2))
A__SEL(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(x0, x1))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(x0), mark(y2))
A__SEL(s(dbls(x0)), cons(y1, y2)) → A__SEL(a__dbls(mark(x0)), mark(y2))

The TRS R consists of the following rules:

a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(nil) → nil
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__indx(nil, X) → nil
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__from(X) → cons(X, from(s(X)))
mark(dbl(X)) → a__dbl(mark(X))
mark(dbls(X)) → a__dbls(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__from(X) → from(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.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

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

A__SEL(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbl(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbl(mark(y0)), a__dbls(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(x0))
A__SEL(s(sel(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbl(mark(x0)))
A__SEL(s(dbl(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbl(y0)), cons(y1, from(x0))) → A__SEL(a__dbl(mark(y0)), a__from(x0))
A__SEL(s(dbl(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbl(mark(y0)), cons(x0, x1))
A__SEL(s(dbl(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(dbl(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbl(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(indx(x0, x1)), cons(y1, y2)) → A__SEL(a__indx(mark(x0), x1), mark(y2))
A__SEL(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(x0, x1))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(x0), mark(y2))
A__SEL(s(dbls(x0)), cons(y1, y2)) → A__SEL(a__dbls(mark(x0)), mark(y2))

The TRS R consists of the following rules:

a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(nil) → nil
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__indx(nil, X) → nil
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__from(X) → cons(X, from(s(X)))
mark(dbl(X)) → a__dbl(mark(X))
mark(dbls(X)) → a__dbls(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__from(X) → from(X)

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

A__SEL(s(indx(y0, y1)), cons(y2, from(x0))) → A__SEL(a__indx(mark(y0), y1), a__from(x0))
A__SEL(s(indx(y0, y1)), cons(y2, s(x0))) → A__SEL(a__indx(mark(y0), y1), s(x0))
A__SEL(s(indx(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__indx(mark(y0), y1), cons(x0, x1))
A__SEL(s(indx(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbls(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbl(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__indx(mark(y0), y1), a__indx(mark(x0), x1))
A__SEL(s(indx(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__indx(mark(y0), y1), a__sel(mark(x0), mark(x1)))
A__SEL(s(indx(y0, y1)), cons(y2, 0)) → A__SEL(a__indx(mark(y0), y1), 0)
A__SEL(s(indx(y0, y1)), cons(y2, nil)) → A__SEL(a__indx(mark(y0), y1), nil)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

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

A__SEL(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(indx(y0, y1)), cons(y2, from(x0))) → A__SEL(a__indx(mark(y0), y1), a__from(x0))
A__SEL(s(dbl(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbl(mark(y0)), a__dbls(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbl(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbl(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(x0))
A__SEL(s(indx(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__indx(mark(y0), y1), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbl(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(indx(y0, y1)), cons(y2, 0)) → A__SEL(a__indx(mark(y0), y1), 0)
A__SEL(s(dbl(y0)), cons(y1, from(x0))) → A__SEL(a__dbl(mark(y0)), a__from(x0))
A__SEL(s(dbl(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbl(mark(y0)), cons(x0, x1))
A__SEL(s(dbl(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(indx(y0, y1)), cons(y2, s(x0))) → A__SEL(a__indx(mark(y0), y1), s(x0))
A__SEL(s(dbl(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbl(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__indx(mark(y0), y1), cons(x0, x1))
A__SEL(s(sel(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbls(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__indx(mark(y0), y1), a__indx(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(x0, x1))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(x0), mark(y2))
A__SEL(s(indx(y0, y1)), cons(y2, nil)) → A__SEL(a__indx(mark(y0), y1), nil)
A__SEL(s(dbls(x0)), cons(y1, y2)) → A__SEL(a__dbls(mark(x0)), mark(y2))

The TRS R consists of the following rules:

a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(nil) → nil
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__indx(nil, X) → nil
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__from(X) → cons(X, from(s(X)))
mark(dbl(X)) → a__dbl(mark(X))
mark(dbls(X)) → a__dbls(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__from(X) → from(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.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

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

A__SEL(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(indx(y0, y1)), cons(y2, from(x0))) → A__SEL(a__indx(mark(y0), y1), a__from(x0))
A__SEL(s(dbl(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbl(mark(y0)), a__dbls(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbl(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(x0))
A__SEL(s(sel(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbl(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__indx(mark(y0), y1), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbl(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbl(y0)), cons(y1, from(x0))) → A__SEL(a__dbl(mark(y0)), a__from(x0))
A__SEL(s(dbl(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbl(mark(y0)), cons(x0, x1))
A__SEL(s(dbl(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(dbl(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbl(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__indx(mark(y0), y1), cons(x0, x1))
A__SEL(s(sel(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbls(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__indx(mark(y0), y1), a__indx(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(x0, x1))
A__SEL(s(s(x0)), cons(y1, y2)) → A__SEL(s(x0), mark(y2))
A__SEL(s(dbls(x0)), cons(y1, y2)) → A__SEL(a__dbls(mark(x0)), mark(y2))

The TRS R consists of the following rules:

a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(nil) → nil
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__indx(nil, X) → nil
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__from(X) → cons(X, from(s(X)))
mark(dbl(X)) → a__dbl(mark(X))
mark(dbls(X)) → a__dbls(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__from(X) → from(X)

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

A__SEL(s(s(y0)), cons(y1, from(x0))) → A__SEL(s(y0), a__from(x0))
A__SEL(s(s(y0)), cons(y1, dbls(x0))) → A__SEL(s(y0), a__dbls(mark(x0)))
A__SEL(s(s(y0)), cons(y1, indx(x0, x1))) → A__SEL(s(y0), a__indx(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, 0)) → A__SEL(s(y0), 0)
A__SEL(s(s(y0)), cons(y1, s(x0))) → A__SEL(s(y0), s(x0))
A__SEL(s(s(y0)), cons(y1, sel(x0, x1))) → A__SEL(s(y0), a__sel(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, nil)) → A__SEL(s(y0), nil)
A__SEL(s(s(y0)), cons(y1, cons(x0, x1))) → A__SEL(s(y0), cons(x0, x1))
A__SEL(s(s(y0)), cons(y1, dbl(x0))) → A__SEL(s(y0), a__dbl(mark(x0)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

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

A__SEL(s(indx(y0, y1)), cons(y2, from(x0))) → A__SEL(a__indx(mark(y0), y1), a__from(x0))
A__SEL(s(s(y0)), cons(y1, indx(x0, x1))) → A__SEL(s(y0), a__indx(mark(x0), x1))
A__SEL(s(dbl(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbl(mark(y0)), a__dbls(mark(x0)))
A__SEL(s(s(y0)), cons(y1, s(x0))) → A__SEL(s(y0), s(x0))
A__SEL(s(indx(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbl(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbl(mark(x0)))
A__SEL(s(dbl(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbl(y0)), cons(y1, from(x0))) → A__SEL(a__dbl(mark(y0)), a__from(x0))
A__SEL(s(dbl(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbl(mark(y0)), cons(x0, x1))
A__SEL(s(dbl(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbl(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__indx(mark(y0), y1), cons(x0, x1))
A__SEL(s(sel(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(s(y0)), cons(y1, 0)) → A__SEL(s(y0), 0)
A__SEL(s(indx(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbls(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__indx(mark(y0), y1), a__indx(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, dbl(x0))) → A__SEL(s(y0), a__dbl(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, from(x0))) → A__SEL(s(y0), a__from(x0))
A__SEL(s(s(y0)), cons(y1, dbls(x0))) → A__SEL(s(y0), a__dbls(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(x0))
A__SEL(s(indx(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__indx(mark(y0), y1), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbl(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, sel(x0, x1))) → A__SEL(s(y0), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(x0, x1))
A__SEL(s(s(y0)), cons(y1, nil)) → A__SEL(s(y0), nil)
A__SEL(s(s(y0)), cons(y1, cons(x0, x1))) → A__SEL(s(y0), cons(x0, x1))
A__SEL(s(dbls(x0)), cons(y1, y2)) → A__SEL(a__dbls(mark(x0)), mark(y2))

The TRS R consists of the following rules:

a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(nil) → nil
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__indx(nil, X) → nil
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__from(X) → cons(X, from(s(X)))
mark(dbl(X)) → a__dbl(mark(X))
mark(dbls(X)) → a__dbls(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__from(X) → from(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.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

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

A__SEL(s(indx(y0, y1)), cons(y2, from(x0))) → A__SEL(a__indx(mark(y0), y1), a__from(x0))
A__SEL(s(s(y0)), cons(y1, indx(x0, x1))) → A__SEL(s(y0), a__indx(mark(x0), x1))
A__SEL(s(dbl(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbl(mark(y0)), a__dbls(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbl(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbl(mark(x0)))
A__SEL(s(dbl(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbl(y0)), cons(y1, from(x0))) → A__SEL(a__dbl(mark(y0)), a__from(x0))
A__SEL(s(dbl(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbl(mark(y0)), cons(x0, x1))
A__SEL(s(dbl(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbl(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(indx(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__indx(mark(y0), y1), cons(x0, x1))
A__SEL(s(sel(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbls(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__indx(mark(y0), y1), a__indx(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, dbl(x0))) → A__SEL(s(y0), a__dbl(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, from(x0))) → A__SEL(s(y0), a__from(x0))
A__SEL(s(s(y0)), cons(y1, dbls(x0))) → A__SEL(s(y0), a__dbls(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(x0))
A__SEL(s(indx(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__indx(mark(y0), y1), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbl(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, sel(x0, x1))) → A__SEL(s(y0), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(x0, x1))
A__SEL(s(dbls(x0)), cons(y1, y2)) → A__SEL(a__dbls(mark(x0)), mark(y2))
A__SEL(s(s(y0)), cons(y1, cons(x0, x1))) → A__SEL(s(y0), cons(x0, x1))

The TRS R consists of the following rules:

a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(nil) → nil
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__indx(nil, X) → nil
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__from(X) → cons(X, from(s(X)))
mark(dbl(X)) → a__dbl(mark(X))
mark(dbls(X)) → a__dbls(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__from(X) → from(X)

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

A__SEL(s(dbls(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbls(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(dbls(y0)), cons(y1, from(x0))) → A__SEL(a__dbls(mark(y0)), a__from(x0))
A__SEL(s(dbls(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbls(mark(y0)), a__dbls(mark(x0)))
A__SEL(s(dbls(y0)), cons(y1, 0)) → A__SEL(a__dbls(mark(y0)), 0)
A__SEL(s(dbls(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbls(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbls(y0)), cons(y1, s(x0))) → A__SEL(a__dbls(mark(y0)), s(x0))
A__SEL(s(dbls(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbls(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(dbls(y0)), cons(y1, nil)) → A__SEL(a__dbls(mark(y0)), nil)
A__SEL(s(dbls(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbls(mark(y0)), cons(x0, x1))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

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

A__SEL(s(dbls(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbls(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(indx(y0, y1)), cons(y2, from(x0))) → A__SEL(a__indx(mark(y0), y1), a__from(x0))
A__SEL(s(s(y0)), cons(y1, indx(x0, x1))) → A__SEL(s(y0), a__indx(mark(x0), x1))
A__SEL(s(dbls(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbls(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(dbl(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbl(mark(y0)), a__dbls(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbl(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbl(mark(x0)))
A__SEL(s(dbl(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbl(y0)), cons(y1, from(x0))) → A__SEL(a__dbl(mark(y0)), a__from(x0))
A__SEL(s(dbl(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbl(mark(y0)), cons(x0, x1))
A__SEL(s(dbl(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbl(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(dbls(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbls(mark(y0)), a__dbls(mark(x0)))
A__SEL(s(dbls(y0)), cons(y1, from(x0))) → A__SEL(a__dbls(mark(y0)), a__from(x0))
A__SEL(s(indx(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__indx(mark(y0), y1), cons(x0, x1))
A__SEL(s(sel(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(dbls(y0)), cons(y1, 0)) → A__SEL(a__dbls(mark(y0)), 0)
A__SEL(s(sel(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(dbls(y0)), cons(y1, s(x0))) → A__SEL(a__dbls(mark(y0)), s(x0))
A__SEL(s(indx(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbls(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__indx(mark(y0), y1), a__indx(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, dbl(x0))) → A__SEL(s(y0), a__dbl(mark(x0)))
A__SEL(s(s(y0)), cons(y1, from(x0))) → A__SEL(s(y0), a__from(x0))
A__SEL(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, dbls(x0))) → A__SEL(s(y0), a__dbls(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(x0))
A__SEL(s(indx(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__indx(mark(y0), y1), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbls(y0)), cons(y1, nil)) → A__SEL(a__dbls(mark(y0)), nil)
A__SEL(s(dbls(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbls(mark(y0)), cons(x0, x1))
A__SEL(s(dbl(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(dbls(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbls(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, sel(x0, x1))) → A__SEL(s(y0), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(x0, x1))
A__SEL(s(s(y0)), cons(y1, cons(x0, x1))) → A__SEL(s(y0), cons(x0, x1))

The TRS R consists of the following rules:

a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(nil) → nil
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__indx(nil, X) → nil
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__from(X) → cons(X, from(s(X)))
mark(dbl(X)) → a__dbl(mark(X))
mark(dbls(X)) → a__dbls(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__from(X) → from(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.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

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

A__SEL(s(dbls(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbls(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(indx(y0, y1)), cons(y2, from(x0))) → A__SEL(a__indx(mark(y0), y1), a__from(x0))
A__SEL(s(s(y0)), cons(y1, indx(x0, x1))) → A__SEL(s(y0), a__indx(mark(x0), x1))
A__SEL(s(dbls(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbls(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(dbl(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbl(mark(y0)), a__dbls(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbl(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbl(mark(x0)))
A__SEL(s(dbl(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbl(y0)), cons(y1, from(x0))) → A__SEL(a__dbl(mark(y0)), a__from(x0))
A__SEL(s(dbl(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbl(mark(y0)), cons(x0, x1))
A__SEL(s(dbl(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbl(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(dbls(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbls(mark(y0)), a__dbls(mark(x0)))
A__SEL(s(dbls(y0)), cons(y1, from(x0))) → A__SEL(a__dbls(mark(y0)), a__from(x0))
A__SEL(s(sel(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(indx(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__indx(mark(y0), y1), cons(x0, x1))
A__SEL(s(sel(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbls(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__indx(mark(y0), y1), a__indx(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, dbl(x0))) → A__SEL(s(y0), a__dbl(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, from(x0))) → A__SEL(s(y0), a__from(x0))
A__SEL(s(s(y0)), cons(y1, dbls(x0))) → A__SEL(s(y0), a__dbls(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__from(x0))
A__SEL(s(indx(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__indx(mark(y0), y1), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbls(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbls(mark(y0)), cons(x0, x1))
A__SEL(s(dbl(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(dbls(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbls(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, sel(x0, x1))) → A__SEL(s(y0), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(x0, x1))
A__SEL(s(s(y0)), cons(y1, cons(x0, x1))) → A__SEL(s(y0), cons(x0, x1))

The TRS R consists of the following rules:

a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(nil) → nil
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__indx(nil, X) → nil
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__from(X) → cons(X, from(s(X)))
mark(dbl(X)) → a__dbl(mark(X))
mark(dbls(X)) → a__dbls(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__from(X) → from(X)

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

A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), from(x0))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(x0, from(s(x0))))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

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

A__SEL(s(dbls(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbls(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), from(x0))
A__SEL(s(indx(y0, y1)), cons(y2, from(x0))) → A__SEL(a__indx(mark(y0), y1), a__from(x0))
A__SEL(s(s(y0)), cons(y1, indx(x0, x1))) → A__SEL(s(y0), a__indx(mark(x0), x1))
A__SEL(s(dbl(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbl(mark(y0)), a__dbls(mark(x0)))
A__SEL(s(dbls(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbls(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbl(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbl(mark(x0)))
A__SEL(s(dbl(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbl(y0)), cons(y1, from(x0))) → A__SEL(a__dbl(mark(y0)), a__from(x0))
A__SEL(s(dbl(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbl(mark(y0)), cons(x0, x1))
A__SEL(s(dbl(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbl(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__indx(mark(y0), y1), cons(x0, x1))
A__SEL(s(sel(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(dbls(y0)), cons(y1, from(x0))) → A__SEL(a__dbls(mark(y0)), a__from(x0))
A__SEL(s(dbls(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbls(mark(y0)), a__dbls(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbls(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__indx(mark(y0), y1), a__indx(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, dbl(x0))) → A__SEL(s(y0), a__dbl(mark(x0)))
A__SEL(s(s(y0)), cons(y1, from(x0))) → A__SEL(s(y0), a__from(x0))
A__SEL(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, dbls(x0))) → A__SEL(s(y0), a__dbls(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(x0, from(s(x0))))
A__SEL(s(indx(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__indx(mark(y0), y1), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbls(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbls(mark(y0)), cons(x0, x1))
A__SEL(s(dbl(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(dbls(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbls(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, sel(x0, x1))) → A__SEL(s(y0), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(x0, x1))
A__SEL(s(s(y0)), cons(y1, cons(x0, x1))) → A__SEL(s(y0), cons(x0, x1))

The TRS R consists of the following rules:

a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(nil) → nil
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__indx(nil, X) → nil
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__from(X) → cons(X, from(s(X)))
mark(dbl(X)) → a__dbl(mark(X))
mark(dbls(X)) → a__dbls(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__from(X) → from(X)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

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

A__SEL(s(dbls(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbls(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(indx(y0, y1)), cons(y2, from(x0))) → A__SEL(a__indx(mark(y0), y1), a__from(x0))
A__SEL(s(s(y0)), cons(y1, indx(x0, x1))) → A__SEL(s(y0), a__indx(mark(x0), x1))
A__SEL(s(dbls(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbls(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(dbl(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbl(mark(y0)), a__dbls(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbl(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbl(mark(x0)))
A__SEL(s(dbl(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbl(y0)), cons(y1, from(x0))) → A__SEL(a__dbl(mark(y0)), a__from(x0))
A__SEL(s(dbl(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbl(mark(y0)), cons(x0, x1))
A__SEL(s(dbl(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbl(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(dbls(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbls(mark(y0)), a__dbls(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__indx(mark(y0), y1), cons(x0, x1))
A__SEL(s(sel(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(dbls(y0)), cons(y1, from(x0))) → A__SEL(a__dbls(mark(y0)), a__from(x0))
A__SEL(s(sel(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbls(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__indx(mark(y0), y1), a__indx(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, dbl(x0))) → A__SEL(s(y0), a__dbl(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, from(x0))) → A__SEL(s(y0), a__from(x0))
A__SEL(s(s(y0)), cons(y1, dbls(x0))) → A__SEL(s(y0), a__dbls(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(x0, from(s(x0))))
A__SEL(s(indx(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__indx(mark(y0), y1), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbls(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbls(mark(y0)), cons(x0, x1))
A__SEL(s(dbl(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(dbls(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbls(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, sel(x0, x1))) → A__SEL(s(y0), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(x0, x1))
A__SEL(s(s(y0)), cons(y1, cons(x0, x1))) → A__SEL(s(y0), cons(x0, x1))

The TRS R consists of the following rules:

a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(nil) → nil
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__indx(nil, X) → nil
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__from(X) → cons(X, from(s(X)))
mark(dbl(X)) → a__dbl(mark(X))
mark(dbls(X)) → a__dbls(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__from(X) → from(X)

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

A__SEL(s(dbl(y0)), cons(y1, from(x0))) → A__SEL(a__dbl(mark(y0)), cons(x0, from(s(x0))))
A__SEL(s(dbl(y0)), cons(y1, from(x0))) → A__SEL(a__dbl(mark(y0)), from(x0))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ DependencyGraphProof

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

A__SEL(s(dbls(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbls(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(indx(y0, y1)), cons(y2, from(x0))) → A__SEL(a__indx(mark(y0), y1), a__from(x0))
A__SEL(s(s(y0)), cons(y1, indx(x0, x1))) → A__SEL(s(y0), a__indx(mark(x0), x1))
A__SEL(s(dbl(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbl(mark(y0)), a__dbls(mark(x0)))
A__SEL(s(dbls(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbls(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbl(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbl(mark(x0)))
A__SEL(s(dbl(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbl(y0)), cons(y1, from(x0))) → A__SEL(a__dbl(mark(y0)), cons(x0, from(s(x0))))
A__SEL(s(dbl(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbl(mark(y0)), cons(x0, x1))
A__SEL(s(dbl(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbl(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(dbls(y0)), cons(y1, from(x0))) → A__SEL(a__dbls(mark(y0)), a__from(x0))
A__SEL(s(sel(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(indx(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__indx(mark(y0), y1), cons(x0, x1))
A__SEL(s(dbls(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbls(mark(y0)), a__dbls(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbls(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__indx(mark(y0), y1), a__indx(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, dbl(x0))) → A__SEL(s(y0), a__dbl(mark(x0)))
A__SEL(s(s(y0)), cons(y1, from(x0))) → A__SEL(s(y0), a__from(x0))
A__SEL(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, dbls(x0))) → A__SEL(s(y0), a__dbls(mark(x0)))
A__SEL(s(dbl(y0)), cons(y1, from(x0))) → A__SEL(a__dbl(mark(y0)), from(x0))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(x0, from(s(x0))))
A__SEL(s(indx(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__indx(mark(y0), y1), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbls(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbls(mark(y0)), cons(x0, x1))
A__SEL(s(dbl(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(dbls(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbls(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, sel(x0, x1))) → A__SEL(s(y0), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(x0, x1))
A__SEL(s(s(y0)), cons(y1, cons(x0, x1))) → A__SEL(s(y0), cons(x0, x1))

The TRS R consists of the following rules:

a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(nil) → nil
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__indx(nil, X) → nil
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__from(X) → cons(X, from(s(X)))
mark(dbl(X)) → a__dbl(mark(X))
mark(dbls(X)) → a__dbls(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__from(X) → from(X)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ DependencyGraphProof

                                                                                                        ↳ QDP

                                                                                                          ↳ Narrowing

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

A__SEL(s(dbls(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbls(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(indx(y0, y1)), cons(y2, from(x0))) → A__SEL(a__indx(mark(y0), y1), a__from(x0))
A__SEL(s(s(y0)), cons(y1, indx(x0, x1))) → A__SEL(s(y0), a__indx(mark(x0), x1))
A__SEL(s(dbls(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbls(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(dbl(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbl(mark(y0)), a__dbls(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbl(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbl(mark(x0)))
A__SEL(s(dbl(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbl(y0)), cons(y1, from(x0))) → A__SEL(a__dbl(mark(y0)), cons(x0, from(s(x0))))
A__SEL(s(dbl(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbl(mark(y0)), cons(x0, x1))
A__SEL(s(dbl(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbl(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__indx(mark(y0), y1), cons(x0, x1))
A__SEL(s(sel(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(dbls(y0)), cons(y1, from(x0))) → A__SEL(a__dbls(mark(y0)), a__from(x0))
A__SEL(s(dbls(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbls(mark(y0)), a__dbls(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbls(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__indx(mark(y0), y1), a__indx(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, dbl(x0))) → A__SEL(s(y0), a__dbl(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, from(x0))) → A__SEL(s(y0), a__from(x0))
A__SEL(s(s(y0)), cons(y1, dbls(x0))) → A__SEL(s(y0), a__dbls(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(x0, from(s(x0))))
A__SEL(s(indx(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__indx(mark(y0), y1), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbls(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbls(mark(y0)), cons(x0, x1))
A__SEL(s(dbl(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(dbls(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbls(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, sel(x0, x1))) → A__SEL(s(y0), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(x0, x1))
A__SEL(s(s(y0)), cons(y1, cons(x0, x1))) → A__SEL(s(y0), cons(x0, x1))

The TRS R consists of the following rules:

a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(nil) → nil
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__indx(nil, X) → nil
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__from(X) → cons(X, from(s(X)))
mark(dbl(X)) → a__dbl(mark(X))
mark(dbls(X)) → a__dbls(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__from(X) → from(X)

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

A__SEL(s(indx(y0, y1)), cons(y2, from(x0))) → A__SEL(a__indx(mark(y0), y1), cons(x0, from(s(x0))))
A__SEL(s(indx(y0, y1)), cons(y2, from(x0))) → A__SEL(a__indx(mark(y0), y1), from(x0))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ DependencyGraphProof

                                                                                                        ↳ QDP

                                                                                                          ↳ Narrowing

                                                                                                            ↳ QDP

                                                                                                              ↳ DependencyGraphProof

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

A__SEL(s(dbls(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbls(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, indx(x0, x1))) → A__SEL(s(y0), a__indx(mark(x0), x1))
A__SEL(s(dbl(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbl(mark(y0)), a__dbls(mark(x0)))
A__SEL(s(dbls(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbls(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbl(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, from(x0))) → A__SEL(a__indx(mark(y0), y1), from(x0))
A__SEL(s(sel(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbl(mark(x0)))
A__SEL(s(dbl(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbl(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbl(mark(y0)), cons(x0, x1))
A__SEL(s(dbl(y0)), cons(y1, from(x0))) → A__SEL(a__dbl(mark(y0)), cons(x0, from(s(x0))))
A__SEL(s(dbl(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbl(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(dbls(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbls(mark(y0)), a__dbls(mark(x0)))
A__SEL(s(dbls(y0)), cons(y1, from(x0))) → A__SEL(a__dbls(mark(y0)), a__from(x0))
A__SEL(s(sel(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(indx(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__indx(mark(y0), y1), cons(x0, x1))
A__SEL(s(sel(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbls(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__indx(mark(y0), y1), a__indx(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, dbl(x0))) → A__SEL(s(y0), a__dbl(mark(x0)))
A__SEL(s(s(y0)), cons(y1, from(x0))) → A__SEL(s(y0), a__from(x0))
A__SEL(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, dbls(x0))) → A__SEL(s(y0), a__dbls(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, from(x0))) → A__SEL(a__indx(mark(y0), y1), cons(x0, from(s(x0))))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(x0, from(s(x0))))
A__SEL(s(indx(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__indx(mark(y0), y1), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbls(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbls(mark(y0)), cons(x0, x1))
A__SEL(s(dbl(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(dbls(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbls(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, sel(x0, x1))) → A__SEL(s(y0), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(x0, x1))
A__SEL(s(s(y0)), cons(y1, cons(x0, x1))) → A__SEL(s(y0), cons(x0, x1))

The TRS R consists of the following rules:

a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(nil) → nil
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__indx(nil, X) → nil
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__from(X) → cons(X, from(s(X)))
mark(dbl(X)) → a__dbl(mark(X))
mark(dbls(X)) → a__dbls(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__from(X) → from(X)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ DependencyGraphProof

                                                                                                        ↳ QDP

                                                                                                          ↳ Narrowing

                                                                                                            ↳ QDP

                                                                                                              ↳ DependencyGraphProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

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

A__SEL(s(dbls(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbls(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, indx(x0, x1))) → A__SEL(s(y0), a__indx(mark(x0), x1))
A__SEL(s(dbls(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbls(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(dbl(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbl(mark(y0)), a__dbls(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbl(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbl(mark(x0)))
A__SEL(s(dbl(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbl(y0)), cons(y1, from(x0))) → A__SEL(a__dbl(mark(y0)), cons(x0, from(s(x0))))
A__SEL(s(dbl(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbl(mark(y0)), cons(x0, x1))
A__SEL(s(dbl(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbl(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__indx(mark(y0), y1), cons(x0, x1))
A__SEL(s(sel(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(dbls(y0)), cons(y1, from(x0))) → A__SEL(a__dbls(mark(y0)), a__from(x0))
A__SEL(s(dbls(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbls(mark(y0)), a__dbls(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbls(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__indx(mark(y0), y1), a__indx(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, dbl(x0))) → A__SEL(s(y0), a__dbl(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, from(x0))) → A__SEL(s(y0), a__from(x0))
A__SEL(s(s(y0)), cons(y1, dbls(x0))) → A__SEL(s(y0), a__dbls(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, from(x0))) → A__SEL(a__indx(mark(y0), y1), cons(x0, from(s(x0))))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(x0, from(s(x0))))
A__SEL(s(indx(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__indx(mark(y0), y1), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbls(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbls(mark(y0)), cons(x0, x1))
A__SEL(s(dbl(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(dbls(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbls(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, sel(x0, x1))) → A__SEL(s(y0), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(x0, x1))
A__SEL(s(s(y0)), cons(y1, cons(x0, x1))) → A__SEL(s(y0), cons(x0, x1))

The TRS R consists of the following rules:

a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(nil) → nil
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__indx(nil, X) → nil
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__from(X) → cons(X, from(s(X)))
mark(dbl(X)) → a__dbl(mark(X))
mark(dbls(X)) → a__dbls(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__from(X) → from(X)

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ DependencyGraphProof

                                                                                                        ↳ QDP

                                                                                                          ↳ Narrowing

                                                                                                            ↳ QDP

                                                                                                              ↳ DependencyGraphProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

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

A__SEL(s(dbls(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbls(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, from(x0))) → A__SEL(s(y0), from(x0))
A__SEL(s(s(y0)), cons(y1, indx(x0, x1))) → A__SEL(s(y0), a__indx(mark(x0), x1))
A__SEL(s(dbl(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbl(mark(y0)), a__dbls(mark(x0)))
A__SEL(s(dbls(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbls(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbl(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbl(mark(x0)))
A__SEL(s(dbl(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbl(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbl(mark(y0)), cons(x0, x1))
A__SEL(s(dbl(y0)), cons(y1, from(x0))) → A__SEL(a__dbl(mark(y0)), cons(x0, from(s(x0))))
A__SEL(s(dbl(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbl(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(dbls(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbls(mark(y0)), a__dbls(mark(x0)))
A__SEL(s(dbls(y0)), cons(y1, from(x0))) → A__SEL(a__dbls(mark(y0)), a__from(x0))
A__SEL(s(sel(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(indx(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__indx(mark(y0), y1), cons(x0, x1))
A__SEL(s(sel(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbls(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__indx(mark(y0), y1), a__indx(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, dbl(x0))) → A__SEL(s(y0), a__dbl(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, dbls(x0))) → A__SEL(s(y0), a__dbls(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, from(x0))) → A__SEL(a__indx(mark(y0), y1), cons(x0, from(s(x0))))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(x0, from(s(x0))))
A__SEL(s(indx(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__indx(mark(y0), y1), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbls(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbls(mark(y0)), cons(x0, x1))
A__SEL(s(s(y0)), cons(y1, from(x0))) → A__SEL(s(y0), cons(x0, from(s(x0))))
A__SEL(s(dbl(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(dbls(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbls(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, sel(x0, x1))) → A__SEL(s(y0), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(x0, x1))
A__SEL(s(s(y0)), cons(y1, cons(x0, x1))) → A__SEL(s(y0), cons(x0, x1))

The TRS R consists of the following rules:

a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(nil) → nil
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__indx(nil, X) → nil
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__from(X) → cons(X, from(s(X)))
mark(dbl(X)) → a__dbl(mark(X))
mark(dbls(X)) → a__dbls(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__from(X) → from(X)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ DependencyGraphProof

                                                                                                        ↳ QDP

                                                                                                          ↳ Narrowing

                                                                                                            ↳ QDP

                                                                                                              ↳ DependencyGraphProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

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

A__SEL(s(dbls(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbls(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, indx(x0, x1))) → A__SEL(s(y0), a__indx(mark(x0), x1))
A__SEL(s(dbl(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbl(mark(y0)), a__dbls(mark(x0)))
A__SEL(s(dbls(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbls(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbl(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbl(mark(x0)))
A__SEL(s(dbl(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbl(y0)), cons(y1, from(x0))) → A__SEL(a__dbl(mark(y0)), cons(x0, from(s(x0))))
A__SEL(s(dbl(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbl(mark(y0)), cons(x0, x1))
A__SEL(s(dbl(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbl(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__indx(mark(y0), y1), cons(x0, x1))
A__SEL(s(sel(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(dbls(y0)), cons(y1, from(x0))) → A__SEL(a__dbls(mark(y0)), a__from(x0))
A__SEL(s(dbls(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbls(mark(y0)), a__dbls(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbls(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__indx(mark(y0), y1), a__indx(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, dbl(x0))) → A__SEL(s(y0), a__dbl(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, dbls(x0))) → A__SEL(s(y0), a__dbls(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, from(x0))) → A__SEL(a__indx(mark(y0), y1), cons(x0, from(s(x0))))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(x0, from(s(x0))))
A__SEL(s(indx(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__indx(mark(y0), y1), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbls(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbls(mark(y0)), cons(x0, x1))
A__SEL(s(s(y0)), cons(y1, from(x0))) → A__SEL(s(y0), cons(x0, from(s(x0))))
A__SEL(s(dbl(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(dbls(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbls(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, sel(x0, x1))) → A__SEL(s(y0), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(x0, x1))
A__SEL(s(s(y0)), cons(y1, cons(x0, x1))) → A__SEL(s(y0), cons(x0, x1))

The TRS R consists of the following rules:

a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(nil) → nil
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__indx(nil, X) → nil
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__from(X) → cons(X, from(s(X)))
mark(dbl(X)) → a__dbl(mark(X))
mark(dbls(X)) → a__dbls(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__from(X) → from(X)

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

A__SEL(s(dbls(y0)), cons(y1, from(x0))) → A__SEL(a__dbls(mark(y0)), from(x0))
A__SEL(s(dbls(y0)), cons(y1, from(x0))) → A__SEL(a__dbls(mark(y0)), cons(x0, from(s(x0))))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ DependencyGraphProof

                                                                                                        ↳ QDP

                                                                                                          ↳ Narrowing

                                                                                                            ↳ QDP

                                                                                                              ↳ DependencyGraphProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                                                                                                            ↳ QDP

                                                                                                                              ↳ DependencyGraphProof

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

A__SEL(s(dbls(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbls(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, indx(x0, x1))) → A__SEL(s(y0), a__indx(mark(x0), x1))
A__SEL(s(dbls(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbls(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(dbl(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbl(mark(y0)), a__dbls(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbl(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbl(mark(x0)))
A__SEL(s(dbl(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbl(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbl(mark(y0)), cons(x0, x1))
A__SEL(s(dbl(y0)), cons(y1, from(x0))) → A__SEL(a__dbl(mark(y0)), cons(x0, from(s(x0))))
A__SEL(s(dbl(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbl(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(dbls(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbls(mark(y0)), a__dbls(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(indx(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__indx(mark(y0), y1), cons(x0, x1))
A__SEL(s(sel(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbls(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__indx(mark(y0), y1), a__indx(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, dbl(x0))) → A__SEL(s(y0), a__dbl(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, dbls(x0))) → A__SEL(s(y0), a__dbls(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, from(x0))) → A__SEL(a__indx(mark(y0), y1), cons(x0, from(s(x0))))
A__SEL(s(dbls(y0)), cons(y1, from(x0))) → A__SEL(a__dbls(mark(y0)), from(x0))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(x0, from(s(x0))))
A__SEL(s(indx(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__indx(mark(y0), y1), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbls(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbls(mark(y0)), cons(x0, x1))
A__SEL(s(s(y0)), cons(y1, from(x0))) → A__SEL(s(y0), cons(x0, from(s(x0))))
A__SEL(s(dbl(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(dbls(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbls(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbls(y0)), cons(y1, from(x0))) → A__SEL(a__dbls(mark(y0)), cons(x0, from(s(x0))))
A__SEL(s(s(y0)), cons(y1, sel(x0, x1))) → A__SEL(s(y0), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(x0, x1))
A__SEL(s(s(y0)), cons(y1, cons(x0, x1))) → A__SEL(s(y0), cons(x0, x1))

The TRS R consists of the following rules:

a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(nil) → nil
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__indx(nil, X) → nil
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__from(X) → cons(X, from(s(X)))
mark(dbl(X)) → a__dbl(mark(X))
mark(dbls(X)) → a__dbls(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__from(X) → from(X)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ DependencyGraphProof

                                                                                                        ↳ QDP

                                                                                                          ↳ Narrowing

                                                                                                            ↳ QDP

                                                                                                              ↳ DependencyGraphProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                                                                                                            ↳ QDP

                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                ↳ AND

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                  ↳ QDP

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

A__SEL(s(s(y0)), cons(y1, from(x0))) → A__SEL(s(y0), cons(x0, from(s(x0))))
A__SEL(s(dbl(y0)), cons(y1, from(x0))) → A__SEL(a__dbl(mark(y0)), cons(x0, from(s(x0))))
A__SEL(s(indx(y0, y1)), cons(y2, from(x0))) → A__SEL(a__indx(mark(y0), y1), cons(x0, from(s(x0))))
A__SEL(s(dbls(y0)), cons(y1, from(x0))) → A__SEL(a__dbls(mark(y0)), cons(x0, from(s(x0))))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(x0, from(s(x0))))

The TRS R consists of the following rules:

a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(nil) → nil
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__indx(nil, X) → nil
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__from(X) → cons(X, from(s(X)))
mark(dbl(X)) → a__dbl(mark(X))
mark(dbls(X)) → a__dbls(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__from(X) → from(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.

A__SEL(s(indx(y0, y1)), cons(y2, from(x0))) → A__SEL(a__indx(mark(y0), y1), cons(x0, from(s(x0))))
A__SEL(s(dbls(y0)), cons(y1, from(x0))) → A__SEL(a__dbls(mark(y0)), cons(x0, from(s(x0))))

The remaining pairs can at least be oriented weakly.

A__SEL(s(s(y0)), cons(y1, from(x0))) → A__SEL(s(y0), cons(x0, from(s(x0))))
A__SEL(s(dbl(y0)), cons(y1, from(x0))) → A__SEL(a__dbl(mark(y0)), cons(x0, from(s(x0))))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(x0, from(s(x0))))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(A__SEL(x₁, x₂)) = x₁
POL(a__dbl(x₁)) = x₁
POL(a__dbls(x₁)) = 0
POL(a__from(x₁)) = 0
POL(a__indx(x₁, x₂)) = 0
POL(a__sel(x₁, x₂)) = 1
POL(cons(x₁, x₂)) = 0
POL(dbl(x₁)) = x₁
POL(dbls(x₁)) = 0
POL(from(x₁)) = 0
POL(indx(x₁, x₂)) = 0
POL(mark(x₁)) = 1
POL(nil) = 0
POL(s(x₁)) = 1
POL(sel(x₁, x₂)) = 0

The following usable rules [17] were oriented:

a__from(X) → from(X)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__dbls(nil) → nil
mark(dbl(X)) → a__dbl(mark(X))
a__from(X) → cons(X, from(s(X)))
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__indx(nil, X) → nil
mark(from(X)) → a__from(X)
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(dbls(X)) → a__dbls(mark(X))
mark(cons(X1, X2)) → cons(X1, X2)
mark(nil) → nil
mark(s(X)) → s(X)
mark(0) → 0

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ DependencyGraphProof

                                                                                                        ↳ QDP

                                                                                                          ↳ Narrowing

                                                                                                            ↳ QDP

                                                                                                              ↳ DependencyGraphProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                                                                                                            ↳ QDP

                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                ↳ AND

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                  ↳ QDP

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

A__SEL(s(dbl(y0)), cons(y1, from(x0))) → A__SEL(a__dbl(mark(y0)), cons(x0, from(s(x0))))
A__SEL(s(s(y0)), cons(y1, from(x0))) → A__SEL(s(y0), cons(x0, from(s(x0))))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(x0, from(s(x0))))

The TRS R consists of the following rules:

a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(nil) → nil
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__indx(nil, X) → nil
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__from(X) → cons(X, from(s(X)))
mark(dbl(X)) → a__dbl(mark(X))
mark(dbls(X)) → a__dbls(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__from(X) → from(X)

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

A__SEL(s(s(x0)), cons(y_1, from(s(y_2)))) → A__SEL(s(x0), cons(s(y_2), from(s(s(y_2)))))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ DependencyGraphProof

                                                                                                        ↳ QDP

                                                                                                          ↳ Narrowing

                                                                                                            ↳ QDP

                                                                                                              ↳ DependencyGraphProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                                                                                                            ↳ QDP

                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                ↳ AND

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Instantiation

                                                                                                                                  ↳ QDP

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

A__SEL(s(dbl(y0)), cons(y1, from(x0))) → A__SEL(a__dbl(mark(y0)), cons(x0, from(s(x0))))
A__SEL(s(s(x0)), cons(y_1, from(s(y_2)))) → A__SEL(s(x0), cons(s(y_2), from(s(s(y_2)))))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(x0, from(s(x0))))

The TRS R consists of the following rules:

a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(nil) → nil
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__indx(nil, X) → nil
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__from(X) → cons(X, from(s(X)))
mark(dbl(X)) → a__dbl(mark(X))
mark(dbls(X)) → a__dbls(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__from(X) → from(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule A__SEL(s(dbl(y0)), cons(y1, from(x0))) → A__SEL(a__dbl(mark(y0)), cons(x0, from(s(x0)))) we obtained the following new rules:

A__SEL(s(dbl(x0)), cons(s(y_1), from(s(s(y_2))))) → A__SEL(a__dbl(mark(x0)), cons(s(s(y_2)), from(s(s(s(y_2))))))
A__SEL(s(dbl(x0)), cons(y_3, from(s(y_4)))) → A__SEL(a__dbl(mark(x0)), cons(s(y_4), from(s(s(y_4)))))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ DependencyGraphProof

                                                                                                        ↳ QDP

                                                                                                          ↳ Narrowing

                                                                                                            ↳ QDP

                                                                                                              ↳ DependencyGraphProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                                                                                                            ↳ QDP

                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                ↳ AND

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Instantiation

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Instantiation

                                                                                                                                  ↳ QDP

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

A__SEL(s(s(x0)), cons(y_1, from(s(y_2)))) → A__SEL(s(x0), cons(s(y_2), from(s(s(y_2)))))
A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(x0, from(s(x0))))
A__SEL(s(dbl(x0)), cons(s(y_1), from(s(s(y_2))))) → A__SEL(a__dbl(mark(x0)), cons(s(s(y_2)), from(s(s(s(y_2))))))
A__SEL(s(dbl(x0)), cons(y_3, from(s(y_4)))) → A__SEL(a__dbl(mark(x0)), cons(s(y_4), from(s(s(y_4)))))

The TRS R consists of the following rules:

a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(nil) → nil
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__indx(nil, X) → nil
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__from(X) → cons(X, from(s(X)))
mark(dbl(X)) → a__dbl(mark(X))
mark(dbls(X)) → a__dbls(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__from(X) → from(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule A__SEL(s(sel(y0, y1)), cons(y2, from(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(x0, from(s(x0)))) we obtained the following new rules:

A__SEL(s(sel(x0, x1)), cons(y_5, from(s(y_6)))) → A__SEL(a__sel(mark(x0), mark(x1)), cons(s(y_6), from(s(s(y_6)))))
A__SEL(s(sel(x0, x1)), cons(s(s(y_3)), from(s(s(s(y_4)))))) → A__SEL(a__sel(mark(x0), mark(x1)), cons(s(s(s(y_4))), from(s(s(s(s(y_4)))))))
A__SEL(s(sel(x0, x1)), cons(s(y_1), from(s(s(y_2))))) → A__SEL(a__sel(mark(x0), mark(x1)), cons(s(s(y_2)), from(s(s(s(y_2))))))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ DependencyGraphProof

                                                                                                        ↳ QDP

                                                                                                          ↳ Narrowing

                                                                                                            ↳ QDP

                                                                                                              ↳ DependencyGraphProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                                                                                                            ↳ QDP

                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                ↳ AND

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Instantiation

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Instantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Instantiation

                                                                                                                                  ↳ QDP

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

A__SEL(s(sel(x0, x1)), cons(y_5, from(s(y_6)))) → A__SEL(a__sel(mark(x0), mark(x1)), cons(s(y_6), from(s(s(y_6)))))
A__SEL(s(s(x0)), cons(y_1, from(s(y_2)))) → A__SEL(s(x0), cons(s(y_2), from(s(s(y_2)))))
A__SEL(s(sel(x0, x1)), cons(s(y_1), from(s(s(y_2))))) → A__SEL(a__sel(mark(x0), mark(x1)), cons(s(s(y_2)), from(s(s(s(y_2))))))
A__SEL(s(sel(x0, x1)), cons(s(s(y_3)), from(s(s(s(y_4)))))) → A__SEL(a__sel(mark(x0), mark(x1)), cons(s(s(s(y_4))), from(s(s(s(s(y_4)))))))
A__SEL(s(dbl(x0)), cons(s(y_1), from(s(s(y_2))))) → A__SEL(a__dbl(mark(x0)), cons(s(s(y_2)), from(s(s(s(y_2))))))
A__SEL(s(dbl(x0)), cons(y_3, from(s(y_4)))) → A__SEL(a__dbl(mark(x0)), cons(s(y_4), from(s(s(y_4)))))

The TRS R consists of the following rules:

a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(nil) → nil
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__indx(nil, X) → nil
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__from(X) → cons(X, from(s(X)))
mark(dbl(X)) → a__dbl(mark(X))
mark(dbls(X)) → a__dbls(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__from(X) → from(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule A__SEL(s(s(x0)), cons(y_1, from(s(y_2)))) → A__SEL(s(x0), cons(s(y_2), from(s(s(y_2))))) we obtained the following new rules:

A__SEL(s(s(x0)), cons(s(s(s(y_5))), from(s(s(s(s(y_6))))))) → A__SEL(s(x0), cons(s(s(s(s(y_6)))), from(s(s(s(s(s(y_6))))))))
A__SEL(s(s(x0)), cons(s(s(y_3)), from(s(s(s(y_4)))))) → A__SEL(s(x0), cons(s(s(s(y_4))), from(s(s(s(s(y_4)))))))
A__SEL(s(s(x0)), cons(s(y_1), from(s(s(y_2))))) → A__SEL(s(x0), cons(s(s(y_2)), from(s(s(s(y_2))))))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ DependencyGraphProof

                                                                                                        ↳ QDP

                                                                                                          ↳ Narrowing

                                                                                                            ↳ QDP

                                                                                                              ↳ DependencyGraphProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                                                                                                            ↳ QDP

                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                ↳ AND

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Instantiation

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Instantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Instantiation

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                  ↳ QDP

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

A__SEL(s(sel(x0, x1)), cons(y_5, from(s(y_6)))) → A__SEL(a__sel(mark(x0), mark(x1)), cons(s(y_6), from(s(s(y_6)))))
A__SEL(s(s(x0)), cons(s(s(s(y_5))), from(s(s(s(s(y_6))))))) → A__SEL(s(x0), cons(s(s(s(s(y_6)))), from(s(s(s(s(s(y_6))))))))
A__SEL(s(s(x0)), cons(s(s(y_3)), from(s(s(s(y_4)))))) → A__SEL(s(x0), cons(s(s(s(y_4))), from(s(s(s(s(y_4)))))))
A__SEL(s(s(x0)), cons(s(y_1), from(s(s(y_2))))) → A__SEL(s(x0), cons(s(s(y_2)), from(s(s(s(y_2))))))
A__SEL(s(dbl(x0)), cons(s(y_1), from(s(s(y_2))))) → A__SEL(a__dbl(mark(x0)), cons(s(s(y_2)), from(s(s(s(y_2))))))
A__SEL(s(sel(x0, x1)), cons(s(s(y_3)), from(s(s(s(y_4)))))) → A__SEL(a__sel(mark(x0), mark(x1)), cons(s(s(s(y_4))), from(s(s(s(s(y_4)))))))
A__SEL(s(sel(x0, x1)), cons(s(y_1), from(s(s(y_2))))) → A__SEL(a__sel(mark(x0), mark(x1)), cons(s(s(y_2)), from(s(s(s(y_2))))))
A__SEL(s(dbl(x0)), cons(y_3, from(s(y_4)))) → A__SEL(a__dbl(mark(x0)), cons(s(y_4), from(s(s(y_4)))))

The TRS R consists of the following rules:

a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(nil) → nil
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__indx(nil, X) → nil
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__from(X) → cons(X, from(s(X)))
mark(dbl(X)) → a__dbl(mark(X))
mark(dbls(X)) → a__dbls(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__from(X) → from(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule A__SEL(s(dbl(x0)), cons(y_3, from(s(y_4)))) → A__SEL(a__dbl(mark(x0)), cons(s(y_4), from(s(s(y_4))))) we obtained the following new rules:

A__SEL(s(dbl(x0)), cons(s(s(s(y_5))), from(s(s(s(s(y_6))))))) → A__SEL(a__dbl(mark(x0)), cons(s(s(s(s(y_6)))), from(s(s(s(s(s(y_6))))))))
A__SEL(s(dbl(x0)), cons(s(s(y_3)), from(s(s(s(y_4)))))) → A__SEL(a__dbl(mark(x0)), cons(s(s(s(y_4))), from(s(s(s(s(y_4)))))))
A__SEL(s(dbl(x0)), cons(s(s(s(s(y_1)))), from(s(s(s(s(s(y_2)))))))) → A__SEL(a__dbl(mark(x0)), cons(s(s(s(s(s(y_2))))), from(s(s(s(s(s(s(y_2)))))))))
A__SEL(s(dbl(x0)), cons(s(y_3), from(s(s(y_4))))) → A__SEL(a__dbl(mark(x0)), cons(s(s(y_4)), from(s(s(s(y_4))))))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ DependencyGraphProof

                                                                                                        ↳ QDP

                                                                                                          ↳ Narrowing

                                                                                                            ↳ QDP

                                                                                                              ↳ DependencyGraphProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                                                                                                            ↳ QDP

                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                ↳ AND

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Instantiation

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Instantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Instantiation

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Instantiation

                                                                                                                                  ↳ QDP

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

A__SEL(s(sel(x0, x1)), cons(y_5, from(s(y_6)))) → A__SEL(a__sel(mark(x0), mark(x1)), cons(s(y_6), from(s(s(y_6)))))
A__SEL(s(s(x0)), cons(s(s(s(y_5))), from(s(s(s(s(y_6))))))) → A__SEL(s(x0), cons(s(s(s(s(y_6)))), from(s(s(s(s(s(y_6))))))))
A__SEL(s(dbl(x0)), cons(s(s(s(y_5))), from(s(s(s(s(y_6))))))) → A__SEL(a__dbl(mark(x0)), cons(s(s(s(s(y_6)))), from(s(s(s(s(s(y_6))))))))
A__SEL(s(s(x0)), cons(s(s(y_3)), from(s(s(s(y_4)))))) → A__SEL(s(x0), cons(s(s(s(y_4))), from(s(s(s(s(y_4)))))))
A__SEL(s(dbl(x0)), cons(s(s(y_3)), from(s(s(s(y_4)))))) → A__SEL(a__dbl(mark(x0)), cons(s(s(s(y_4))), from(s(s(s(s(y_4)))))))
A__SEL(s(s(x0)), cons(s(y_1), from(s(s(y_2))))) → A__SEL(s(x0), cons(s(s(y_2)), from(s(s(s(y_2))))))
A__SEL(s(dbl(x0)), cons(s(s(s(s(y_1)))), from(s(s(s(s(s(y_2)))))))) → A__SEL(a__dbl(mark(x0)), cons(s(s(s(s(s(y_2))))), from(s(s(s(s(s(s(y_2)))))))))
A__SEL(s(sel(x0, x1)), cons(s(y_1), from(s(s(y_2))))) → A__SEL(a__sel(mark(x0), mark(x1)), cons(s(s(y_2)), from(s(s(s(y_2))))))
A__SEL(s(sel(x0, x1)), cons(s(s(y_3)), from(s(s(s(y_4)))))) → A__SEL(a__sel(mark(x0), mark(x1)), cons(s(s(s(y_4))), from(s(s(s(s(y_4)))))))
A__SEL(s(dbl(x0)), cons(s(y_1), from(s(s(y_2))))) → A__SEL(a__dbl(mark(x0)), cons(s(s(y_2)), from(s(s(s(y_2))))))

The TRS R consists of the following rules:

a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(nil) → nil
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__indx(nil, X) → nil
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__from(X) → cons(X, from(s(X)))
mark(dbl(X)) → a__dbl(mark(X))
mark(dbls(X)) → a__dbls(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__from(X) → from(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule A__SEL(s(sel(x0, x1)), cons(y_5, from(s(y_6)))) → A__SEL(a__sel(mark(x0), mark(x1)), cons(s(y_6), from(s(s(y_6))))) we obtained the following new rules:

A__SEL(s(sel(x0, x1)), cons(s(s(s(y_5))), from(s(s(s(s(y_6))))))) → A__SEL(a__sel(mark(x0), mark(x1)), cons(s(s(s(s(y_6)))), from(s(s(s(s(s(y_6))))))))
A__SEL(s(sel(x0, x1)), cons(s(s(s(s(y_1)))), from(s(s(s(s(s(y_2)))))))) → A__SEL(a__sel(mark(x0), mark(x1)), cons(s(s(s(s(s(y_2))))), from(s(s(s(s(s(s(y_2)))))))))
A__SEL(s(sel(x0, x1)), cons(s(s(s(s(s(y_3))))), from(s(s(s(s(s(s(y_4))))))))) → A__SEL(a__sel(mark(x0), mark(x1)), cons(s(s(s(s(s(s(y_4)))))), from(s(s(s(s(s(s(s(y_4))))))))))
A__SEL(s(sel(x0, x1)), cons(s(y_5), from(s(s(y_6))))) → A__SEL(a__sel(mark(x0), mark(x1)), cons(s(s(y_6)), from(s(s(s(y_6))))))
A__SEL(s(sel(x0, x1)), cons(s(s(y_3)), from(s(s(s(y_4)))))) → A__SEL(a__sel(mark(x0), mark(x1)), cons(s(s(s(y_4))), from(s(s(s(s(y_4)))))))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ DependencyGraphProof

                                                                                                        ↳ QDP

                                                                                                          ↳ Narrowing

                                                                                                            ↳ QDP

                                                                                                              ↳ DependencyGraphProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                                                                                                            ↳ QDP

                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                ↳ AND

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Instantiation

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Instantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Instantiation

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Instantiation

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                  ↳ QDP

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

A__SEL(s(dbl(x0)), cons(s(s(s(y_5))), from(s(s(s(s(y_6))))))) → A__SEL(a__dbl(mark(x0)), cons(s(s(s(s(y_6)))), from(s(s(s(s(s(y_6))))))))
A__SEL(s(s(x0)), cons(s(s(y_3)), from(s(s(s(y_4)))))) → A__SEL(s(x0), cons(s(s(s(y_4))), from(s(s(s(s(y_4)))))))
A__SEL(s(sel(x0, x1)), cons(s(s(s(s(y_1)))), from(s(s(s(s(s(y_2)))))))) → A__SEL(a__sel(mark(x0), mark(x1)), cons(s(s(s(s(s(y_2))))), from(s(s(s(s(s(s(y_2)))))))))
A__SEL(s(sel(x0, x1)), cons(s(s(y_3)), from(s(s(s(y_4)))))) → A__SEL(a__sel(mark(x0), mark(x1)), cons(s(s(s(y_4))), from(s(s(s(s(y_4)))))))
A__SEL(s(dbl(x0)), cons(s(y_1), from(s(s(y_2))))) → A__SEL(a__dbl(mark(x0)), cons(s(s(y_2)), from(s(s(s(y_2))))))
A__SEL(s(sel(x0, x1)), cons(s(s(s(s(s(y_3))))), from(s(s(s(s(s(s(y_4))))))))) → A__SEL(a__sel(mark(x0), mark(x1)), cons(s(s(s(s(s(s(y_4)))))), from(s(s(s(s(s(s(s(y_4))))))))))
A__SEL(s(s(x0)), cons(s(s(s(y_5))), from(s(s(s(s(y_6))))))) → A__SEL(s(x0), cons(s(s(s(s(y_6)))), from(s(s(s(s(s(y_6))))))))
A__SEL(s(sel(x0, x1)), cons(s(s(s(y_5))), from(s(s(s(s(y_6))))))) → A__SEL(a__sel(mark(x0), mark(x1)), cons(s(s(s(s(y_6)))), from(s(s(s(s(s(y_6))))))))
A__SEL(s(dbl(x0)), cons(s(s(y_3)), from(s(s(s(y_4)))))) → A__SEL(a__dbl(mark(x0)), cons(s(s(s(y_4))), from(s(s(s(s(y_4)))))))
A__SEL(s(s(x0)), cons(s(y_1), from(s(s(y_2))))) → A__SEL(s(x0), cons(s(s(y_2)), from(s(s(s(y_2))))))
A__SEL(s(dbl(x0)), cons(s(s(s(s(y_1)))), from(s(s(s(s(s(y_2)))))))) → A__SEL(a__dbl(mark(x0)), cons(s(s(s(s(s(y_2))))), from(s(s(s(s(s(s(y_2)))))))))
A__SEL(s(sel(x0, x1)), cons(s(y_1), from(s(s(y_2))))) → A__SEL(a__sel(mark(x0), mark(x1)), cons(s(s(y_2)), from(s(s(s(y_2))))))

The TRS R consists of the following rules:

a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(nil) → nil
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__indx(nil, X) → nil
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__from(X) → cons(X, from(s(X)))
mark(dbl(X)) → a__dbl(mark(X))
mark(dbls(X)) → a__dbls(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__from(X) → from(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule A__SEL(s(dbl(x0)), cons(s(y_1), from(s(s(y_2))))) → A__SEL(a__dbl(mark(x0)), cons(s(s(y_2)), from(s(s(s(y_2)))))) we obtained the following new rules:

A__SEL(s(dbl(x0)), cons(s(s(s(s(s(s(y_5)))))), from(s(s(s(s(s(s(s(y_6)))))))))) → A__SEL(a__dbl(mark(x0)), cons(s(s(s(s(s(s(s(y_6))))))), from(s(s(s(s(s(s(s(s(y_6)))))))))))
A__SEL(s(dbl(x0)), cons(s(s(s(y_5))), from(s(s(s(s(y_6))))))) → A__SEL(a__dbl(mark(x0)), cons(s(s(s(s(y_6)))), from(s(s(s(s(s(y_6))))))))
A__SEL(s(dbl(x0)), cons(s(s(y_3)), from(s(s(s(y_4)))))) → A__SEL(a__dbl(mark(x0)), cons(s(s(s(y_4))), from(s(s(s(s(y_4)))))))
A__SEL(s(dbl(x0)), cons(s(s(s(s(s(y_3))))), from(s(s(s(s(s(s(y_4))))))))) → A__SEL(a__dbl(mark(x0)), cons(s(s(s(s(s(s(y_4)))))), from(s(s(s(s(s(s(s(y_4))))))))))
A__SEL(s(dbl(x0)), cons(s(s(s(s(y_1)))), from(s(s(s(s(s(y_2)))))))) → A__SEL(a__dbl(mark(x0)), cons(s(s(s(s(s(y_2))))), from(s(s(s(s(s(s(y_2)))))))))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ DependencyGraphProof

                                                                                                        ↳ QDP

                                                                                                          ↳ Narrowing

                                                                                                            ↳ QDP

                                                                                                              ↳ DependencyGraphProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                                                                                                            ↳ QDP

                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                ↳ AND

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Instantiation

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Instantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Instantiation

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Instantiation

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                  ↳ QDP

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

A__SEL(s(s(x0)), cons(s(s(y_3)), from(s(s(s(y_4)))))) → A__SEL(s(x0), cons(s(s(s(y_4))), from(s(s(s(s(y_4)))))))
A__SEL(s(dbl(x0)), cons(s(s(s(y_5))), from(s(s(s(s(y_6))))))) → A__SEL(a__dbl(mark(x0)), cons(s(s(s(s(y_6)))), from(s(s(s(s(s(y_6))))))))
A__SEL(s(sel(x0, x1)), cons(s(s(s(s(y_1)))), from(s(s(s(s(s(y_2)))))))) → A__SEL(a__sel(mark(x0), mark(x1)), cons(s(s(s(s(s(y_2))))), from(s(s(s(s(s(s(y_2)))))))))
A__SEL(s(sel(x0, x1)), cons(s(s(s(s(s(y_3))))), from(s(s(s(s(s(s(y_4))))))))) → A__SEL(a__sel(mark(x0), mark(x1)), cons(s(s(s(s(s(s(y_4)))))), from(s(s(s(s(s(s(s(y_4))))))))))
A__SEL(s(sel(x0, x1)), cons(s(s(y_3)), from(s(s(s(y_4)))))) → A__SEL(a__sel(mark(x0), mark(x1)), cons(s(s(s(y_4))), from(s(s(s(s(y_4)))))))
A__SEL(s(dbl(x0)), cons(s(s(s(s(s(s(y_5)))))), from(s(s(s(s(s(s(s(y_6)))))))))) → A__SEL(a__dbl(mark(x0)), cons(s(s(s(s(s(s(s(y_6))))))), from(s(s(s(s(s(s(s(s(y_6)))))))))))
A__SEL(s(sel(x0, x1)), cons(s(s(s(y_5))), from(s(s(s(s(y_6))))))) → A__SEL(a__sel(mark(x0), mark(x1)), cons(s(s(s(s(y_6)))), from(s(s(s(s(s(y_6))))))))
A__SEL(s(s(x0)), cons(s(s(s(y_5))), from(s(s(s(s(y_6))))))) → A__SEL(s(x0), cons(s(s(s(s(y_6)))), from(s(s(s(s(s(y_6))))))))
A__SEL(s(dbl(x0)), cons(s(s(y_3)), from(s(s(s(y_4)))))) → A__SEL(a__dbl(mark(x0)), cons(s(s(s(y_4))), from(s(s(s(s(y_4)))))))
A__SEL(s(s(x0)), cons(s(y_1), from(s(s(y_2))))) → A__SEL(s(x0), cons(s(s(y_2)), from(s(s(s(y_2))))))
A__SEL(s(dbl(x0)), cons(s(s(s(s(s(y_3))))), from(s(s(s(s(s(s(y_4))))))))) → A__SEL(a__dbl(mark(x0)), cons(s(s(s(s(s(s(y_4)))))), from(s(s(s(s(s(s(s(y_4))))))))))
A__SEL(s(dbl(x0)), cons(s(s(s(s(y_1)))), from(s(s(s(s(s(y_2)))))))) → A__SEL(a__dbl(mark(x0)), cons(s(s(s(s(s(y_2))))), from(s(s(s(s(s(s(y_2)))))))))
A__SEL(s(sel(x0, x1)), cons(s(y_1), from(s(s(y_2))))) → A__SEL(a__sel(mark(x0), mark(x1)), cons(s(s(y_2)), from(s(s(s(y_2))))))

The TRS R consists of the following rules:

a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(nil) → nil
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__indx(nil, X) → nil
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__from(X) → cons(X, from(s(X)))
mark(dbl(X)) → a__dbl(mark(X))
mark(dbls(X)) → a__dbls(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__from(X) → from(X)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule A__SEL(s(sel(x0, x1)), cons(s(y_1), from(s(s(y_2))))) → A__SEL(a__sel(mark(x0), mark(x1)), cons(s(s(y_2)), from(s(s(s(y_2)))))) we obtained the following new rules:

A__SEL(s(sel(x0, x1)), cons(s(s(s(y_5))), from(s(s(s(s(y_6))))))) → A__SEL(a__sel(mark(x0), mark(x1)), cons(s(s(s(s(y_6)))), from(s(s(s(s(s(y_6))))))))
A__SEL(s(sel(x0, x1)), cons(s(s(s(s(s(s(s(y_3))))))), from(s(s(s(s(s(s(s(s(y_4))))))))))) → A__SEL(a__sel(mark(x0), mark(x1)), cons(s(s(s(s(s(s(s(s(y_4)))))))), from(s(s(s(s(s(s(s(s(s(y_4))))))))))))
A__SEL(s(sel(x0, x1)), cons(s(s(s(s(y_1)))), from(s(s(s(s(s(y_2)))))))) → A__SEL(a__sel(mark(x0), mark(x1)), cons(s(s(s(s(s(y_2))))), from(s(s(s(s(s(s(y_2)))))))))
A__SEL(s(sel(x0, x1)), cons(s(s(s(s(s(s(y_5)))))), from(s(s(s(s(s(s(s(y_6)))))))))) → A__SEL(a__sel(mark(x0), mark(x1)), cons(s(s(s(s(s(s(s(y_6))))))), from(s(s(s(s(s(s(s(s(y_6)))))))))))
A__SEL(s(sel(x0, x1)), cons(s(s(s(s(s(y_3))))), from(s(s(s(s(s(s(y_4))))))))) → A__SEL(a__sel(mark(x0), mark(x1)), cons(s(s(s(s(s(s(y_4)))))), from(s(s(s(s(s(s(s(y_4))))))))))
A__SEL(s(sel(x0, x1)), cons(s(s(y_5)), from(s(s(s(y_6)))))) → A__SEL(a__sel(mark(x0), mark(x1)), cons(s(s(s(y_6))), from(s(s(s(s(y_6)))))))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ DependencyGraphProof

                                                                                                        ↳ QDP

                                                                                                          ↳ Narrowing

                                                                                                            ↳ QDP

                                                                                                              ↳ DependencyGraphProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                                                                                                            ↳ QDP

                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                ↳ AND

                                                                                                                                  ↳ QDP

                                                                                                                                    ↳ QDPOrderProof

                                                                                                                                      ↳ QDP

                                                                                                                                        ↳ Instantiation

                                                                                                                                          ↳ QDP

                                                                                                                                            ↳ Instantiation

                                                                                                                                              ↳ QDP

                                                                                                                                                ↳ Instantiation

                                                                                                                                                  ↳ QDP

                                                                                                                                                    ↳ Instantiation

                                                                                                                                                      ↳ QDP

                                                                                                                                                        ↳ Instantiation

                                                                                                                                                          ↳ QDP

                                                                                                                                                            ↳ Instantiation

                                                                                                                                                              ↳ QDP

                                                                                                                                                                ↳ Instantiation

                                                                                                                                                                  ↳ QDP

                                                                                                                                                                    ↳ Instantiation

                                                                                                                                                                      ↳ QDP

                                                                                                                                  ↳ QDP

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

A__SEL(s(sel(x0, x1)), cons(s(s(s(s(s(s(s(y_3))))))), from(s(s(s(s(s(s(s(s(y_4))))))))))) → A__SEL(a__sel(mark(x0), mark(x1)), cons(s(s(s(s(s(s(s(s(y_4)))))))), from(s(s(s(s(s(s(s(s(s(y_4))))))))))))
A__SEL(s(dbl(x0)), cons(s(s(s(y_5))), from(s(s(s(s(y_6))))))) → A__SEL(a__dbl(mark(x0)), cons(s(s(s(s(y_6)))), from(s(s(s(s(s(y_6))))))))
A__SEL(s(s(x0)), cons(s(s(y_3)), from(s(s(s(y_4)))))) → A__SEL(s(x0), cons(s(s(s(y_4))), from(s(s(s(s(y_4)))))))
A__SEL(s(sel(x0, x1)), cons(s(s(s(s(y_1)))), from(s(s(s(s(s(y_2)))))))) → A__SEL(a__sel(mark(x0), mark(x1)), cons(s(s(s(s(s(y_2))))), from(s(s(s(s(s(s(y_2)))))))))
A__SEL(s(sel(x0, x1)), cons(s(s(s(s(s(s(y_5)))))), from(s(s(s(s(s(s(s(y_6)))))))))) → A__SEL(a__sel(mark(x0), mark(x1)), cons(s(s(s(s(s(s(s(y_6))))))), from(s(s(s(s(s(s(s(s(y_6)))))))))))
A__SEL(s(sel(x0, x1)), cons(s(s(y_3)), from(s(s(s(y_4)))))) → A__SEL(a__sel(mark(x0), mark(x1)), cons(s(s(s(y_4))), from(s(s(s(s(y_4)))))))
A__SEL(s(sel(x0, x1)), cons(s(s(s(s(s(y_3))))), from(s(s(s(s(s(s(y_4))))))))) → A__SEL(a__sel(mark(x0), mark(x1)), cons(s(s(s(s(s(s(y_4)))))), from(s(s(s(s(s(s(s(y_4))))))))))
A__SEL(s(dbl(x0)), cons(s(s(s(s(s(s(y_5)))))), from(s(s(s(s(s(s(s(y_6)))))))))) → A__SEL(a__dbl(mark(x0)), cons(s(s(s(s(s(s(s(y_6))))))), from(s(s(s(s(s(s(s(s(y_6)))))))))))
A__SEL(s(s(x0)), cons(s(s(s(y_5))), from(s(s(s(s(y_6))))))) → A__SEL(s(x0), cons(s(s(s(s(y_6)))), from(s(s(s(s(s(y_6))))))))
A__SEL(s(sel(x0, x1)), cons(s(s(s(y_5))), from(s(s(s(s(y_6))))))) → A__SEL(a__sel(mark(x0), mark(x1)), cons(s(s(s(s(y_6)))), from(s(s(s(s(s(y_6))))))))
A__SEL(s(dbl(x0)), cons(s(s(y_3)), from(s(s(s(y_4)))))) → A__SEL(a__dbl(mark(x0)), cons(s(s(s(y_4))), from(s(s(s(s(y_4)))))))
A__SEL(s(s(x0)), cons(s(y_1), from(s(s(y_2))))) → A__SEL(s(x0), cons(s(s(y_2)), from(s(s(s(y_2))))))
A__SEL(s(dbl(x0)), cons(s(s(s(s(y_1)))), from(s(s(s(s(s(y_2)))))))) → A__SEL(a__dbl(mark(x0)), cons(s(s(s(s(s(y_2))))), from(s(s(s(s(s(s(y_2)))))))))
A__SEL(s(dbl(x0)), cons(s(s(s(s(s(y_3))))), from(s(s(s(s(s(s(y_4))))))))) → A__SEL(a__dbl(mark(x0)), cons(s(s(s(s(s(s(y_4)))))), from(s(s(s(s(s(s(s(y_4))))))))))

The TRS R consists of the following rules:

a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(nil) → nil
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__indx(nil, X) → nil
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__from(X) → cons(X, from(s(X)))
mark(dbl(X)) → a__dbl(mark(X))
mark(dbls(X)) → a__dbls(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__from(X) → from(X)

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ DependencyGraphProof

                                ↳ QDP

                                  ↳ Narrowing

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ QDP

                                          ↳ Narrowing

                                            ↳ QDP

                                              ↳ DependencyGraphProof

                                                ↳ QDP

                                                  ↳ Narrowing

                                                    ↳ QDP

                                                      ↳ DependencyGraphProof

                                                        ↳ QDP

                                                          ↳ Narrowing

                                                            ↳ QDP

                                                              ↳ DependencyGraphProof

                                                                ↳ QDP

                                                                  ↳ Narrowing

                                                                    ↳ QDP

                                                                      ↳ DependencyGraphProof

                                                                        ↳ QDP

                                                                          ↳ Narrowing

                                                                            ↳ QDP

                                                                              ↳ DependencyGraphProof

                                                                                ↳ QDP

                                                                                  ↳ Narrowing

                                                                                    ↳ QDP

                                                                                      ↳ DependencyGraphProof

                                                                                        ↳ QDP

                                                                                          ↳ Narrowing

                                                                                            ↳ QDP

                                                                                              ↳ DependencyGraphProof

                                                                                                ↳ QDP

                                                                                                  ↳ Narrowing

                                                                                                    ↳ QDP

                                                                                                      ↳ DependencyGraphProof

                                                                                                        ↳ QDP

                                                                                                          ↳ Narrowing

                                                                                                            ↳ QDP

                                                                                                              ↳ DependencyGraphProof

                                                                                                                ↳ QDP

                                                                                                                  ↳ Narrowing

                                                                                                                    ↳ QDP

                                                                                                                      ↳ DependencyGraphProof

                                                                                                                        ↳ QDP

                                                                                                                          ↳ Narrowing

                                                                                                                            ↳ QDP

                                                                                                                              ↳ DependencyGraphProof

                                                                                                                                ↳ AND

                                                                                                                                  ↳ QDP

                                                                                                                                  ↳ QDP

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

A__SEL(s(dbls(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbls(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, indx(x0, x1))) → A__SEL(s(y0), a__indx(mark(x0), x1))
A__SEL(s(dbl(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbl(mark(y0)), a__dbls(mark(x0)))
A__SEL(s(dbls(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbls(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbl(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, dbl(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbl(mark(x0)))
A__SEL(s(dbl(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbl(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbl(mark(y0)), cons(x0, x1))
A__SEL(s(dbl(y0)), cons(y1, dbl(x0))) → A__SEL(a__dbl(mark(y0)), a__dbl(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__indx(mark(y0), y1), cons(x0, x1))
A__SEL(s(sel(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__indx(mark(x0), x1))
A__SEL(s(dbls(y0)), cons(y1, dbls(x0))) → A__SEL(a__dbls(mark(y0)), a__dbls(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__sel(mark(y0), mark(y1)), a__dbls(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, dbls(x0))) → A__SEL(a__indx(mark(y0), y1), a__dbls(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, indx(x0, x1))) → A__SEL(a__indx(mark(y0), y1), a__indx(mark(x0), x1))
A__SEL(s(s(y0)), cons(y1, dbl(x0))) → A__SEL(s(y0), a__dbl(mark(x0)))
A__SEL(s(sel(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), a__sel(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, dbls(x0))) → A__SEL(s(y0), a__dbls(mark(x0)))
A__SEL(s(indx(y0, y1)), cons(y2, sel(x0, x1))) → A__SEL(a__indx(mark(y0), y1), a__sel(mark(x0), mark(x1)))
A__SEL(s(dbls(y0)), cons(y1, cons(x0, x1))) → A__SEL(a__dbls(mark(y0)), cons(x0, x1))
A__SEL(s(dbl(y0)), cons(y1, indx(x0, x1))) → A__SEL(a__dbl(mark(y0)), a__indx(mark(x0), x1))
A__SEL(s(dbls(y0)), cons(y1, sel(x0, x1))) → A__SEL(a__dbls(mark(y0)), a__sel(mark(x0), mark(x1)))
A__SEL(s(s(y0)), cons(y1, sel(x0, x1))) → A__SEL(s(y0), a__sel(mark(x0), mark(x1)))
A__SEL(s(sel(y0, y1)), cons(y2, cons(x0, x1))) → A__SEL(a__sel(mark(y0), mark(y1)), cons(x0, x1))
A__SEL(s(s(y0)), cons(y1, cons(x0, x1))) → A__SEL(s(y0), cons(x0, x1))

The TRS R consists of the following rules:

a__dbl(0) → 0
a__dbl(s(X)) → s(s(dbl(X)))
a__dbls(nil) → nil
a__dbls(cons(X, Y)) → cons(dbl(X), dbls(Y))
a__sel(0, cons(X, Y)) → mark(X)
a__sel(s(X), cons(Y, Z)) → a__sel(mark(X), mark(Z))
a__indx(nil, X) → nil
a__indx(cons(X, Y), Z) → cons(sel(X, Z), indx(Y, Z))
a__from(X) → cons(X, from(s(X)))
mark(dbl(X)) → a__dbl(mark(X))
mark(dbls(X)) → a__dbls(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(indx(X1, X2)) → a__indx(mark(X1), X2)
mark(from(X)) → a__from(X)
mark(0) → 0
mark(s(X)) → s(X)
mark(nil) → nil
mark(cons(X1, X2)) → cons(X1, X2)
a__dbl(X) → dbl(X)
a__dbls(X) → dbls(X)
a__sel(X1, X2) → sel(X1, X2)
a__indx(X1, X2) → indx(X1, X2)
a__from(X) → from(X)

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